Search Results for decimal*

Biographies

1. Al-Uqlidisi biography
   
   • The aim of the first part is to introduce the Hindu numerals, to explain a place value system and to describe addition, multiplication and other arithmetic operations on integers and fractions in both decimal and sexagesimal notation.
     
   • Al-Uqlidisi's work is historically important as it is the earliest known text offering a direct treatment of decimal fractions.
     
   • At one time it was thought that Stevin was the first to propose decimal fractions.
     
   • Further research showed that decimal fractions appeared in the work of al-Kashi, who was then credited with this extremely important contribution.
     
   • The most remarkable idea in this work is that of decimal fraction.
     
   • Al-Uqlidisi uses decimal fractions as such, appreciates the importance of a decimal sign, and suggests a good one.
     
   • 1436/7) who treated decimal fractions in his "Miftah al-Hisab", but al-Uqlidisi, who lived five centuries earlier, is the first Muslim mathematician so far known to write about decimal fractions.
     
   • Following Saidan's paper, some historians went even further in attributing to al-Uqlidisi the complete credit for giving the first complete description and applications of decimal fractions.
     
   • Rashed, however, although he does not wish to minimise the importance of al-Uqlidisi's contribution to decimal fractions, sees it as [The development of Arabic mathematics : between arithmetic and algebra (London, 1994).',2)">2]:- .
     
   • We put the resulting fraction in front of this number and we move it to the unit place after marking it [with the ' sign he uses for the decimal point] thus.
     
   • Saidan (writing in [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]) sees in this passage that al-Uqlidisi has fully understood the idea of decimal fractions, saying that earlier authors:- .
     
   • rather mechanically transformed the decimal fraction obtained into the sexagesimal system, without showing any sign of comprehension of the decimal idea.
     
   • That said, in the passage just quoted, three basic ideas emerge whose intuitive resonance may have misled historians; what they thought was a theoretical exposition was merely understood implicitly, and, as a result, they have overestimated the author's contribution to decimal fractions.
     
   • The two points of view are almost impossible to decide between since what we are looking at is the development of the idea of decimal fractions by different mathematicians, each contributing to its understanding.
     

2. Al-Kashi biography
   
   • He produced his Treatise on the Circumference in July 1424, a work in which he calculated 2π to nine sexagesimal places and translated this into sixteen decimal places.
     
   • This was an achievement far beyond anything which had been obtained before, either by the ancient Greeks or by the Chinese (who achieved 6 decimal places in the 5th century).
     
   • It would be almost 200 years before van Ceulen surpassed Al-Kashi's accuracy with 20 decimal places.
     
   • Al-Kashi uses decimal fractions in calculating the total surface area of types of muqarnas.
     
   • We mentioned above al-Kashi's use of decimal fractions and it is through his use of these that he has attained considerable fame.
     
   • The generally held view that Stevin had been the first to introduce decimal fractions was shown to be false in 1948 when P Luckey (see [Die Rechnenkunst bei Gamsid b.
     
   • Masud al-Kasi (Wiesbaden, 1951).',4)">4]) showed that in the Key to Arithmetic al-Kashi gives as clear a description of decimal fractions as Stevin does.
     
   • However, to claim that al-Kashi is the inventor of decimal fractions, as was done by many mathematicians following the work of Luckey, would be far from the truth since the idea had been present in the work of several mathematicians of al-Karaji's school, in particular al-Samawal.
     
   • (1) The analogy between both systems of fractions; the sexagesimal and the decimal systems.
     
   • (2) The usage of decimal fractions no longer for approaching algebraic real numbers, but for real numbers such as π.
     
   • Al-Kashi can no longer be considered as the inventor of decimal fractions; it remains nonetheless, that in his exposition the mathematician, far from being a simple compiler, went one step beyond al-Samawal and represents an important dimension in the history of decimal fractions.
     

3. Stevin biography
   
   • In 1585 he published La Theinde (The tenth), a twenty-nine page booklet in which he presented an elementary and thorough account of decimal fractions.
     
   • Although he did not invent decimals (they had been used by the Arabs and the Chinese long before Stevin's time) he did introduce their use in mathematics in Europe.
     
   • Stevin states that the universal introduction of decimal coinage, measures and weights would only be a matter of time (but he probably would be amazed to know that in the 21st century some countries still resist adopting decimal systems).
     
   • It was titled Disme, The Arts of Tenths or Decimal Arithmetike and it was this translation which inspired Thomas Jefferson to propose a decimal currency for the United States (note that one tenth of a dollar is still called a dime).
     

4. Shanks biography
   
   • In this coal mining area near Durham, England, he ran a boarding school but he used his leisure hours working on mathematics, particularly on calculating the decimal expansion of π.
     
   • In the same year William Rutherford gave 440 decimal places in the expansion of π and, later in the same year, Shanks, in a collaboration with Rutherford, gave 530 places.
     
   • This was a busy year for Shanks, for also in 1853 he gave 607 decimal places in the expansion of π which had been independently checked as correct to the first 500 of those places.
     
   • At this point Shanks rested in his calculations of the decimal expansion of π, but he continued to write mathematical works.
     
   • which had been discovered by Machin in 1706 and used by him to correctly calculate to 100 decimal places.
     
   • Shanks also calculated e and Euler's constant γ to a great many decimal places.
     

5. Pitiscus biography
   
   • The tables give the values to five or six decimal places.
     
   • He recomputed all the tangents and secants between 83° and 90° to eleven decimal places and 86 pages of Opus Palatinum de triangulis was reprinted incorporating Pitiscus' corrections.
     
   • The Thesaurus mathematicus was eventually published in 1613 and contained a table of sines by Rheticus calculated for every 10'' to fifteen decimal places; a calculation of the sine at 1'' intervals for the first and last degree of the quadrant, again by Rheticus to fifteen decimal places; values for the basic sines from which the others were calculated to 22 decimal places by Pitiscus; and sines to 22 decimal places by Pitiscus for each tenth, thirtieth, and fiftieth second in the first 35 minutes.
     

6. Al-Umawi biography
   
   • Although he only gives these special cases, the general rule which they all obey is the following: take a number n written in decimