Search Results for numbers

Biographies

1. Kaprekar biography
   
   • Although astrology requires no deep mathematics, it does require a considerable ability to calculate with numbers, and Kaprekar's father certainly gave his son a love of calculating.
     
   • The fascination for numbers which Kaprekar had as a child continued throughout his life.
     
   • He was a good school teacher, using his own love of numbers to motivate his pupils, and was often invited to speak at local colleges about his unique methods.
     
   • The same is the case with me in so far as numbers are concerned.
     
   • Kaprekar's name today is well-known and many mathematicians have found themselves intrigued by the ideas about numbers which Kaprekar found so addictive.
     
   • Rearrange the digits to form the largest and smallest numbers with these digits, namely 7643 and 3467, and subtract the smaller from the larger to obtain 4167.
     
   • Exactly 77 four digit numbers stabilize to 0 under the Kaprekar process, the remainder will stabilize to 6174.
     
   • Anyone interested could experiment with numbers with more than 4 digits and see if they stabilise to a single number (other than 0).
     
   • Of course from this observation we see that there are infinitely many Kaprekar numbers (certainly 9, 99, 999, 9999, ..
     
   • are all Kaprekar numbers).
     
   • The first few Kaprekar numbers are: .
     
   • It was shown in 2000 that Kaprekar numbers are in one-one correspondence with the unitary divisors of 10n - 1 (x is a unitary divisor of z if z = xy where x and y are coprime).
     
   • Of course we have looked at Kaprekar numbers to base 10.
     
   • A paper by Kaprekar describing properties of these numbers is [J.
     
   • Next we describe Kaprekar's 'self-numbers' or 'Swayambhu' (see [Puzzles of the Self-Numbers (311 Devlal Camp, Devlali, India, 1959).',5)">5]).
     
   • The self-numbers are .
     
   • Now Kaprekar makes other remarks about self-numbers in [Puzzles of the Self-Numbers (311 Devlal Camp, Devlali, India, 1959).',5)">5].
     
   • For example he notes that certain numbers are generated by more than a single number - these he calls junction numbers.
     
   • He remarks that numbers exist with more than 2 generators.
     
   • 6 (1938), 68.',4)">4] and [Demlo Numbers (Khareswada, Devlali, India, 1948).',6)">6] look at 'Demlo numbers'.
     
   • We will not give the definition of these numbers but we note that the name comes from the station where he was changing trains on the Bombay to Thane line in 1923 when he had the idea to study numbers of that type.
     
   • For the final type of numbers which we will consider that were examined by Kaprekar we look at Harshad numbers (from the Sanskrit meaning "great joy").
     
   • These are numbers divisible by the sum of their digits.
     
   • ., 9 must be Harshad numbers, and the next ones are .
     
   • It will be noticed that 80, 81 are a pair of consecutive numbers which are both Harshad, while 110, 111, 112 are three consecutive numbers all Harshad.
     
   • It was proved in 1994 that no 21 consecutive numbers can all be Harshad numbers.
     
   • It is possible to have 20 consecutive Harshad numbers but one has to go to numbers greater than 1044363342786 before such a sequence is found.
     
   • are all Harshad numbers.
     
   • The self-numbers which are also Harshad numbers are: .
     
   • Note that 2007 (the year in which this article was written) is both a self-numbers and a Harshad number.
     
   • Harshad numbers for bases other than 10 are also interesting and we can ask whether any number is a Harshad number for every base.
     
   • The are only four such numbers 1, 2, 4, and 6.
     
   • We have taken quite a while to look at a selection of different properties of numbers investigated by Kaprekar.
     
   • One has to understand that this was despite the fact that Kaprekar lived in the cheapest possible way, being only interested in spending his waking hours experimenting with numbers.
     
   • International fame only came in 1975 when Martin Gardener wrote about Kaprekar and his numbers in his 'Mathematical Games' column in the March issue of Scientific American.
     

2. Cantor biography
   
   • Cantor published a paper on trigonometric series in 1872 in which he defined irrational numbers in terms of convergent sequences of rational numbers.
     
   • Dedekind published his definition of the real numbers by "Dedekind cuts" also in 1872 and in this paper Dedekind refers to Cantor's 1872 paper which Cantor had sent him.
     
   • In 1873 Cantor proved the rational numbers countable, i.e.
     
   • they may be placed in one-one correspondence with the natural numbers.
     
   • He also showed that the algebraic numbers, i.e.
     
   • the numbers which are roots of polynomial equations with integer coefficients, were countable.
     
   • However his attempts to decide whether the real numbers were countable proved harder.
     
   • He had proved that the real numbers were not countable by December 1873 and published this in a paper in 1874.
     
   • Liouville established in 1851 that transcendental numbers exist.
     
   • Twenty years later, in this 1874 work, Cantor showed that in a certain sense 'almost all' numbers are transcendental by proving that the real numbers were not countable while he had proved that the algebraic numbers were countable.
     
   • those which are in 1-1 correspondence with the natural numbers.
     
   • The major achievement of the Grundlagen was its presentation of the transfinite numbers as an autonomous and systematic extension of the natural numbers.
     
   • I realise that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers.
     
   • Mathematical worries began to trouble Cantor at this time, in particular he began to worry that he could not prove the continuum hypothesis, namely that the order of infinity of the real numbers was the next after that of the natural numbers.
     
   • Cantor published a rather strange paper in 1894 which listed the way that all even numbers up to 1000 could be written as the sum of two primes.
     
   • However, it was not to be, but the second paper describes his theory of well-ordered sets and ordinal numbers.
     
   • History Topics: The real numbers: Stevin to Hilbert .
     
   • History Topics: The real numbers: Attempts to understand .
     

3. Numbers biography
   
   • Annie Hutton Numbers .
     
   • Annie Hutton Numbers began her education at Mrs Steele's Private School in Upper Gray Street in Edinburgh.
     
   • While on the staff of the University, Numbers undertook research towards the degree of Ph.D.
     
   • Annie Hutton Numbers joined the Edinburgh Mathematical Society in January 1917 when living at 16 West Savile Terrace, Edinburgh.
     
   • http://www-history.mcs.st-andrews.ac.uk/Biographies/Numbers.html .
     

4. Fibonacci biography
   
   • His book on commercial arithmetic Di minor guisa is lost as is his commentary on Book X of Euclid's Elements which contained a numerical treatment of irrational numbers which Euclid had approached from a geometric point of view.
     
   • A problem in the third section of Liber abaci led to the introduction of the Fibonacci numbers and the Fibonacci sequence for which Fibonacci is best remembered today:- .
     
   • This sequence, in which each number is the sum of the two preceding numbers, has proved extremely fruitful and appears in many different areas of mathematics and science.
     
   • There are also problems involving perfect numbers, problems involving the Chinese remainder theorem and problems involving summing arithmetic and geometric series.
     
   • Fibonacci treats numbers such as √10 in the fourth section, both with rational approximations and with geometric constructions.
     
   • Fibonacci first notes that square numbers can be constructed as sums of odd numbers, essentially describing an inductive construction using the formula n2 + (2n+1) = (n+1)2.
     
   • I thought about the origin of all square numbers and discovered that they arose from the regular ascent of odd numbers.
     
   • For unity is a square and from it is produced the first square, namely 1; adding 3 to this makes the second square, namely 4, whose root is 2; if to this sum is added a third odd number, namely 5, the third square will be produced, namely 9, whose root is 3; and so the sequence and series of square numbers always rise through the regular addition of odd numbers.
     
   • Thus when I wish to find two square numbers whose addition produces a square number, I take any odd square number as one of the two square numbers and I find the other square number by the addition of all the odd numbers from unity up to but excluding the odd square number.
     
   • For example, I take 9 as one of the two squares mentioned; the remaining square will be obtained by the addition of all the odd numbers below 9, namely 1, 3, 5, 7, whose sum is 16, a square number, which when added to 9 gives 25, a square number.
     
   • Fibonacci numbers and the Euclidean algorithm .
     
   • Continued fractions and Fibonacci numbers .
     
   • History Topics: Prime numbers .
     
   • History Topics: The real numbers: Pythagoras to Stevin .
     
   • R Knott (Fibonacci numbers and other links) .
     

5. Bombelli biography
   
   • Bombelli's Algebra gives a thorough account of the algebra then known and includes Bombelli's important contribution to complex numbers.
     
   • Before looking at his remarkable contribution to complex numbers we should remark that Bombelli first wrote down how to calculate with negative numbers.
     
   • Bombelli is explicitly working with signed numbers.
     
   • Bombelli, himself, did not find working with complex numbers easy at first, writing in [R Bombelli L\'Algebra, Books I-V (Milan, 1966).',2)">2] (see also [The emergence of number (Singapore, 1980).',3)">3]):- .
     
   • Bombelli was the first person to write down the rules for addition, subtraction and multiplication of complex numbers.
     
   • After giving this description of multiplication of complex numbers, Bombelli went on to give rules for adding and subtracting them.
     
   • He then showed that, using his calculus of complex numbers, correct real solutions could be obtained from the Cardan-Tartaglia formula for the solution to a cubic even when the formula gave an expression involving the square roots of negative numbers.
     
   • Jayawardene writes in [Dictionary of Scientific Biography (New York 1970-1990).',1)">1] that in his treatment of complex numbers Bombelli:- .
     
   • Thus we have an engineer, Bombelli, making practical use of complex numbers perhaps because they gave him useful results, while Cardan found the square roots of negative numbers useless.
     
   • Bombelli is the first to give a treatment of any complex numbers..
     
   • It is remarkable how thorough he is in his presentation of the laws of calculation of complex numbers..
     
   • It seems to be quite fair to describe Bombelli as the inventor of complex numbers.
     
   • Nobody before him had given rules for working with such numbers, nor had they suggested that working with such numbers might prove useful.
     
   • imaginaries had been used long before Bombelli's book, and it is therefore not quite justified to call him the "first discoverer" of complex numbers.
     
   • I think that Bombelli's Algebra is one of the most remarkable achievements of 16th century mathematics, and he must be credited with understanding the importance of complex numbers at a time when clearly nobody else did.
     

6. Theon of Smyrna biography
   
   • This work is a handbook for philosophy students to show how prime numbers, geometrical numbers such as squares, progressions, music and astronomy are interrelated.
     
   • In the section on numbers Theon adopts a Pythagorean approach, writing about odd numbers, even numbers, prime numbers, composite numbers, square numbers, oblong numbers, triangular numbers, polygonal numbers, circular numbers, spherical numbers, solid numbers with three factors, pyramidal numbers, perfect numbers, deficient numbers and abundant numbers.
     

7. Ribenboim biography
   
   • During 1951 up to the spring of 1952 he attended lecture courses on Lie groups by Delsarte, algebraic numbers by Dieudonne and the theory of distributions by Laurent Schwartz.
     
   • Also in 1972 he published Algebraic numbers.
     
   • The first of these is devoted to ramification theory in Galois extensions and the second to a proof of the theorem by Kronecker and Heinrich Weber on the abelian extensions of the field of rational numbers.
     
   • Are there functions defining prime numbers?; .
     
   • Heuristic and probabilistic results about prime numbers.
     
   • More recent books by Ribenboim are: The theory of classical valuations (1999); My numbers, my friends (2000); Classical theory of algebraic numbers (2001) In the Preface to the first of these Ribenboim explains the what he intends to study:- .
     
   • Let us end by quoting from two reviews of My numbers, my friends , a book which both in title and content say much about Ribenboim's mathematical loves.
     
   • This is a book by a man who loves numbers.
     
   • In 10 long essays the author deals with number theoretic properties of numbers which have captivated both professional and amateur mathematicians.
     
   • Examples of such numbers are Fibonacci numbers, class numbers, prime numbers, consecutive powers, powerful numbers and 1093.
     
   • So when 'My Numbers, My Friends' came out, I was excited.
     
   • And this book more than rewarded my anticipation! Ribenboim is now retired, and treats us to a delightful survey of a few of his favourite numbers ..
     

8. Gelfond biography
   
   • During 1929-30 he taught mathematics at Moscow Technological College but already he had published some important papers: The arithmetic properties of entire functions (1929); Transcendental numbers (1929); and An outline of the history and the present state of the theory of transcendental numbers (1930).
     
   • These papers by Gelfond represent a major step forward in the study of transcendental numbers.
     
   • In the second of the 1929 papers Gelfond applied this result to prove that certain numbers are transcendental, so solving a special case of Hilbert's Seventh Problem.
     
   • After his return to Russia, Gelfond taught mathematics from 1931 at Moscow State University where he held chairs of analysis, theory of numbers and the history of mathematics.
     
   • Gelfond developed basic techniques in the study of transcendental numbers, that is numbers that are not the solution of an algebraic equation with rational coefficients.
     
   • In addition to his important work in the number theory of transcendental numbers, Gelfond made significant contributions to the theory of interpolation and the approximation of functions of a complex variable.
     
   • Returning to Gelfond's contributions to transcendental numbers which we mentioned above, in 1929 he conjectured that:- .
     
   • If am, 1 ≤ m ≤ n and bm,1 ≤ m ≤ n are algebraic numbers such that { ln(am), 1 ≤ m ≤ n } are linearly independent over Q, then .
     
   • Gelfond addressed the Second All-Union Mathematics Congress in Leningrad in 1934) on Transcendental numbers.
     
   • His major contributions to transcendental numbers is set out in Transcendentnye algebraicheskie chisla (Transcendental and algebraic numbers) (1952).
     
   • to show the contemporary state of the theory of transcendental numbers, to exhibit the fundamental methods of this theory, to present the historical course of development of these methods, and to show the connections which exist between this theory and other problems in the theory of numbers.
     
   • In 1962 Gelfond published the book Elementary methods in the analytic theory of numbers written jointly with Linnik.
     

9. Thabit biography
   
   • played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-euclidean geometry.
     
   • We shall examine in more detail Thabit's work in these areas, in particular his work in number theory on amicable numbers.
     
   • Perfect numbers are those numbers n with S(n) = n while m and n are amicable if S(n) = m, and S(m) = n.
     
   • In Book on the determination of amicable numbers Thabit claims that Pythagoras began the study of perfect and amicable numbers.
     
   • This claim, probably first made by Iamblichus in his biography of Pythagoras written in the third century AD where he gave the amicable numbers 220 and 284, is almost certainly false.
     
   • However Thabit then states quite correctly that although Euclid and Nicomachus studied perfect numbers, and Euclid gave a rule for determining them ([The development of Arabic mathematics : between arithmetic and algebra (London, 1994).',6)">6] or [Entre arithmetique et algebre: Recherches sur l\'histoire des mathematiques arabes (Paris, 1984).',7)">7]):- .
     
   • neither of these authors either mentioned or showed interest in [amicable numbers].
     
   • Since the matter of [amicable numbers] has occurred to my mind, and since I have derived a proof for them, I did not wish to write the rule without proving it perfectly because they have been neglected by [Euclid and Nicomachus].
     
   • If pn-1, pn, and qn are prime numbers, then a = 2npn-1pn and b = 2nqn are amicable numbers while a is abundant and b is deficient.
     
   • 12 (3) (1985), 269-273.',13)">13] Hogendijk shows that Thabit was probably the first to discover the pair of amicable numbers 17296, 18416.
     
   • The Greeks had dealt with geometric quantities but had not thought of them in the same way as numbers to which the usual rules of arithmetic could be applied.
     
   • Thabit's concept of number follows that of Plato and he argues that numbers exist, whether someone knows them or not, and they are separate from numerable things.
     
   • History Topics: Perfect numbers .
     

10. Al-Baghdadi biography
    
    • He also considers the arithmetic of irrational numbers and business arithmetic.
      
    • Al-Baghdadi gives an interesting discussion of abundant numbers, deficient numbers, perfect numbers and equivalent numbers.
      
    • First al-Baghdadi defines perfect numbers (those number n with S(n) = n), abundant numbers (those number n with S(n) > n), and deficient numbers (those number n with S(n) < n).
      
    • Of course these properties of numbers had been studied by the ancient Greeks.
      
    • Nicomachus had made claims about perfect numbers in around 100 AD which were accepted, seemingly without question, in Europe up to the 16th century.
      
    • He who affirms that all perfect numbers end with the figure 6 or 8 are right.
      
    • Next al-Baghdadi goes on to define equivalent numbers, and appears to be the first to study them.
      
    • Two numbers m and n are called equivalent if S(m) = S(n).
      
    • The results that al-Baghdadi gives on amicable numbers are only a slight variations on results given earlier by Thabit ibn Qurra.
      
    • Then if pn-1, pn, and qn are prime, then a = 2npn-1pn and b = 2nqn are amicable numbers while a is abundant and b is deficient.
      

11. Mahler biography
    
    • It was through lectures by Emmy Noether that he learnt about p-adic numbers which were to be one of the major topics of his research throughout his life.
      
    • he developed a new method in transcendental theory, found his celebrated classification of transcendental numbers and pioneered diophantine approximation in p-adic fields.
      
    • I almost immediately posed him the following problem: An integer is called powerful if p | m implies p2 | m; are there infinitely many consecutive powerful numbers? Mahler immediately answered: Trivially, yes! x2 - 8 y2 = 1 has infinitely many solutions.
      
    • He worked on transcendence of numbers, showing in 1946 that .
      
    • He also classified real and complex numbers into classes which are algebraically independent.
      
    • Other major themes of his work were rational approximations of algebraic numbers, p-adic numbers, p-adic Diophantine approximation, geometry of numbers (a term coined by Minkowski to describe the mathematics of packings and coverings) and measure on polynomials.
      
    • For example Lectures on diophantine approximations : g-adic numbers and Roth's theorem (1961) was prepared from notes by R P Bambah of lectures given by Mahler at the University of Notre Dame in autumn 1957 and was described as an "extremely valuable contribution".
      
    • In the Preface to Introduction to p-adic numbers and their functions (1973) Mahler writes:- .
      
    • This set of notes contains an elementary introduction to the theory of p-adic numbers and their analysis.
      
    • These numbers were introduced by K Hensel some eighty years ago and have slowly become of importance in more and more parts of mathematics.
      
    • Nevertheless, while many recent books on algebra have short chapters or paragraphs on the subject, a really good introduction to p-adic numbers from the standpoint of elementary analysis does not seem to exist.
      
    • Mahler's Lectures on transcendental numbers (1976) was based on lectures given twenty years earlier.
      
    • History Topics: The real numbers: Attempts to understand .
      

12. Al-Farisi biography
    
    • Al-Farisi's most impressive work in number theory is on amicable numbers.
      
    • The numbers m and n are called amicable if S(n) = m, and S(m) = n.
      
    • In Tadhkira al-ahbab fi bayan al-tahabb (Memorandum for friends on the proof of amicability) al-Farisi gave a new proof of the following theorem by Thabit ibn Qurra on amicable numbers: .
      
    • If pn-1, pn, and qn are prime numbers, then a = 2npn-1pn and b = 2nqn are amicable numbers.
      
    • In fact al-Farisi's approach is based on the unique factorisation of an integer into powers of prime numbers, and, according to Rashed, he states and attempts to prove this, the so-called fundamental theorem of arithmetic, in this work.
      
    • At the end of his treatise al-Farisi gives the pairs of amicable numbers 220, 284 and 17296, 18416, obtained from using Thabit's rule with n = 2 and n = 4 respectively.
      
    • To check that Thabit's theorem gives amicable numbers with n = 4, al-Farisi has to show that p3, p4, and q4 are prime numbers.
      
    • There is no doubt that al-Farisi proved these to be amicable numbers long before Euler.
      
    • However, al-Farisi was probably not the first to discover these amicable numbers.
      
    • Al-Farisi saw the relation between polygonal numbers and the binomial coefficients and he presented arguments, using an early type of mathematical induction, which showed a relation between triangular numbers, the sums of triangular numbers, the sums of the sums of triangular number, etc., and the combinations of n objects taken k at a time.
      

13. Pythagoras biography
    
    • having been brought up in the study of mathematics, thought that things are numbers ..
      
    • Pythagoras noticed that vibrating strings produce harmonious tones when the ratios of the lengths of the strings are whole numbers, and that these ratios could be extended to other instruments.
      
    • Pythagoras studied properties of numbers which would be familiar to mathematicians today, such as even and odd numbers, triangular numbers, perfect numbers etc.
      
    • However to Pythagoras numbers had personalities which we hardly recognise as mathematics today [The philosophers of Greece (Albany, N.Y., 1981).',3)">3]:- .
      
    • This went against Pythagoras's philosophy the all things are numbers, since by a number he meant the ratio of two whole numbers.
      
    • However, because of his belief that all things are numbers it would be a natural task to try to prove that the hypotenuse of an isosceles right angled triangle had a length corresponding to a number.
      
    • In addition to his beliefs about numbers, geometry and astronomy described above, he held [Encyclopaedia Britannica.
      
    • History Topics: Perfect numbers .
      
    • History Topics: Prime numbers .
      
    • History Topics: The real numbers: Pythagoras to Stevin .
      

14. Nicomachus biography
    
    • This is obvious from his writings on numbers and music, but we are also told this by Porphyry who says that he was one of the leading members of the Pythagorean School.
      
    • An example of this we look more closely at the results which Nicomachus quotes on perfect numbers.
      
    • He states that the nth perfect number has n digits, and that all perfect numbers end in 6 and 8 alternately.
      
    • These statements must be merely false deductions from the fact that there were four perfect numbers known to Nicomachus, namely 6, 28, 496 and 8128.
      
    • To illustrate Nicomachus's rather strange approach to numbers, giving the moral properties, we look at his description of abundant numbers and deficient numbers.
      
    • Nicomachus writes of these numbers in Introduction to Arithmetic (see [History of Mathematics : History of Problems (Paris, 1997), 389-410.',6)">6], or [Nicomachus of Gerasa, Introduction to Arithmetic (New York, 1926).',3)">3] for a different translation):- .
      
    • He then continues his description of abundant numbers as resembling an animal:- .
      
    • Nicomachus also wrote two volumes Theologoumena arithmetikes (The Theology of Numbers) which was completely concerned with mystic properties of numbers.
      
    • History Topics: Perfect numbers .
      
    • History Topics: The real numbers: Pythagoras to Stevin .
      

15. Bhaskara II biography
    
    • Given that he was building on the knowledge and understanding of Brahmagupta it is not surprising that Bhaskaracharya understood about zero and negative numbers.
      
    • In dealing with numbers Bhaskaracharya, like Brahmagupta before him, handled efficiently arithmetic involving negative numbers.
      
    • We follow Ifrah who explains these two methods due to Bhaskaracharya in [A universal history of numbers : From prehistory to the invention of the computer (London, 1998).',4)">4].
      
    • To multiply 325 by 243 Bhaskaracharya writes the numbers thus: .
      
    • Finally add the three numbers below the second line to obtain the answer 78975.
      
    • Bhaskaracharya, like many of the Indian mathematicians, considered squaring of numbers as special cases of multiplication which deserved special methods.
      
    • Then Bhaskaracharya's problem is to find the total number of numbers of the form (*) that satisfy .
      
    • The topics are: positive and negative numbers; zero; the unknown; surds; the kuttaka; indeterminate quadratic equations; simple equations; quadratic equations; equations with more than one unknown; quadratic equations with more than one unknown; operations with products of several unknowns; and the author and his work.
      
    • Having explained how to do arithmetic with negative numbers, Bhaskaracharya gives problems to test the abilities of the reader on calculating with negative and affirmative quantities:- .
      
    • Example: Tell quickly the result of the numbers three and four, negative or affirmative, taken together; that is, affirmative and negative, or both negative or both affirmative, as separate instances; if thou know the addition of affirmative and negative quantities.
      
    • Negative numbers are denoted by placing a dot above them:- .
      
    • If this were true then 0 times ∞ must be equal to every number n, so all numbers are equal.
      

16. Thomae biography
    
    • In 1866 he submitted work On the introduction of ideal numbers to the University of Gottingen as his habilitation and began lecturing there.
      
    • Thomae was the first to attempt to introduce "trans-Archimedean numbers" but Cantor argued that these were unworthy of the name of magnitude or quantity.
      
    • the whole of pure mathematics is concerned with relations between numbers.
      
    • He then went to construct the rational numbers using Weierstrass's approach, then continued with a construction of the real numbers using the Cauchy sequence type of definition already published by Cantor and Heine.
      
    • Only the positive integers had a concrete existence, while all other numbers were interpreted as signs.
      
    • Following Hankel's ideas, Thomae wrote in his book that these numbers should be:- .
      
    • The formal conception of numbers requires of itself more modest limitations than does the logical conception.
      
    • It does not ask, what are and what shall the numbers be, but it asks, what does one require of numbers in arithmetic.
      
    • The rules of chess are arbitrary; the system of rules for arithmetic is such that by means of simple axioms the numbers can be related to intuitive manifolds, so that they are of essential service in the knowledge of nature.
      
    • History Topics: The real numbers: Stevin to Hilbert .
      

17. Siegel biography
    
    • It extended an idea first noted by Liouville, then pushed forward by Thue who proved that, given a rational number q and any e > 0 there are only finitely many rational numbers p/q (in their lowest terms) such that .
      
    • Siegel improved this by showing that there are only finitely many rational numbers p/q such that if q is an algebraic number of degree n .
      
    • Student numbers rapidly built up after Siegel was appointed.
      
    • It was at this time that the student numbers reached a maximum, then they began to drop again.
      
    • Approximation of algebraic numbers by rationals and applications thereof to Diophantine equations.
      
    • Zeta functions including applications to class numbers.
      
    • Geometry of numbers and its applications to algebraic number theory.
      
    • Hardy-Littlewood method, including Waring-type problems for algebraic numbers.
      
    • Siegel is especially famed for his work on the theory of numbers where he held an eminent role.
      
    • Just now Lang has published another book on algebraic numbers which, in my opinion, is still worse than the former one.
      
    • Siegel enjoyed teaching, however, even elementary courses, and he published textbooks on the theory of numbers, celestial mechanics, and the theory of functions of several complex variables.
      

18. Conway biography
    
    • Having solved the open problem posed by Davenport on writing numbers as the sums of fifth powers, Conway began to become interested in infinite ordinals.
      
    • Another topic for which Conway is famous is discovery of surreal numbers, which again dates from around 1970.
      
    • There is a common belief that complex numbers are the end of the road in the development of numbers and Conway's discovery is the perfect illustration of how even the number systems are part of the continually evolving subject.
      
    • Analysing the situation Conway discovered that certain games behaved like numbers and surreal numbers were born.
      
    • The name surreal numbers was not invented by Conway, however, but by Donald Knuth who was so impressed with Conway's discovery that he wrote Surreal Numbers (1974) in the form of a novelette aimed at introducing the ideas of mathematical research to students.
      
    • Let us mention that, in addition to his innovative contributions to group theory and his creation of surreal numbers mentioned above, he has done leading research in knot theory, number theory, game theory, quadratic forms, coding theory, and tilings.
      
    • On numbers and games (1976):- .
      
    • (with R K Guy) The book of numbers (1996):- .
      

19. Al-Khwarizmi biography
    
    • Early in the book al-Khwarizmi describes the natural numbers in terms that are almost funny to us who are so familiar with the system, but it is important to understand the new depth of abstraction and understanding here [Muhammad ibn Musa Al-Khwarizmi : Algebra (London, 1831).',11)">11]:- .
      
    • Having introduced the natural numbers, al-Khwarizmi introduces the main topic of this first section of his book, namely the solution of equations.
      
    • Squares equal to numbers.
      
    • Roots equal to numbers.
      
    • Squares and roots equal to numbers; e.g.
      
    • Squares and numbers equal to roots; e.g.
      
    • Roots and numbers equal to squares; e.g.
      
    • For example, two applications of "al-muqabala" reduces 50 + 3 x + x2 = 29 + 10 x to 21 + x2 = 7 x (one application to deal with the numbers and a second to deal with the roots).
      
    • In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers.
      
    • The work on arithmetic first introduced the Hindu numbers to Europe, as the very name algorism signifies; and the work on algebra ..
      

20. Euler biography
    
    • Nowhere else could he have been surrounded by such a group of eminent scientists, including the analyst, geometer Jakob Hermann, a relative; Daniel Bernoulli, with whom Euler was connected not only by personal friendship but also by common interests in the field of applied mathematics; the versatile scholar Christian Goldbach, with whom Euler discussed numerous problems of analysis and the theory of numbers; F Maier, working in trigonometry; and the astronomer and geographer J-N Delisle.
      
    • Goldbach asked Euler, in 1729, if he knew of Fermat's conjecture that the numbers 2n + 1 were always prime if n is a power of 2.
      
    • In 1737 he proved the connection of the zeta function with the series of prime numbers giving the famous relation .
      
    • Here the sum is over all natural numbers n while the product is over all prime numbers.
      
    • By 1739 Euler had found the rational coefficients C in ζ(2n) = Cπ2n in terms of the Bernoulli numbers.
      
    • Perhaps more significant than the result here was the fact that he introduced a proof involving numbers of the form a + b√-3 for integers a and b.
      
    • He published his full theory of logarithms of complex numbers in 1751.
      
    • History Topics: Prime numbers .
      
    • History Topics: The real numbers: Stevin to Hilbert .
      

21. Al-Karaji biography
    
    • This shows that for every number composed of two numbers, if we multiple each of them by itself once - since the two extremes are 'one' and 'one' - and if we multiply each one by the other twice - since the intermediate term is 'two' - we obtain the square of this number.
      
    • If we then add 'two' from the second column to ''one' below it we have 'three' which is written under the 'three', then we write 'one' under this 'three'; we thus obtain a third column whose numbers are 'one', 'three', 'three', and 'one'.
      
    • Other results obtained by al-Karaji include summing the first n natural numbers, the squares of the first n natural numbers and the cubes of these numbers.
      
    • He proved that the sum of the first n natural numbers was ½ n(n + 1).
      
    • The sum of the squares of the numbers that follow one another in natural order from one is equal to the sum of these numbers and the product of each of them by its predecessor.
      
    • Al-Karaji also considered sums of the cubes of the first n natural numbers writing (in Rashed and Ahmad's translation, see for example [The development of Arabic mathematics : between arithmetic and algebra (London, 1994).',5)">5]):- .
      
    • If we want to add the cubes of the numbers that follow one another in their natural order we multiply their sum by itself.
      

22. Goldbach biography
    
    • It has been checked by computer for numbers up to at least 4 × 1014.
      
    • 7 (1954), 625-629.',14)">14]) they discuss Fermat numbers, Mersenne numbers, perfect numbers, the representation of natural numbers as a sum of four squares, Waring's problem (which Euler solved before Waring), polynomials representing numerous primes, Fermat's Last Theorem, and the representation of any odd numbers in the form 2n2 + p where p is prime.
      
    • In 1856 Moritz A Stern, a professor of mathematics at Gottingen, found two numbers which could not be written as twice a square plus a prime, namely 5777 and 5993.
      
    • No other examples of numbers failing to satisfy this conjecture of Goldbach seem to be known.
      
    • History Topics: Prime numbers .
      
    • History Topics: The real numbers: Stevin to Hilbert .
      

23. Hensel biography
    
    • In 1897 the Weierstrass method of power-series development for algebraic functions led him to the invention of the p-adic numbers.
      
    • The p-adic numbers can be regarded as a completion of the rational numbers in a different way from the usual completion which leads to the real numbers.
      
    • During the last decade of the 19th century Kurt Hensel started his investigations on p-adic numbers ..
      
    • He was motivated by the analogies of the number field case and the function field case, e.g., by the observation that prime numbers p and linear factors z-c play similar roles in these theories.
      
    • It was not until 1921 that the full potential of the p-adic numbers was demonstrated by Hasse when he formulated the local-global principle.
      
    • He showed, at least for quadratic forms, that an equation has a rational solution if and only if it has a solution in the p-adic numbers for each prime p and a solution in the reals.
      
    • It was in this book that he developed his great idea of p-adic numbers into a systematic theory.
      

24. Brahmagupta biography
    
    • He also gives arithmetical rules in terms of fortunes (positive numbers) and debts (negative numbers):- .
      
    • Positive or negative numbers when divided by zero is a fraction the zero as denominator.
      
    • Zero divided by negative or positive numbers is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator.
      
    • However it is a brilliant attempt to extend arithmetic to negative numbers and zero.
      
    • We give three examples of the methods he presents in the Brahmasphuta siddhanta and in doing so we follow Ifrah in [A universal history of numbers : From prehistory to the invention of the computer (London, 1998).',4)">4].
      
    • Now add the three numbers below the line .
      
    • Brahmagupta gives the sum of the squares of the first n natural numbers as n(n+1)(2n+1)/6 and the sum of the cubes of the first n natural numbers as (n(n+1)/2)2.
      

25. Voronoy biography
    
    • He wrote in his diary (see for example [The St Petersburg School of the Theory of Numbers (Moscow-Leningrad, 1947).',2)">2]):- .
      
    • After graduating from St Petersburg in 1889, writing a dissertation on Bernoulli numbers, Voronoy decided to remain there and work for his teaching qualification.
      
    • In the essay I am now presenting, results from the general theory of algebraic integers are applied to the particular case of numbers depending on the root of an irreducible equation x3 = rx + s.
      
    • Using the form taken by the integers, it is not difficult to find a form embracing all the integral numbers divisible by a given ideal number, or, in other words, to find the ideal corresponding to that given ideal number.
      
    • Later Voronoy worked on the theory of numbers, in particular he worked on algebraic numbers and the geometry of numbers.
      
    • Delone writes in [The St Petersburg School of the Theory of Numbers (Moscow-Leningrad, 1947).',2)">2] about Voronoy's work of 1907-08:- .
      
    • Voronoy's memoir on parallelohedra represents one of the deepest investigations in the geometry of numbers in the world's literature, and the originality of the methods employed in the purely geometrical first part stamps the memoir with the imprint of genius.
      

26. Bennett biography
    
    • Bennett's first paper, entitled On the Residues of Powers of Numbers for any Composite Modulus, Real or Complex, was published in 1892 in the Philosophical Transactions of the Royal Society.
      
    • In the simple cases, when the modulus is a real number which is an odd prime, a power of an odd prime, or double the power of an odd prime, we know that there exist primitive roots of the modulus: that is, that there are numbers whose successive powers have for their rests the complete set of numbers less than, and prime to, the modulus.
      
    • It is also known that in general, when the modulus is any composite number, though primitive roots do not exist, there may be laid down a set of numbers, which will here be called generators, the products of powers of which give the complete set of rests prime to the modulus.
      
    • The principal object of Part I is to investigate the relations which must subsist among any such set of generators; to determine the most general form that they can take; to show how to form any such set of generators, and, conversely, to furnish tests for the efficiency, as generators, of any given set of numbers.
      
    • Other results which are obtained as instrumental in effecting these objects, such as the determination of the number of numbers that belong to any exponent, may also possess independent interest.
      
    • The object of Part II is to make, for complex numbers, an investigation which shall be as nearly as possible parallel to that of Part I for real numbers.
      

27. Ramanujan biography
    
    • He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series.
      
    • He began to study the Bernoulli numbers, although this was entirely his own independent discovery.
      
    • After publication of a brilliant research paper on Bernoulli numbers in 1911 in the Journal of the Indian Mathematical Society he gained recognition for his work.
      
    • He is a young man of quite exceptional capacity in mathematics and especially in work relating to numbers.
      
    • He wrote to Hill on 12 November 1912 sending some of Ramanujan's work and a copy of his 1911 paper on Bernoulli numbers.
      
    • Ramanujan's dissertation was on Highly composite numbers and consisted of seven of his papers published in England.
      
    • Despite many brilliant results, some of his theorems on prime numbers were completely wrong.
      
    • MacMahon had produced tables of the value of p(n) for small numbers n, and Ramanujan used this numerical data to conjecture some remarkable properties some of which he proved using elliptic functions.
      

28. Feldman biography
    
    • Gelfond, Feldman's supervisor, had extended Borel's result to numbers of the form αβ, where α, β are algebraic numbers.
      
    • Feldman proved in his thesis Borel type results (called the measure of transcendence) for logarithms of algebraic numbers, obtaining estimates for the lower bound depending (as did Gelfond) on both the degree of P and the maximum modulus of its coefficients.
      
    • In addition to his work on the measure of transcendence of numbers, Feldman also produced many results strengthening Liouville's theorem on the rational approximation of algebraic numbers.
      
    • For example in 1960 Feldman published two papers The measure of transcendency of the number π and Approximation by algebraic numbers to logarithms of algebraic numbers which were reviewed together by Mahler:- .
      
    • The last part of the book describes Alan Baker's work on linear forms in the logarithms of algebraic numbers and its applications to Diophantine equations and to the determination of imaginary quadratic fields with class number 1 or 2.
      

29. Diophantus biography
    
    • Diophantus, often known as the 'father of algebra', is best known for his Arithmetica, a work on the solution of algebraic equations and on the theory of numbers.
      
    • The reviewer, familiar with the Arabic text of this manuscript, does not doubt that this manuscript is the translation from the Greek text written in Alexandria but the great difference between the Greek books of Diophantus's Arithmetic combining questions of algebra with deep questions of the theory of numbers and these books containing only algebraic material make it very probable that this text was written not by Diophantus but by some one of his commentators (perhaps Hypatia?).
      
    • The reason why there were three cases to Diophantus, while today we have only one case, is that he did not have any notion for zero and he avoided negative coefficients by considering the given numbers a, b, c to all be positive in each of the three cases above.
      
    • Fragments of another of Diophantus's books On polygonal numbers, a topic of great interest to Pythagoras and his followers, has survived.
      
    • One such lemma is that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers, i.e.
      
    • given any numbers a, b then there exist numbers c, d such that a3 - b3= c3 + d3.
      

30. Dedekind biography
    
    • He attended courses by Dirichlet on the theory of numbers, on potential theory, on definite integrals, and on partial differential equations.
      
    • His idea was that every real number r divides the rational numbers into two subsets, namely those greater than r and those less than r.
      
    • Dedekind's brilliant idea was to represent the real numbers by such divisions of the rationals.
      
    • One remarkable piece of work was his redefinition of irrational numbers in terms of Dedekind cuts which, as we mentioned above, first came to him as early as 1858.
      
    • presented a logical theory of number and of complete induction, presented his principal conception of the essence of arithmetic, and dealt with the role of the complete system of real numbers in geometry in the problem of the continuity of space.
      
    • Among other things, he provides a definition independent of the concept of number for the infiniteness or finiteness of a set by using the concept of mapping and treating the recursive definition, which is so important to the theory of ordinal numbers.
      
    • Introduction to Richard Dedekind - the man and the numbers .
      
    • History Topics: The real numbers: Stevin to Hilbert .
      

31. Stevin biography
    
    • He also made a strong plea that all numbers such as square roots, irrational numbers, surds, negative numbers etc should all be treated as numbers and not distinguished as being different in nature.
      
    • Particularly important was Stevin's acceptance of negative numbers but he did not accept the 'new' imaginary numbers and this was to hold back their development.
      
    • History Topics: The real numbers: Pythagoras to Stevin .
      
    • History Topics: The real numbers: Stevin to Hilbert .
      

32. Haselgrove biography
    
    • His dissertation was Some Theorems in the Analytic Theory of Numbers.
      
    • The first was A Note on Fermat's Last Theorem and the Mersenne Numbers in the January/February issue of 1949 and the second was Telepathy Experiment in the October issue of 1950.
      
    • This was A connection between the zeros and the mean values of z(s) (1949) followed by Some theorems in the analytic theory of numbers (1951), On Ingham's Tauberian theorem for partitions (1952), and (with H N V Temperley) Asymptotic formulae in the theory of partitions (1954).
      
    • The conjecture of Polya claims that for every x > 1 there are at least as many numbers less than or equal to x having an odd number of prime factors as there are numbers with an even number of prime factors.
      
    • R S Lehman and W G Spohn had verified the conjecture for all numbers x up to 800,000 but Haselgrove found a counterexample using methods based on those developed by Ingham with the help of computations carried out on the EDSAC 1 computer at Cambridge.
      
    • at Manchester he published papers on Dirichlet functions and the Riemann hypothesis, ray paths in the ionosphere, numerical integration using quasi-random numbers, two-point boundary-value problems, and some geometrical puzzles.
      
    • Student numbers were relatively small at first, but the course provided a source of research students in Numerical Analysis.
      

33. Eudoxus biography
    
    • The method of comparing two lengths x and y by finding a length t so that x = m × t and y = n × t for whole numbers m and n failed to work for lines of lengths 1 and √2 as the Pythagoreans had shown.
      
    • A number of authors have discussed the ideas of real numbers in the work of Eudoxus and compared his ideas with those of Dedekind, in particular the definition involving 'Dedekind cuts' given in 1872.
      
    • corresponds exactly to the modern theory of irrationals due to Dedekind, and that it is word for word the same as Weierstrass's definition of equal numbers.
      
    • It then demonstrates the radical originality, relative to this theory, of the definition of real numbers on the basis of the set of rationals proposed by Dedekind.
      
    • This work developed directly out of his work on the theory of proportion since he was now able to compare irrational numbers.
      
    • It was also based on earlier ideas of approximating the area of a circle by Antiphon where Antiphon took inscribed regular polygons with increasing numbers of sides.
      
    • History Topics: The real numbers: Pythagoras to Stevin .
      

34. Euclid biography
    
    • In particular book seven is a self-contained introduction to number theory and contains the Euclidean algorithm for finding the greatest common divisor of two numbers.
      
    • Book eight looks at numbers in geometrical progression but van der Waerden writes in [Encyclopaedia Britannica.
      
    • Book ten deals with the theory of irrational numbers and is mainly the work of Theaetetus.
      
    • History Topics: Perfect numbers .
      
    • History Topics: Prime numbers .
      
    • History Topics: The real numbers: Pythagoras to Stevin .
      
    • History Topics: The real numbers: Stevin to Hilbert .
      

35. Domninus biography
    
    • The Manual of Introductory Arithmetic studies numbers, means, and proportion.
      
    • an examination of numbers in themselves, an examination of numbers in relation to other numbers, the theory of numbers both in themselves and in relation to others, the theory of means and proportions, and the theory of numbers as figures.
      
    • It is a sketch of the elements of the theory of numbers, very concise and well arranged, and is interesting because it indicates a serious attempt at a reaction against the Introductio arithmetica of Nicomachus and a return to the doctrine of Euclid.
      

36. Yang Hui biography
    
    • Yang also gave formulae for the sum of certain series, for example he found the sum of the squares of the natural numbers from m2 to (m+n)2 and showed that .
      
    • In 1275 two further works by Yang appeared; the Practical mathematical rules for surveying and Continuation of ancient mathematical methods for elucidating strange properties of numbers, both being works of two chapters.
      
    • Here is a problem taken from Chapter 2 of Continuation of ancient mathematical methods for elucidating strange properties of numbers.
      
    • These are the numbers to be found; 6 Wenzhou oranges and 6 green oranges respectively.
      
    • Firstly it is important to realise that he presents them as a good way to interest people in numbers, and he does not claim any magic properties.
      
    • Each circle has a central number and four other numbers, in the north, south, east and west positions, on its circumference.
      
    • Adding the central number and the four numbers on the circumference gives 65 for each of the seven circles.
      

37. Kronecker biography
    
    • Kronecker did not attract great numbers of students.
      
    • We have already indicated that Kronecker's primary contributions were in the theory of equations and higher algebra, with his major contributions in elliptic functions, the theory of algebraic equations, and the theory of algebraic numbers.
      
    • Kronecker believed that mathematics should deal only with finite numbers and with a finite number of operations.
      
    • It appears that, from the early 1870s, Kronecker was opposed to the use of irrational numbers, upper and lower limits, and the Bolzano-Weierstrass theorem, because of their non-constructive nature.
      
    • Another consequence of his philosophy of mathematics was that to Kronecker transcendental numbers could not exist.
      
    • In that year he argued against the theory of irrational numbers used by Dedekind, Cantor and Heine giving the arguments by which he opposed:- .
      
    • Lindemann had proved that π is transcendental in 1882, and in a lecture given in 1886 Kronecker complimented Lindemann on a beautiful proof but, he claimed, one that proved nothing since transcendental numbers did not exist.
      

38. Argand biography
    
    • Argand is famed for his geometrical interpretation of the complex numbers where i is interpreted as a rotation through 90°.
      
    • However, the fact that his name is associated with this geometrical interpretation of complex numbers is only as a result of a rather strange sequence of events.
      
    • The first to publish this geometrical interpretation of complex numbers was Caspar Wessel.
      
    • However, Argand was not a professional mathematician either, so when he published his geometrical interpretation of complex numbers in 1806 it was in a book which he published privately at his own expense.
      
    • In September 1813 Jacques Francais published a work in which he gave a geometric representation of complex numbers, with interesting applications, based on Argand's ideas.
      
    • In this correspondence Jacques Francais and Argand argued in favour of the validity of the geometric representation, while Servois argued that complex numbers must be handled using pure algebra.
      
    • Certainly Argand was the first to state the theorem in the case where the coefficients were complex numbers.
      

39. Cantelli biography
    
    • He proved the strong law of large numbers, a result which was proved independently by Mazurkiewicz.
      
    • On this topic he published Sulla legge dei grandi numeri (On the law of large numbers) in 1916.
      
    • Cantelli, in his work on the law of large numbers, was developing ideas which had been first suggested by Jacob Bernoulli in the 17th century.
      
    • The expression "law of large numbers" was introduced somewhat later by Poisson who studied the weak law of large numbers (as did Chebyshev in his thesis).
      
    • Around the time that Cantelli worked on the law of large numbers, Borel was also interested in the topic.
      

40. Meray biography
    
    • Meray is remembered for having anticipated, clearly and with only minor differences of style, Cantor's theory of irrational numbers, one of the main steps in the arithmetisation of analysis.
      
    • In 1869 Meray was the first to publish an arithmetical theory of irrational numbers in his paper Remarques sur la nature des quantites definies par la condition de servir de limites a des variables donnees.
      
    • Others such as Martin Ohm (1829), Bolzano (1835) and Hamilton (1833) had published work on irrational numbers but none of these earlier authors gave a rigorous account.
      
    • Meray's is the earliest coherent and rigorous theory of the irrational numbers to appear in print.
      
    • Hermann Laurent, in his review, ignored Meray's irrational numbers [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
      
    • History Topics: The real numbers: Stevin to Hilbert .
      

41. Napier biography
    
    • Seeing there is nothing (right well-beloved Students of the Mathematics) that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.
      
    • But amongst all, none more profitable than this which together with the hard and tedious multiplications, divisions, and extractions of roots, doth also cast away from the work itself even the very numbers themselves that are to be multiplied, divided and resolved into roots, and putteth other numbers in their place which perform as much as they can do, only by addition and subtraction, division by two or division by three.
      
    • He described a method of multiplication using "numbering rods" with numbers marked off on them.
      
    • To multiply numbers the bones were placed side by side and the appropriate products read off.
      
    • History Topics: The real numbers: Pythagoras to Stevin .
      

42. Weierstrass biography
    
    • In his 1863/64 course on The general theory of analytic functions Weierstrass began to formulate his theory of the real numbers.
      
    • In his 1863 lectures he proved that the complex numbers are the only commutative algebraic extension of the real numbers.
      
    • Its contents were: numbers, the function concept with Weierstrass's power series approach, continuity and differentiability, analytic continuation, points of singularity, analytic functions of several variables, in particular Weierstrass's "preparation theorem", and contour integrals.
      
    • The standards of rigour that Weierstrass set, defining, for example, irrational numbers as limits of convergent series, strongly affected the future of mathematics.
      
    • History Topics: The real numbers: Stevin to Hilbert .
      

43. Frege biography
    
    • What are numbers? What is the nature of arithmetical truth? .
      
    • The main thrust of this volume was to develop the rules of number theory and in the later volumes Frege intended to extend the work to the real numbers.
      
    • This second volume gives Frege's development of the real numbers which he constructed straight from the integers without taking the route of first defining the rational numbers.
      
    • In particular he strongly criticised Cantor's and Dedekind's theories of irrational numbers.
      
    • History Topics: The real numbers: Stevin to Hilbert .
      

44. Bernoulli Jacob biography
    
    • By 1689 he had published important work on infinite series and published his law of large numbers in probability theory.
      
    • The law of large numbers is a mathematical interpretation of this result.
      
    • The Bernoulli numbers appear in the book in a discussion of the exponential series.
      
    • probability as a measurable degree of certainty; necessity and chance; moral versus mathematical expectation; a priori an a posteriori probability; expectation of winning when players are divided according to dexterity; regard of all available arguments, their valuation, and their calculable evaluation; law of large numbers ..
      
    • Karl Dilcher (A bibliography of books and articles about Bernoulli numbers) .
      
    • Kevin Brown (Bernoulli numbers) .
      

45. Aryabhata I biography
    
    • First we look at the system for representing numbers which Aryabhata invented and used in the Aryabhatiya.
      
    • The higher numbers are denoted by these consonants followed by a vowel to obtain 100, 10000, ..
      
    • In fact the system allows numbers up to 1018to be represented with an alphabetical notation.
      
    • Ifrah in [A universal history of numbers : From prehistory to the invention of the computer (London, 1998).',3)">3] argues that Aryabhata was also familiar with numeral symbols and the place-value system.
      
    • He writes in [A universal history of numbers : From prehistory to the invention of the computer (London, 1998).',3)">3]:- .
      
    • This supposition is based on the following two facts: first, the invention of his alphabetical counting system would have been impossible without zero or the place-value system; secondly, he carries out calculations on square and cubic roots which are impossible if the numbers in question are not written according to the place-value system and zero.
      

46. Chung biography
    
    • In [The man who loved only numbers (London, 1998).
      
    • Wilf writes (see [The man who loved only numbers (London, 1998).
      
    • She had found her first original results in Ramsey theory and it led to the publication of her first paper On the Ramsey numbers N(3, 3, ..
      
    • There she presented a paper On triangular and cyclic Ramsey numbers with k colors which was published in the Proceedings of the Conference in the following year.
      
    • Also in 1975 Chung published her first joint paper with Ron Graham On multicolor Ramsey numbers for complete bipartite graphs which appeared in the Journal of Combinatorial Theory.
      
    • Graham has said (see [The man who loved only numbers (London, 1998).
      

47. Jeans biography
    
    • An excellent mathematics teacher at the school encouraged Jeans' interest in the subject but from the time he was a young child he had shown a fascination with numbers.
      
    • Several stories about his remarkable abilities as a child indicate both an interest and curiosity about numbers and an outstanding memory.
      
    • His interest in numbers was early and deep-seated: he not only factorised cab-numbers, but retained in his memory the numbers that he encountered ..
      
    • Although he would not return again to pure mathematics, Jeans wrote a paper on the theory of numbers while an undergraduate.
      

48. Sierpinski biography
    
    • I was awarded a gold medal by the university for work in a competition on the theory of numbers.
      
    • These books were The theory of irrational numbers (1910), Outline of Set Theory (1912) and The theory of numbers (1912).
      
    • Borel had proved such numbers exist but Sierpinski was the first to give an example.
      
    • He retired in 1960 as professor at the University of Warsaw but he continued to give a seminar on the theory of numbers at the Polish Academy of Sciences up to 1967.
      
    • History Topics: The real numbers: Attempts to understand .
      

49. Al-Umawi biography
    
    • He considers the sum of the first n polygonal numbers, that is 1 + (r - 1)d summed from r = 1 to r = n.
      
    • These sums of polygonal numbers are called pyramidal numbers and al-Umawi then considers the sums of the first n pyramidal numbers.
      
    • None of these results are hard to prove today (try them!) with our understanding of the decimal representation of numbers.
      
    • One has to remember that these results are about decimal representations rather than about numbers themselves and show how an understanding of the decimal system was progressing at a time when Christian Europe (if I may call it that) had little interest in anything beyond the mathematics of the ancient Greeks.
      

50. Liouville biography
    
    • Another important area which Liouville is remembered for today is that of transcendental numbers.
      
    • However his contributions were great and led him to prove the existence of a transcendental number in 1844 when he constructed an infinite class of such numbers using continued fractions.
      
    • In 1851 he published results on transcendental numbers removing the dependence on continued fractions.
      
    • History Topics: The real numbers: Stevin to Hilbert .
      
    • History Topics: The real numbers: Attempts to understand .
      

51. Lehmer Derrick biography
    
    • To give an indication of his father's work during the time that Dick was growing up, let us mention that DNL published Factor table for the first ten millions when Dick was four years old, and List of prime numbers from 1 to 10006721 when he was nine.
      
    • The chapter headings are: Lucas' functions; Tests for primality; Continued fractions; Bernoulli numbers and polynomials; Diophantine equations; Numerical functions; Matrices; Power residues; Analytic number theory; Partitions; Modular forms; Cyclotomy; Combinatorics; Sieves; Equation solving; Computing techniques; and Miscellaneous.
      
    • His most famous monograph was Guide to Tables in the Theory of Numbers.
      
    • A descriptive account is given of existing tables in the theory of numbers; this is set forth in such a way as to indicate clearly what each table contains.
      
    • He was a pioneer in the application of mechanical methods, including digital computers, to the solution of problems in number theory and he talked about some of the methods used to factorise numbers including: factor tables, trial division, Legendre's method, factor stencils, the continued fraction method, Fermat's method, methods based on quadratic forms, and Shanks' method.
      

52. Davenport biography
    
    • There he was influenced by Mordell to become interested in both the geometry of numbers and Diophantine approximation.
      
    • At this time Davenport worked mainly on the geometry of numbers and on Diophantine approximation; he also acquired a lasting interest in problems of packing and covering.
      
    • Davenport worked on number theory, in particular the geometry of numbers, Diophantine approximation and the analytic theory of numbers.
      
    • in recognition of his many distinguished contributions to the theory of numbers.
      

53. Robinson biography
    
    • [Robinson's] theory is based on the metamathematical fact that the system of real numbers is incomplete.
      
    • Thus, there exist extensions of the field of real numbers that possess all the properties of the system of real numbers that are formulated in the lower predicate calculus in terms of some given set of relations.
      
    • A non-standard model for the system of real numbers has the feature of being a non-Archimedean totally ordered field which contains a copy of the real number system.
      
    • This book, which appeared just 250 years after Leibniz's death, presents a rigorous and efficient theory of infinitesimals obeying, as Leibniz wanted, the same laws as the ordinary numbers.
      

54. Vinogradov biography
    
    • While he was successfully preparing for the Master's examination with its very broad syllabus, Vinogradov was working on very difficult problems in the theory of numbers ..
      
    • The importance of trigonometric sums in the theory of numbers was first shown by Weyl in 1916.
      
    • His methods reached their height in Some theorems concerning the theory of prime numbers written in 1937 which provides a partial solution to the Goldbach conjecture.
      
    • For example, in what is probably his most celebrated piece of work [Some theorems concerning the theory of prime numbers (1937)], he was able to combine the bilinear form technique with the Hardy-Littlewood method so as to reduce the Goldbach ternary problem to that of checking a finite number of cases.
      
    • Two of his monographsThe method of trigonometric sums in the theory of numbers, and Special variants of the method of trigonometric sums are also in the book.
      

55. Cramer Harald biography
    
    • He began to produce a series of papers on analytic number theory, and he addressed the Scandinavian Congress of Mathematicians in 1922 on Contributions to the analytic theory of numbers detailing his work on the topic up to that time.
      
    • Cramer's work in prime numbers is put into the context of the history of prime number theory from Eratosthenes to the mid 1990s in [Harald Cramer Symposium, Stockholm, 1993, Scand.
      
    • Such highlights as the probabilistic method in the study of asymptotic properties of prime numbers, the spectral analysis of stationary processes, the mathematical foundation of inference and the fundamental work on risk theory all add up to a brilliant career as a scientist.
      
    • For these mathematicians numbers were a necessary form of human thought, and the science of numbers was a central humanistic discipline with a cultural value of its own, completely independent of its role as auxiliary science in technical or other areas.
      

56. Narayana biography
    
    • In the Ganita Kaumudi Narayana considers the mathematical operation on numbers.
      
    • Like many other Indian writers of arithmetics before him he considered an algorithm for multiplying numbers and he then looked at the special case of squaring numbers.
      
    • One of the unusual features of Narayana's work Karmapradipika is that he gave seven methods of squaring numbers which are not found in the work of other Indian mathematicians.
      
    • The thirteenth chapter of Ganita Kaumudi was called Net of Numbers and was devoted to number sequences.
      

57. Rolle biography
    
    • Find four numbers the difference of any two being a perfect square, in addition the sum of the first three numbers being a perfect square.
      
    • Ozanam stated that the smallest of the four numbers with these properties would have at least 50 digits, but Rolle found four numbers satisfying the conditions with each number having seven digits.
      
    • It seems strange today to realise that this was not the current practice at the time but was in opposition to the ordering of the real numbers used by Descartes and others.
      

58. Roth Klaus biography
    
    • He solved the major open problem of approximating algebraic numbers by rationals in 1955.
      
    • For any irrational number r it is easy to see that there are infinitely many rational numbers a/b with .
      
    • For a given r let μ(r) be the upper bound the exponents e such that there are infinitely many rational numbers a/b with .
      
    • Speaking of Roth's solution to this problem of approximating algebraic numbers Davenport said [2]:- .
      
    • of natural numbers satisfying .
      

59. Cardan biography
    
    • One of the first problems that Cardan hit was that the formula sometimes involved square roots of negative numbers even though the answer was a 'proper' number.
      
    • Indeed Cardan gives precisely the conditions here for the formula to involve square roots of negative numbers.
      
    • Tartaglia by this time greatly regretted telling Cardan the method and tried to confuse him with his reply (although in fact Tartaglia, like Cardan, would not have understood the complex numbers now entering into mathematics):- .
      
    • It is to Cardan's credit that, although one could not expect him to understand complex numbers, he does present the first calculation with complex numbers in Ars Magna.
      

60. Wright biography
    
    • Wright's best known mathematical contribution was his joint authorship of An introduction to the theory of numbers written with Hardy.
      
    • These topics are: prime numbers; congruences and the quadratic reciprocity law; continued fractions; irrational, algebraic and transcendental numbers; quadratic fields; arithmetical functions, their order of magnitude and the Dirichlet or power series which generate them; partitions and representations of numbers as sums of squares, cubes and higher powers; Diophantine approximation; and the geometry of numbers.
      

61. Lehmer Emma biography
    
    • For example around 1930 the Lehmers applied to the Carnegie Institution for funds to construct a computer to factorise numbers.
      
    • To perform the operation with pencil and paper one must start with the million or so numbers among which the solution is known to lie.
      
    • Actually to write out all these numbers is obviously impossible.
      
    • We therefore propose to construct a machine which will canvass a million numbers in about three minutes.
      
    • Another joint project which Lehmer undertook with her husband was assisting Vandiver with his work on Fermat's Last Theorem, and together the Lehmers computed many Bernoulli numbers which Vandiver required in his work.
      

62. Kushyar biography
    
    • He discusses decimal numbers in the main body of the text, relegating sexagesimal numbers to a separate treatment in tables.
      
    • Topics considered include addition and subtraction of decimal numbers followed by multiplication and division of decimal numbers.
      
    • Kushyar gives methods to construct exact square roots, as well as approximate methods to calculate the square roots of non-square numbers.
      

63. Lehmer Derrick N biography
    
    • Lehmer published Factors in 1909, and List of prime numbers from 1 to 10006721 in 1914.
      
    • The history of factor tables really begins in the seventeenth century, starting perhaps with a table by Cataldi (Bologna, 1603), which gave all of the factors of all the numbers up to 750 ..
      
    • Following on from the idea of factor stencils, Lehmer came up with another mechanical device to factor numbers.
      
    • The immediate purpose of the machine is to determine the factors of large numbers which are greater than two millions.
      
    • Meshing with these gears are thirty other gears with a varying number of teeth, depending on the prime numbers from 1 to 127.
      

64. Minkowski biography
    
    • This lecture is particularly interesting, for it contains the first example of the method which Minkowski would develop some years later in his famous "geometry of numbers".
      
    • His most original achievement, however, was his 'geometry of numbers' which he initiated in 1890.
      
    • It gave an elementary account of his work on the geometry of numbers and of its applications to the theories of Diophantine approximation and of algebraic numbers.
      
    • Work on the geometry of numbers led on to work on convex bodies and to questions about packing problems, the ways in which figures of a given shape can be placed within another given figure.
      

65. Al-Banna biography
    
    • He also studied fractional numbers and learnt much of the impressive contributions that the Arabs had made to mathematics over the preceding 400 years.
      
    • Two "firsts" for al-Banna are that he seems to have been the first to consider a fraction as a ratio between two numbers (see [Deuxieme Colloque Maghrebin sur l\'Histoire des Mathematiques Arabes (Tunis, 1990), A97-A109.',12)">12] for more details) and he is the first to use the expression almanakc (in Arabic al-manakh meaning weather) in a work containing astronomical and meteorological data.
      
    • in our opinion, there is something more fundamental than [the Pascal triangle] results; it is precisely the combinatorial appearance of ibn al-Banna's exposition, together with the relation he partially establishes between polygonal numbers and combinations.
      
    • It concern, in the first place, triangular numbers and combinations of p objects in twos, and then polygonal numbers of order 4 and combinations of p objects in threes.
      

66. Richard Jules biography
    
    • Jules Richard is remembered mainly, however, for Richard's paradox involving the set of real numbers which can be defined in a finite number of words.
      
    • Basically the paradox comes about from the fact that the real numbers are uncountable, yet one can only ever describe countably many real numbers using the English language.
      
    • (Actually, of course, Richard used French but since we are writing this biography in English, we will have to explain the paradox in English.) Examples of English descriptions or real numbers are "one third", "the base of natural logs", and "the ratio of the circumference of a circle to its diameter", etc.
      
    • Richard then created a list of all real numbers which could be described in English.
      

67. Turing biography
    
    • In 1936 he published On Computable Numbers, with an application to the Entscheidungsproblem.
      
    • He showed that π was computable, but since only countably many real numbers are computable, most real numbers are not computable.
      
    • It is impossible to decide (using another Turing machine) whether a Turing machine with a given table of instructions will output an infinite sequence of numbers.
      
    • History Topics: The real numbers: Attempts to understand .
      

68. Blum biography
    
    • An inductive inference machine produces, from any enumeration of a partial function, a certain output sequence of numbers.
      
    • Calculus uses real numbers rather than counting numbers because it's measuring things in the real world.
      
    • The theory of computer science deals with counting numbers but not real numbers.
      

69. Al-Haytham biography
    
    • One of them is the canonical method: we multiply the numbers mentioned that divide the number sought by each other; we add one to the product; this is the number sought.
      
    • Another contribution by ibn al-Haytham to number theory was his work on perfect numbers.
      
    • But this partial failure should not eclipse the essential: a deliberate attempt to characterise the set of perfect numbers.
      
    • History Topics: Perfect numbers .
      

70. Vandiver biography
    
    • For example he published two short papers in 1902 A Problem Connected with Mersenne's Numbers and Applications of a Theorem Regarding Circulants.
      
    • Over the next few years he published papers such as Theory of finite algebras (1912), Note on Fermat's last theorem (1914), and Symmetric functions formed by systems of elements of a finite algebra and their connection with Fermat's quotient and Bernoulli's numbers (1917).
      
    • He also worked during the summer with Dickson at Chicago on his classic treatise History of the Theory of Numbers.
      
    • It is his life-long work on Fermat's Last Theorem for which he is best known, but Vandiver also wrote papers on cyclotomic fields, Bernoulli numbers, the reciprocity laws, finite fields, techniques for factorisation, semigroups, semirings, and algebras.
      

71. Tartaglia biography
    
    • For mathematicians of this time there was more than one type of cubic equation and Fior had only been shown by del Ferro how to solve one type, namely 'unknowns and cubes equal to numbers' or (in modern notation) x3 + ax = b.
      
    • As negative numbers were not used this led to a number of other cases, even for equations without a square term.
      
    • In fact Tartaglia had also discovered how to solve one type of cubic equation since his friend Zuanne da Coi had set two problems which had led Tartaglia to a general solution of a different type from that which Fior could solve, namely 'squares and cubes equal to numbers' or (in modern notation) x3 + ax2 = b.
      
    • However, in the early hours of 13 February 1535, inspiration came to Tartaglia and he discovered the method to solve 'squares and cubes equal to numbers'.
      

72. Russell biography
    
    • Like Gottlob Frege, Russell's basic idea for defending logicism was that numbers may be identified with classes of classes and that number-theoretic statements may be explained in terms of quantifiers and identity.
      
    • During the 1950s and 1960s, Russell became something of an inspiration to large numbers of idealistic youth as a result of his continued anti-war and anti-nuclear protests.
      
    • History Topics: The real numbers: Stevin to Hilbert .
      
    • History Topics: The real numbers: Attempts to understand .
      

73. Yano biography
    
    • He soon learnt of the methods concerning curvature and Betti numbers developed by Bochner at Princeton during the war years.
      
    • Not only was he able to attend the International Congress of Mathematicians in Harvard in 1950 but he was also able to publish papers on curvature and Betti numbers.
      
    • Bochner suggested to Yano that he write up all the known results on this topic and this resulted in the joint publication of the monograph Curvature and Betti numbers published by Princeton University Press in 1953.
      
    • Although the book is self-contained it continues to develop the material treated in Yano's earlier books Curvature and Betti numbers and Theory of Lie derivatives and its applications.
      

74. Catalan biography
    
    • He defined the numbers, called today the 'Catalan numbers', while considering the solution of the problem of dissecting a polygon into triangles by means of non-intersecting diagonals.
      
    • Two consecutive whole numbers, other than 8 and 9, cannot be consecutive powers; otherwise said, the equation xm - yn = 1 in which the unknowns are positive integers only admits a single solution.
      
    • Catalan numbers .
      

75. Couturat biography
    
    • For him the actual infinite was a generalisation of number, in the same way that negative numbers, fractions, irrational numbers and complex numbers had all been seen at extending the concept of number.
      
    • Infinite numbers, he claimed, were necessary in order to maintain the continuity of magnitudes.
      

76. Saunderson biography
    
    • Each number from 0 to 9 was represented by the position of a large and small peg in a square array, and numbers with 2, 3 or larger numbers of digits were represented by placing 2, 3 or a larger number of squares in a horizontal row.
      
    • The introduction gives the reader the necessary arithmetical skills to begin the study of algebra, teaching the reader to carry out the standard arithmetical operations, take roots of numbers, calculate with fractions and become skilled in problems of proportion.
      
    • In every right-triangle, if the double product of the legs be either added or subtracted from the square of the hypotenuse, both the sum and the remainder will be square numbers.
      

77. Sankara biography
    
    • The system is based on writing numbers using the letters of the Indian alphabet.
      
    • Let us quote from [A universal history of numbers : From prehistory to the invention of the computer (London, 1998).',1)">1]:- .
      
    • the numerical attribution of syllables corresponds to the following rule, according to the regular order of succession of the letters of the Indian alphabet: the first nine letters represent the numbers 1 to 9 while the tenth corresponds to zero; the following nine letters also receive the values 1 to 9 whilst the following letter has the value zero; the next five represent the first five units; and the last eight represent the numbers 1 to 8.
      

78. Mirsky biography
    
    • During this time he took a passionate interest in the theory of numbers and became a great admirer of Edmund Landau.
      
    • The theory of numbers, where he studied r-free numbers, i.e.
      
    • numbers not divisible by the r th power of any integer.
      

79. Varahamihira biography
    
    • In [A universal history of numbers : From prehistory to the invention of the computer (London, 1998).',1)">1] Ifrah notes that Varahamihira was one of the most famous astrologers in Indian history.
      
    • His work Brihatsamhita (The Great Compilation) discusses topics such as [A universal history of numbers : From prehistory to the invention of the computer (London, 1998).',1)">1]:- .
      
    • He wrote the numbers n in a column with n = 1 at the bottom.
      
    • He then put the numbers r in rows with r = 1 at the left-hand side.
      

80. Wessel biography
    
    • This report already contains Wessel's brilliant mathematical innovation, namely the geometric interpretation of complex numbers.
      
    • Wessel's fame as a mathematician rests solely on this paper, which was published in 1799, giving for the first time a geometrical interpretation of complex numbers.
      
    • Of course it is not unreasonable to call the geometrical interpretation of complex numbers the Argand diagram since it was Argand's work which was influential.
      
    • However more is claimed for Wessel's single mathematical paper than the first geometric interpretation of complex numbers.
      

81. Al-Khazin biography
    
    • Al-Khujandi claimed to have proved that x3 + y3 = z3 is impossible for whole numbers x, y, z which of course is the n = 3 case of Fermat's Last Theorem.
      
    • that what Abu Muhammad al-Khujandi advanced - may God have mercy on him - in his demonstration that the sum of two cubic numbers is not a cube is defective and incorrect.
      
    • In modern notation the problem is given a natural number a, find natural numbers x, y, z so that x2 + a = y2 and x2 - a = z2.
      
    • Al-Khazin proves that the existence of x, y, z with these properties is equivalent to the existence of natural numbers u, v with a = 2uv, and u2 + v2 is a square (in fact u2 + v2 = x2).
      

82. Jordan biography
    
    • Volumes 1 and 2 contain Jordan's papers on finite groups, Volume 3 contains his papers on linear and multilinear algebra and on the theory of numbers, while Volume 4 contains papers on the topology of polyhedra, differential equations, and mechanics.
      
    • The treatise contains the 'Jordan normal form' theorem for matrices, not over the complex numbers but over a finite field.
      
    • Generalising a result of Fuchs on linear differential equations, Jordan was led to study the finite subgroups of the general linear group of n × n matrices over the complex numbers.
      
    • Another generalisation, this time of work by Hermite on quadratic forms with integral coefficients, led Jordan to consider the special linear group of n × n matrices of determinant 1 over the complex numbers acting on the vector space of complex polynomials in n indeterminates of degree m.
      

83. Lowenheim biography
    
    • The result was called a paradox since it was believed that certain sets of axioms characterised the real numbers, and now Lowenheim's result showed that the same axioms must hold in a countable subset of the real numbers.
      
    • If one examines the case of the real numbers more closely, then the axioms for an ordered field are all first-order sentences.
      
    • Lowenheim's result then shows that the real numbers contain a countable ordered field which then cannot satisfy the least-upper-bound axiom which is a second-order sentence.
      

84. Cataldi biography
    
    • He is, however, best known for his work on perfect numbers and on continued fractions.
      
    • His contributions to perfect numbers were made in 1603.
      
    • This gives the perfect numbers 6, 28, 496 and 8128 by taking n = 2, 3, 5, 7 respectively.
      
    • He used no clever tricks, merely checked that these numbers were prime by dividing each by all primes up to their square roots.
      
    • In fact Cataldi calculated a list of all primes up to 750 and a list of the factorisation of all numbers up to 800.
      
    • By showing that 217 - 1 = 131071 and 219 - 1 = 524287 were prime, Cataldi had, in fact, found the sixth and seventh perfect numbers 8589869056 and 137438691328.
      
    • Cataldi found square roots of numbers by use of an infinite series leading to an early investigation into continued fractions.
      
    • History Topics: Perfect numbers .
      
    • History Topics: Prime numbers .
      

85. Chebyshev biography
    
    • In particular the paper he published from his thesis examined Poisson's weak law of large numbers.
      
    • Chebyshev's work on prime numbers included the determination of the number of primes not exceeding a given number, published in 1848, and a proof of Bertrand's conjecture.
      
    • In 1867 he published a paper On mean values which used Bienayme's inequality to give a generalised law of large numbers.
      
    • History Topics: Prime numbers .
      

86. Li Shanlan biography
    
    • Li wrote Duoji bilei (Summing finite series) (published in 1867 as part of his collected works) where, in Chapter 4, he gave fascinating formulae relating binomial coefficients, Stirling numbers, Eulerian numbers and many others.
      
    • ',3)">3] Li's method of writing the sum of the pth powers of the first n natural numbers as sums of binomial coefficients is given.
      
    • Li also wrote on prime numbers.
      

87. Riemann biography
    
    • On one occasion he lent Bernhard Legendre's book on the theory of numbers and Bernhard read the 900 page book in six days.
      
    • Here the sum is over all natural numbers n while the product is over all prime numbers.
      
    • History Topics: Prime numbers .
      

88. Gentzen biography
    
    • The idea of levels, probably first introduced by Weyl, considers number theory as the first level since it deals with the natural numbers, analysis as the second level since it deals with the real numbers, and set theory as the third level where the full extent of Cantor's cardinal and ordinal numbers would be studied.
      
    • History Topics: The real numbers: Attempts to understand .
      

89. Hypsicles biography
    
    • If there are as many numbers as we please beginning from 1 and increasing by the same common difference, then, when the common difference is 1, the sum of all the numbers is a triangular number; when 2 a square; when 3, a pentagonal number [and so on].
      
    • We do not know for certain that Hypsicles wrote a text on polygonal numbers, but it is fairly certain that he did write such a text which has been lost.
      
    • This work on polygonal numbers is related to the ideas on arithmetic progressions that appear in another work by Hypsicles, making it more likely that indeed Hypsicles had indeed done original work on this topic.
      

90. Artin biography
    
    • Artin himself proved that when O is the field of algebraic numbers, the subfield K of real algebraic numbers solves the problem and, moreover, it is the unique solution up to automorphisms of the field O.
      
    • Given any integer g not 1 or -1, and g not a power of some other integer, then Artin conjectured that there are infinitely many prime numbers p such that g is a primitive root modulo p in the sense of Gauss.
      
    • More precisely: the set of those prime numbers has positive density, which can be written down and computed explicitly.
      

91. Cassels biography
    
    • Cassels has worked on every aspect of the theory of numbers, particularly on the theory of rational quadratic forms and local fields.
      
    • His mathematical publications started in about 1947 with a series of papers on the geometry of numbers, in particular papers on theorems of Khinchin and of Davenport, and on a problem of Mahler.
      
    • After further papers on subgroups of infinite abelian groups and normal numbers he wrote a series of eight papers on Arithmetic on curves of genus 1.
      
    • Then in 1959 he published another book, An introduction to the geometry of numbers.
      
    • for his numerous important contributions to the theory of numbers.
      
    • Ian Cassels has made many distinguished contributions to the theory of numbers; possibly his most important work is on the arithmetic of elliptic curves, published in a series of papers between 1959 and 1964.
      
    • His work includes numerous papers on Diophantine Approximation and the Geometry of Numbers, and seminal contributions to the theory of quadratic forms and sums of squares.
      
    • He has written excellent books on Diophantine Approximation, Geometry of Numbers, Algebraic Number Theory and Rational Quadratic Forms.
      

92. Lax Anneli biography
    
    • After much persuasion, I agreed and wrote 'The Lore of Large Numbers', Number 6 in the series currently still in print, but horribly out of date! .
      
    • 'The Geometry of Numbers' was derived essentially from a raw manuscript left incomplete by C D Olds (1912 - 1979) who was a professor of mathematics at San Jose State University.
      
    • In the end I'm sure Anneli would have been pleased with the final result: a fine introduction to the geometry of numbers ..
      
    • The first NML, 'Numbers: Rational and Irrational', has been reprinted 14 times and has sold over 40 000 copies.
      

93. Ore biography
    
    • The paper Systematic computations on amicable numbers was written by Ore in collaboration with J Alanen and J Stemple.
      
    • The first pair of amicable numbers beyond the classical (220; 284) was obtained by Fermat in 1636.
      
    • The computations also produce the perfect numbers < 106.
      
    • Altogether there are 42 pairs of amicable numbers below 106.
      

94. Smithies biography
    
    • In fact he had published Weierstrass's theory of the real numbers in 1975 before he retired.
      
    • It is an interesting account which first shows how Weierstrass constructed the rational numbers from the natural numbers.
      
    • It then shows that essentially the same technique allowed Weierstrass to construct the real numbers from the rationals.
      

95. Robinson Raphael biography
    
    • As an example of another of his early papers let us say a little about The approximation of irrational numbers by fractions with odd or even terms which he published in the Duke Mathematical Journal in 1940.
      
    • The paper looks at a problem first studied by Hurwitz in 1891, namely to approximate an irrational number x by rational numbers A/B subject to the condition of | x - A/B | < 1/mB2 for various values of m.
      
    • He gave his results in Mersenne and Fermat numbers published in the Proceedings of the American Mathematical Society in 1954.
      
    • These showed that these Mersenne numbers were all composite except for the seventeen values: n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, for which 2n - 1 is a prime.
      

96. Erdos biography
    
    • Princeton found him [The man who loved only numbers (London, 1998).',3)">3]:- .
      
    • Another result on prime numbers associated with Erdos is the Prime Number Theorem, namely:- .
      
    • Erdos did receive the Cole Prize of the American Mathematical Society in 1951 for his many papers on the theory of numbers, and in particular for the paper On a new method in elementary number theory which leads to an elementary proof of the prime number theorem published in the Proceedings of the National Academy of Sciences in 1949.
      
    • History Topics: The real numbers: Attempts to understand .
      

97. Hasse biography
    
    • Hensel's work on p-adic numbers was to have a major influence on the direction of Hasse's research.
      
    • In October 1920 Hasse discovered the 'local-global' principle which shows that a quadratic form that represents 0 non-trivially over the p-adic numbers for each prime p, and over the real numbers, represents 0 non-trivially over the rationals.
      

98. Sridhara biography
    
    • Following this algorithms are given for carrying out the elementary arithmetical operations, squaring, cubing, and square and cube root extraction, carried out with natural numbers.
      
    • After giving the rules for computing with natural numbers, Sridhara gives rules for operating with rational fractions.
      
    • There are sections of the book devoted to arithmetic and geometric progressions, including progressions with a fractional numbers of terms, and formulae for the sum of certain finite series are given.
      

99. Sokhotsky biography
    
    • His lectures, especially on higher algebra, the theory of numbers, and the theory of definite integrals, were extremely successful.
      
    • Other topics which Sokhotsky studied included Zolotarev's theory of divisibility of algebraic numbers in The application of the principle of the greatest divisor to the theory of divisibility of algebraic numbers (1898).
      

100. Lawson biography
     
     • In this field, the lecturer said that there were three unsatisfactory features of the present system, the neglect of form, which he regarded as the fundamental thing in algebra; defects in the treatment of the laws of formal algebra; and the difficulties experienced in applying the now commonly accepted treatment of negative numbers.
       
     • Mr Lawson gave an account of his method of introducing the idea of negative numbers to his pupils, and criticised some of the methods used in the text-books.
       
     • In the subsequent discussion, in which a number of members of the teaching profession took part, most attention was directed to the question of negative numbers, and the lecturer's views met with some criticism, to which he replied.
       

101. Chuquet biography
     
     • The first part deals with arithmetic and includes work on fractions, progressions, perfect numbers, proportion etc.
       
     • In this work negative numbers, used as coefficients, exponents and solutions, appear for the first time.
       
     • Zero is used and his rules for arithmetical operations includes zero and negative numbers.
       

102. Bolzano biography
     
     • Bolzano opens this notebook of Miscellanea mathematica with notes on irrational and transcendental numbers and functions.
       
     • But he was reading and recording his ideas on a host of other subjects as well, including the problem of how best to approach the proper mathematical understanding of zero; Legendre's work on surfaces, convexity, concavity, and conditions for congruity; analysis of other geometric concepts, including lengths, areas, volumes, and spheres; trigonometric formulas and spherical trigonometry; imaginary and exponential numbers; definition of the differential and discussion of the infinite and various opinions about it, as well as aspects of maxima and minima.
       
     • History Topics: The real numbers: Stevin to Hilbert .
       

103. Goodstein biography
     
     • Goodstein then did research at Cambridge on transfinite numbers under Littlewood's supervision.
       
     • he presided over an expansion from a mathematics staff of six to 23 at the time of his retirement in 1977, with a corresponding increase in student numbers.
       
     • Goodstein worked on mathematical logic, in particular ordinal numbers, recursive arithmetic, analysis, and the philosophy of mathematics.
       

104. Zermelo biography
     
     • can be put in 1-1 correspondence with the natural numbers) or has the cardinality of the continuum (i.e.
       
     • can be put in 1-1 correspondence with the real numbers).
       
     • The set of natural numbers with the usual ordering is therefore a well ordered set but the set of integers is not well ordered with the usual ordering since the subset of negative integers has no least element.
       

105. Francais Jacques biography
     
     • In September 1813 Francais published a work in which he gave a geometric representation of complex numbers with interesting applications.
       
     • Although Wessel had published an account of the geometric representation of complex numbers in 1799, and then Argand had done so again in 1806, the idea was still little known among mathematicians.
       
     • In this argument Francais and Argand believed in the validity of the geometric representation, while Servois argued that complex numbers must be handled using pure algebra.
       

106. Xenocrates biography
     
     • Diogenes Laertius gives the titles of two mathematics books by Xenocrates, namely On numbers and The theory of numbers.
       
     • He agreed with Pythagoras regarding the importance of numbers in philosophy and attributed to Pythagoras an atomic view of acoustics where sound, perceived as a single entity, consists of discrete sounds.
       

107. Ferro biography
     
     • Firstly, in the middle of the 16th century in Europe, zero was not in use; secondly negative numbers were not in use; and thirdly there was no understanding of a quadratic having two roots.
       
     • Mathematicians in the time of del Ferro knew that the problem of solving the general cubic could be reduced to solving the two cases x3 + mx = n and x3 = mx + n, where m and n are positive numbers.
       
     • On unknowns and cubes equal to numbers.
       

108. Mahavira biography
     
     • I glean from the great ocean of the knowledge of numbers a little of its essence, in the manner in which gems are picked from the sea, gold from the stony rock and the pearl from the oyster shell; and I give out according to the power of my intelligence, the Sara Samgraha, a small work on arithmetic, which is however not small in importance.
       
     • He examined methods of squaring numbers which, although a special case of multiplying two numbers, can be computed using special methods.
       

109. Ollerenshaw biography
     
     • Kathleen was good at mathematics, and had always found pleasure in patterns and numbers.
       
     • At the age of nine, a new headmistress who had studied mathematics at Cambridge increased her passion for numbers, insisting on [1]:- .
       
     • Critical lattices relate to whole numbers in two or more dimensions and lead, by geometrical methods, to solutions concerned with 'close packing', for example, how best to stack tins in a cupboard or oranges in a box.
       

110. Hilbert biography
     
     • Hilbert's problems included the continuum hypothesis, the well ordering of the reals, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of Dirichlet's principle and many more.
       
     • History Topics: The real numbers: Stevin to Hilbert .
       
     • History Topics: The real numbers: Attempts to understand .
       

111. Abu'l-Wafa biography
     
     • Abu'l-Wafa's text is of this second type with no numerals; all the numbers are written in words and all calculations are performed mentally.
       
     • Of particular interest is the reference to negative numbers in Part II of Abu'l-Wafa's treatise, and this particular aspect is studied in detail in [Istor.-Mat.
       
     • This seems to be the only place that negative numbers have been found in medieval Arabic mathematics.
       

112. Mersenne biography
     
     • He tried to find a formula that would represent all primes but, although he failed in this, his work on numbers of the form .
       
     • History Topics: Perfect numbers .
       
     • History Topics: Prime numbers .
       

113. Ferrers biography
     
     • Prove that the number of ways in which any number x can be composed of n numbers (not necessarily different from each other), is equal to the number of ways in which x can be composed of n and numbers not exceeding n, the order in which the numbers occur not being considered.
       

114. Hamilton biography
     
     • On 4 November 1833 Hamilton read a paper to the Royal Irish Academy expressing complex numbers as algebraic couples, or ordered pairs of real numbers.
       
     • History Topics: The real numbers: Stevin to Hilbert .
       

115. Vijayanandi biography
     
     • Many Indian astronomers proposed different values for these integral numbers of revolutions.
       
     • However Vijayanandi changed these numbers to 488211 and 232234.
       
     • The reasons for giving the new numbers is unclear.
       

116. Hemchandra biography
     
     • The answer lies in his contribution to the Fibonacci numbers which was made fifty years before Fibonacci wrote Liber Abaci with its famous rabbit problem.
       
     • Before we rush to try to change the name of the Fibonacci numbers into Hemchandra number it is worth noting that Gopala had studied these numbers in about 1135 and Indian mathematicians as early as the 7th century had looked at sequences which are produced by the familiar Fibonacci rule.
       

117. Fogels biography
     
     • In particular he showed that if any countable set has an arithmetic where the elements have unique decompositions into primes, then it is isomorphic to the arithmetic of the natural numbers.
       
     • These restrict the tools available so that only rational numbers with bounded denominators can occur in a proof, which also cannot use differentiation and integration and other infinite tools.
       
     • However, it presented some new interesting connections of the Riemann hypothesis with the theory of prime numbers.
       

118. Tarski biography
     
     • In A decision method for elementary algebra and geometry Tarski showed that the first-order theory of the real numbers under addition and multiplication is decidable which is in contrast, in a way which is really surprising to non-experts, to the results of Godel and Church who showed that the first-order theory of the natural numbers under addition and multiplication is undecidable.
       
     • Cardinal Algebras presents a study of algebras satisfying certain properties which capture the arithmetic of cardinal numbers.
       

119. Linnik biography
     
     • Returning to Leningrad State University, he submitted his thesis Representation of Big Numbers by Positive Ternary Quadratic Forms and, due to the high quality of the work, he was awarded the higher degree of D.Sc.
       
     • From that time on he undertook research in three areas, namely probability, mathematical statistics and the analytic theory of numbers.
       
     • An important feature of the method used in this paper, which was largely responsible for its success, is the use of arguments from the study of trigonometric sums in the theory of numbers.
       

120. Hankel biography
     
     • He worked on the theory of complex numbers, the theory of functions and the history of mathematics.
       
     • Beginning with a revised statement of George Peacock's principle of permanence of formal laws, he developed complex numbers as well as such higher algebraic systems as Mobius' barycentric calculus, some of Hermann Grassmann's algebras, and W R Hamilton's quaternions.
       
     • History Topics: The real numbers: Stevin to Hilbert .
       

121. Riesz Marcel biography
     
     • W H J Fuchs, reviewing [Marcel Riesz: Clifford Numbers and Spinors , (Kluwer, 1993).',2)">2], writes about how Riesz's interests developed through the 1920s:- .
       
     • W H J Fuchs, reviewing [Marcel Riesz: Clifford Numbers and Spinors , (Kluwer, 1993).',2)">2], writes:- .
       
     • He gave an important series of lectures Clifford numbers and spinors at the University of Maryland between October 1957 and January 1958.
       

122. Mathews biography
     
     • In his two volume work Theory of numbers (1892) topics covered included Gauss's theory of quadratic forms and their development by mathematicians such as Dirichlet, Eisenstein and Smith.
       
     • The book also discusses prime numbers and Riemann's memoir on primes but, since it was written two or three years before the prime number theorem was proved, this part of the work became dated rather quickly.
       
     • In addition to his treatises and many papers on the classical theory of numbers, Mathews also wrote some articles for Encyclopaedia Britannica, in particular writing the article on universal algebra and the one on number.
       

123. Al-Samawal biography
     
     • Al-Samawal could not have described arithmetic operations on powers of the unknown without having developed an understanding of negative numbers.
       
     • Multiplication of negative numbers was also completely understood by al-Samawal.
       
     • Book 3 contains a description of how to carry out arithmetic with irrational numbers.
       

124. Blichfeldt biography
     
     • Some of the many topics that he covered were diophantine approximations, orders of linear homogeneous groups, theory of geometry of numbers, approximate solutions of the integers of a set of linear equations, low-velocity fire angle, finite collineation groups, and characteristic roots.
       
     • Blichfeldt wrote papers on the geometry of numbers and he has an important book Finite Collineation Groups.
       
     • In [Yearbook : surveys of mathematics 1980 (Mannheim, 1980), 9-41.',3)">3] Hlawka looks at Blichfeldt's contributions to the geometry of numbers, in particular looking at Blichfeldt's principle.
       

125. Aitken biography
     
     • Aitken had an incredible memory (he knew π to 2000 places) and could instantly multiply, divide and take roots of large numbers.
       
     • Familiarity with numbers acquired by innate faculty sharpened by assiduous practice does give insight into the profounder theorems of algebra and analysis.
       
     • For the latter Professor Aitken would ask for members of the class to give him numbers for which he would then write down the reciprocal, the square root, the cube root or other appropriate expression.
       

126. Zeuthen biography
     
     • In this work Zeuthen adhered closely to Chasles's theory of the characteristics of conic systems but also presented new points of view: for the elementary systems under consideration, he first ascertained the numbers for point or line conics in order to employ them to determine the characteristics.
       
     • Caveing, in [Centaurus 38 (2-3) (1996), 277-292.',3)">3], looks at Zeuthen's ideas on the discovery of irrational numbers.
       
     • History Topics: The real numbers: Pythagoras to Stevin .
       

127. Fermat biography
     
     • This topic did not interest Huygens but Fermat tried hard and in New Account of Discoveries in the Science of Numbers sent to Huygens via Carcavi in 1659, he revealed more of his methods than he had done to others.
       
     • The handicap imposed by the awkward notations operated less severely in Fermat's favourite field of study, the theory of numbers, but here, unfortunately, he found no correspondent to share his enthusiasm.
       
     • History Topics: Prime numbers .
       

128. La Roche biography
     
     • Here is his description of numbers, powers and the unknown (a thing).
       
     • And a number may be considered as a continuous quantity, in other words, a linear number, which may be designated a thing or as primary, and such numbers are marked by the apposition of unity above them in this manner 121 or 131 , etc., or such numbers are indicated also by a character after them, like 12.p.
       

129. Weyl biography
     
     • Also over this period Weyl also made contributions on the uniform distribution of numbers modulo 1 which are fundamental in analytic number theory.
       
     • These include Elementary Theory of Invariants (1935), The classical groups (1939), Algebraic Theory of Numbers (1940), Philosophy of Mathematics and Natural Science (1949), Symmetry (1952), and The Concept of a Riemannian Surface (1955).
       
     • History Topics: The real numbers: Attempts to understand .
       

130. Pauli biography
     
     • He is best known for the Pauli exclusion principle , proposed in 1925, which states that no two electrons in an atom can have the same four quantum numbers.
       
     • He found that four quantum numbers are in general needed in order to define the energy state of an electron.
       
     • Three quantum numbers only can be related to the revolution of the electron round the nucleus.
       

131. Kummer biography
     
     • The clarity and vividness of his presentations brought him great numbers of students - as many as 250 were counted at his lectures.
       
     • In 1843 Kummer, realising that attempts to prove Fermat's Last Theorem broke down because the unique factorisation of integers did not extend to other rings of complex numbers, attempted to restore the uniqueness of factorisation by introducing 'ideal' numbers.
       

132. Cauchy biography
     
     • He achieved real fame however when he submitted a paper to the Institute solving one of Fermat's claims on polygonal numbers made to Mersenne.
       
     • History Topics: The real numbers: Stevin to Hilbert .
       
     • History Topics: The real numbers: Attempts to understand .
       

133. Rankin biography
     
     • He published four papers on The difference between consecutive prime numbers between this time and 1950.
       
     • Rankin wrote over 100 research papers, mostly on the theory of numbers and the theory of functions.
       
     • We should emphasise that his remarkable contributions to the theory of numbers have played a majr part in the modern developemnt of the topic.
       

134. Kruskal Martin biography
     
     • Kruskal's later work studied soliton equations, asymptotic analysis, and surreal numbers.
       
     • Analysing asymptotic series also led Kruskal to become interested in surreal numbers, generalisations of real numbers introduced by John Conway.
       

135. Faulhaber biography
     
     • Faulhaber did not discover the Bernoulli numbers but Jacob Bernoulli refers to Faulhaber in Ars Conjectandi published in Basel in 1713, eight years after Jacob Bernoulli died, where the Bernoulli numbers (so named by De Moivre) appear.
       
     • One cannot help thinking that nobody has ever checked these numbers since Faulhaber himself wrote them down, until today.
       

136. Briggs biography
     
     • This gave the logarithms of the natural numbers from 1 to 20,000 and 90,000 to 100,000 computed to 14 decimal places.
       
     • In this book Briggs suggested that the logs of the missing numbers might be computed by a team of people and he even offered to supply specially designed paper for the purpose.
       
     • The completed tables were printed at Gouda, in the Netherlands, in 1628 in an edition by Vlacq in which Vlacq had added the logarithms of the natural numbers from 20,000 to 90,000.
       

137. Al-Biruni biography
     
     • He then studies the Indian systems of writing and numbers before going on to examine the geography of the country.
       
     • These include: theoretical and practical arithmetic, summation of series, combinatorial analysis, the rule of three, irrational numbers, ratio theory, algebraic definitions, method of solving algebraic equations, geometry, Archimedes' theorems, trisection of the angle and other problems which cannot be solved with ruler and compass alone, conic sections, stereometry, stereographic projection, trigonometry, the sine theorem in the plane, and solving spherical triangles.
       
     • Al-Biruni displayed the results as combinations of integers and numbers of the form 1/n, n = 2, 3, 4, ..
       

138. Hildebrant biography
     
     • The Real Numbers; 9.
       
     • Ordinal Numbers; 10 Cardinal Numbers.
       

139. Stirling biography
     
     • Stirling numbers of the first kind .
       
     • Stirling numbers of the second kind .
       
     • Bell numbers .
       

140. Feigenbaum biography
     
     • I love numbers and always as an amusement, and more seriously than that, invented new algorithms to calculate them.
       
     • For someone who cares for numbers, much of the tedium was eliminated.
       
     • When Feigenbaum first found 4.669 in August 1975, which he only found to three places due to the limit of the accuracy of his HP65, he spend some time trying to see if it was a simple combination of 'well-known' numbers.
       

141. Theodorus biography
     
     • Theodorus is remembered by mathematicians for his contribution to the development of irrational numbers and it is this aspect of his work which Plato describes (see for example [A History of Greek Mathematics I (Oxford, 1921), 203-204, 209-212.',5)">5]):- .
       
     • This proof generalises easily (for a modern mathematicians thinking in terms of numbers rather than lengths) to show √n is irrational for any non-square n.
       
     • History Topics: The real numbers: Pythagoras to Stevin .
       

142. Honda biography
     
     • Honda's next three papers all considered the problem of class numbers of algebraic number fields.
       
     • Further work on the class numbers of algebraic number fields saw Honda prove that there are infinitely many real quadratic fields whose class numbers are divisible by 3 and also to classify those n for which the class number of Q(3√n) is a multiple of 3.
       

143. Al-Kindi biography
     
     • Al-Kindi wrote many works on arithmetic which included manuscripts on Indian numbers, the harmony of numbers, lines and multiplication with numbers, relative quantities, measuring proportion and time, and numerical procedures and cancellation.
       

144. Chowla biography
     
     • He wrote on additive number theory (lattice points, partitions, Waring's problem), analysis, Bernoulli numbers, class invariants, definite integrals, elliptic integrals, infinite series, the Weierstrass approximation theorem), analytic number theory (Dirichlet L-functions, primes, Riemann and Epstein zeta functions), binary quadratic forms and class numbers, combinatorial problems (block designs, difference sets, Latin squares), Diophantine equations and Diophantine approximation, elementary number theory (arithmetic functions, continued fractions, and Ramanujan's tau function), and exponential and character sums (Gauss sums, Kloosterman sums, trigonometric sums).
       
     • Among a long list of other results we mention just a very few such as his generalisation of Wolstenholme's theorem; his work on classes of quintics not soluble by radicals; his closed form for the Bernoulli numbers; and his work on the length of the period of the continued fraction expansion of √N.
       

145. Hammersley biography
     
     • Hammersley published a variety of papers in 1951 including A theorem on multiple integrals, On a certain type of integral associated with circular cylinders, The sums of products of the natural numbers, and The total length of the edges of the polyhedron.
       
     • One of the areas which typified Hammersley's mathematical contributions was in Monte Carlo methods, a technique to estimate a quantity through computations involving random numbers.
       

146. Lah biography
     
     • The binding coefficients L(n, k) = n! (n-1)!/(k-1)!/(n-k)!/k! were called unsigned Lah numbers by J Riordan in his book An Introduction to Combinatorial Analysis (1958).
       
     • The unsigned Lah numbers L(n, k) have an interesting combinatorial interpretation.
       

147. Shnirelman biography
     
     • Using these ideas of compactness of a sequence of natural numbers he was able to prove a weak form of the Goldbach conjecture showing that every number is the sum of ≤ 20 primes.
       
     • Later significant contributions by Shnirelman include his two papers On the additive properties of numbers, and On addition of sequences published in 1940 after his death.
       

148. Robinson Julia biography
     
     • In her thesis Definability and decision problems in arithmetic Robinson proved that the arithmetic of rational numbers is undecidable by giving an arithmetical definition of the integers in the rationals.
       
     • In addition to work on Hilbert's Tenth Problem, Robinson also wrote other important mathematics papers: on general recursive functions (1950), on primitive recursive functions (1955), on the undecidability of algebraic rings and fields (1959) and on decision problems for algebraic rings in 1962 in which she showed that rings of integers of various fields of algebraic numbers are undecidable.
       

149. Pappus biography
     
     • Book I covered arithmetic (and is lost) while Book II is partly lost but the remaining part deals with Apollonius's method for dealing with large numbers.
       
     • The method expresses numbers as powers of a myriad, that is as powers of 10000.
       

150. Rychlik biography
     
     • He did excellent work on algebra and number theory, for example he generalised Hensel's ideas on g-adic numbers in 1914, later approaching them via sequences and limits unlike the 'generalised decimal expansion' approach of Hensel.
       
     • Other works by Rychlik on Bolzano from this later period of his research career include Theory of real numbers in the manuscripts left by Bolzano (Czech) (1956), Theorie der reellen Zahlen im Bolzano's handschriftlichen Nachlasse (1957), Betrachtungen aus der Logik im Bolzano's handschriftlichen Nachlasse (Czech) (1958), Betrachtungen aus der Logik in Bolzanos handschriftlichem Nachlasse (1958), La theorie des nombres reels de Bolzano d'apres ses manuscrits inedits (Russian) (1958), and Theorie der reellen Zahlen in Bolzanos handschriftlichem Nachlasse (1962).
       

151. Lucas biography
     
     • He gave the well-known formula for the Fibonacci numbers .
       
     • History Topics: Prime numbers .
       

152. Kulik biography
     
     • His as yet unpublished manuscript is the result of a twenty-year hobby, and covers all numbers up to 100,000,000.
       
     • Similar statements are made in many other books, see for example [History of the theory of numbers I (Washington, D.C., 1919), 351-352.',2)">2].
       

153. Bellavitis biography
     
     • According to Bellavitis, the plane does not just provide a means to represent complex numbers.
       
     • Starting in 1832 Bellavitis developed geometrically the algebra of complex numbers.
       

154. Cole biography
     
     • His main research contributions are to number theory, in particular to prime numbers, and to group theory.
       
     • His contributions to factoring large numbers was published in 1903.
       

155. Ahmes biography
     
     • The Recto contains division of 2 by the odd numbers 3 to 101 in unit fractions and the numbers 1 to 9 by 10.
       

156. Behrend biography
     
     • In the same year of 1948 he also published Generalization of an inequality of Heilbronn and Rohrbach and Some remarks on the distribution of sequences of real numbers, with Some remarks on the construction of continuous non-differentiable functions being published in the Proceedings of the London Mathematical Society in the following year.
       
     • In A contribution to the theory of magnitudes and the foundations of analysis (1956) Behrend characterised the additive semigroup of positive real numbers, the "magnitudes" of the title.
       

157. Drach biography
     
     • In 1945 Drach published Sur quelques points de theorie des nombres et sur la theorie generale des courbes algegriques in which he used the method of descent to prove theorems concerning numbers represented by the sums of two, three and four squares and by the sum of three triangular numbers.
       

158. Collatz biography
     
     • The problem asks if, for every starting value m, the sequence a(i) always reaches 1? The problem remains unsolved, but before you try a few small numbers yourself looking for a counterexample, let us say that the conjecture has been verified for all numbers m up to about 1014 .
       

159. Ulam biography
     
     • While Ulam was at Los Alamos, he developed the 'Monte-Carlo method' which searched for solutions to mathematical problems using a statistical sampling method with random numbers.
       
     • Ulam's writing include A collection of mathematical problems (1960), Sets numbers and universes (1974) and Adventures of a Mathematician (1976).
       

160. Budan de Boislaurent biography
     
     • Budan is considered an amateur mathematician and he is best remembered for his discovery of a rule which gives necessary conditions for a polynomial equation to have n real roots between two given numbers.
       
     • Budan's goal was to solve Lagrange's problem - between which real numbers do real roots lie? - purely by methods of elementary arithmetic.
       

161. Paman biography
     
     • He next defines the "first State of x" which would in today's terminology be a neighbourhood of 0 in the positive real numbers.
       
     • Paman's explanation looks like our present definition of infimum and supremum, but whereas our infimum and supremum given a set (of real numbers) gives a real number, the Maximinus and Minimajus of a quantity is a quantity of the same kind.
       

162. Widman biography
     
     • He considered computation with irrational numbers and polynomials to be part of algebra, preparing his students for this study by first introducing them to fractions and proportion.
       
     • The book consists of three parts: the first section is on counting with whole numbers, the second is on proportion, while the third section is on geometry.
       

163. Eisenstein biography
     
     • He was working on a variety of topics at this time including quadratic forms and cubic forms, the reciprocity theorem for cubic residues, quadratic partition of prime numbers and reciprocity laws.
       
     • Despite his health problems Eisenstein published one treatise after another on quadratic partition of prime numbers and reciprocity laws.
       

164. Weldon biography
     
     • The contention 'that numbers mean nothing and do not exist in Nature' is a very serious thing, which will have to be fought.
       
     • Biometrika will include (a) memoirs on variation, inheritance, and selection in animals and plants, based upon the examination of statistically large numbers of specimens (this will of course include statistical investigations of anthropometry); (b) those developments of statistical theory which are applicable to biological problems; (c) numerical tables and graphical solutions tending to reduce the labour of statistical arithmetic; (d) abstracts of memoirs, dealing with these subjects, which are published elsewhere; and (e) notes on current biometric work and unsolved problems.
       

165. Daubechies biography
     
     • The numbers became very large very quickly but I would keep going quite a while.
       
     • It was fascinating, again, to see how fast these numbers grew.
       

166. Bienayme biography
     
     • Bienayme published the Bienayme-Chebyshev inequality which was used to give a very simple and precise demonstration of the generalised law of large numbers.
       
     • He argued with Cauchy over the least squares method and, in 1842, he criticised Poisson's law of large numbers.
       

167. Jeffery Ralph biography
     
     • He published a paper based on the results of his thesis in the Annals of Mathematics, then published further papers such as The uniform approximation of a summable function by step functions (1931), Non-absolutely convergent integrals with respect to functions of bounded variation (1932), Relative summability (1932), Sets of k-extent in n-dimensional space (1933), and Derived numbers with respect to functions of bounded variation (1934).
       
     • The author states that his main purpose in this book is to present the contents of chapter VI (The inversion of derivatives) and of chapter VII (Derived numbers and derivatives).
       

168. Qin Jiushao biography
     
     • The novelty here is that the coefficients are not numbers but are functions of lengths in the figure which are left as unspecified.
       
     • However, Qin is happy to look at problems where the numbers concerned are rational.
       

169. Koksma biography
     
     • One then finds a discussion of Minkowski's analysis, his 'Geometry of Numbers' and applications to homogeneous and non-homogeneous linear forms.
       
     • The distribution of real numbers (mod 1) covers two chapters and one finds accounts of the investigations of Weyl, Vinogradov, van der Corput, and others.
       

170. Simson biography
     
     • In 1753 Simson noted that, as the Fibonacci numbers increased in magnitude, the ratio between adjacent numbers approached the golden ratio, whose value is .
       

171. Lang biography
     
     • Other books by Lang include Introduction to algebraic geometry (1958), Abelian varieties (1959), Diophantine geometry (1962), Introduction to differentiable manifolds (1962), Algebraic numbers (1964), Linear algebra (1966), Introduction to diophantine approximations (1966), Introduction to transcendental numbers (1966), Algebraic structures (1967), Algebraic number theory (1970), Introduction to algebraic and abelian functions (1972), Differential manifolds (1972), Elliptic functions (1973), SL(R) (1975), Introduction to modular forms (1976), Complex analysis (1977), Cyclotomic fields (1978), Elliptic curves: Diophantine analysis (1978), Undergraduate analysis (1983), Complex multiplication (1983), Riemann-Roch algebra (1985), The beauty of doing mathematics.
       

172. Mordell biography
     
     • Together with Davenport and Mahler, Mordell initiated great advances in the geometry of numbers while he held the Chair of Pure Mathematics at Manchester.
       
     • for his distinguished researches in pure mathematics, especially for his discoveries in the theory of numbers.
       

173. Jordanus biography
     
     • He also used letters to replace numbers and was able to state general algebraic theorems but this early use of algebraic notation was not used by subsequent writers.
       
     • For example, separate 10 into two numbers whose product is 21.
       

174. Khinchin biography
     
     • Around 1922 Khinchin took up new mathematical interests when he began to study the theory of numbers and probability theory.
       
     • With these ideas he also strengthened the law of large numbers due to Borel.
       

175. Leibniz biography
     
     • In this work Leibniz aimed to reduce all reasoning and discovery to a combination of basic elements such as numbers, letters, sounds and colours.
       
     • Leibniz discussed logarithms of negative numbers with Johann Bernoulli, see [Akten des II.
       

176. Ricci Giovanni biography
     
     • During that time he published work on the Goldbach conjecture which concerns writing numbers as the sum of primes, and also on Hilbert's Seventh Problem which asks whether or not aő was transcendental when a and b are algebraic.
       
     • We will look briefly, however, at some of his interesting expository articles: Figure, reticoli e computo di nodi (1948) discussed classical lattice-point problems and puts these in historical perspective; La differenza di numeri primi consecutivi (1952) looks at the the history of the various problems concerned with the difference between consecutive primes; and Aritmetica additiva: aspetti e problemi (1954) is an expository account of classical and modern results in the additive theory of numbers.
       

177. Godel biography
     
     • It was an injustice which infuriated Godel; in fact he always took such injustices as personal even although large numbers suffered in the same way.
       
     • History Topics: The real numbers: Attempts to understand .
       

178. Babbage biography
     
     • The store was to hold 1000 numbers each of 50 digits, but Babbage designed the analytic engine to effectively have infinite storage.
       
     • elaborations on the points made by Menabrea, together with some complicated programs of her own, the most complex of these being one to calculate the sequence of Bernoulli numbers.
       

179. Poincare biography
     
     • the Diophantine problem of finding the points with rational coordinates on a curve f (x, y) = 0, where the coefficients of f are rational numbers.
       
     • History Topics: The real numbers: Attempts to understand .
       

180. Grosswald biography
     
     • He wrote a number of important books: Topics from the theory of numbers (1966), Bessel polynomials (1978), Dedekind sums (written jointly with Hans Rademacher) (1972), and Representations of integers as sums of squares (1985).
       
     • First we look at Topics from the theory of numbers.
       

181. Zeckendorf biography
     
     • Eduourd Zeckendorf was an amateur mathematician whose name is given to the property that every positive integer can be represented uniquely as the sum of non-consecutive Fibonacci numbers, the sequence defined by .
       
     • This is called Zeckendorf's theorem, and the subsequence of Fibonacci numbers which add up to a given integer is called its Zeckendorf representation.
       

182. Archimedes biography
     
     • He invented a system for expressing large numbers.
       
     • The Sandreckoner is a remarkable work in which Archimedes proposes a number system capable of expressing numbers up to 8 × 1063in modern notation.
       

183. Clausen biography
     
     • The first to show that not all the Fermat numbers were prime was Euler in 1732 when he showed that 2n + 1 where n = 25 was not prime.
       
     • Clausen also gave a new method of factorising numbers.
       

184. Grassmann biography
     
     • where the aj are real numbers, defines addition and multiplication by real numbers [in what is now the usual way] and formally proves the linear space properties for these operations.
       

185. Baker Alan biography
     
     • From this work he generated a large category of transcendental numbers not previously identified and showed how the underlying theory could be used to solve a wide range of Diophantine problems.
       
     • Among his famous books are Transcendental number theory (1975), Transcendence theory : advances and applications (1977), A concise introduction to the theory of numbers (1984) and (with Gisbert Wustholz) Logarithmic forms and Diophantine geometry (2007).
       

186. Morley biography
     
     • Show that on a chess-board the number of squares visible is 204, and the number of rectangles (including squares) visible is 1296; and that, on a similar board with n squares in each side, the number of squares is the sum of the first n square numbers, and the number of rectangles (including squares) is the sum of the first n cube numbers.
       

187. Tamarkin biography
     
     • In 1905 Tamarkin and Friedmann wrote a paper on Bernoulli numbers which they submitted to Hilbert for publication in Mathematische Annalen.
       
     • Tamarkin and Friedmann worked on another paper at this time and submitted Some formulas pertaining to the theory of the function [x] and Bernoulli numbers to Crelle's Journal.
       

188. De Moivre biography
     
     • De Moivre first published this result in a Latin pamphlet dated 13 November 1733 (see [XXX : Abraham de Moivre\'s 1733 derivation of the normal curve : a bibliographical note, Biometrika 59 (1972), 677-680.',4)">4] for an interesting discussion) aiming to improve on Jacob Bernoulli's law of large numbers.
       
     • which took trigonometry into analysis, and was important in the early development of the theory of complex numbers.
       

189. Knapowski biography
     
     • Among Knapowski's other number theory papers we mention: On prime numbers in arithmetical progression (1958), On the Mobius function (1958), Contributions to the theory of the distribution of prime numbers in arithmetical progressions (1961, 1962), On Linnik's theorem concerning exceptional L-zeros (1961), and Further developments in the comparative prime number theory (8 papers).
       

190. Li Zhi biography
     
     • Here the numbers which in our notation correspond to the coefficients of the equation are placed above each other so that the coefficient of x is placed above the constant, the coefficient of x2 is placed above the coefficient of x etc.
       
     • When he takes π = 3 it is not because he is obtaining the best approximate answer that he can, rather it is the method of solving the problem which is important and he is better able to illustrate this with "nice" numbers.
       

191. Jourdain biography
     
     • He wrote a number of articles, between 1906 and 1913, explaining and evaluating Cantor's set theory under the title Development of the Theory of Transfinite Numbers.
       
     • Other papers which he wrote on mathematical logic and the foundations of set theory include On the question of the existence of transfinite numbers which was published in the Proceedings of the London Mathematical Society in 1907.
       

192. Adleman biography
     
     • In that same year, Adleman, along with R S Rumely and C Pomerance, published a paper describing a 'nearly polynomial time' deterministic algorithm for the problem of distinguishing prime numbers from composite ones.
       
     • (This notion is essentially the content of the well-known "Church's thesis".) In other words, one could program a Turing machine to produce Watson-Crick complementary strings, factor numbers, play chess and so on.
       

193. Servois biography
     
     • Servois worked in projective geometry, functional equations and complex numbers.
       
     • Servois was critical of Argand's geometric interpretation of the complex numbers.
       

194. Higman biography
     
     • The first speaker was G H Hardy who addressed the Invariant Society on round numbers.
       
     • In this work, among other results, he classified group rings over the rational numbers without non-trivial units.
       

195. Du Bois-Reymond biography
     
     • Although Cantor proved that the real numbers are uncountable one year earlier he did not find the much clearer diagonal argument until some years later.
       
     • Also in this book he discussed the real numbers, the continuum, and space:- .
       

196. Wolf biography
     
     • In the following year he devised a system which is now known as 'Wolf's sunspot numbers'.
       
     • He continued to publish reports on sunspot numbers until his death.
       

197. Eichler biography
     
     • Later books by Eichler include: Einfuhrung in die Theorie der algebraischen Zahlen und Funktionen (1963) translated into English as Introduction to the theory of algebraic numbers and functions (1966); Projective varieties and modular forms (1971); and (coauthored with Don Zagier) The theory of Jacobi forms (1985).
       
     • However, the modular forms gain by being placed in the context of algebraic function theory and serve to illustrate it as well, while the elementary substratum common to algebraic numbers and functions is well known.
       

198. Zolotarev biography
     
     • In a short eleven year career Zolotarev produced fundamental work in approximation theory, quadratic forms, algebraic numbers and elliptic integrals.
       
     • Let us first comment on his work on algebraic numbers.
       

199. Knopp biography
     
     • Chapter I: Complex numbers and their geometric representation.
       
     • Volume 1 covers numbers, functions, limits, analytic geometry, algebra, set theory; volume 2 covers differential calculus, infinite series, elements of differential geometry and of function theory; and volume 3 covers integral calculus and its applications, function theory, differential equations.
       

200. Al-Kashi biography
     
     • (2) The usage of decimal fractions no longer for approaching algebraic real numbers, but for real numbers such as π.
       

201. Wiles biography
     
     • about ten years ago, G Frey suggested and K Ribet proved (building on ideas of B Mazur and J-P Serre) that Fermat's Last Theorem follows from the Shimura-Taniyama conjecture that every elliptic curve defined over the rational numbers is modular.
       
     • Using Mazur's deformation theory of Galois representations, recent results on Serre's conjecture on the modularity of Galois representations, and deep arithmetical properties of Hecke algebras, Wiles (with one key step due jointly to Wiles and R Taylor) succeeded in proving that all semistable elliptic curves defined over the rational numbers are modular.
       

202. Molin biography
     
     • In his doctoral thesis On higher complex numbers which was examined in 1892, Molien classified the complex semisimple algebras; later Cartan classified the real semisimple algebras and Wedderburn in 1907 gave the result for semisimple algebras over an arbitrary field.
       
     • Molien studied how many times a given irreducible representation of a finite group appears in a complete reduction of the representation of the group on the vector space of homogeneous polynomials of degree n over the complex numbers.
       

203. Martin biography
     
     • In his writings and problem-solving, Martin dealt mostly with Diophantine analysis, probability, elliptic integrals, logarithms, and properties of numbers and triangles.
       
     • He did publish one research level article, namely On fifth power numbers whose sum is a fifth power to the International Mathematical Congress in Chicago in 1893.
       

204. Karsten biography
     
     • He wrote an important article in 1768 Von den Logarithmen vermeinter Grossen in which he discussed logarithms of negative and imaginary numbers, giving a geometric interpretation of logarithms of complex numbers as hyperbolic sectors, based on the similarity of the equations of the circle and of the equilateral hyperbola.
       

205. Guinand biography
     
     • Guinand worked on summation formulae and prime numbers, the Riemann zeta function, general Fourier type transformations, geometry and some papers on a variety of topics such as computing, air navigation, calculus of variations, the binomial theorem, determinants and special functions.
       
     • [In an important paper in 1948] the main application of the general result yields a deep-seated connection between the distribution of the prime numbers and the location of the zeros of the Riemann zeta function on (or near to it if the Riemann hypothesis is false) the critical line in the complex plane..
       

206. Keller biography
     
     • This is an account of results in the theory of numbers obtained by geometric methods up to 1951.
       
     • Among the chapter headings are convex bodies in lattices, star bodies, linear forms, minima of homogeneous forms, inhomogeneous forms, definite quadratic forms, continued fractions and algebraic numbers.
       

207. Stewartson biography
     
     • He spoke on boundary layer theory, in particular flows at high Reynolds numbers, in his inaugural lecture in London.
       
     • Stewartson studied rotating fluid flows, shear layers, magnetohydrodynamics, the triple-deck theory, and flow at both high and low Reynolds numbers.
       

208. Recorde biography
     
     • The book discusses operations with Arabic numerals, computation with counters, proportion, the 'rule of three', all arithmetic being studied for the natural numbers in the first version which had second and third editons in 1549 and 1550.
       
     • In 1552 Recorde published a second enlarged version of The Grounde of Artes extending the work of the first edition to rational as well as whole numbers and including such topics as 'false position'.
       

209. Thom George biography
     
     • As a natural consequence, the numbers attending provincial schools have diminished everywhere, in some old established schools, to the extent of fifty per cent.
       
     • The numbers of pupils fluctuate considerably from year to year, so that it is difficult to know exactly how we stand, but it is a coincidence worth noting, that three years ago the enrolment was exactly the same (405) as it was in the upper school in 1878 and 1879, the year Dr Thom came.
       

210. Renyi biography
     
     • Renyi went to Russia as a postdoctoral student and, between October 1946 and June 1947, worked with Yuri Vladimirovich Linnik on the theory of numbers, in particular working on the Goldbach conjecture [19]:- .
       
     • In the hands of writers like Linnik, Erdos and Renyi, the theory of numbers is not clearly distinguished from the theory of probability.
       

211. Meissel biography
     
     • Meissel's mathematical interests covered the following fields: number theory (in particular, properties of prime numbers), theta functions, elliptic functions, spherical trigonometry, hydrodynamics, ordinary differential equations, asymptotic expansions, and Bessel functions.
       
     • He worked on prime numbers and found, in the 1870s, a method for computing individual values of π(x), the counting function for the number of primes less than or equal to x.
       

212. Stampioen biography
     
     • Stampioen let this cube root be A + √B, where a, b, A and B are all natural numbers.
       
     • equating the natural numbers gives a = A3 + 3AB and equating the surds gives √b = (3A2 + B)√B.
       

213. Jung biography
     
     • Let K be a field of algebraic functions of two variables over the field of complex numbers.
       
     • On the whole, the book contains a very large mass of information about algebraic surfaces over the field of complex numbers.
       

214. Vallee Poussin biography
     
     • Other than the prime number theorem, Vallee Poussin's only contributions to prime numbers were contained in two papers on the Riemann zeta function which he published in 1916.
       
     • History Topics: Prime numbers .
       

215. Polya biography
     
     • Mathematics is about numbers.
       
     • Numbers are an abstraction.
       

216. Study biography
     
     • Study became a leader in the geometry of complex numbers.
       
     • With Corrado Segre, Study was one of the leading pioneers in the geometry of complex numbers.
       

217. Eratosthenes biography
     
     • Eratosthenes also worked on prime numbers.
       
     • History Topics: Prime numbers .
       

218. Schmidt biography
     
     • It is so much more difficult to stick to this virtue, proven with numbers and figures, against humans and friends.
       
     • Schmidt defined a space H whose elements are square summable sequences of complex numbers.
       

219. Suetuna biography
     
     • The Analytical theory of numbers was originally published in the form of lecture notes but in 1950 a revised edition was published which incorporated recent developments of the theory.
       
     • This book, based mainly on the Riemann zeta-functions and L-functions, is a unique exposition of the analytical theory of numbers in a modern sense as can be seen from the chapter headings: I) Riemann's zeta-functions; II) Hecke's L-functions; III) Dirichlet's L-functions; and IV) Artin's L-series.
       

220. Remak biography
     
     • He had broad interests, working on mathematical economics as well as group theory and the geometry of numbers.
       
     • He may, without doubt, be called a leading scholar in the splendid and important field of geometry of numbers.
       

221. Bethe biography
     
     • Recollections of both his own and of relatives include his ability to compute square roots at the age of four, a full understanding of fractions by the age of five, and the ability to find prime numbers by the age of seven.
       
     • But mathematics and the word of numbers was not his only forte.
       

222. Schroder biography
     
     • His work on ordered sets and ordinal numbers is fundamental to the subject.
       
     • he put forward mathematics as 'the doctrine of numbers', rather than of magnitudes; and he stressed the algebraic bent by seeking 'absolute algebra' of which common algebra was an example.
       
     • In arithmetic, letters are numbers, but here, they are arbitrary concepts.
       

223. Von Neumann biography
     
     • He could answer any question put to him (who has number such and such?) or recite names, addresses, and numbers in order.
       
     • He published a definition of ordinal numbers when he was 20, the definition is the one used today.
       

224. Bolyai biography
     
     • In addition to his work on geometry, Bolyai developed a rigorous geometric concept of complex numbers as ordered pairs of real numbers.
       

225. Edmonds biography
     
     • In 1957 Edmonds published Sums of powers of the natural numbers.
       
     • are possible for C a constant and p, q non-negative integers? In Sums of powers of the natural numbers she showed that al-Karaji's formula (*) is the only one that exists.
       

226. Ezra biography
     
     • Of the most interest to us in this archive devoted to the history of mathematics is ibn Ezra's work on numbers.
       
     • He wrote three treatises on numbers which helped to bring the Indian symbols and ideas of decimal fractions to the attention of some of the learned people in Europe.
       

227. Regiomontanus biography
     
     • The first computed in 1467 was Tables of directions which was based on sexagesimal numbers, while in the following year in Buda he computed tables of sines to a decimal base.
       
     • History Topics: Perfect numbers .
       

228. Bortkiewicz biography
     
     • Bortkiewicz was interested in the law of small numbers and he used the divergence coefficient Q, deducing its expectation and standard deviation.
       
     • He published a work The Law of Small Numbers in 1898.
       

229. Mackey biography
     
     • He then produced a series of important papers on group representations including On induced representations of groups (1951), Induced representations of locally compact groups (1952), and Symmetric and anti symmetric Kronecker squares and intertwining numbers of induced representations of finite groups (1953).
       
     • For better or worse I chose to de-emphasize the intuitive geometric aspects of the subject and to present it as a deductive system starting with axioms for the real and complex numbers.
       

230. Seki biography
     
     • Seki also discovered Bernoulli numbers before Jacob Bernoulli.
       
     • He studied equations treating both positive and negative roots but had no concept of complex numbers.
       

231. Kurepa biography
     
     • After introducing the fundamental concepts and elementary operations in Chapter 1, he looks at cardinal numbers in the second chapter, then partially ordered sets and ordinal numbers in the third.
       

232. Roberts biography
     
     • Among the subjects to which his principal papers related were plane and solid geometry, theory of numbers, and link motion.
       
     • In theory of numbers he was interested in the Pellian equation and similar problems.
       

233. Hamel biography
     
     • He is perhaps best known for the Hamel basis, published in 1905, when he made an early and explicit use of the Axiom of Choice to construct a basis for the real numbers as a vector space over the rational numbers.
       

234. Goldstine biography
     
     • It has ushered in a new era in calculations with discrete numbers and has not only swept away the bottleneck of many existing and known forms of computation, but has also opened up new vistas of numerical thought by providing the means of doing what has never yet been attempted because of prohibitive time and cost.
       
     • The machine is to make full use of the flexible and compact coding of problems which is possible when orders as well as numbers are stored in the high speed memory and can be operated on and modified according to the progress of the computation.
       

235. Tietze biography
     
     • Topics outside topology which Tietze worked on included ruler and compass constructions, continued fractions, partitions, the distribution of prime numbers, and differential geometry.
       
     • Heinrich Tietze on Numbers .
       

236. Xiahou Yang biography
     
     • One significant idea which appears in the text concerns representation of numbers in the decimal notation.
       
     • Although Xiahou Yang has no symbol for 0 in an empty place, there is good evidence from his description of moving numbers to the right and left that he at least has a virtual zero in mind despite the lack of a symbol.
       

237. Panini biography
     
     • In particular he suggests that algebraic reasoning, the Indian way of representing numbers by words, and ultimately the development of modern number systems in India, are linked through the structure of language.
       
     • Then he put the finishing touches to the theory by suggesting that Panini in the eighth century BC (earlier than most historians place Panini) was the first to come up with the idea of using letters of the alphabet to represent numbers.
       

238. Cesaro biography
     
     • Sur diverses questions d'arithmetique was the first of a series which Cesaro wrote on the theory of numbers.
       
     • the number of common divisors of two numerals, determination of the values of the sum totals of their squares, the probability of incommensurability of three arbitrary numbers, and so on; to these he attempted to apply obtained results in the theory of Fourier series.
       

239. Takagi biography
     
     • hope that the reader has understood that the essential point in algebra does not lie in the nature of the elements (which are not necessarily numbers) but in the way elements are composed.
       
     • The 500 page work developed real numbers via Dedekind cuts.
       

240. Dickson biography
     
     • The 3-volume History of the Theory of Numbers (1919-23) is another famous work still much consulted today.
       
     • Dickson published Modern Elementary Theory of Numbers in 1939.
       

241. Bernstein Felix biography
     
     • This is a vital result in the study of cardinal numbers, indeed a vital result in the development of set theory.
       
     • In 1905 Bernstein published another important article on transfinite ordinal numbers Uber die Reihe der Transfiniten Ordnungszahlen which appeared in Mathematische Annalen.
       

242. Smith biography
     
     • From 1859 to 1865 he prepared a report in five parts on the Theory of Numbers.
       
     • the most complete and elegant monument ever erected to the theory of numbers.
       

243. Andrews biography
     
     • He already knew that he wanted to undertake research in number theory but at the time he began his course it was prime numbers which fascinated him.
       
     • Andrews had come across these ideas before, since Joy, before they were married, had given him a book which contained G H Hardy's A Mathematician's Apology and in that he had encountered the partition formula that Hardy and Ramanujan had discovered in 1916 for the number of different ways an integer can be expressed as a sum of natural numbers.
       

244. Peschl biography
     
     • The titles of the chapters are: Algebra and geometry of complex numbers; Fundamental topological concepts, sets, sequences of complex numbers and infinite series; Functions, real and complex differentiability and holomorphy; Integral theorems and their consequences; Winding number and curves homologous to zero; Taylor development of holomorphic functions; Elementary transcendental functions; Laurent series, isolated singularities and residue calculus; Holomorphic and meromorphic functions obtained by limiting processes; Analytic continuation; and Conformal mappings.
       

245. Hecke biography
     
     • discovered new connections between prime numbers and analytic functions and new rules for the representation of natural numbers through positive integral quadratic forms of an even number of variables.
       

246. Todd John biography
     
     • These papers included A characterisation of algebraic numbers (1940), Matrices with finite period (1940), Matrices of finite period (1941), Inversion in groups (1941) and Infinite powers of matrices (1942).
       
     • He studied methods for evaluating mathematical functions, generating random numbers (for Monte Carlo calculations), conformal mappings, and computations with matrices.
       

247. Barlow biography
     
     • In addition to these articles, Barlow also published several important books, for example in 1811 he published An elementary investigation of the theory of numbers and three years later he published A new mathematical and philosophical dictionary.
       
     • These soon became known as Barlow's Tables and this work gives factors, squares, cubes, square roots, reciprocals and hyperbolic logarithms of all numbers from 1 to 10 000.
       

248. Gleason biography
     
     • Chapters I to VI cover elementary logic and set theory; Chapters VII to X deal with the various "number systems" from the natural integers to the complex numbers; Chapter XI briefly returns to set theory (countable sets, cardinal numbers and the axiom of choice); finally, the last four chapters deal, respectively, with limits of complex sequences, infinite series and products, metric spaces, and the elementary theory of holomorphic functions of one variable (Cauchy integral excluded, but the logarithmic function is defined and studied).
       

249. Markov biography
     
     • Especially remarkable is his research relating to the theorem of Jacob Bernoulli known as the Law of Large Numbers, to two fundamental theorems of probability theory due to Chebyshev, and to the method of least squares.
       
     • Markov showed his disapproval of the celebration but holding celebrations of his own - he celebrated 200 years of the Law of Large Numbers! The Russian Revolution began early in 1917 as food supplies ran low.
       

250. Ledermann biography
     
     • Other books which Ledermann has written for undergraduates include Complex numbers (1960), Integral calculus (1964), Multiple integrals (1966), Introduction to group theory (1973), and Introduction to group characters (1977).
       
     • Ledermann: "Complex Numbers" .
       

251. Parry biography
     
     • Parry's first paper On the b-expansions of real numbers was published in 1960.
       
     • There he worked on entropy theory showing, amongst other things, that each aperiodic measure-preserving transformation could be viewed as the shift on the realisation space of a stationary, countable state, stochastic process indexed by the integers or the natural numbers.
       

252. Hirzebruch biography
     
     • Basically, the role of topology in number theory has progressed beyond the local methods such as p-adic theory to global methods such as intersection numbers of homology classes.
       
     • the systematic study of Hilbert modular-forms and-surfaces and their relation to class numbers.
       

253. Zorn biography
     
     • His proof uses his maximum principle rather than using ordinal numbers as had been done in previous proofs of the result.
       
     • He proved the uniqueness of the Cayley numbers (or octonians) in 1933 by showing that it was the only alternative, quadratic, real nonassociative algebra without zero divisors.
       

254. Levi biography
     
     • He gives formulas for the sum of squares and the sum of cubes of natural numbers as well as studying the binomial coefficients.
       
     • In 1342, at the request of the bishop of Meaux, he wrote The Harmony of Numbers which contains a proof that (1,2), (2,3), (3,4) and (8,9) are the only pairs of consecutive numbers whose only factors are 2 or 3.
       

255. Gauss biography
     
     • His teacher, Buttner, and his assistant, Martin Bartels, were amazed when Gauss summed the integers from 1 to 100 instantly by spotting that the sum was 50 pairs of numbers each pair summing to 101.
       
     • History Topics: Prime numbers .
       

256. Ingham biography
     
     • His book On the distribution of prime numbers published in 1932 was his only book and it is a classic.
       
     • Ingham's work was on the Riemann zeta function, the theory of numbers, the theory of series and Tauberian theorems.
       

257. Al-Sijzi biography
     
     • Given a triangle and three given numbers, find a point inside the triangle where the lines to the three vertices divide the triangle into three triangles having areas proportional to the three given numbers.
       

258. Mazur Barry biography
     
     • More recently he has written an outstanding popular work on complex numbers entitled Imagining numbers (2003).
       

259. Yule biography
     
     • The correlation of lengths or measurements on portions of the body form examples of the first kind; of numbers of children in families, petals or other parts of flowers, are examples of the second.
       
     • The last chapters discuss interpolation and graduation, index numbers, and time series.
       

260. Korkin biography
     
     • Aleksandr Nikolaevich Korkin's father, Nikolay Ivanovich Korkin, was [The St Petersburg school in the theory of numbers (American Mathematical Society, London Mathematical Society, Providence, R.I.
       
     • Delone writes in [The St Petersburg school in the theory of numbers (American Mathematical Society, London Mathematical Society, Providence, R.I.
       

261. Dieudonne biography
     
     • Let us not pass judgement on whether the text is too sophisticated to fulfil its intended purpose but we do note that in introducing the real numbers in the first chapter Dieudonne assumes they are an ordered field in which the intermediate value theorem is valid for polynomials of degree 3.
       
     • In the first volume Dieudonne has contributed an article on Jordan's work on finite groups and in the second volume an interesting 116-page introduction to Jordan's work on linear and multilinear algebra and on the theory of numbers.
       

262. Knuth biography
     
     • In the first Knuth describes an imaginary number system using the imaginary number 2i as its base, giving methods for the addition, subtraction and multiplication of the numbers.
       

263. Carlitz biography
     
     • In 1927 Artin made a major contribution to the theory of noncommutative rings, called hypercomplex numbers at this time.
       

264. Mazurkiewicz biography
     
     • He proved the strong law of large numbers in 1922, a result which was proved independently by Cantelli.
       

265. Verhulst biography
     
     • There he worked on the theory of numbers, and, influenced by Quetelet, he became interested in social statistics.
       

266. Wedderburn biography
     
     • In this paper On hypercomplex numbers which appeared in the Proceedings of the London Mathematical Society, he showed that every semisimple algebra is a direct sum of simple algebras and that a simple algebra was a matrix algebra over a division ring.
       

267. Steggall biography
     
     • His research interests were in the theory of numbers and in kinematical geometry, particularly the geometry of the triangle.
       

268. Gosset biography
     
     • Gosset discovered the form of the t distribution by a combination of mathematical and empirical work with random numbers, an early application of the Monte-Carlo method.
       

269. James Ralph biography
     
     • In the paper On the sieve method of Viggo Brun James takes the sieve method which had been used by Buchstab in 1940 to establish results on prime numbers, and applies it to obtain results about infinite subsets of primes, such as primes in arithmetic progression.
       

270. Springer biography
     
     • As in the case of the group of all non-singular linear transformations these invariants consist of irreducible polynomials and systems of non-negative integral numbers, but apart from these also equivalence classes of hermitian forms and of quadratic forms have to be included.
       

271. Padoa biography
     
     • He gave the important lecture Essay of an algebraic theory of whole numbers, preceded by a logical introduction to any deductive theory at the International Congress of Philosophy in Paris in 1900.
       

272. Huntington biography
     
     • He gave axioms for a group, an abelian group, a boolean algebra, geometry, the real number field, and the complex numbers.
       

273. Picken biography
     
     • Teacher, for example The Arithmetic and Algebra of the Natural Numbers (1946).
       

274. Rey Pastor biography
     
     • It was here that his true vocation for mathematics was awoken and he published his first paper in 1905, entitled Sobre los numeros consecutivos cuyo suma es a la vez cuadrado y cubo perfecto (Consecutive numbers whose sum is both the square and the perfect cube).
       

275. Bessel-Hagen biography
     
     • This said, of course, most students were also conscripted so they did not have large numbers to teach.
       

276. Pell Alexander biography
     
     • Certainly after this meeting Degaev left for Tbilisi to work on the Tbilisi-Baku railway while Sudeikin was able to arrest large numbers of the Narodnaia Volia in St Petersburg.
       

277. Jyesthadeva biography
     
     • All the terms are then divided by the odd numbers 1, 3, 5, ..
       

278. Wang Yuan biography
     
     • Wang Yuan fell in love with analytic number theory and gave a series of lectures to the graduate seminar based on Ingham's book The distribution of prime numbers.
       

279. Noether Emmy biography
     
     • For Emmy Noether, relationships among numbers, functions, and operations became transparent, amenable to generalisation, and productive only after they have been dissociated from any particular objects and have been reduced to general conceptual relationships.
       

280. Krieger biography
     
     • To qualify for the Master's Degree she took graduate level courses on: Modular Elliptic Functions given by Jacques Chapelon; Minimum Principles of Mechanics given by John Lighton Synge; The Theory of Sets given by Samuel Beatty; The Theory of Numbers given by John Charles Fields; and The Theory of Functions given by W J Webber.
       

281. Anderson biography
     
     • He especially believed that statistics, based on the law of large numbers and the sorting out of random deviations, is the only substitute for experimentation, which is impossible in economics.
       

282. Chrystal biography
     
     • Chrystal's mathematical publications cover many topics including non-euclidean geometry, line geometry, determinants, conics, optics, differential equations, and partitions of numbers.
       

283. Schur biography
     
     • They were unaware at that time that the second cohomology group with coefficients in the nonzero complex numbers is the Schur multiplier, and therefore that Schur had made some of the first steps forty years earlier.
       

284. Wang Xiaotong biography
     
     • Suppose we forget the numbers for a moment and look at the data we are given.
       

285. Salmon biography
     
     • His last mathematics publication was in 1873 on periods of the recurring decimals of the reciprocals of prime numbers.
       

286. Bouvelles biography
     
     • This was Liber de XII numeris which studied perfect numbers.
       

287. Rademacher biography
     
     • Perhaps his most famous result, obtained in 1936 when he was in the United States, is his proof of the asymptotic formula for the growth of the partition function (the number of representations of a number as a sum of natural numbers).
       

288. Gluskin biography
     
     • By September Kiev had fallen and large numbers of Soviet troops surrendered [Semigroup Forum 32 (3) (1985), 221-231.',5)">5]:- .
       

289. Allardice biography
     
     • For example at the meeting held on Friday 14 March 1884 he read a paper on the geometry of the spherical surface; at the meeting on Friday 8 January 1886 he discussed a problem of symmetry in an algebraical function; on 11 February 1887 he communicated a note on a theorem in algebra; on 11 January 1889 he contributed a note on a formula in quaternions; on 13 December 1889 he discussed some theorems in the theory of numbers; on 13 November 1891 his paper Barycentric Calculus of Mobius was read by John Alison; on 14 December 1901 his paper Four Circles Touching a Common Circle was communicated to the meeting by Mr George Duthie; and on 13 January 1911 his paper On the envelope of the directrices of a system of similar conics through three points was communicated by E D Williamson.
       

290. Sun Zi biography
     
     • So the numbers to be multiplier are placed in the top and bottom of the three rows of the counting board and multiplications by single digits and additions take place in constructing the product in the middle row.
       

291. De Morgan biography
     
     • In 1849 he published Trigonometry and double algebra in which he gave a geometric interpretation of complex numbers.
       

292. Selberg biography
     
     • Selberg is also well known for his elementary proof of the prime number theorem, with a generalisation to prime numbers in an arbitrary arithmetic progression.
       

293. Girard Albert biography
     
     • History Topics: Prime numbers .
       

294. Fine Nathan biography
     
     • In this article Fine proved that here exist rational numbers a and b that are never sides of a rational triangle, and also there exists a rational triangle of any given rational area.
       

295. Hadamard biography
     
     • History Topics: Prime numbers .
       

296. Sleszynski biography
     
     • ensures the convergence of the continued fraction K(an / bn), where an and bn are complex numbers; a result now known as the Pringsheim criterion.
       

297. Karlin biography
     
     • Large numbers of papers and books on a wide range of topics continued to flow from his pen.
       

298. Mostowski biography
     
     • Many of Mostowski's wartime results - on the hierarchy of projective sets, on arithmetically definable sets of natural numbers, and on consequences of the axiom of constructibility in descriptive set theory - were lost when his apartment was destroyed during the uprising.
       

299. Zu Chongzhi biography
     
     • Of course, it is not unreasonable to ask where the numbers 144 and 391 came from.
       

300. Mendelsohn biography
     
     • He wrote papers on a wide variety of combinatorial problems, for example: Symbolic solution of card matching problems (1946), Applications of combinatorial formulae to generalizations of Wilson's theorem (1949), Representations of positive real numbers by infinite sequences of integers (1952), A problem in combinatorial analysis (1953), The asymptotic series for a certain class of permutation problems (1956), and Some elementary properties of ill conditioned matrices and linear equations (1956).
       

301. Cassini biography
     
     • Impressively he correctly proposed that the rings were composed of large numbers of tiny satellites each orbiting the planet.
       

302. Klein biography
     
     • There he, and his colleague Brill, taught advanced courses to large numbers of excellent students and Klein's great talent at teaching was fully expressed.
       

303. Gopel biography
     
     • Gopel's doctoral dissertation studied periodic continued fractions of the roots of integers and derived a representation of the numbers by quadratic forms.
       

304. White biography
     
     • Now, why would it not be possible to combine with this miscellaneous program (which ought by all means to be kept up) something more akin to university models? Would not a series of three or six lectures on nearly related topics, if well chosen, prove attractive and useful to larger numbers? .
       

305. Fraenkel biography
     
     • Fraenkel's first work was on Hensel's p-adic numbers and on the theory of rings.
       

306. Brill biography
     
     • There he was joined by Klein in 1875 and the two taught advanced courses to large numbers of excellent students.
       

307. Williamson biography
     
     • The general trend in this programme was to pass from the ordinary field of complex numbers to more restricted and specialised fields.
       

308. Clavius biography
     
     • History Topics: The real numbers: Pythagoras to Stevin .
       

309. Schutzenberger biography
     
     • The range of Schutzenberger's contributions is so vast that it is almost impossible to do them justice in a biography of this type - his list of publications numbers over 250.
       

310. Feller biography
     
     • Other papers written by Feller while still at Brown University include: On the time distribution of so-called random events (1940), On the integral equation of renewal theory (1941), On A C Aitken's method of interpolation (1943), The fundamental limit theorems in probability (1945) and Note on the law of large numbers and "fair" games (1945).
       

311. Maschke biography
     
     • He was a teacher of great ability and his courses were made more valuable by his all-round culture, by his originality of thought, and by his personal interest in the large numbers of young mathematicians who attended his lectures.
       

312. Woodhouse biography
     
     • Woodhouse was interested in the theoretical foundations of the calculus, the importance of notation, the nature of imaginary numbers and other similar topics.
       

313. Bendixson biography
     
     • The proof of the theorem which Bendixson gave uses Cantor's notion of transfinite numbers.
       

314. Aiken biography
     
     • whereas accounting machines handle only positive numbers, scientific machines must be able to handle negative ones as well; that scientific machines must be able to handle such functions as logarithms, sines, cosines and a whole lot of other functions; the computer would be most useful for scientists if, once it was set in motion, it would work through the problem frequently for numerous numerical values without intervention until the calculation was finished; and that the machine should compute lines instead of columns, which is more in keeping with the sequence of mathematical events.
       

315. Dirac biography
     
     • In 1937, the same year that he married, Dirac published his first paper on large numbers and cosmological matters.
       

316. Finsler biography
     
     • Combinatorially, Finsler considers sets as generalised numbers to which one may apply arithmetical techniques.
       

317. Banu Musa biography
     
     • The Greeks had not thought of areas and volumes as numbers, but had only compared ratios of areas etc.
       

318. Mertens biography
     
     • Here Mertens defines M(n) to be the sum of the numbers m(i) where i runs from 1 to n and where m is the Mobius function.
       

319. Escher biography
     
     • At high school in Arnhem, I was extremely poor at arithmetic and algebra because I had, and still have, great difficulty with the abstractions of numbers and letters.
       

320. Hoyle biography
     
     • She also provided Fred with his early education, in particular teaching him numbers.
       

321. Descartes biography
     
     • History Topics: Perfect numbers .
       

322. Baire biography
     
     • While at Montpellier he wrote a paper on irrational numbers and limits.
       

323. Aleksandrov Aleksandr biography
     
     • Delone's interests in the geometry of numbers and the structure of crystals soon began to attract Aleksandrov at least as much as his work in physics which was supervised by V A Fok.
       

324. Al-Jayyani biography
     
     • Among the similarities between al-Jayyani's treatise and that of Regiomontanus are the definition of ratios as numbers, the lack of a tangent function, and a similar method of solving a spherical triangle when all sides are unknown.
       

325. Legendre biography
     
     • History Topics: Prime numbers .
       

326. Speiser biography
     
     • These topics include possible number systems, the questions of "fixed" vs "floating" point and complementation, the arithmetic processes, the grouping of numbers to achieve higher than normal precisions, conversion between number systems, the structure of finite approximation methods, error analysis, programming and coding as well as the physical organs of a machine.
       

327. Witt biography
     
     • Witt joined Hasse's seminar on congruence function fields and p-adic numbers; he was appointed as Hasse's assistant.
       

328. Zhang Heng biography
     
     • Some historians believe that Zhang understood the difference between rational and irrational numbers but this seems to be stretching things a bit too far.
       

329. Duarte biography
     
     • His main three books are: Monograph on the numbers π and e.
       

330. Coates biography
     
     • Then, together with A Baker, he extended Mahler's work on fractional parts of powers of rational numbers.
       

331. Lame biography
     
     • In fact he believed that he had solved the whole problem at one stage but he had overlooked the lack of unique factorisation in certain subrings of the complex numbers.
       

332. Beatty biography
     
     • Define two sequences [nR] and [nS] where n runs through the natural numbers and, for any real number x, [x] denotes the greatest integer less than or equal to x.
       

333. Tunstall biography
     
     • since he had explained the calculating of numbers in such an excellent manner.
       

334. Barrow biography
     
     • Geometry is the basic mathematical science, for it includes arithmetic, and mathematical numbers are simply the signs of geometrical magnitude.
       

335. Gronwall biography
     
     • Gronwall's work contains classical analysis (Fourier series, Gibbs phenomenon, summability theory, Laplace and Legendre series), differential and integral equations, analytic number theory (transcendental numbers, divisor function, L-function of Dirichlet), complex function theory (Dirichlet L-series, conformal mappings, univalent functions), differential geometry, mathematical physics (problems of elasticity, ballistics, induction, potential theory, kinetic theory of gases, optics), nomography, atomic physics (wave mechanics of hydrogen and helium atom, lattice theory of crystals) and physical chemistry where he is especially known as a very important contributor.
       

336. Aaboe biography
     
     • In Some Seleucid mathematical tables (Extended reciprocals and squares of regular numbers) (1965) Aaboe looks at eight mathematical cuneiform tablets, three of which had not previously been published.
       

337. Lovelace biography
     
     • So also was the algebraic working out of the different problems, except, indeed, that relating to the numbers of Bernoulli, which I had offered to do to save Lady Lovelace the trouble.
       

338. Tarry biography
     
     • First we note that a magic square of order n contains the numbers 1 to n2 in an n × n array such that each row, each column, and the two main diagonals all have the same sum.
       

339. Bombieri biography
     
     • The award was made for his major contributions to the study of the prime numbers, to the study of univalent functions and the local Bieberbach conjecture, to the theory of functions of several complex variables, and to the theory of partial differential equations and minimal surfaces.
       

340. Gray Andrew biography
     
     • His organizational exertions on numerous committees during the First World War, the death of a son in 1915, and the demands of vastly increased student numbers after the war undermined Gray's already uncertain health and he resigned his chair in 1923.
       

341. Neyman biography
     
     • Here, a wilful, persistent and distinguished director has succeeded, step by step over a fifteen year period, against the wish of his department chairman and dean, in converting a small 'laboratory' or institute into, in terms of numbers of students taught, an enormously expensive unit; and he then argues that the unit should be renamed a 'department' because no additional expense will be incurred.
       

342. Schwerdtfeger biography
     
     • In 1962 he published Geometry of complex numbers : Circle Geometry, Mobius Transformations, Non-Euclidean Geometry which:- .
       

343. Thomson James biography
     
     • These lectures he continued for some years till, the mathematical classes having increased in numbers, he was compelled by pressure of work to discontinue them.
       

344. Griffiths Brian biography
     
     • We give the titles of a few of his mathematical education article which give an overview of his interests in that topic: Pure mathematicians as teachers of applied mathematicians (1968); Mathematics Education today (1975); Successes and failures of mathematical curricula in the past two decades (1980); Simplification and complexity in mathematics education (1983); The implicit function theorem: technique versus understanding (1984); A critical analysis of university examinations in mathematics (1984); Cubic equations, or where did the examination question come from? (1994); The British Experience of Teaching Geometry since 1900 (1998); and The Divine Proportion, matrices and Fibonacci numbers (2008).
       

345. Hausdorff biography
     
     • He studied the Gaussian law of errors, limit theorems and problems of moments, and set theory and the strong law of large numbers, which he based on measure theory.
       

346. Poinsot biography
     
     • In addition Poinsot worked on number theory and on this topic he studied Diophantine equations, how to express numbers as the difference of two squares and primitive roots.
       

347. Lanczos biography
     
     • Cornelius Lanczos was born Kornel Lowy but when there was a reaction in Hungary against German names he, along with large numbers of his countrymen, changed his name from the German form and became Kornel Lanczos (or rather, Lanczos Kornel since Hungarians put the family name first).
       

348. Heine biography
     
     • History Topics: The real numbers: Stevin to Hilbert .
       

349. Xu Yue biography
     
     • There is no doubt that one of the main aims of the text is to introduce a notation which will allow the representation of large numbers.
       

350. Faraday biography
     
     • After the regular hours of business, he was chiefly employed in drawing and copying from the Artist's Repository, a work published in numbers which he took in weekly.
       

351. Gunter biography
     
     • He made a mechanical device, Gunter's scale, to multiply numbers based on the logs using a single scale and a pair of dividers.
       

352. Savart biography
     
     • For example, he would use, in combination, wheels with numbers of teeth which bore a simple relationship to each other.
       

353. Dyson biography
     
     • Foreign languages came easily to Dyson and when he became interested in number theory in 1938 he decided to read An introduction to the theory of numbers by Vinogradov.
       

354. Arnold biography
     
     • Very young children start thinking about [old merchant] problems even before they have any knowledge of numbers.
       

355. Al-Uqlidisi biography
     
     • A dust board was used because the methods required the moving of numbers around in the calculation and rubbing some out as the calculation proceeded.
       

356. Avicenna biography
     
     • The topics dealt with in the geometry section of the encyclopaedia are: lines, angles, and planes; parallels; triangles; constructions with ruler and compass; areas of parallelograms and triangles; geometric algebra; properties of circles; proportions without mentioning irrational numbers; proportions relating to areas of polygons; areas of circles; regular polygons; and volumes of polyhedra and the sphere.
       

357. Democritus biography
     
     • He wrote On numbers, On geometry, On tangencies, On mappings, On irrationals but none of these works survive.
       

358. Peacock biography
     
     • He investigated the basic properties of numbers, such as the distributive property, that underlie the subject of algebra.
       

359. James Ralph biography
     

360. Bianchi biography
     
     • In particular he wrote Lectures on differential geometry (1894), Lectures on the theory of groups of substitutions (1900), Lectures on the theory of continuous groups (1918), Lectures on the theory of functions of a complex variable (1901) and Lectures on the theory of algebraic numbers (1923).
       

361. Rutherford biography
     
     • Rutherford's papers in the 1940s included On the relations between the numbers of standard tableaux, On the matrix representation of complex symbols, On substitutional equations, Some continuant determinants arising in physics and chemistry, On commuting matrices and commutative algebras; these being published either by the Edinburgh Mathematical Society or by the Royal Society of Edinburgh.
       

362. Littlewood biography
     
     • in recognition of his mathematical discoveries and supreme insight in the analytic theory of numbers.
       

363. Turnbull biography
     
     • As a result of careful scrutiny it has been established that Gregory made several remarkable and unsuspected discoveries, particularly in the calculus and the theory of numbers, which he never published.
       

364. Dase biography
     
     • He would then write the numbers involved on a blackboard, and after carrying out mental calculations, write down a provisional answer on the board.
       
     • He multiplied and divided large numbers in his head, but when the numbers were very large he required considerable time.
       
     • Schumacher once gave him the numbers 79532853 and 93758479 to be multiplied.
       
     • He multiplied mentally two numbers each of 20 figures in 6 minutes; 40 figures in 40 minutes; and 100 figures in 83/4 hours, which last calculation must have made his exhibitions somewhat tiresome to the onlookers.
       
     • He constructed 7 figure tables of natural logarithms of the numbers from 1 to 1,005,000 in 1847 (this task was completed over a period of three years).
       
     • In 1849 Dase proposed to the Academy of Sciences in Hamburg that he would compute the factors of all the numbers from 7 million to 10 million.
       
     • With small numbers, everybody that possesses any readiness in reckoning, sees the answer to such a question [the divisibility of a number] at once directly, for greater numbers with more or less trouble; this trouble grows in an increasing relation as the numbers grow, till even a practiced reckoner requires hours, yes days, for a single number; for still greater numbers, the solution by special calculation is entirely impractical.
       
     • When he died in 1861, he had completed computing factors of all the numbers between 7 million and 8 million, and also, with the exception of a small portion, of the next million numbers.
       

365. Pringsheim biography
     
     • He also suggested that the paradoxes of the infinitary calculus arose from transferring properties of real numbers to infinite-dimensional domains where they fail, and agreed with Cantor that any use of infinitesimals in analysis would necessarily lead to inconsistencies.
       

366. Mengoli biography
     
     • Other interesting results about series in Novae quadraturae arithmeticae include a study of the sum of reciprocals of the triangular numbers n(n+1)/2.
       

367. Nielsen biography
     
     • He turned to number theory and studied Bernoulli numbers in Traite elementaire des nombres de Bernoulli (Gauthier-Villars, Paris, 1923) and Fermat's equation writing good textbooks on these topics.
       

368. Zorawski biography
     
     • Russia decided to break up the Reds by drafting large numbers of them into the Russian army.
       

369. Khayyam biography
     
     • History Topics: The real numbers: Pythagoras to Stevin .
       

370. Brauer Alfred biography
     
     • Brauer made major contributions to number theory, for example on the non-existence of odd perfect numbers of certain forms, and the Khinchin conjecture which was later proved and extended by Henry B Mann.
       

371. Pontryagin biography
     
     • His main tool was to use link numbers which had been introduced by Brouwer and, by 1932, he had produced the most significant of these duality results when he proved the duality between the homology groups of bounded closed sets in Euclidean space and the homology groups in the complement of the space.
       

372. Boutroux biography
     
     • Boutroux's topics range from rational numbers to an analysis of the notion of a function.
       

373. Shatunovsky biography
     
     • In particular he produced good work in group theory, the theory of numbers and geometry.
       

374. Rutishauser biography
     
     • These topics include possible number systems, the questions of "fixed" vs "floating" point and complementation, the arithmetic processes, the grouping of numbers to achieve higher than normal precisions, conversion between number systems, the structure of finite approximation methods, error analysis, programming and coding as well as the physical organs of a machine.
       

375. Liu Hui biography
     
     • In Chapter 8 he looks at simultaneous linear equations and computes with both positive and negative numbers.
       

376. Bernoulli Daniel biography
     
     • History Topics: The real numbers: Stevin to Hilbert .
       

377. Broglie biography
     
     • And I realised that, on the one hand, the Hamilton-Jacobi theory pointed somewhat in that direction, for it can be applied to particles and, in addition, it represents a geometrical optics; on the other hand, in quantum phenomena one obtains quantum numbers, which are rarely found in mechanics but occur very frequently in wave phenomena and in all problems dealing with wave motion.
       

378. Wald biography
     
     • seasonal corrections to time series, approximate formulas for economic index numbers, indifference surfaces, the existence and uniqueness of solutions of extended forms of the Walrasian system of equations of production, the Cournot duopoly problem, and finally, in his much used work written with Mann (1943), stochastic difference equations.
       

379. Schatten biography
     
     • .? "Then," Schatten would say, "I come along with a big bag of numbers over my shoulder, and hang one number on each hook." This of course was accompanied by suitable gestures for emphasis.
       

380. Shafarevich biography
     
     • While still in his teens, he read mathematical treatises independently and at the age of 15 he was reading David Hilbert's celebrated report on the theory of Numbers.
       

381. Scherk biography
     
     • Another contribution by Scherk is still important today, namely his work on the distribution of the prime numbers.
       

382. Caramuel biography
     
     • In a work in 1670 he expounded the general principle of numbers to base n pointing out the benefits of some other bases than 10.
       

383. Hay biography
     
     • Until high school, I was not particularly mathematically inclined - indeed, I was much better at verbal subjects; combinatorial aspects of numbers and equations have never been my strong point ..
       

384. Pacioli biography
     
     • History Topics: Perfect numbers .
       

385. Thymaridas biography
     
     • Thymaridas was a Pythagorean and a number theorist who wrote on prime numbers.
       

386. Galois biography
     
     • He organised some mathematics classes in higher algebra which attracted 40 students to the first meeting but after that the numbers quickly fell off.
       

387. Calderwood biography
     
     • She was in the same class as Annie Numbers.
       

388. Cartwright biography
     
     • This was a difficult time to enter university since, World War I having just ended, there were large numbers of men returning from the army who were either restarting the university studies they had begun before the war or were taking up their studies for the first time.
       

389. Chernoff biography
     
     • A list of other topics treated follows: D-optimality and the Kiefer-Wolfowitz equivalence theorem; hypothesis-testing in a treatment which is largely, although not whole-heartedly, decision-theoretical; the large-sample evaluation of risk in terms of the Chernoff bounds (a term not used in the text) and the various information numbers; optimisation of sample size in the case of low-cost experimentation; the sequential probability ratio test, no-overshoot approximations, optimality; the Chernoff "procedure A" for sequential design, and its asymptotic optimality; adjacent hypotheses, and the Schwarz boundaries; testing for the sign of a normal mean, with a general consideration of dynamic programming ideas, and of helpful asymptotics; some discussion of one- and two-armed bandits.
       

390. Sommerfeld biography
     
     • Hilbert, Hurwitz and Lindemann all lectured to Sommerfeld and, after attending a course by Hilbert on the theory of ideal numbers, he felt that abstract pure mathematics was the right subject for him.
       

391. Farey biography
     
     • The standard reference for the Farey sequence is [An introduction to the theory of numbers (New York, 1945).',2)">2] in which Hardy writes:- .
       

392. David biography
     
     • In the second David examines the signs of the deviations from expected numbers to see whether they change sufficiently often.
       

393. Cohn biography
     
     • Other books by Cohn include Skew field constructions (1977), Algebraic numbers and algebraic functions (1991), Elements of linear algebra (1994) and Skew fields published as Volume 57 in the Encyclopedia of Mathematics and its Applications.
       

394. Alison biography
     
     • It is no mean tribute to Mr Alison's skill and zeal that under his administration Watson' s has remained true to the high ideal's of his great predecessor Dr Ogilvie, and has steadily grown in numbers, efficiency, and reputation.
       

395. Frohlich biography
     
     • This paper, Artin root numbers and normal integral bases for quaternion fields, is described by the authors of [Biographical Memoirs of Fellows of the Royal Society of London 51 (2005), 149-168.',1)">1] as:- .
       

396. Hsiung biography
     
     • The second of Mu-Han and Tu Shih's sons, C Y Hsiung, also went on the become a professor of mathematics writing books such as Elementary theory of numbers (1992).
       

397. Dodgson biography
     
     • Soon enough, jokes, puzzles, games, questions-and-answers, tricks with numbers and with words, and mental exercises became for him a means of everyday amusement and for his family and friends source of fun and diversion.
       

398. Carleman biography
     
     • If (an), n ≥ 1, is a sequence of positive numbers, then .
       

399. Herigone biography
     
     • He also introduced a code by which numbers were translated into words to aid memorising them.
       

400. Wu Wen-Tsun biography
     
     • Shanghai fell to the Japanese on 26 November 1937, large numbers having been killed on each side.
       

401. Feynman biography
     
     • He studied a lot of mathematics in his own time including trigonometry, differential and integral calculus, and complex numbers long before he met these topics in his formal education.
       

402. Manin biography
     
     • In the 1960s it was said (in a certain connection) that the most important discovery of recent years in physics was the complex numbers.
       

403. Von Staudt biography
     
     • He turned to projective geometry and Bernoulli numbers.
       

404. Betti biography
     
     • Betti published a memoir on topology in 1871 which contained what we now call the "Betti numbers".
       

405. Poisson biography
     
     • He also introduced the expression "law of large numbers".
       

406. Lalla biography
     
     • Ifrah writes in [A universal history of numbers : From prehistory to the invention of the computer (London, 1998).',2)">2]:- .
       

407. Landau biography
     
     • In addition he gave lecture courses on the foundations of mathematics, irrational numbers, and set theory.
       

408. Archytas biography
     
     • Another interesting mathematical discovery due to Archytas is that there can be no number which is a geometric mean between two numbers in the ratio (n+1) : n.
       

409. Boole biography
     
     • He published around 50 papers and was one of the first to investigate the basic properties of numbers, such as the distributive property, that underlie the subject of algebra.
       

410. Aepinus biography
     
     • During this period he undertook research in several different areas of mathematics including algebraic equations, solving partial differential equations, and on negative numbers.
       

411. Al-Mahani biography
     
     • However, he was led to an equation involving cubes, squares and numbers which he failed to solve after giving it lengthy meditation.
       

412. Margulis biography
     
     • The different approaches to this and related conjectures (and theorems) involve analytic number theory, the theory of Lie groups and algebraic groups, ergodic theory, representation theory, reduction theory, geometry of numbers and some other topics.
       

413. Madhava biography
     
     • All the terms are then divided by the odd numbers 1, 3, 5, ..
       

414. Stifel biography
     
     • History Topics: The real numbers: Pythagoras to Stevin .
       

415. Bohr Harald biography
     
     • Harald Bohr worked on Dirichlet series, and applied analysis to the theory of numbers.
       

416. Maseres biography
     
     • He rejected negative numbers and that part of algebra which is not arithmetic, despite writing 150 years after Viete and Harriot.
       

417. Kostrikin biography
     
     • This text, intended for 19- or 20-year old students, includes such topics as: general properties of mappings and of binary relations; some properties of simple groups; theory of representations; elements of the theory of finite fields; fields of algebraic numbers; as well as traditional subjects in a first course in algebra.
       

418. Jia Xian biography
     
     • It is clear from Yang Hui's description that Jia Xian understood the method of generating the triangle, namely adding the numbers in the two positions above in order to find the number in the position below.
       

419. Frobenius biography
     
     • Ideas from a paper by Dedekind in 1885 made an important contribution and Frobenius was able to construct a complete set of representations by complex numbers.
       

420. Lagny biography
     
     • In 1733 he examined the continued fraction expansion of the quotient of two integers and, as an example, considered adjacent Fibonacci numbers as the worst case expansion for the Euclidean algorithm in his paper Analyse generale ou Methodes nouvelles pour resoudre les problemes de tous les genres et de tous les degres a l'infini.
       

421. Kneser Hellmuth biography
     
     • Large numbers of mathematicians like myself [EFR] who have benefited from visits to this unique conference centre must have said a quiet thank you to Suss, Kneser and their colleagues.
       

422. Zarankiewicz biography
     
     • His work on triangular numbers inspired Sierpinski to further work on this topic while Zarankiewicz also worked jointly with Kuratowski on topology.
       

423. Cheng Dawei biography
     
     • His book is an encyclopaedic hotch-potch of ideas which contains everything from A to Z relating to the Chinese mystique of numbers (magic squares, ..
       

424. Kirkman biography
     
     • For example the Cayley numbers and generalisations are discussed.
       

425. Friedmann biography
     
     • In 1905 Friedmann and Tamarkin wrote a paper on Bernoulli numbers and submitted the paper to Hilbert for publication in Mathematische Annalen.
       

426. Wilkinson biography
     
     • As well as the large numbers of papers on his theoretical work on numerical analysis, Wilkinson developed computer software, working on the production of libraries of numerical routines.
       

427. Segre Beniamino biography
     
     • Segre's contributions to geometry are many but, particularly in the latter part of his life, he is remembered for his study of geometries over fields other than the complex numbers.
       

428. Van der Pol biography
     
     • Of van der Pol's papers on the theory of numbers [An electro-mechanical investigation of the Riemann Zeta function in the critical strip (1947)] is perhaps the best known.
       

429. Berwick biography
     
     • He presumes in the reader a considerable background of knowledge - usually more than is suggested by the introductory paragraphs of his papers - and he was himself of course a master of all the classical theory of algebraic numbers.
       

430. Peano biography
     
     • In 1889 Peano published his famous axioms, called Peano axioms, which defined the natural numbers in terms of sets.
       

431. Al-Khujandi biography
     
     • that what Abu Muhammad al-Khujandi advanced - may God have mercy on him - in his demonstration that the sum of two cubic numbers is not a cube, is defective and incorrect.
       

432. Schubert biography
     
     • We should note that Frege was highly critical of Schubert's approach to numbers.
       

433. Bhaskara I biography
     
     • ',12)">12], [Ganita 23 (1) (1972), 57-79',13)">13] and [Ganita 23 (2) (1972), 41-50.',14)">14] Shukla discusses some features of Bhaskara's mathematics such as: numbers and symbolism, the classification of mathematics, the names and solution methods of equations of the first degree, quadratic equations, cubic equations and equations with more than one unknown, symbolic algebra, unusual and special terms in Bhaskara's work, weights and measures, the Euclidean algorithm method of solving linear indeterminate equations, examples given by Bhaskara I illustrating Aryabhata I's rules, certain tables for solving an equation occurring in astronomy, and reference made by Bhaskara I to the works of earlier Indian mathematicians.
       

434. Bell biography
     
     • Bell numbers .
       

435. Bernstein Sergi biography
     
     • He generalised Lyapunov's conditions for the central limit theorem, studied generalisations of the law of large numbers, worked on Markov processes and stochastic processes.
       

436. Lakatos biography
     
     • as well as having great philosophical and historical value, was circulated in offprint form in enormous numbers.
       

437. Fagnano Giulio biography
     
     • He improved Bombelli's work on complex numbers giving a famous formula .
       

438. Borel biography
     
     • History Topics: The real numbers: Attempts to understand .
       

439. Delone biography
     
     • The mathematical topics that Delone studied include algebra, the geometry of numbers.
       

440. Newman biography
     
     • It was these lectures which introduced Turing to the concept of 'decidability' that in turn inspired Turing's famous paper,On Computable Numbers, with an application to the Entscheidungsproblem.nwhich was published with considerable help from Newman.
       

441. Bryson biography
     
     • It is unclear how quite how Bryson continued the argument but it seems likely that he was saying that by taking polygons with larger and larger numbers of sides then the difference the inscribed and circumscribed polygons could be made as small as we like so that a polygon intermediate between them will equal the circle to whatever degree of accuracy we chose.
       

442. Al-Tusi Sharaf biography
     
     • In Aleppo al-Tusi taught various mathematical topics including the science of numbers, astronomical tables and astrology.
       

443. Rees biography
     
     • It may be because the Graduate Dean is a woman, or it may be for completely objective reasons, that ours is proving an ideal university to draw into advanced graduate work the most obvious source of unused talent in a society that desperately needs additional numbers of persons with training through the doctorate, namely women.
       

444. Janovskaja biography
     
     • There she studied mathematics under Timchenko, who we mentioned above, and also Samuil Osipovich Shatunovsky who was interested in a wide variety of mathematical topics including group theory, the theory of numbers, and geometry.
       

445. Hurwitz biography
     
     • The lectures contained Weierstrass's version of the arithmetisation of analysis including his "construction" of the real numbers, the ε, δ approach to analysis and his theory of complex functions based on power series.
       

446. Parseval biography
     
     • It also only worked, he noted, when certain imaginary parts of two complex numbers cancelled out.
       

447. Fuss biography
     
     • They concern the construction of telescopes, experiments and formulas for the sounds generated by closed pipes, population statistics and the reliability of mortality tables, the project of constructing a wooden bridge of extraordinary width across the Neva, and Euler's numeri idonei for testing large prime numbers.
       

448. Durell biography
     
     • Further topics encompass the special hyperbolic functions; projection and finite series; complex numbers; de Moivre's theorem and its applications; one- and many-valued functions of a complex variable; and roots of equations.
       

449. Lindemann biography
     
     • Lambert had proved in 1761 that π was irrational but this was not enough to prove the impossibility of squaring the circle with ruler and compass since certain algebraic numbers can be constructed with ruler and compass.
       

450. Van Vleck biography
     
     • By following Van Vleck's own steps in deriving consequences of his zero-one law, a result ("the extended Van Vleck theorem") is given which is directly comparable to Borel's law of normal numbers.
       

451. Burali-Forti biography
     
     • History Topics: The real numbers: Attempts to understand .
       

452. Gordan biography
     
     • One would have to say that this lack of numbers is more than made up for by the remarkable quality of that one student who would do so much to set algebra on the path it is still on today.
       

453. Plucker biography
     
     • In this way of specifying coordinates, a point has a linear equation, namely that of all lines through the point while a line has a pair of numbers namely the x and y coordinates of where it cuts the axes.
       

454. Matsushima biography
     
     • (3) vector bundle valued harmonic forms - Betti numbers of locally symmetric spaces; cohomology of vector bundles over locally symmetric spaces; second fundamental forms and curvature forms; .
       

455. Tits biography
     
     • The classical families (A through D) soon led to groups over fields other than the real or complex numbers, and a comprehensive study was published by Dickson in 1901.
       

456. Fine biography
     
     • The first part deals with arithmetic, particularly with whole numbers, common fractions and sexagesimal fractions.
       

457. Schmetterer biography
     
     • He became interested in mathematics when he was about twelve years old, his interest beginning when he read a book on analysis which defined logarithms of complex numbers.
       

458. Ito biography
     
     • With clear definition of real numbers formulated at the end of the19th century, differential and integral calculus had developed into an authentic mathematical system.
       

459. Maurolico biography
     
     • Maurolico also worked on geometry, the theory of numbers (L E Dickson notes some of his results), optics, conics and mechanics, writing important books on these topics.
       

460. Herstein biography
     
     • The second paper proves a conjecture that the solubility of groups of odd order is equivalent to a condition on the group ring of a group, while the third paper takes methods from the study of Banach rings and topological groups to prove results about group rings over the complex numbers.
       

461. Kolmogorov biography
     
     • These included his versions of the strong law of large numbers and the law of the iterated logarithm, some generalisations of the operations of differentiation and integration, and a contribution to intuitional logic.
       

462. Fefferman biography
     
     • I like to lie down on the sofa for hours at a stretch thinking intently about shapes, relationships and change - rarely about numbers as such.
       

463. Zassenhaus biography
     
     • Zassenhaus worked on a broad range of topics and, in addition to those mentioned above, he worked on nearfields, the theory of orders, representation theory, the geometry of numbers and the history of mathematics.
       

464. Kaplansky biography
     
     • Similarly his many other books are beautiful introductions to various areas of algebra and have been enjoyed for their clarity, style and beauty by large numbers of undergraduate and graduate students.
       

465. Jarnik biography
     
     • During the decade 1939-49 he wrote a series of papers dealing with the geometry of numbers, in particular dealing with Minkowski's inequality for convex bodies.
       

466. Al-Qalasadi biography
     
     • A dust board was used because the methods required the moving of numbers around in the calculation and rubbing some out as the calculation proceeded.
       

467. Hertz Heinrich biography
     
     • He was a true friend to his friends, a respected teacher to his students, who had begun to gather around him in large numbers, some of the coming from great distances; and to his family a loving husband and father.
       

468. Mercator Nicolaus biography
     
     • Further publications followed: Rationes mathematicae subductae (1653) sets out to distinguish between rational and irrational numbers pointing out that in music rational ratios lead to harmony while irrational leads to dissonance.
       

469. Camus biography
     
     • As examiner, Camus was able to base his examinations on the three parts of his Cours which were in print and this had, as one might expect, the result that his books sold widely and in great numbers.
       

470. Rogers biography
     
     • His early work was on number theory and he wrote on Diophantine inequalities and the geometry of numbers.
       

471. Wittgenstein biography
     
     • History Topics: The real numbers: Pythagoras to Stevin .
       

472. Apollonius biography
     
     • Apollonius also wrote a work on the cylindrical helix and another on irrational numbers which is mentioned by Proclus.
       

473. Savile biography
     
     • Other requirements for the astronomy professor was to teach spherics, calculation with sexagesimal numbers, optics, geography and navigation.
       

474. Kloosterman biography
     
     • His solution of this case appeared in his paper On the representation of numbers in the form ax2 + by2 + cz2 + dt2 which was published in Acta Mathematica in 1926.
       

475. Plato biography
     
     • History Topics: The real numbers: Pythagoras to Stevin .
       

476. Stiefel biography
     
     • These topics include possible number systems, the questions of "fixed" vs "floating" point and complementation, the arithmetic processes, the grouping of numbers to achieve higher than normal precisions, conversion between number systems, the structure of finite approximation methods, error analysis, programming and coding as well as the physical organs of a machine.
       

477. Killing biography
     
     • Furthermore, he had permitted complex numbers into the calculations to facilitate the analysis, but eventually, for his classification of space forms, he must deal with the "real" case.
       

478. Ford biography
     
     • A major paper based on his thesis Rational approximations to irrational complex numbers was published in the Transactions of the American Mathematical Society in 1918.
       

479. Steinhaus biography
     
     • To Steinhaus mathematics was a mirror of reality and life much in the same way as poetry is a mirror, and he liked to "play" with numbers, sets, and curves, the way a poet plays with words, phrases, and sounds.
       

480. Schramm biography
     
     • In 1994 Schramm published a joint paper with Greg Kuperberg entitled Average kissing numbers for non-congruent sphere packings.
       

481. Gateaux biography
     
     • In [Lecons d\'Analyse fonctionnelle (Gauthier-Villars, 1922).',12)">12], Levy has finally admitted that the right formulation for these problems is in a probabilistic framework, and it is impressive to see how in the book (and in particular in Chapter VI), Levy makes use of probability theory to justify the passages to the limit by means of the law of large numbers.
       

482. Turan biography
     
     • One was A problem in the elementary theory of numbers which appeared in the American Mathematical Monthly.
       

483. Zuse biography
     
     • It was entirely mechanical, with an arithmetic unit composed of large numbers of mechanical switches, and a memory consisting of layers of metal bars between layers of glass.
       

484. Chasles biography
     
     • In January 1813, after the disaster of the Russian campaign, Napoleon called up more men to fill the dwindling numbers in his armies.
       

485. Geminus biography
     
     • The occurrence of a Latin name in a centre of Greek culture need not surprise us, since Romans settled in such centres in large numbers during the last century BC.
       

486. Ramanathan biography
     
     • He began publishing papers in 1941 when On Demlo numbers appeared in print.
       

487. Frechet biography
     
     • He defines a functional operation as a numerically valued function defined on arbitrary objects which he wants to include points, lines, functions, numbers, surfaces etc.
       

488. Ward Seth biography
     
     • We have conceived it requisite to examine all the books of our public library (every one taking his part) and to make a catalogue or index of the matters and that very particularly in philosophy, physics, mathematics and indeed in all other faculties, that so that great numbers of books may be serviceable and a man may at once see where he may find whatever is there concerning the argument he is upon, and this is our present business which we hope to dispatch this Lent.
       

489. Magnitsky biography
     
     • A wide variety of students might use it, and in straightforward pragmatic terms, Magnitskii declared that the science of numbers was useful to merchants, for those in charge of financial matters, for the keepers of church funds, for property owners and stewards, for all manner of craftsmen whether they were constructing buildings, sailing ships, measuring granaries, levying taxes, or performing some military duty.
       

490. Harriot biography
     
     • Harriot and binary numbers .
       

491. Straus biography
     
     • Algebraic equations satisfied by roots of natural numbers.
       

492. Hironaka biography
     
     • This university was founded in 1897 to train small numbers of selected students as academics.
       

493. Olive biography
     
     • She published papers such as Binomial functions and combinatorial mathematics (1979), A combinatorial approach to generalized powers (1980), Binomial functions with the Stirling property (1981), Some functions that count (1983), Taylor series revisited (1984), Catalan numbers revisited (1985), A special class of infinite matrices (1987), and The ballot problem revisited (1988).
       

494. Kurschak biography
     
     • The rationals are not complete with this metric and their completion is the field of p-adic numbers.
       

495. Prufer biography
     
     • In addition to his work on abelian groups, Prufer also worked on algebraic numbers, publishing the paper Neue Begrundung der algebraischen Zahlentheorie in 1925, and knot theory.
       

496. Aristotle biography
     
     • History Topics: The real numbers: Pythagoras to Stevin .
       

497. Perron biography
     
     • In addition to the work on continued fractions mentioned above, which in fact ran to three editions the last being a two volume version in 1954/57, he published an important text on irrational numbers in 1921.
       

498. Rudin biography
     
     • I'm not really interested in numbers.
       

499. Steinitz biography
     
     • The direction of his mathematics was also much influenced by Heinrich Weber and by Hensel's results on p-adic numbers in 1899.
       

500. Van der Waerden biography
     
     • His doctoral thesis De algebraiese grondslagen der meetkunde van het aantal (The algebraic foundations of the geometry of numbers) was submitted to the University of Amsterdam and he defended it in the grand hall of the University on 24 March 1926.
       
     • History Topics: The real numbers: Pythagoras to Stevin .
       

501. Milnor biography
     
     • The reason that Milnor could use them to distinguish the differential properties of manifolds is because they have arithmetic properties, involving the Bernoulli numbers, which reflect in a deep and not fully understood way these differential properties.
       

502. Caratheodory biography
     
     • 1: Numbers, Point sets, Functions (1939), and Funktionentheorie, a 2 volume work published in 1950.
       

503. Lorenz Edward biography
     
     • Later papers include: A very narrow spectral band (1984) which investigates the spectral properties of the Lorenz system; The local structure of a chaotic attractor in four dimensions (1984); Lyapunov numbers and the local structure of attractors (1985); On the existence of a slow manifold (1986); Atmospheric models as dynamical systems (1986); Computational chaos - a prelude to computational instability (1989); and The slow manifold - what is it? (1992).
       

504. James Ralph biography
     
     • In the paper On the sieve method of Viggo Brun James takes the sieve method which had been used by Buchstab in 1940 to establish results on prime numbers, and applies it to obtain results about infinite subsets of primes, such as primes in arithmetic progression.
       

505. Jonquieres biography
     
     • He also worked on algebra, in particular the theory of equations, and, in the latter part of his life, on the theory of numbers where he examined Diophantine equations and the distribution of primes.
       

506. Tannery Jules biography
     
     • Tannery made an impressive contribution to the Bulletin, writing large numbers of reviews.
       

507. Thue biography
     
     • In 1909 he produced an important paper, published in Crelle's Journal, on algebraic numbers showing that, for example, y3 - 2x2 = 1 cannot be satisfied by infinitely many pairs of integers.
       

508. Al-Nasawi biography
     
     • The book is composed of four separate treatises, each dealing with a particular class of numbers.
       

509. Moser Leo biography
     
     • Here are a small selection of his papers (some co-authored with Joachim Lambek, some with Max Wyman, some single author papers): On the different distances determined by n points (1952), Note on a combinatorial formula of Mendelsohn (1953), On the distribution of Pythagorean triangles (1955), An asymptotic formula for the Bell numbers (1955), and Rational analogues of the logarithm function (1956).
       

510. Wallis biography
     
     • History Topics: The real numbers: Stevin to Hilbert .
       

511. Krasnosel'skii biography
     
     • Following this he achieved a remarkable publication record with papers (all written in Russian) such as On the deficiency numbers of closed operators (1947), (with M G Krein) On the centre of a general dynamical system (1947), (with M G Krein) Fundamental theorems on the extension of Hermitian operators and certain of their applications to the theory of orthogonal polynomials and the problem of moments (1947), On the extension of Hermitian operators with a nondense domain of definition (1948), On self-adjoint extensions of Hermitian operators (1949), (with M G Krein) On a proof of the theorem on category of a projective space (1949), and On a fixed point principle for completely continuous operators in functional spaces (1950).
       

512. Alling biography
     
     • The third book 'Foundations of analysis over surreal number fields' appeared in 1987, and includes an account of Conway's theory of surreal numbers.
       

513. Dupre biography
     
     • He won an honourable mention for the 1858 Grand Prix of the Academy of Sciences with a paper on Legendre's theory of numbers.
       

514. Dirichlet biography
     
     • History Topics: Prime numbers .
       

515. Skolem biography
     
     • With these he defined prime numbers and developed a considerable amount of number theory.
       

516. Mandelbrot biography
     
     • The problem was both one of geometry concerning the nature of the line thought of as built up of points and of arithmetic thought of as the theory of the real numbers.
       

517. Bertrand biography
     
     • History Topics: Prime numbers .
       

518. Leger biography
     
     • is counted) to recognise that the worst case of the Euclidean algorithm occurs when the inputs are consecutive Fibonacci numbers.
       

519. Titchmarsh biography
     
     • in recognition of his distinguished researches on the Riemann zeta-function, analytic theory of numbers, Fourier analysis and eigenfunction expansions.
       

520. Ajima biography
     
     • He produced log tables which were designed for taking 10th roots and powers of numbers.
       

521. Jacobsthal biography
     
     • In the thesis he gives, among other things, a very beautiful proof that every prime number p of the form 4n + 1 can be written as a sum of two square numbers.
       

History Topics

1. Perfect numbers
   
   • Perfect numbers .
     
   • It is not known when perfect numbers were first studied and indeed the first studies may go back to the earliest times when numbers first aroused curiosity.
     
   • It is quite likely, although not certain, that the Egyptians would have come across such numbers naturally given the way their methods of calculation worked, see for example [Centaurus 20 (4) (1976), 269-275.',17)">17] where detailed justification for this idea is given.
     
   • Perfect numbers were studied by Pythagoras and his followers, more for their mystical properties than for their number theoretic properties.
     
   • Before we begin to look at the history of the study of perfect numbers, we define the concepts which are involved.
     
   • The four perfect numbers 6, 28, 496 and 8128 seem to have been known from ancient times and there is no record of these discoveries.
     
   • The first recorded mathematical result concerning perfect numbers which is known occurs in Euclid's Elements written around 300BC.
     
   • However, although numbers are represented by line segments and so have a geometrical appearance, there are significant number theory results in the Elements.
     
   • If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes a prime, and if the sum multiplied into the last make some number, the product will be perfect.
     
   • Now Euclid gives a rigorous proof of the Proposition and we have the first significant result on perfect numbers.
     
   • The next significant study of perfect numbers was made by Nicomachus of Gerasa.
     
   • Around 100 AD Nicomachus wrote his famous text Introductio Arithmetica which gives a classification of numbers based on the concept of perfect numbers.
     
   • Nicomachus divides numbers into three classes: the superabundant numbers which have the property that the sum of their aliquot parts is greater than the number; deficient numbers which have the property that the sum of their aliquot parts is less than the number; and perfect numbers which have the property that the sum of their aliquot parts is equal to the number (see [History of Mathematics : History of Problems (Paris, 1997), 389-410.',8)">8], or [Nicomachus, Introduction to arithmetic (New York, 1926).',1)">1] for a different translation):- .
     
   • Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed to one another; as for those that occupy the middle position between the two, they are said to be perfect.
     
   • Now satisfied with the moral considerations of numbers, Nicomachus goes on to provide biological analogies in which he describes superabundant numbers as being like an animal with (see [History of Mathematics : History of Problems (Paris, 1997), 389-410.',8)">8], or [Nicomachus, Introduction to arithmetic (New York, 1926).',1)">1]):- .
     
   • Deficient numbers are compared to animals with:- .
     
   • Nicomachus goes on to describe certain results concerning perfect numbers.
     
   • (2) All perfect numbers are even.
     
   • (3) All perfect numbers end in 6 and 8 alternately.
     
   • (4) Euclid's algorithm to generate perfect numbers will give all perfect numbers i.e.
     
   • (5) There are infinitely many perfect numbers.
     
   • Let us look in more detail at Nicomachus's description of the algorithm to generate perfect numbers which is assertion (4) above (see [History of Mathematics : History of Problems (Paris, 1997), 389-410.',8)">8], or [Nicomachus, Introduction to arithmetic (New York, 1926).',1)">1]):- .
     
   • There exists an elegant and sure method of generating these numbers, which does not leave out any perfect numbers and which does not include any that are not; and which is done in the following way.
     
   • However, it is probable that this methods of generating perfect numbers was part of the general mathematical tradition handed down from before Euclid's time and continuing till Nicomachus wrote his treatise.
     
   • Whether the five assertions of Nicomachus were based on any more than this algorithm and the fact the there were four perfect numbers known to him 6, 28, 496 and 8128, it is impossible to say, but it does seem unlikely that anything more lies behind the unproved assertions.
     
   • Some of the assertions are made in this quote about perfect numbers which follows the description of the algorithm [Nicomachus, Introduction to arithmetic (New York, 1926).',1)">1]:- .
     
   • The Arab mathematicians were also fascinated by perfect numbers and Thabit ibn Qurra wrote the Treatise on amicable numbers in which he examined when numbers of the form 2np, where p is prime, can be perfect.
     
   • Ibn al-Haytham proved a partial converse to Euclid's proposition in the unpublished work Treatise on analysis and synthesis when he showed that perfect numbers satisfying certain conditions had to be of the form 2k-1(2k - 1) where 2k - 1 is prime.
     
   • Among the many Arab mathematicians to take up the Greek investigation of perfect numbers with great enthusiasm was Ismail ibn Ibrahim ibn Fallus (1194-1239) who wrote a treatise based on the Introduction to arithmetic by Nicomachus.
     
   • He accepted Nicomachus's classification of numbers but the work is purely mathematical, not containing the moral comments of Nicomachus.
     
   • Ibn Fallus gave, in his treatise, a table of ten numbers which were claimed to be perfect, the first seven are correct and are in fact the first seven perfect numbers, the remaining three numbers are incorrect.
     
   • At the beginning of the renaissance of mathematics in Europe around 1500 the assertions of Nicomachus were taken as truths, nothing further being known concerning perfect numbers not even the work of the Arabs.
     
   • Charles de Bovelles, a theologian and philosopher, published a book on perfect numbers in 1509.
     
   • It has also been found in a manuscript written around 1458, while both the fifth and sixth perfect numbers have been found in another manuscript written by the same author probably shortly after 1460.
     
   • Nicomachus's claim that perfect numbers ended in 6 and 8 alternately still stood however.
     
   • This was not noticed until 1977 and therefore did not influence progress on perfect numbers.
     
   • The next step forward came in 1603 when Cataldi found the factors of all numbers up to 800 and also a table of all primes up to 750 (there are 132 such primes).
     
   • This result by Cataldi showed that Nicomachus's assertion that perfect numbers ended in 6 and 8 alternately was false since the fifth and sixth perfect numbers both ended in 6.
     
   • As the reader will have already realised, the history of perfect numbers is littered with errors and Cataldi, despite having made the major advance of finding two new perfect numbers, also made some false claims.
     
   • He writes in Utriusque Arithmetices that the exponents p = 2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37 give perfect numbers 2p-1(2p - 1).
     
   • Many mathematicians were interested in perfect numbers and tried to contribute to the theory.
     
   • I think I am able to prove that there are no even numbers which are perfect apart from those of Euclid; and that there are no odd perfect numbers, unless they are composed of a single prime number, multiplied by a square whose root is composed of several other prime number.
     
   • But I can see nothing which would prevent one from finding numbers of this sort.
     
   • For example, if 22021 were prime, in multiplying it by 9018009 which is a square whose root is composed of the prime numbers 3, 7, 11, 13, one would have 198585576189, which would be a perfect number.
     
   • But, whatever method one might use, it would require a great deal of time to look for these numbers..
     
   • The treatise would never be written, partly because Fermat never got round to writing his results up properly, but also because he did not achieve the substantial results on perfect numbers he had hoped.
     
   • In June 1640 Fermat wrote to Mersenne telling him about his discoveries concerning perfect numbers.
     
   • The numbers less by one than the double progression, like .
     
   • let them be called the radicals of perfect numbers, since whenever they are prime, they produce them.
     
   • Put above these numbers in natural progression 1, 2, 3, 4, 5, etc., which are called their exponents.
     
   • Here are three beautiful propositions which I have found and proved without difficulty, I shall call them the foundations of the invention of perfect numbers.
     
   • Certainly Fermat found his Little Theorem as a consequence of his investigations into perfect numbers.
     
   • In fact assuming that perfect numbers are of the form 2p-1(2p - 1) where p is prime, the question readily translates into asking whether 237 - 1 is prime.
     
   • Mersenne was very interested in the results that Fermat sent him on perfect numbers and soon produced a claim of his own which was to fascinate mathematicians for a great many years.
     
   • The next person to make a major contribution to the question of perfect numbers was Euler.
     
   • This verifies the fourth assertion of Nicomachus at least in the case of even numbers.
     
   • It also leads to an easy proof that all even perfect numbers end in either a 6 or 8 (but not alternately).
     
   • Euler also tried to make some headway on the problem of whether odd perfect numbers existed.
     
   • However, as with most others whose contribution we have examined, Euler made predictions about perfect numbers which turned out to be wrong.
     
   • The search for perfect numbers had now become an attempt to check whether Mersenne was right with his claims in Cogitata physica mathematica.
     
   • Mathematicians such as Peter Barlow wrote in his book Theory of Numbers published in 1811, that the perfect number 230(231 - 1):- .
     
   • This, of course, turned out to be yet one more false assertion about perfect numbers! .
     
   • Lucas was also able to verify that one of the numbers in Mersenne's list was correct when he showed that 2127 - 1 is a Mersenne prime and so 2126(2127- 1) is indeed a perfect number.
     
   • Lucas made another important advance which, as modified by Lehmer in 1930, is the basis of computer searches used today to find Mersenne primes, and so to find perfect numbers.
     
   • Of course if this conjecture were true it would solve the still open question of whether there are an infinite number of Mersenne primes (and also solve the still open question of whether there are infinitely many perfect numbers).
     
   • In October 1903 Cole presented a paper On the factorisation of large numbers to a meeting of the American Mathematical Society.
     
   • Without speaking a work he multiplied the two numbers together to get 147573952589676412927 and sat down to applause from the audience.
     
   • We have followed the progress of finding even perfect numbers but there was also attempts to show that an odd perfect number could not exist.
     
   • Today 46 perfect numbers are known, 288(289- 1) being the last to be discovered by hand calculations in 1911 (although not the largest found by hand calculations), all others being found using a computer.
     
   • In fact computers have led to a revival of interest in the discovery of Mersenne primes, and therefore of perfect numbers.
     
   • If you wonder why we have not included the number in decimal form, then let me say that it contains about 150 times as many characters as this whole article on perfect numbers.
     
   • Perfect numbers etc .
     
   • More about perfect numbers at Singapore .
     
   • A printout of the first 22 perfect numbers is at Math Forum .
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/Perfect_numbers.html .
     

2. Real numbers 2
   
   • The real numbers: Stevin to Hilbert .
     
   • Details of the earlier contributions are examined in some detail in our article: The real numbers: Pythagoras to Stevin .
     
   • He still only considers finite decimal expansions and realises that with these one can approximate numbers (which for him are constructed from positive integers by addition, subtraction, multiplication, division and taking nth roots) as closely as one wishes.
     
   • Now, as for other incommensurable quantities, though this proportion cannot be accurately expressed in absolute numbers, yet by continued approximation it may; so as to approach nearer to it than any difference assignable.
     
   • This leads into the study of infinite series but without the necessary machinery to prove that these infinite series converged to a limit, he was never going to be able to progress much further in studying real numbers.
     
   • Real numbers became very much associated with magnitudes.
     
   • All could be measured by real numbers.
     
   • By the beginning of the nineteenth century a more rigorous approach to mathematics, principally by Cauchy and Bolzano, began to provide the machinery to put the real numbers on a firmer footing.
     
   • though Cauchy implicitly assumed several forms of the completeness axiom for the real numbers, he did not fully understand the nature of completeness or the related topological properties of sets of real numbers or of points in space.
     
   • Cauchy did not have explicit formulations for the completeness of the real numbers.
     
   • Though Cauchy understood that a real number could be obtained as the limit of rationals, he did not develop his insight into a definition of real numbers or a detailed description of the properties of real numbers.
     
   • Cauchy, in Cours d'analyse (1821), did not worry too much about the definition of the real numbers.
     
   • He does say that a real number is the limit of a sequence of rational numbers but he is assuming here that the real numbers are known.
     
   • Then the product AB will be the limit of the sequence of rational numbers Ab, Ab', Ab'', ..
     
   • Bolzano, on the other hand, showed that bounded Cauchy sequence of real numbers had a least upper bound in 1817.
     
   • He later worked out his own theory of real numbers which he did not publish.
     
   • His definition of a real number was made in terms of convergent sequences of rational numbers and is explained in [Casopis Pest.
     
   • As Bolzano's contributions were unpublished they had little influence in the development of the theory of the real numbers.
     
   • Cauchy himself does not seem to have understood the significance of his own "Cauchy sequence" criterion for defining the real numbers.
     
   • It was Weierstrass, Heine, Meray, Cantor and Dedekind who, after convergence and uniform convergence were better understood, were able to give rigorous definitions of the real numbers.
     
   • Up to this time there was no proof that numbers existed that were not the roots of polynomial equations with rational coefficients.
     
   • Clearly √2 is the root of a polynomial equation with rational coefficients, namely x2 = 2, and it is easy to see that all roots of rational numbers arise as solutions of such equations.
     
   • Liouville's interest in transcendental numbers stemmed from reading a correspondence between Goldbach and Daniel Bernoulli.
     
   • However his contributions led him to prove the existence of a transcendental number in 1844 when he constructed an infinite class of such numbers using continued fractions.
     
   • These were the first numbers to be proved transcendental.
     
   • In 1851 he published results on transcendental numbers removing the dependence on continued fractions.
     
   • One of the first people to attempt to give a rigorous definition of the real numbers was Hamilton.
     
   • He tried another approach of defining numbers given by some law, say x goesto x2.
     
   • Even if one got round this problem he is only defining numbers given by a law.
     
   • It is unclear whether he thought that all real numbers would arise in this way.
     
   • Weierstrass gave his own theory of real numbers in his Berlin lectures beginning in 1865 but this work was not published.
     
   • Therefore, the question of the existence of numbers can only refer to the thinking subject or to those objects of thought whose relations are represented by numbers.
     
   • In his 1867 monograph Hankel addressed the question of whether there were other "number systems" which had essentially the same rules as the real numbers.
     
   • Two years after the publication of Hankel's monograph, Meray published Remarques sur la nature des quantites in which he considered Cauchy sequences of rational numbers which, if they did not converge to a rational limit, had what he called a "fictitious limit".
     
   • He then considered the real numbers to consist of the rational numbers and his fictitious limits.
     
   • Heine's system has become one of the two standard ways of defining the real numbers today.
     
   • Essentially Heine looks at Cauchy sequences of rational numbers.
     
   • to be equivalent if the sequence of rational numbers a1 - b1, a2 - b2 , a3 - b3 , a4 - b4 , ..
     
   • Cantor also published his version of the real numbers in 1872 which followed a similar method to that of Heine.
     
   • His numbers were Cauchy sequences of rational numbers and he used the term "determinate limit".
     
   • Similarly Cantor realised that if he wants the line to represent the real numbers then he has to introduce an axiom to recover the connection between the way the real numbers are now being defined and the old concept of measurement.
     
   • If this distance has a rational relation to the unit of measure, then it is expressed by a rational quantity in the domain of rational numbers; otherwise, if the point is one known through a construction, it is always possible to give a sequence of rationals a1 , a2 , a3 , ..
     
   • When he realised that others like Heine and Cantor were about to publish their versions of a rigorous definition of the real numbers he decided that he too should publish his ideas.
     
   • This resulted in yet another 1872 publication giving a definition of the real numbers.
     
   • Dedekind considered all decompositions of the rational numbers into two sets A1 , A2 so that a1 < a2 for all a1 in A1 and a2 in A2.
     
   • He claimed that the real numbers defined in this way had a right to exist because:- .
     
   • He wanted to develop a theory of real numbers based on a purely logical base and attacked the philosophy behind the constructions which had been published.
     
   • The formal conception of numbers requires of itself more modest limitations than does the logical conception.
     
   • It does not ask, what are and what shall the numbers be, but it asks, what does one require of numbers in arithmetic.
     
   • Hilbert had taken a totally different approach to defining the real numbers in 1900.
     
   • He defined the real numbers to be a system with eighteen axioms.
     
   • The Archimedean axiom stated that given positive numbers a and b then it is possible to add a to itself a finite number of times so that the sum exceed b.
     
   • This was totally new since all other methods built the real numbers from the known rational numbers.
     
   • Hilbert's numbers were unconnected with any known system.
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/Real_numbers_2.html .
     

3. Real numbers 1
   
   • The real numbers: Pythagoras to Stevin .
     
   • "All right: the concept of number is defined for you as the logical sum of these individual interrelated concepts: cardinal numbers, rational numbers, real numbers etc.; and, in the same way the concept of a game is the logical sum of a corresponding set of sub-concepts." - It need not be so.
     
   • We should begin a discussion of real numbers by looking at the concepts of magnitude and number in ancient Greek times.
     
   • (the natural numbers in the terminology of today) in a geometrical way, not as lengths of a line as we do, but rather in the form of discrete points.
     
   • Magnitudes, being distinct entities from numbers, had to have a separate definition and indeed Nicomachus makes such a parallel definition for magnitudes.
     
   • [In the time of Pythagoras] since all other things seemed in their whole nature to be modelled on numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number.
     
   • And all the properties of numbers and scales which they could show to agree with the attributes and parts and the whole arrangement of the heavens, they fitted into their scheme ..
     
   • the Pythagoreans say that things are what they are by intimating numbers ..
     
   • the Pythagoreans take objects themselves to be numbers and do not treat mathematical objects as distinct from them ..
     
   • All numbers, essentially by definition, were, as we have seen, (positive integer) multiples of a base unit but ratios of lengths were shown not to have the property of being ratios of numbers (integers).
     
   • Heimonen, in [Historia Mathematica 23 (1996), 355-375.',10)" onmouseover="window.status='Click to see reference';return true">10], looks at the views of different historians concerning the discovery of the irrational numbers:- .
     
   • Before continuing to describe advances in ideas concerning numbers, it should be mentioned at this stage that the Egyptians and the Babylonians had a different notion of number to that of the ancient Greeks.
     
   • The Babylonians looked at reciprocals and also at approximations to irrational numbers, such as √2, long before Greek mathematicians considered approximations.
     
   • The Egyptians also looked at approximating irrational numbers.
     
   • It says that a : b = c : d if given any natural numbers n and m we have .
     
   • For example for magnitudes a and b and natural numbers n and m he proves:- .
     
   • In Book VII Euclid studies numbers.
     
   • It was a unit and the numbers 2, 3, 4, ..
     
   • Various properties of numbers are assumed but are not listed as axioms.
     
   • He then introduces proportion for numbers and shows essentially that for numbers a, b, c, d that a : b = c : d precisely when the least numbers with ratio a : b is equal to the least numbers with ratio c : d.
     
   • So where does Euclid's Elements leave us with respect to numbers.
     
   • Basically numbers were 1, 2, 3, ..
     
   • and ratios of numbers were used which (although not considered to be numbers) basically allowed manipulation with what we call rationals.
     
   • By the sixteenth century rational numbers and roots of numbers were becoming accepted as numbers although there was still a sharp distinction between these different types of numbers.
     
   • It is rightly disputed whether irrational numbers are true numbers or false.
     
   • Because in studying geometrical figures, where rational numbers desert us, irrationals take their place, and show precisely what rational numbers are unable to show ..
     
   • However, he goes on to argue that, as they are not proportional to rational numbers, they cannot be true numbers even if they are correct.
     
   • He ends up arguing that all irrational numbers result from radical expressions.
     
   • One has to understand here that in fact it was in a sense fortuitous that his invention led to a much deeper understanding of numbers for he certainly did not introduce the notation with that in mind.
     
   • Stevin made a number of other important advances in the study of the real numbers.
     
   • He argued strongly in L'Arithmetique (1585) that all numbers such as square roots, irrational numbers, surds, negative numbers etc should all be treated as numbers and not distinguished as being different in nature.
     
   • Thesis 2:nnnThat any given numbers can be square, cubes, fourth powers etc.
     
   • Thesis 4:nnnThat there are no absurd, irrational, irregular, inexplicable or surd numbers.
     
   • It is a very common thing amongst authors of arithmetics to treat numbers like √8 and similar ones, which they call absurd, irrational, irregular, inexplicable or surds etc and which we deny to be the case for number which turns up.
     
   • His first thesis was to argue against the Greek idea that 1 is not a number but a unit and the numbers 2, 3, 4, ..
     
   • The other three theses were encouraging people to treat different types of numbers, which were at that time treated separately, as a single entity - namely a number.
     
   • even though it were possible for us to subtract by due process several hundred thousand times the smaller magnitude from the larger and continue that for several thousands of years, nevertheless if the two given numbers were incommensurable one would labour eternally, always ignorant of what could still happen in the end.
     
   • Further progress in the development of the real numbers only became possible after ideas of convergence were put on a firm basis.
     
   • However, there was a strong influence in the other direction too, since progress in rigorous analysis required a deeper understanding of the real numbers.
     
   • This is studied further in the article The real numbers: Stevin to Hilbert.
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/Real_numbers_1.html .
     

4. Prime numbers
   
   • Prime numbers .
     
   • Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians.
     
   • The mathematicians of Pythagoras's school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties.
     Go directly to this paragraph
     
   • They understood the idea of primality and were interested in perfect and amicable numbers.
     Go directly to this paragraph
     
   • A pair of amicable numbers is a pair like 220 and 284 such that the proper divisors of one number sum to the other and vice versa.
     
   • You can see more about these numbers in the History topics article Perfect numbers.
     
   • In Book IX of the Elements, Euclid proves that there are infinitely many prime numbers.
     Go directly to this paragraph
     
   • The mathematician Euler (much later in 1747) was able to show that all even perfect numbers are of this form.
     Go directly to this paragraph
     
   • It is not known to this day whether there are any odd perfect numbers.
     Go directly to this paragraph
     
   • There is then a long gap in the history of prime numbers during what is usually called the Dark Ages.
     
   • He devised a new method of factorising large numbers which he demonstrated by factorising the number 2027651281 = 44021 × 46061.
     
   • Fermat's Little Theorem is the basis for many other results in Number Theory and is the basis for methods of checking whether numbers are prime which are still in use on today's electronic computers.
     
   • In one of his letters to Mersenne he conjectured that the numbers 2n + 1 were always prime if n is a power of 2.
     Go directly to this paragraph
     
   • Numbers of this form are called Fermat numbers and it was not until more than 100 years later that Euler showed that the next case 232 + 1 = 4294967297 is divisible by 641 and so is not prime.
     Go directly to this paragraph
     
   • These are often called Mersenne numbers Mn because Mersenne studied them.
     
   • Not all numbers of the form 2n - 1 with n prime are prime.
     
   • For many years numbers of this form provided the largest known primes.
     Go directly to this paragraph
     
   • In 1952 the Mersenne numbers M521, M607, M1279, M2203 and M2281 were proved to be prime by Robinson using an early computer and the electronic age had begun.
     
   • As mentioned above he factorised the 5th Fermat Number 232 + 1, he found 60 pairs of the amicable numbers referred to above, and he stated (but was unable to prove) what became known as the Law of Quadratic Reciprocity.
     
   • formed by summing the reciprocals of the prime numbers, is also divergent.
     
   • For example in the 100 numbers immediately before 10 000 000 there are 9 primes, while in the 100 numbers after there are only 2 primes.
     Go directly to this paragraph
     
   • Gauss (who was a prodigious calculator) told a friend that whenever he had a spare 15 minutes he would spend it in counting the primes in a 'chiliad' (a range of 1000 numbers).
     Go directly to this paragraph
     
   • There are still many open questions (some of them dating back hundreds of years) relating to prime numbers.
     
   • Are there infinitely many prime Fermat numbers? Indeed, are there any prime Fermat numbers after the fourth one? .
     
   • Perfect numbers, etc.
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/Prime_numbers.html .
     

5. Real numbers 3
   
   • The real numbers: Attempts to understand .
     
   • Were the real numbers consistent? Would an inconsistency appear one day and much of the mathematical building come tumbling down? Some of the intuitive difficulties that began to be felt revolved around the fact that the real numbers were not countable, that is, they could not be put in 1-1 correspondence with the natural numbers.
     
   • Cantor proved that the real numbers were not countable in 1874.
     
   • He produced his famous "diagonal argument" in 1890 which gave a second, more striking, proof that the real numbers were not countable.
     
   • To do this he assumed that the real numbers were countable, that is they could be listed in order.
     
   • Hence the real numbers are not countable.
     
   • Let us start with the 100 two digit numbers.
     
   • , 51 becomes Z, then code all the punctuation marks, and then make all the remaining numbers up to 99 translate to an empty space.
     
   • The first thing to notice is that all descriptions of real numbers in English (let us forget about words in other languages) must appear in c, since every possible sentence occurs in c.
     
   • There are only a countable number of such descriptions of real numbers in English so all real numbers (except a tiny countable subset) can never be described in English.
     
   • His proof of this involved showing that the non-normal numbers formed a subset of the reals of measure zero.
     
   • There were still an uncountable number of non-normal numbers, however, which was easily seen by taking the subset of all real numbers with no digit equal to 1.
     
   • The first of these was derived from the fact that the ordinal numbers themselves formed an ordered set whose order type had to be an ordinal number.
     
   • Poincare (1908) and Weyl (1918) complained that analysis had to be based on a concept of the real numbers which eliminated the non-constructive features.
     
   • He showed that a formal theory which includes the arithmetic of the natural numbers had to lead to statements which could neither be proved nor disproved within the theory.
     
   • Although this topic of research is still an active one, most mathematicians accept the uncountable world of Cantor and the non-constructive system of real numbers.
     
   • Write the numbers 1, 2, 3, ..
     
   • The answer is that despite "knowing" that such numbers must be absolutely normal, no proof of this has yet been found.
     
   • The problem is that we can only use some finite process to describe a real number so only such numbers are accessible.
     
   • In 1936 Turing published a paper called On computable numbers.
     
   • Rather than look at the real numbers which can be described in English, Turing looked at a very precise description of a number, namely one which can be output digit by digit by a computer.
     
   • He then took Richard's paradox and ran through it again, this time with computable numbers.
     
   • Hence computable numbers are countable.
     
   • Although the "English descriptions" of Richard's paradox must hold the key to the paradox, in this case our "computable numbers" are very precise and not subject to the same difficulties.
     
   • Do we really have the ultimate paradox which shows that the real numbers are inconsistent? No! So where is the error in our paradox? The error lies in the fact that when we run the computer programmes we do not know whether they will ever output an n-th digit.
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/Real_numbers_3.html .
     

6. Real numbers 2 references
   
   • References for: The real numbers: Stevin to Hilbert .
     
   • V S Albis Gonzalez, and L I Soriano-Lleras, The work of Indalecio Lievano on the foundations of real numbers, Historia Math.
     
   • R P Burn, Irrational numbers in English language textbooks, 1890-1915 : constructions and postulates for the completeness of the real numbers, Historia Math.
     
   • J R Chicano Requena, The founding of analysis in the nineteenth century: a model for the real numbers (Catalan), Butl.
     
   • R P Coelho, Old and new aspects of the theory of real numbers.
     
   • Irrational numbers.
     
   • Irrational numbers.
     
   • J J da Silva, Wittgenstein on irrational numbers, in Wittgenstein's philosophy of mathematics, Kirchberg am Wechsel, 1992 (Holder-Pichler-Tempsky, Vienna, 1993), 93--99.
     
   • J G Dhombres, Real numbers from Cauchy to Robinson, Southeast Asian Bull.
     
   • P Djugak, The limit concept and irrational numbers: ideas of Charles Meray and Karl Weierstrass (Russian), inStudies in the history of mathematics, No.
     
   • S Giuntini, A discussion concerning the nature of zero and the relation between imaginary and real numbers (Italian), Boll.
     
   • M Lopez Pellicer, Constructions of real numbers (Spanish), in History of mathematics in the XIXth century, Part 2 (Spanish), Madrid, 1993, Real Acad.
     
   • A I Markusevic, Remarks on the paper: "Cantor's theory of real numbers" by F A Medvedev (Russian), Istor.-Mat.
     
   • F A Medvedev, Cantor's theory of real numbers (Russian), Istor.-Mat.
     
   • F A Medvedev, On the problem of completeness in the theories of real numbers (Russian), Voprosy Istor.
     
   • L C Mejlbo, Addendum to: "Some fundamental theorems about real numbers and their history" (Danish), Nordisk Mat.
     
   • L C Mejlbo, Some fundamental theorems about real numbers and their history (Danish), Nordisk Mat.
     
   • K Rychlik, Theory of real numbers in the manuscripts left by Bolzano (Czech), Casopis Pest.
     
   • P M Simons, Frege's theory of real numbers, Hist.
     
   • P M Simons, Frege's theory of real numbers, in Frege's philosophy of mathematics (Harvard Univ.
     
   • J Simsa, Development of the concept of real numbers (Czech), in Mathematics in the 16th and 17th centuries (Czech), Jev’cko, 1997 (Prometheus, Prague, 1999), 259-282.
     
   • F Smithies, Weierstrass's theory of the real numbers, Bull.
     
   • J E Snow, Views on the real numbers and the continuum, Rev.
     
   • B van Rootselaar, Bolzano's theory of real numbers, Arch.
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Real_numbers_2.html .
     

7. Real numbers 2 references
   
   • References for: The real numbers: Stevin to Hilbert .
     
   • V S Albis Gonzalez, and L I Soriano-Lleras, The work of Indalecio Lievano on the foundations of real numbers, Historia Math.
     
   • R P Burn, Irrational numbers in English language textbooks, 1890-1915 : constructions and postulates for the completeness of the real numbers, Historia Math.
     
   • J R Chicano Requena, The founding of analysis in the nineteenth century: a model for the real numbers (Catalan), Butl.
     
   • R P Coelho, Old and new aspects of the theory of real numbers.
     
   • Irrational numbers.
     
   • Irrational numbers.
     
   • J J da Silva, Wittgenstein on irrational numbers, in Wittgenstein's philosophy of mathematics, Kirchberg am Wechsel, 1992 (Holder-Pichler-Tempsky, Vienna, 1993), 93--99.
     
   • J G Dhombres, Real numbers from Cauchy to Robinson, Southeast Asian Bull.
     
   • P Djugak, The limit concept and irrational numbers: ideas of Charles Meray and Karl Weierstrass (Russian), inStudies in the history of mathematics, No.
     
   • S Giuntini, A discussion concerning the nature of zero and the relation between imaginary and real numbers (Italian), Boll.
     
   • M Lopez Pellicer, Constructions of real numbers (Spanish), in History of mathematics in the XIXth century, Part 2 (Spanish), Madrid, 1993, Real Acad.
     
   • A I Markusevic, Remarks on the paper: "Cantor's theory of real numbers" by F A Medvedev (Russian), Istor.-Mat.
     
   • F A Medvedev, Cantor's theory of real numbers (Russian), Istor.-Mat.
     
   • F A Medvedev, On the problem of completeness in the theories of real numbers (Russian), Voprosy Istor.
     
   • L C Mejlbo, Addendum to: "Some fundamental theorems about real numbers and their history" (Danish), Nordisk Mat.
     
   • L C Mejlbo, Some fundamental theorems about real numbers and their history (Danish), Nordisk Mat.
     
   • K Rychlik, Theory of real numbers in the manuscripts left by Bolzano (Czech), Casopis Pest.
     
   • P M Simons, Frege's theory of real numbers, Hist.
     
   • P M Simons, Frege's theory of real numbers, in Frege's philosophy of mathematics (Harvard Univ.
     
   • J Simsa, Development of the concept of real numbers (Czech), in Mathematics in the 16th and 17th centuries (Czech), Jev’cko, 1997 (Prometheus, Prague, 1999), 259-282.
     
   • F Smithies, Weierstrass's theory of the real numbers, Bull.
     
   • J E Snow, Views on the real numbers and the continuum, Rev.
     
   • B van Rootselaar, Bolzano's theory of real numbers, Arch.
     
   • [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Real_numbers_2.html] .
     

8. Real numbers 3 references
   
   • References for: The real numbers: Attempts to understand .
     
   • G Chaitin, How real are real numbers?, www.
     
   • J R Chicano Requena, The founding of analysis in the nineteenth century: a model for the real numbers (Catalan), Butl.
     
   • R P Coelho, Old and new aspects of the theory of real numbers.
     
   • J J da Silva, Wittgenstein on irrational numbers, in Wittgenstein's philosophy of mathematics, Kirchberg am Wechsel, 1992 (Holder-Pichler-Tempsky, Vienna, 1993), 93--99.
     
   • J G Dhombres, Real numbers from Cauchy to Robinson, Southeast Asian Bull.
     
   • P Djugak, The limit concept and irrational numbers: ideas of Charles Meray and Karl Weierstrass (Russian), inStudies in the history of mathematics, No.
     
   • M Lopez Pellicer, Constructions of real numbers (Spanish), in History of mathematics in the XIXth century, Part 2 (Spanish), Madrid, 1993, Real Acad.
     
   • A I Markusevic, Remarks on the paper: "Cantor's theory of real numbers" by F A Medvedev (Russian), Istor.-Mat.
     
   • F A Medvedev, Cantor's theory of real numbers (Russian), Istor.-Mat.
     
   • F A Medvedev, On the problem of completeness in the theories of real numbers (Russian), Voprosy Istor.
     
   • L C Mejlbo, Addendum to: "Some fundamental theorems about real numbers and their history" (Danish), Nordisk Mat.
     
   • L C Mejlbo, Some fundamental theorems about real numbers and their history (Danish), Nordisk Mat.
     
   • P M Simons, Frege's theory of real numbers, Hist.
     
   • P M Simons, Frege's theory of real numbers, in Frege's philosophy of mathematics (Harvard Univ.
     
   • J Simsa, Development of the concept of real numbers (Czech), in Mathematics in the 16th and 17th centuries (Czech), Jev’cko, 1997 (Prometheus, Prague, 1999), 259-282.
     
   • F Smithies, Weierstrass's theory of the real numbers, Bull.
     
   • J E Snow, Views on the real numbers and the continuum, Rev.
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Real_numbers_3.html .
     

9. Real numbers 3 references
   
   • References for: The real numbers: Attempts to understand .
     
   • G Chaitin, How real are real numbers?, www.
     
   • J R Chicano Requena, The founding of analysis in the nineteenth century: a model for the real numbers (Catalan), Butl.
     
   • R P Coelho, Old and new aspects of the theory of real numbers.
     
   • J J da Silva, Wittgenstein on irrational numbers, in Wittgenstein's philosophy of mathematics, Kirchberg am Wechsel, 1992 (Holder-Pichler-Tempsky, Vienna, 1993), 93--99.
     
   • J G Dhombres, Real numbers from Cauchy to Robinson, Southeast Asian Bull.
     
   • P Djugak, The limit concept and irrational numbers: ideas of Charles Meray and Karl Weierstrass (Russian), inStudies in the history of mathematics, No.
     
   • M Lopez Pellicer, Constructions of real numbers (Spanish), in History of mathematics in the XIXth century, Part 2 (Spanish), Madrid, 1993, Real Acad.
     
   • A I Markusevic, Remarks on the paper: "Cantor's theory of real numbers" by F A Medvedev (Russian), Istor.-Mat.
     
   • F A Medvedev, Cantor's theory of real numbers (Russian), Istor.-Mat.
     
   • F A Medvedev, On the problem of completeness in the theories of real numbers (Russian), Voprosy Istor.
     
   • L C Mejlbo, Addendum to: "Some fundamental theorems about real numbers and their history" (Danish), Nordisk Mat.
     
   • L C Mejlbo, Some fundamental theorems about real numbers and their history (Danish), Nordisk Mat.
     
   • P M Simons, Frege's theory of real numbers, Hist.
     
   • P M Simons, Frege's theory of real numbers, in Frege's philosophy of mathematics (Harvard Univ.
     
   • J Simsa, Development of the concept of real numbers (Czech), in Mathematics in the 16th and 17th centuries (Czech), Jev’cko, 1997 (Prometheus, Prague, 1999), 259-282.
     
   • F Smithies, Weierstrass's theory of the real numbers, Bull.
     
   • J E Snow, Views on the real numbers and the continuum, Rev.
     
   • [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Real_numbers_3.html] .
     

10. Greek numbers
    
    • We should say immediately that the ancient Greeks had different systems for cardinal numbers and ordinal numbers so we must look carefully at what we mean by Greek number systems.
      
    • Here are the symbols for the numbers 5, 10, 100, 1000, 10000.
      
    • Here is 1-10 in Greek acrophonic numbers.
      
    • 1-10 in Greek acrophonic numbers.
      
    • If base 10 is used with an additive system without intermediate symbols then many characters are required to express certain numbers.
      
    • We have already seen that that Greek acrophonic numbers had a special symbol for 5.
      
    • The next point worth noting is that this number system did not really consist of abstract numbers in the way we think of numbers today.
      
    • We know that the ancient Greeks had a somewhat different idea because the numbers were used in slightly different forms depending to what the number referred.
      
    • Sometimes when these letters are written to represent numbers, a bar was put over the symbol to distinguish it from the corresponding letter.
      
    • Now numbers were formed by the additive principle.
      
    • Larger numbers were constructed in the same sort of way.
      
    • Now this number system is compact but without modification is has the major drawback of not allowing numbers larger than 999 to be expressed.
      
    • The numbers between 1000 and 9000 were formed by adding a subscript or superscript iota to the symbols for 1 to 9.
      
    • How did the Greeks represent numbers greater than 9999? Well they based their numbers larger than this on the myriad which was 10000.
      
    • For most purposes this number system could represent all the numbers which might arise in normal day to day life.
      
    • In fact numbers as large as 71755875 would be unlikely to arise very often.
      
    • The idea which Apollonius used to extend the system to larger numbers was to work with powers of the myriad.
      
    • The first octet for Archimedes consisted of numbers up to 108 while the second octet was the numbers from 108 up to 1016.
      
    • http://www-history.mcs.st-andrews.ac.uk/HistTopics/Greek_numbers.html .
      

11. Perfect numbers references
    
    • References for: Perfect numbers .
      
    • O V Kuzhel, Development of the concept of number : Divisibility criteria : Perfect numbers (Ukrainian), Vidavnice Obednannja "Visca Skola" (Kiev, 1974).
      
    • R Shoemaker, Perfect numbers (Washington, 1973).
      
    • M Crubellier and J Sip, Looking for perfect numbers, History of Mathematics : History of Problems (Paris, 1997), 389-410.
      
    • S Drake, The rule behind "Mersenne's numbers", Physis-Riv.
      
    • N Miura, Charles de Bovelles and perfect numbers, Historia Sci.
      
    • K P Moesgaard, Tychonian observations, perfect numbers, and the date of creation : Longomontanus's solar and precessional theories, J.
      
    • M L Nankar, History of perfect numbers, Ganita Bharati 1 (1-2) (1979), 7-8.
      
    • Perfect numbers and imperfect mathematicians (Persian), Bull.
      
    • T N Sinha, Perfect numbers as a source of fundamentals of number theory, Math.
      
    • C M Taisbak, Perfect numbers : A mathematical pun? An analysis of the last theorem in the ninth book of Euclid's Elements, Centaurus 20 (4) (1976), 269-275.
      
    • H S Uhler, A brief history of the investigations on Mersenne numbers and the latest immense primes, Scripta Math.
      
    • A M Vaidya, Comment on : "History of perfect numbers", Ganita Bharati 1 (3-4) (1979), 22.
      
    • S Wagon, Perfect numbers, Math.
      
    • http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Perfect_numbers.html .
      

12. Perfect numbers references
    
    • References for: Perfect numbers .
      
    • O V Kuzhel, Development of the concept of number : Divisibility criteria : Perfect numbers (Ukrainian), Vidavnice Obednannja "Visca Skola" (Kiev, 1974).
      
    • R Shoemaker, Perfect numbers (Washington, 1973).
      
    • M Crubellier and J Sip, Looking for perfect numbers, History of Mathematics : History of Problems (Paris, 1997), 389-410.
      
    • S Drake, The rule behind "Mersenne's numbers", Physis-Riv.
      
    • N Miura, Charles de Bovelles and perfect numbers, Historia Sci.
      
    • K P Moesgaard, Tychonian observations, perfect numbers, and the date of creation : Longomontanus's solar and precessional theories, J.
      
    • M L Nankar, History of perfect numbers, Ganita Bharati 1 (1-2) (1979), 7-8.
      
    • Perfect numbers and imperfect mathematicians (Persian), Bull.
      
    • T N Sinha, Perfect numbers as a source of fundamentals of number theory, Math.
      
    • C M Taisbak, Perfect numbers : A mathematical pun? An analysis of the last theorem in the ninth book of Euclid's Elements, Centaurus 20 (4) (1976), 269-275.
      
    • H S Uhler, A brief history of the investigations on Mersenne numbers and the latest immense primes, Scripta Math.
      
    • A M Vaidya, Comment on : "History of perfect numbers", Ganita Bharati 1 (3-4) (1979), 22.
      
    • S Wagon, Perfect numbers, Math.
      
    • [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Perfect_numbers.html] .
      

13. Real numbers 1 references
    
    • References for: The real numbers: Pythagoras to Stevin .
      
    • R P Coelho, Old and new aspects of the theory of real numbers.
      
    • Irrational numbers.
      
    • Irrational numbers.
      
    • S Giuntini, A discussion concerning the nature of zero and the relation between imaginary and real numbers (Italian), Boll.
      
    • I Grattan-Guinness, Numbers, ratios, and proportions in Euclid's Elements : How did he handle them?, Historia Mathematica 23 (1996), 355-375.
      
    • A Heimonen, How were the irrational numbers discovered? (Finnish), Arkhimedes (6) (1997), 10-16.
      
    • M Lopez Pellicer, Constructions of real numbers (Spanish), in History of mathematics in the XIXth century, Part 2 (Spanish), Madrid, 1993, Real Acad.
      
    • L C Mejlbo, Addendum to: "Some fundamental theorems about real numbers and their history" (Danish), Nordisk Mat.
      
    • L C Mejlbo, Some fundamental theorems about real numbers and their history (Danish), Nordisk Mat.
      
    • M Pihl, The place of Theodoros in Plato's "Theaitetos" and the earliest history of irrational numbers (Danish), Mat.
      
    • J Simsa, Development of the concept of real numbers (Czech), in Mathematics in the 16th and 17th centuries (Czech), Jev’cko, 1997 (Prometheus, Prague, 1999), 259-282.
      
    • J E Snow, Views on the real numbers and the continuum, Rev.
      
    • http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Real_numbers_1.html .
      

14. Real numbers 1 references
    
    • References for: The real numbers: Pythagoras to Stevin .
      
    • R P Coelho, Old and new aspects of the theory of real numbers.
      
    • Irrational numbers.
      
    • Irrational numbers.
      
    • S Giuntini, A discussion concerning the nature of zero and the relation between imaginary and real numbers (Italian), Boll.
      
    • I Grattan-Guinness, Numbers, ratios, and proportions in Euclid's Elements : How did he handle them?, Historia Mathematica 23 (1996), 355-375.
      
    • A Heimonen, How were the irrational numbers discovered? (Finnish), Arkhimedes (6) (1997), 10-16.
      
    • M Lopez Pellicer, Constructions of real numbers (Spanish), in History of mathematics in the XIXth century, Part 2 (Spanish), Madrid, 1993, Real Acad.
      
    • L C Mejlbo, Addendum to: "Some fundamental theorems about real numbers and their history" (Danish), Nordisk Mat.
      
    • L C Mejlbo, Some fundamental theorems about real numbers and their history (Danish), Nordisk Mat.
      
    • M Pihl, The place of Theodoros in Plato's "Theaitetos" and the earliest history of irrational numbers (Danish), Mat.
      
    • J Simsa, Development of the concept of real numbers (Czech), in Mathematics in the 16th and 17th centuries (Czech), Jev’cko, 1997 (Prometheus, Prague, 1999), 259-282.
      
    • J E Snow, Views on the real numbers and the continuum, Rev.
      
    • [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Real_numbers_1.html] .
      

15. Infinity
    
    • Pythagoras had argued that "all is number" and his universe was made up of finite natural numbers.
      
    • His idea was that we can never conceive of the natural numbers as a whole.
      
    • Of relevance to our discussion is the remarkable advance made by the Babylonians who introduced the idea of a positional number system which, for the first time, allowed a concise representation of numbers without limit to their size.
      
    • Only a finite number of natural numbers has ever been written down or has ever been conceived.
      
    • How then, one may ask, was Euclid able to prove that the set of prime numbers is infinite in 300 BC? Well the answer is that Euclid did not prove this in the Elements.
      
    • Prime numbers are more than any assigned magnitude of prime numbers.
      
    • So in fact what Euclid proved was that the prime numbers are potentially infinite but in practice, of course, this amounts to the same thing.
      
    • His proof shows that given any finite collection of prime numbers there must be a prime number not in the collection.
      
    • The authors of [SCIAMVS 2 (2001), 9-29.',32)" onmouseover="window.status='Click to see reference';return true">32] have noticed a remarkable way that Archimedes investigates infinite numbers of objects in The Method in the Archimedes palimpsest:- .
      
    • in this case Archimedes discusses actual infinities almost as if they possessed numbers in the usual sense ..
      
    • Such as say that things infinite are past God's knowledge may just as well leap headlong into this pit of impiety, and say that God knows not all numbers.
      
    • Of course it does not work since if it were introduced as Bhaskara II suggests then 0 times infinity must be equal to every number n, so all numbers are equal.
      
    • He then gave another paradox similar to the circle paradox yet this time with numbers so no infinite indivisibles could be inserted to correct the situation.
      
    • On the one hand this showed that there were the same number of squares as there were whole numbers.
      
    • However most numbers were not perfect squares.
      
    • the totality of all numbers is infinite, and that the number of squares is infinite.; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and, finally, the attributes "equal", "greater", and "less" are not applicable to the infinite, but only to finite quantities.
      
    • At this stage the mathematical study of infinity moves into set theory and we refer the reader to the article Beginnings of set theory for more information about Bolzano's contribution and also the treatment of infinity by Cantor who built a theory of different sizes of infinity with his definitions of cardinal and ordinal numbers.
      
    • This book, which appeared just 250 years after Leibniz' death, presents a rigorous and efficient theory of infinitesimals obeying, as Leibniz wanted, the same laws as the ordinary numbers.
      

16. Zero
    
    • Numbers in early historical times were thought of much more concretely than the abstract concepts which are our numbers today.
      
    • Of course their notation for numbers was quite different from ours (and not based on 10 but on 60) but to translate into our notation they would not distinguish between 2106 and 216 (the context would have to show which was intended).
      
    • If this reference to context appears silly then it is worth noting that we still use context to interpret numbers today.
      
    • We can see from this that the early use of zero to denote an empty place is not really the use of zero as a number at all, merely the use of some type of punctuation mark so that the numbers had the correct interpretation.
      
    • In other words Greek mathematicians did not need to name their numbers since they worked with numbers as lengths of lines.
      
    • Numbers which required to be named for records were used by merchants, not mathematicians, and hence no clever notation was needed.
      
    • In fact there is evidence of an empty place holder in positional numbers from as early as 200AD in India but some historians dismiss these as later forgeries.
      
    • Later Indian mathematicians had names for zero in positional numbers yet had no symbol for it.
      
    • Both of the numbers 270 and 50 are denoted almost as they appear today although the 0 is smaller and slightly raised.
      
    • From early times numbers are words which refer to collections of objects.
      
    • Certainly the idea of number became more and more abstract and this abstraction then makes possible the consideration of zero and negative numbers which do not arise as properties of collections of objects.
      
    • Of course the problem which arises when one tries to consider zero and negatives as numbers is how they interact in regard to the operations of arithmetic, addition, subtraction, multiplication and division.
      
    • Brahmagupta attempted to give the rules for arithmetic involving zero and negative numbers in the seventh century.
      
    • However it is a brilliant attempt from the first person that we know who tried to extend arithmetic to negative numbers and zero.
      
    • If this were true then 0 times ∞ must be equal to every number n, so all numbers are equal.
      
    • Ibn Ezra, in the 12th century, wrote three treatises on numbers which helped to bring the Indian symbols and ideas of decimal fractions to the attention of some of the learned people in Europe.
      
    • It is significant that Fibonacci is not bold enough to treat 0 in the same way as the other numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 since he speaks of the "sign" zero while the other symbols he speaks of as numbers.
      

17. Prime numbers references
    
    • References for: Prime numbers .
      
    • B C Berndt, Ramanujan and the theory of prime numbers, Number theory Madras 1987 (Berlin, 1989), 122-139.
      
    • L E Dickson, History of the Theory of Numbers (3 volumes) (New York, 1919-23, reprinted 1966).
      
    • A Granville, Harald Cramer and the distribution of prime numbers, Harald Cramer Symposium, Scand.
      
    • H S Uhler, A brief history of the investigations on Mersenne numbers and the latest immense primes, Scripta Math.
      
    • http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Prime_numbers.html .
      

18. Ring Theory
    
    • Familiar examples of rings such as the real numbers, the complex numbers, the rational numbers, the integers, the even integers, 2 cross 2 real matrices, the integers modulo m for a fixed integer m, will almost certainly be given in the Abstract Algebra book as will many beautiful theorems on rings but what will be missing are the reasons systems satisfying these particular axioms have been singled out for such intensive study.
      
    • Euler's work on the case n = 3 involved extending ordinary integer arithmetic to apply to the ring of numbers of the form a + b√-3 where a, b are integers.
      
    • Complex numbers of the form a + b√-3, where a, b are integers, form a ring.
      
    • A prime number in this ring is defined in an analogous way to a prime integer, namely a number whose only divisors of the form a + b√-3 other than itself are those numbers with multiplicative inverses.
      
    • In this ring 4 can be written as a product of prime numbers in two different ways .
      
    • Gauss had proved around 1801 that numbers of the form a + b√-1, where a, b are integers, could be written uniquely as a product of prime numbers of the form a + b√-1 in an analogous manner to the unique decomposition of an integer as a product of prime integers.
      
    • In fact, numbers of the form a + b +c2 where a, b, c are integers and is a complex cube root of 1, also have unique factorisation, and this can be used to prove the n = 3 case of Fermat's last Theorem.
      
    • The argument following Lame's announcement was settled by Kummer who pointed out that he had published an example in 1844 to show that the uniqueness of such decompositions failed and in 1846 he had restored the uniqueness by introducing "ideal complex numbers".
      
    • The popular story that Kummer invented "ideal complex numbers" in an attempt to correct an error in this proof of Fermat's Last Theorem is almost certainly false; see Edwards [Fermat\'s Last Theorem, (Berlin 1977).
      
    • In 1847, just after Lame's announcement, Kummer used his "ideal complex numbers" to prove Fermat's Last Theorem for all n < 100 except n = 37, 59, 67 and 74.
      
    • Up to this point we are still firmly within the realms of number theory but the genius of Dedekind pinpointed the important properties of the "ideal complex numbers".
      
    • Prime numbers were generalised to prime ideals by Dedekind in 1871.
      
    • It is important to realise that at this stage rings of polynomials and rings of numbers were being studied, but it was to be another 40 years before an axiomatic theory of commutative rings was to be developed bringing these theories together.
      
    • In about 1921 she made the important step, which we commented on earlier, of bringing the two theories of rings of polynomials and rings of numbers under a single theory of abstract commutative rings.
      
    • Hamilton attempted to generalise the complex numbers as a two dimensional algebra over the reals to a three dimensional algebra.
      
    • Hamilton, who introduced the idea of a vector space, felt that this three dimensional analogue of the complex numbers would revolutionise applied mathematics but he struggled unsuccessfully with the idea for many years.
      

19. Prime numbers references
    
    • References for: Prime numbers .
      
    • B C Berndt, Ramanujan and the theory of prime numbers, Number theory Madras 1987 (Berlin, 1989), 122-139.
      
    • L E Dickson, History of the Theory of Numbers (3 volumes) (New York, 1919-23, reprinted 1966).
      
    • A Granville, Harald Cramer and the distribution of prime numbers, Harald Cramer Symposium, Scand.
      
    • H S Uhler, A brief history of the investigations on Mersenne numbers and the latest immense primes, Scripta Math.
      
    • [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Prime_numbers.html] .
      

20. Mental arithmetic
    
    • However, it is clear that multiplying two eight digit numbers in his head was a task which he could accomplish with little effort.
      
    • He could instantly give the product of two numbers each of four digits but hesitated if both numbers exceeded 10,000.
      
    • he worked less quickly when asked to raise numbers of two digits like 37 or 59 to high powers.
      
    • He, however, found it difficult to answer questions about numbers higher than 1,000,000.
      
    • Colburn is also interesting in that he was able to give an idea of how he carried out the calculations, the main method being by factorising the numbers concerned:- .
      
    • One of Bidder's sons was able to multiply two numbers of 15 digits but he was slow, and less accurate, compared with his father.
      
    • Bidder explained how the sound of numbers was much more important to him than their visual representation, something that Aitken was also to emphasise.
      
    • if required to find the product of two numbers each of nine digits which were read to me, I should not require this to be done more than once; but if they were represented in the usual way, and put into my hands, it would probably take me four times to peruse them before it would be in my power to repeat them, and after all they would not be impressed so vividly on my imagination.
      
    • He multiplied two 20 digit numbers in 6 minutes; two 40 digit numbers in 40 minutes; two 100 digit numbers in 8 hours 45 minutes.
      
    • In his spare time, between 1844 and 1847, he calculated the natural logarithms of the first 1005000 numbers to 7 decimal places.
      
    • Numbers filled Aitken's world.
      
    • There are other possibilities: For example, the mental calculator is, or should be, very familiar with the factorisation of numbers; he should know not merely that 23 time 13 is 299, but that 23 times 87 is 2001.
      
    • Aitken, when asked what he believed made him more able to calculate than the average person, put it down to his ability to easily remember numbers.
      
    • The person concerned can 'see' the numbers of objects which have been committed to memory and in some sense read them off as if they had been written on paper.
      
    • However, when asked to recite the digits of π backwards Aitken had, interestingly, no other option but to bring the numbers into a visual form and read them off backwards from his visual image.
      
    • Aitken recited the names and numbers of all the members of his platoon.
      

21. Set theory
    
    • These papers contain Cantor's first ideas on set theory and also important results on irrational numbers.
      Go directly to this paragraph
      
    • Dedekind was working independently on irrational numbers and Dedekind published Continuity and irrational numbers.
      Go directly to this paragraph
      
    • However Cantor examines the set of algebraic real numbers, that is the set of all real roots of equations of the form .
      Go directly to this paragraph
      
    • Cantor proves that the algebraic real numbers are in one-one correspondence with the natural numbers in the following way.
      Go directly to this paragraph
      
    • Putting them in 1-1 correspondence with the natural numbers is now clear but ordering them in order of index and increasing magnitude within each index.
      
    • In the same paper Cantor shows that the real numbers cannot be put into one-one correspondence with the natural numbers using an argument with nested intervals which is more complex than that used today (which is in fact due to Cantor in a later paper of 1891).
      Go directly to this paragraph
      
    • not algebraic) numbers in each interval.
      Go directly to this paragraph
      
    • He proves that the rational numbers have the smallest infinite power and also shows that Rn has the same power as R.
      Go directly to this paragraph
      
    • He only accepted mathematical objects that could be constructed finitely from the intuitively given set of natural numbers.
      Go directly to this paragraph
      
    • Why study such problems when irrational numbers do not exist.
      
    • Ordinal numbers are introduced as the order types of well-ordered sets.
      Go directly to this paragraph
      
    • Multiplication and addition of transfinite numbers are also defined in this work although Cantor was to give a fuller account of transfinite arithmetic in later work.
      Go directly to this paragraph
      
    • In 1885 Cantor continued to extend his theory of cardinal numbers and of order types.
      Go directly to this paragraph
      
    • He extended his theory of order types so that now his previously defined ordinal numbers became a special case.
      Go directly to this paragraph
      
    • It basically revolves round the set of all ordinal numbers.
      Go directly to this paragraph
      

22. Indian numerals
    
    • Now these Brahmi numerals were not just symbols for the numbers between 1 and 9.
      
    • The situation is much more complicated for it was not a place-value system so there were symbols for many more numbers.
      
    • Also there were no special symbols for 2 and 3, both numbers being constructed from the symbol for 1.
      
    • , 9 appear to us to have no obvious link to the numbers they represent.
      
    • In [A universal history of numbers : From prehistory to the invention of the computer (London, 1998).',1)" onmouseover="window.status='Click to see reference';return true">1] Ifrah lists a number of the hypotheses which have been put forward.
      
    • One is that the numerals came from an alphabet in a similar way to the Greek numerals which were the initial letters of the names of the numbers.
      
    • Ifrah proposes a theory of his own in [A universal history of numbers : From prehistory to the invention of the computer (London, 1998).',1)" onmouseover="window.status='Click to see reference';return true">1], namely that:- .
      
    • Many other charters have been found which are dated and use of the place-value system for either the date or some other numbers within the text.
      
    • Another source is the Bakhshali manuscript which contains numbers written in place-value notation.
      
    • The reason, Joseph believes, is due to the Indian fascination with large numbers.
      
    • To see clearly this early Indian fascination with large numbers, we can take a look at the Lalitavistara which is an account of the life of Gautama Buddha.
      
    • It is stories such as this, and many similar ones, which convince Joseph that the fascination of the Indians with large numbers must have driven them to invent a system in which such numbers are easily expressed, namely a place-valued notation.
      
    • The early use of such large numbers eventually led to the adoption of a series of names for successive powers of ten.
      
    • The decimal place-value system developed when a decimal scale came to be associated with the value of the places of the numbers arranged left to right or right to left.
      
    • However, the same story in Lalitavistara convinces Kaplan (see [The nothing that is : a natural history of zero (London, 1999).',3)" onmouseover="window.status='Click to see reference';return true">3]) that the Indians' ideas of numbers came from the Greeks, for to him the story is an Indian version of Archimedes' Sand-reckoner.
      

23. Chinese numerals
    
    • However larger numbers have not been found, the largest number discovered on the Shang bones and tortoise shells being 30000.
      
    • Because we have not illustrated many numbers above here is one further example of a Chinese oracular number.
      
    • Although the representation of the numbers 1, 2, 3, 4 needs little explanation, the question as to why particular symbols are used for the other digits is far less obvious.
      
    • A second theory about the symbols comes from the fact that numbers, and in fact all writing in this Late Shang period, were only used as part of religious ceremonies.
      
    • Numbers were represented by little rods made from bamboo or ivory.
      
    • The most significant property of representing numbers this way on the counting board was that it was a natural place valued system.
      
    • Now the numbers from 1 to 9 had to be formed from the rods and a fairly natural way was found.
      
    • They used both forms of the numbers given in the above illustration.
      
    • The alternating forms of the numbers again helped to show that there was indeed a space.
      
    • Now the Chinese counting board numbers were not just used on a counting board, although this is clearly their origin.
      
    • In particular the "tian yuan" or "coefficient array method" or "method of the celestial unknown" developed out of the counting board representation of numbers.
      
    • In many ways it was similar to the counting board, except instead of using rods to represent numbers, they were represented by beads sliding on a wire.
      
    • Arithmetical rules for the abacus were analogous to those of the counting board (even square roots and cube roots of numbers could be calculated) but it appears that the abacus was used almost exclusively by merchants who only used the operations of addition and subtraction.
      
    • For numbers up to 4 slide the required number of beads in the lower part up to the middle bar.
      
    • For five or above, slide one bead above the middle bar down (representing 5), and 1, 2, 3 or 4 beads up to the middle bar for the numbers 6, 7, 8, or 9 respectively.
      
    • This was to make the intermediate working easier so that in fact numbers bigger than 9 could be stored on a single wire during a calculation, although by the end such "carries" would have to be taken over to the wire to the left.
      

24. Inca mathematics
    
    • The quipu consists of strings which were knotted to represent numbers.
      
    • For larger numbers more knot groups were used, one for each power of 10, in the same way as the digits of the number system we use here are occur in different positions to indicate the number of the corresponding power of 10 in that position.
      
    • Numbers were recorded on strings of a particular colour to identify what that number was recording.
      
    • For example numbers of cattle might be recorded on green strings while numbers of sheep might be recorded on white strings.
      
    • The book [The social life of numbers : A Quechua ontology of numbers and philosophy of arithmetic (Austin, TX, 1997).',6)" onmouseover="window.status='Click to see reference';return true">6] by Urton is interesting for it examines the concept of number as understood by the Inca people.
      
    • The concrete way of conceiving numbers is illustrated by different words used when describing properties of numbers.
      
    • One example given in [The social life of numbers : A Quechua ontology of numbers and philosophy of arithmetic (Austin, TX, 1997).',6)" onmouseover="window.status='Click to see reference';return true">6] is that of even and odd numbers.
      
    • For example separate words occur for the idea of [The social life of numbers : A Quechua ontology of numbers and philosophy of arithmetic (Austin, TX, 1997).',6)" onmouseover="window.status='Click to see reference';return true">6]:- .
      

25. Mathematics and Architecture
    
    • The authors of [Nexus III : architecture and mathematics, Ferrara, 2000 (Pisa, 2000), 87-104.',23)" onmouseover="window.status='Click to see reference';return true">23], however, suggest reasons for the occurrence of many of the nice numbers, in particular numbers close to powers of the golden number, as arising from the building techniques used rather than being deliberate decisions of the architects.
      
    • Even if deep mathematical ideas went into the construction of the pyramids, I think that Ifrah makes a useful contribution to this debate in [The Universal History of Numbers (London, 1998).',4)" onmouseover="window.status='Click to see reference';return true">4] when he writes:- .
      
    • The Pythagorean belief that "all things are numbers" clearly had great significance for architecture so let us consider for a moment what this means.
      
    • Pythagoras saw the connection between music and numbers and clearly understood how the note produced by a string related to its length.
      
    • Numbers for Pythagoras also had geometrical properties.
      
    • The Pythagoreans spoke of square numbers, oblong numbers, triangular numbers etc.
      
    • Geometry was the study of shapes and shapes were determined by numbers.
      
    • It should now be clear what the belief that "all things are numbers" meant to the Pythagoreans and how this was to influence ancient Greek architecture.
      
    • Berger, in [11], makes a study of the way that the Pythagorean ideas of ratios of small numbers were used in the construction of the Temple of Athena Parthenos.
      
    • With this training Bombelli was soon working himself as both engineer and architect employing his mathematical skills both in his work and in his deep investigation of complex numbers.
      

26. Arabic mathematics
    
    • Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects".
      
    • Al-Khwarizmi's successors undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra.
      
    • This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose.
      
    • He discovered a beautiful theorem which allowed pairs of amicable numbers to be found, that is two numbers such that each is the sum of the proper divisors of the other.
      Go directly to this paragraph
      
    • Al-Baghdadi (born 980) looked at a slight variant of Thabit ibn Qurra's theorem, while al-Haytham (born 965) seems to have been the first to attempt to classify all even perfect numbers (numbers equal to the sum of their proper divisors) as those of the form 2k-1(2k - 1) where 2k - 1 is prime.
      Go directly to this paragraph
      
    • Continuing the story of amicable numbers, from which we have taken a diversion, it is worth noting that they play a large role in Arabic mathematics.
      Go directly to this paragraph
      
    • He also gave the pair of amicable numbers 17296, 18416 which have been attributed to Euler, but we know that these were known earlier than al-Farisi, perhaps even by Thabit ibn Qurra himself.
      Go directly to this paragraph
      
    • A dust board was needed because the methods required the moving of numbers around in the calculation and rubbing some out as the calculation proceeded.
      
    • Al-Kashi (born1380) contributed to the development of decimal fractions not only for approximating algebraic numbers, but also for real numbers such as π.
      Go directly to this paragraph
      

27. Egyptian Papyri
    
    • Next check the numbers in the right hand column corresponding to 32, 8, 1 and add them.
      
    • Basically we can think of the method as writing one of the numbers to base 2.
      
    • Now we look for numbers in the right hand column which add up to 1495.
      
    • We see that 1040 + 260 + 130 + 65 = 1495 and we tick the rows in which these numbers occur: .
      
    • Now add the numbers in the left hand column which are in ticked rows: .
      
    • What happens if the numbers do not divide exactly? Then the Egyptian method will yield fractions as the following example shows.
      
    • Now look for the numbers in the right hand column which add to a number n with 1500-65 < n ≤ 1500.
      
    • Now add the numbers in the left hand column which are in ticked rows: .
      
    • The next problem is how to multiply and divide numbers involving fractions.
      
    • The favourite rules which many historians such as Gillings believe guided the scribes in their choice of decomposition of 2/n into unit fractions are (1) prefer small numbers (2) the fewer terms the better, and never more than four (3) prefer even to odd numbers.
      
    • Next find the numbers in the left hand column which add to 30+1/3.
      

28. Mayan mathematics
    
    • In [A universal history of numbers : From prehistory to the invention of the computer (London, 1998).',4)">4] Ifrah argues that the number system we have just introduced was the system of the Mayan priests and astronomers which they used for astronomical and calendar calculations.
      
    • Ifrah writes [A universal history of numbers : From prehistory to the invention of the computer (London, 1998).',4)">4]:- .
      
    • Even though no trace of it remains, we can reasonably assume that the Maya had a number system of this kind, and that intermediate numbers were figured by repeating the signs as many times as was needed.
      
    • The Long Count is based on a year of 360 days, or perhaps it is more accurate to say that it is just a count of days with then numbers represented in the Mayan number system.
      
    • Consider the two examples of Mayan numbers given above.
      
    • Also since the Mayan numbers were not a true positional base 20 system, it fails to have the nice mathematical properties that we expect of a positional system.
      
    • Moving all the numbers one place left would multiply the number by 20 in a true base 20 positional system yet 20 × 67873 = 1357460 which is not equal to 1357100.
      
    • We should also note that the Mayans almost certainly did not have methods of multiplication for their numbers and definitely did not use division of numbers.
      
    • Rhonda Robinson (Mayan numbers) .
      

29. Greek numbers references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      
    • http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Greek_numbers.html .
      

30. Quadratic etc equations
    
    • He has six chapters each devoted to a different type of equation, the equations being made up of three types of quantities namely: roots, squares of roots and numbers i.e.
      Go directly to this paragraph
      
    • x, x2 and numbers.
      Go directly to this paragraph
      
    • Squares equal to numbers.
      
    • Roots equal to numbers.
      
    • Squares and roots equal to numbers, e.g.
      
    • Squares and numbers equal to roots, e.g.
      
    • Roots and numbers equal to squares, e.g.
      
    • However, without the Hindu's knowledge of negative numbers, dal Ferro would not have been able to use his solution of the one case to solve all cubic equations.
      Go directly to this paragraph
      
    • In Ars Magna Cardan gives a calculation with 'complex numbers' to solve a similar problem but he really did not understand his own calculation which he says is as subtle as it is useless.
      Go directly to this paragraph
      
    • The irreducible case of the cubic, namely the case where Cardan's formula leads to the square root of negative numbers, was studied in detail by Rafael Bombelli in 1572 in his work Algebra.
      Go directly to this paragraph
      

31. Greek numbers references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      
    • [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Greek_numbers.html] .
      

32. History overview
    
    • Only when some record of the counting was kept and, therefore, some representation of numbers occurred can mathematics be said to have started.
      
    • It allowed arbitrarily large numbers and fractions to be represented and so proved to be the foundation of more high powered mathematical development.
      
    • A more precise formulation of concepts led to the realisation that the rational numbers did not suffice to measure all lengths.
      Go directly to this paragraph
      
    • A geometric formulation of irrational numbers arose.
      Go directly to this paragraph
      
    • The end of the 19th Century saw Cantor invent set theory almost single handedly while his analysis of the concept of number added to the major work of Dedekind and Weierstrass on irrational numbers .
      Go directly to this paragraph
      
    • For example, work with numbers is clearly hindered by poor notation.
      
    • Try multiplying two numbers together in Roman numerals.
      
    • There is no reason why anyone should introduce negative numbers just to be solutions of equations such as x + 3 = 0.
      
    • In fact there is no real reason why negative numbers should be introduced at all.
      
    • Negative numbers do not have this type of concrete representation on which to build the abstraction.
      

33. Fermat's last theorem
    
    • His method is imaginative, calculating with numbers of the form a + b√-3.
      Go directly to this paragraph
      
    • However numbers of this form do not behave in the same way as the integers, which Euler did not seem to appreciate.
      Go directly to this paragraph
      
    • Sophie Germain proved Case 1 of Fermat's Last Theorem for all n less than 100 and Legendre extended her methods to all numbers less than 197.
      Go directly to this paragraph
      
    • He sketched a proof which involved factorizing xn + yn = zn into linear factors over the complex numbers.
      Go directly to this paragraph
      
    • However Liouville addressed the meeting after Lame and suggested that the problem of this approach was that uniqueness of factorisation into primes was needed for these complex numbers and he doubted if it were true.
      Go directly to this paragraph
      
    • The letter was from Kummer, enclosing an off-print of a 1844 paper which proved that uniqueness of factorization failed but could be 'recovered' by the introduction of ideal complex numbers which he had done in 1846.
      Go directly to this paragraph
      
    • By September 1847 Kummer sent to Dirichlet and the Berlin Academy a paper proving that a prime p is regular (and so Fermat's Last Theorem is true for that prime) if p does not divide the numerators of any of the Bernoulli numbers B2 , B4 , ..
      Go directly to this paragraph
      
    • More powerful techniques were used to prove Fermat's Last Theorem for these numbers.
      Go directly to this paragraph
      
    • This work was done and continued to larger numbers by Kummer, Mirimanoff, Wieferich, Furtwangler, Vandiver and others.
      Go directly to this paragraph
      

34. Fund theorem of algebra
    
    • Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers.
      
    • Cardan was the first to realise that one could work with quantities more general than the real numbers.
      Go directly to this paragraph
      
    • He was able to manipulate with his 'complex numbers' to obtain the right answer yet he in no way understood his own mathematics.
      Go directly to this paragraph
      
    • Bombelli, in his Algebra, published in 1572, was to produce a proper set of rules for manipulating these 'complex numbers'.
      Go directly to this paragraph
      
    • Now he shows that there are complex numbers z1 and w1 so that .
      Go directly to this paragraph
      
    • In this paper he interpreted i as a rotation of the plane through 90° so giving rise to the Argand plane or Argand diagram as a geometrical representation of complex numbers.
      Go directly to this paragraph
      
    • It is worth noting that despite Gauss's insistence that one could not assume the existence of roots which were then to be proved reals he did believe, as did everyone at that time, that there existed a whole hierarchy of imaginary quantities of which complex numbers were the simplest.
      
    • It was in searching for such generalisations of the complex numbers that Hamilton discovered the quaternions around 1843, but of course the quaternions are not a commutative system.
      Go directly to this paragraph
      

35. Debating topics
    
    • The first ideas we present are simply to make people think about numbers, and in particular to encourage the use of the history archive to find birth dates and death dates before making calculations.
      
    • What about negative numbers.
      
    • Building all numbers from the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 is very clever indeed.
      
    • We have looked at how numbers are built from the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
      
    • Does 0 obey the same rules as other numbers? .
      
    • Here are questions about a more advanced topic, namely complex numbers.
      
    • Do we need to introduce negative numbers to get solutions of such equations? .
      
    • Why then did people introduce negative numbers? Why did people introduce i? .
      

36. Babylonian numerals
    
    • However, rather than have to learn 10 symbols as we do to use our decimal numbers, the Babylonians only had to learn two symbols to produce their base 60 positional system.
      
    • This is because the 59 numbers, which go into one of the places of the system, were built from a 'unit' symbol and a 'ten' symbol.
      
    • The Babylonian sexagesimal positional system places numbers with the same convention, so the right most position is for the units up to 59, the position one to the left is for 60 cross n where 1 ≤ n ≤ 59, etc.
      
    • Since two is represented by two characters each representing one unit, and 61 is represented by the one character for a unit in the first place and a second identical character for a unit in the second place then the Babylonian sexagesimal numbers 1,1 and 2 have essentially the same representation.
      
    • The numbers sexagesimal numbers 1 and 1,0, namely 1 and 60 in decimals, had exactly the same representation and now there was no way that spacing could help.
      
    • Returning to empty places in the middle of numbers we can look at actual examples where this happens.
      

37. Indian mathematics
    
    • We shall look briefly at the Indian development of the place-value decimal system of numbers later in this article and in somewhat more detail in the separate article Indian numerals.
      
    • The main topics of Jaina mathematics in around 150 BC were: the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations.
      
    • The main ideas of Jaina mathematics, particularly those relating to its cosmology with its passion for large finite numbers and infinite numbers, continued to flourish with scholars such as Yativrsabha.
      
    • The next figure of major importance at the Ujjain school was Brahmagupta near the beginning of the seventh century AD and he would make one of the most major contributions to the development of the numbers systems with his remarkable contributions on negative numbers and zero.
      
    • It is a sobering thought that eight hundred years later European mathematics would be struggling to cope without the use of negative numbers and of zero.
      

38. function concept
    
    • Indeed if we look at Babylonian mathematics we find tables of squares of the natural numbers, cubes of the natural numbers, and reciprocals of the natural numbers.
      
    • But if we conceive a function, not as a formula, but as a more general relation associating the elements of one set of numbers with the elements of another set, it is obvious that functions in that sense abound throughout the Almagest.
      
    • A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities.
      
    • The value of the function could be given either by an analytic expression or by a condition which offers a means for testing all numbers and selecting one from them, or lastly the dependence may exist but remain unknown.
      

39. Tartaglia versus Cardan
    
    • And the reason why I was able to solve his thirty problems in so short a time is that all thirty concerned work involving the algebra of unknowns and cubes equalling numbers.
      
    • Find me two numbers in double proportion such that when the square of the larger number is multiplied by the smaller, and this product is added to the two original numbers, the result is forty, that is 40.
      
    • Find two other numbers differing in this one.
      
    • Find me two numbers such that when they are added together, they make as much as the cube of the lesser added to the product of its triple with the square of the greater; and the cube of the greater added to its triple times the square of the lesser make 64 more than the sum of these two numbers.
      

40. Mathematical classics
    
    • Sandeng shu (Art of the Three Degrees; Notation of Large Numbers) .
      
    • Duke of Zhu: How great is the art of numbers? Tell me something about the application of the gnomon.
      
    • The commentary does contain mathematics, particularly relating to questions concerning the calendar and large numbers.
      
    • It is a difficult work to understand, in part showing how very large numbers can be constructed using powers of ten.
      
    • The author may have had in mind convincing his reader that it was possible to express arbitrarily large numbers.
      
    • Sandeng shu (Art of the Three Degrees; Notation of Large Numbers) .
      

41. Ten classics
    
    • Sandeng shu (Art of the Three Degrees; Notation of Large Numbers) .
      
    • Duke of Zhu: How great is the art of numbers? Tell me something about the application of the gnomon.
      
    • The commentary does contain mathematics, particularly relating to questions concerning the calendar and large numbers.
      
    • It is a difficult work to understand, in part showing how very large numbers can be constructed using powers of ten.
      
    • The author may have had in mind convincing his reader that it was possible to express arbitrarily large numbers.
      
    • Sandeng shu (Art of the Three Degrees; Notation of Large Numbers) .
      

42. Arabic numerals
    
    • Now in [A universal history of numbers : From prehistory to the invention of the computer (London, 1998).',1)" onmouseover="window.status='Click to see reference';return true">1] (where a longer quote is given) Ifrah tries to determine which Indian work is referred to.
      
    • A dust board was used because the arithmetical methods required the moving of numbers around in the calculation and rubbing some out some of them as the calculation proceeded.
      
    • The numbers were represented by letters but not in the dictionary order.
      
    • The system was known as huruf al jumal which meant "letters for calculating" and also sometimes as abjad which is just the first four numbers (1 = a, 2 = b, j = 3, d = 4).
      
    • The numbers from 1 to 9 were represented by letters, then the numbers 10, 20, 30, ..
      

43. Jaina mathematics
    
    • the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations.
      
    • There was a fascination with large numbers in Indian thought over a long period and this again almost required them to consider infinitely large measures.
      
    • Numbers are calculated in the cases where n = 2, 3 and 4.
      
    • The author then says that one can compute the numbers in the same way for larger n.
      
    • Interestingly here too there is the suggestion that the arithmetic can be extended to various infinite numbers.
      

44. Bolzano publications.html
    
    • Contains Bolzano's Reine Zahlenlehre (Theory of Numbers) which was written between 1820 and 1825.
      
    • It concerns natural numbers and elementary properties of integers and rational numbers.
      
    • The volume also contains Bolzano's theory of the real numbers.
      
    • Although earlier versions of Bolzano's work on real numbers was incorrect, Berg shows here for the first time that Bolzano had realised this himself and proposed a modification which could lead to his work being correct.
      

45. Measurement
    
    • It is associating numbers with physical quantities and so the earliest forms of measurement constitute the first steps towards mathematics.
      
    • Once the step of associating numbers with physical objects has been made, it becomes possible to compare the objects by comparing the associated numbers.
      
    • This leads to the development of methods of working with numbers.
      

46. Babylonian mathematics
    
    • They give squares of the numbers up to 59 and cubes of the numbers up to 32.
      
    • which shows that a table of squares is all that is necessary to multiply numbers, simply taking the difference of the two squares that were looked up in the table then taking a quarter of the answer.
      
    • We still have their reciprocal tables going up to the reciprocals of numbers up to several billion.
      

47. Abstract linear spaces
    
    • In 1814 Argand had represented the complex numbers as points on the plane, that is as ordered pairs of real numbers.
      Go directly to this paragraph
      
    • Hamilton represented the complex numbers as a two dimensional vector space over the reals although of course he did not use these general abstract terms.
      Go directly to this paragraph
      
    • In this work Laguerre aims to unify algebraic systems such as complex numbers, Hamilton's quaternions and notions introduced by Galois and by Cauchy.
      Go directly to this paragraph
      

48. Babylonian Pythagoras
    
    • Of course these numbers are written in Babylonian numerals to base 60.
      
    • Now the Babylonian numbers are always ambiguous and no indication occurs as to where the integer part ends and the fractional part begins.
      
    • Also the rows do not appear in any logical order except that the numbers in column 1 decrease regularly.
      
    • The puzzle then is how the numbers were found and why are these particular Pythagorean triples are given in the table.
      

49. Topology history
    
    • Poincare introduced the concept of homology and gave a more precise definition of the Betti numbers associated with a space than had Betti himself.
      Go directly to this paragraph
      
    • This process really began in 1817 when Bolzano removed the association of convergence with a sequence of numbers and associated convergence with any bounded infinite subset of the real numbers.
      Go directly to this paragraph
      
    • A bounded infinite subset S of the real numbers possesses at least one point of accumulation p, i.e.
      

50. Bourbaki 2
    
    • Chapter V is concerned with the additive group R of real numbers, its subgroups, quotient groups and homomorphisms.
      
    • Chapters VI and VII are concerned with elementary properties of Euclidean spaces, projective spaces, complex numbers, complex projective spaces, quaternions, etc.
      
    • This chapter looks at the use of the real numbers in general topology.
      

51. Ledermann interview
    
    • It was absolutely shattering; we went with our bicycles to the cemeteries and there would be 20,000 or 30,000 graves of allied soldiers with white crosses, sometimes with consecutive numbers, New Zealanders, British, Canadians.
      
    • There would be determinants and matrices, he did not use the term linear algebra, and also the theory of ideals, algebraic numbers and so on, or invariants, or some advanced work on algebraic numbers and ideal theory.
      

52. Babylonian Pythagoras references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      
    • Pythagorean numbers in Babylonian mathematics, Nederl.
      
    • O Schmidt, On Plimpton 322: Pythagorean numbers in Babylonian mathematics, Centaurus 24 (1980), 4-13.
      

53. Inca mathematics references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      
    • G Urton, The social life of numbers : A Quechua ontology of numbers and philosophy of arithmetic (Austin, TX, 1997).
      

54. EMS History
    
    • Nothing in connection with our Universities is more astounding to a foreigner than the fact that there are large numbers of students enrolled every year to begin the first proposition of Euclid, and that, of all the mathematical students within the walls, by far the greater portion have confined their studies to elementary Algebra, Geometry and Trigonometry.
      
    • Other changes took place due to World War I - large numbers of young men had been killed with a major impact on the teaching profession.
      
    • However, schoolteachers did not return in numbers to the meetings and research talks soon became the norm.
      

55. Calculus history
    
    • To the Greeks numbers were ratios of integers so the number line had "holes" in it.
      
    • They got round this difficulty by using lengths, areas and volumes in addition to numbers for, to the Greeks, not all lengths were numbers.
      

56. Indian numerals references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      
    • K W Menninger, Number words and number symbols : A cultural history of numbers (Boston, 1969).
      
    • R K Sarma, A note on the use of words for numbers in ancient Indian mathematics, Math.
      

57. Euclid's definitions
    
    • As we noted in The real numbers: Pythagoras to Stevin, Book V of The Elements considers magnitudes and the theory of proportion of magnitudes.
      
    • When Euclid introduces magnitudes and numbers he gives some definitions but no postulates or common notions.
      
    • When Euclid introduces numbers in Book VII he does make a definition rather similar to the basic ones at the beginning of Book I: .
      

58. Ptolemy mss.html
    
    • The A version of the manuscripts contain twenty-six regional maps which corresponds to the numbers given in the text of the Geography Book 8.2.
      
    • Degrees of longitude and latitude are marked in the margins, and the maps themselves are traversed by parallels and meridians; but while the meridians are fixed by the numbers in the margins, just as on modern maps, with one for every five degrees, the parallels are fixed according to the length of the longest day, at intervals of a quarter, half, or whole hour of difference.
      
    • Planudes and his assistants therefore probably had no pictorial models, and the success of their enterprise is proof that Ptolemy succeeded in his attempt to encode the map in words and numbers.
      

59. Indian numerals references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      
    • K W Menninger, Number words and number symbols : A cultural history of numbers (Boston, 1969).
      
    • R K Sarma, A note on the use of words for numbers in ancient Indian mathematics, Math.
      

60. test.html
    
    • The A version of the manuscripts contain twenty-six regional maps which corresponds to the numbers given in the text of the Geography Book 8.2.
      
    • Degrees of longitude and latitude are marked in the margins, and the maps themselves are traversed by parallels and meridians; but while the meridians are fixed by the numbers in the margins, just as on modern maps, with one for every five degrees, the parallels are fixed according to the length of the longest day, at intervals of a quarter, half, or whole hour of difference.
      
    • Planudes and his assistants therefore probably had no pictorial models, and the success of their enterprise is proof that Ptolemy succeeded in his attempt to encode the map in words and numbers.
      

61. Egyptian numerals
    
    • The numbers appeared thus: .
      
    • These numerals allowed numbers to be written in a far more compact form yet using the system required many more symbols to be memorised.
      
    • With this system numbers could be formed of a few symbols.
      

62. Inca mathematics references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      
    • G Urton, The social life of numbers : A Quechua ontology of numbers and philosophy of arithmetic (Austin, TX, 1997).
      

63. Babylonian Pythagoras references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      
    • Pythagorean numbers in Babylonian mathematics, Nederl.
      
    • O Schmidt, On Plimpton 322: Pythagorean numbers in Babylonian mathematics, Centaurus 24 (1980), 4-13.
      

64. Bourbaki 1
    
    • All three of these structures are present in the concept of the real numbers, for example, and certainly not in an independent way but interlinked in a complex fashion.
      
    • To many this was a major strength of the highly logical approach but to others it was a major weakness in that real numbers, which seem of fundamental importance, could not be introduced until vast areas of algebra and topology had been set up, of course always in the most general form possible [Math.
      
    • the construction of real numbers starting from the rationals is for [Bourbaki] a special case of a more general construction: the completion of a topological group (Chapter 3 Book III).
      

65. Zero references
    
    • G Ifrah, From one to zero : A universal history of numbers (New York, 1987).
      
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      
    • S Giuntini, A discussion concerning the nature of zero and the relation between imaginary and real numbers (Italian), Boll.
      

66. Zero references
    
    • G Ifrah, From one to zero : A universal history of numbers (New York, 1987).
      
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      
    • S Giuntini, A discussion concerning the nature of zero and the relation between imaginary and real numbers (Italian), Boll.
      

67. Babylonian mathematics references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      
    • Reciprocals of regular sexagesimal numbers, J.
      

68. Arabic numerals references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      
    • K W Menninger, Number words and number symbols : A cultural history of numbers (Boston, 1969).
      

69. Bolzano's manuscripts references
    
    • D Laugwitz, Bolzano's infinitesimal numbers, Czechoslovak Math.
      
    • B van Rootselaar, Bolzano's theory of real numbers, Arch.
      

70. Babylonian mathematics references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      
    • Reciprocals of regular sexagesimal numbers, J.
      

71. Arabic numerals references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      
    • K W Menninger, Number words and number symbols : A cultural history of numbers (Boston, 1969).
      

72. Bolzano's manuscripts
    
    • He had worked for many years on Grossenlehre (Theory of quantity) which was intended to be an introduction to mathematics covering many different areas of mathematics such as numbers, elementary geometry, geometry in general, function theory, methodology, and the ideas of quantity and space.
      
    • Specific parts such as Functionenlehre (Theory of functions), and Zahlenlehre (Theory of Numbers) were written, much was in a less complete form with workings and reworkings of parts of his ambitious project.
      

73. Bakhshali manuscript
    
    • No line appears between the numbers as we would write today, however.
      
    • The Bakhshali manuscript describes the rule where the three numbers are written .
      

74. 20th century time
    
    • Dirac tackled a similar problem in his development of his Large Numbers Hypothesis.
      
    • Dirac was forced into this conclusion based on results of the Large Numbers Hypothesis that threw off age calculations of the Moon and Sun.
      

75. Doubling the cube
    
    • Between two cube numbers there are two mean proportional numbers, and the cube has to the cube the ratio triplicate of that which the side has to the side.
      

76. Matrices and determinants
    
    • These were important in that for the first time the definition of the determinant was made in an algorithmic way and the entries in the determinant were not specified so his results applied equally well to cases were the entries were numbers or to where they were functions.
      Go directly to this paragraph
      
    • Eisenstein in 1844 denoted linear substitutions by a single letter and showed how to add and multiply them like ordinary numbers except for the lack of commutativity.
      Go directly to this paragraph
      

77. Pi history
    
    • Moreover he did not even have the advantage of our decimal notation for numbers, so that the calculation of a6 and b6 from (1) and (2) was by no means a trivial task.
      
    • (π and Fibonnaci numbers) .
      

78. Squaring the circle
    
    • This was not enough to prove the impossibility of squaring the circle with ruler and compass since certain algebraic numbers can be constructed with ruler and compass.
      
    • A few years later the Royal Society in London also banned consideration of any further 'proofs' of squaring the circle as large numbers of amateur mathematicians tried to achieve fame by presenting the Society with a solution.
      

79. The number e
    
    • There is a great contrast between the historical developments of these two numbers and in many ways writing a history of e is a much harder task than writing one for π.
      
    • This was in 1618 when, in an appendix to Napier's work on logarithms, a table appeared giving the natural logarithms of various numbers.
      

80. Christianity and Mathematics
    
    • Pythagoras saw the beauty in the theory of numbers and he saw this mathematical beauty translated into musical beauty.
      
    • From there he developed a view of the world based on numbers and shapes.
      

81. Egyptian mathematics
    
    • Most historians believe that the Egyptians did not think of numbers as abstract quantities but always thought of a specific collection of 8 objects when 8 was mentioned.
      
    • To overcome the deficiencies of their system of numerals the Egyptians devised cunning ways round the fact that their numbers were poorly suited for multiplication as is shown in the Rhind papyrus.
      

82. Nine chapters
    
    • The Euclidean algorithm method for finding the greatest common divisor of two numbers is given.
      
    • Negative numbers are used in the matrix and the chapter includes rules to compute with them.
      

83. Bolzano's manuscripts references
    
    • D Laugwitz, Bolzano's infinitesimal numbers, Czechoslovak Math.
      
    • B van Rootselaar, Bolzano's theory of real numbers, Arch.
      

84. Physical world
    
    • The numbers he obtained were approximately the densities of materials such as iron, silver and lead.
      

85. Fractal Geometry
    
    • A typical student will, at various points in her mathematical career -- however long or brief that may be -- encounter the concepts of dimension, complex numbers, and "geometry".
      

86. Egyptian Papyri references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      

87. Egyptian mathematics references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      

88. Egyptian numerals references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      

89. Christianity and Mathematics references
    
    • D C Lindberg and R L Numbers, God and Nature (Berkeley, 1986).
      

90. Mayan mathematics references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      

91. Babylonian numerals references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      

92. Babylonian and Egyptian references
    
    • I : Reciprocals of regular sexagesimal numbers, J.
      

93. Chinese numerals references
    
    • G Ifrah, The universal history of numbers (London, 1998).
      

94. Mathematics and Architecture references
    
    • G Ifrah, The Universal History of Numbers (London, 1998).
      

95. Mathematics and Architecture references
    
    • G Ifrah, The Universal History of Numbers (London, 1998).
      

96. Indian mathematics references
    
    • R K Sarma, A note on the use of words for numbers in ancient Indian mathematics, Math.
      

97. Egyptian Papyri references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      

98. Egyptian mathematics references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      

99. Egyptian numerals references
    
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
      

100. Greek astronomy
     
     • This model, certainly not suggested by any observational evidence, is more likely to have been proposed so that there were 10 heavenly bodies, for 10 was the most perfect of all numbers to the Pythagoreans.
       

101. Christianity and Mathematics references
     
     • D C Lindberg and R L Numbers, God and Nature (Berkeley, 1986).
       

102. Babylonian mathematics references
     
     • I : Reciprocals of regular sexagesimal numbers, J.
       

103. Mayan mathematics references
     
     • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
       

104. Egyptian mathematics references
     
     • I : Reciprocals of regular sexagesimal numbers, J.
       

105. Babylonian numerals references
     
     • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
       

106. Chinese numerals references
     
     • G Ifrah, The universal history of numbers (London, 1998).
       

107. Word problems
     
     • In the 1930s Kurt Godel investigated how symbolic manipulation in formal logic could be simulated by functions on the natural numbers.
       

108. Indian mathematics references
     
     • R K Sarma, A note on the use of words for numbers in ancient Indian mathematics, Math.
       

109. Mathematical games
     
     • Magic squares involve using all the numbers 1, 2, 3, ..
       Go directly to this paragraph
       

110. Mayan mathematics references
     
     • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
       

111. Longitude1
     
     • Since there were many large prizes on offer, large numbers of people tried to win them.
       Go directly to this paragraph
       

112. Hirst's diary
     
     • He talked continuously for that time about his partitions of numbers and strange to say he was less obscure than I expected.
       

113. Chinese overview
     
     • Xiahou Yang (about 400 - about 470) was the supposed author of the Xiahou Yang suanjing (Xiahou Yang's Mathematical Manual) which contains representations of numbers in the decimal notation using positive and negative powers of ten.
       

114. Golden ratio
     
     • just like God cannot be properly defined, nor can be understood through words, likewise this proportion of ours cannot ever be designated through intelligible numbers, nor can it be expressed through any rational quantity, but always remains occult and secret, and is called irrational by the mathematicians.
       

115. Size of the Universe
     
     • However when he studied large numbers of these he discovered that their distribution was connected to the plane of the Milky Way, so this seemed to prove that these were not "island universes" but rather star clusters in the Milky Way which could not be resolved into stars.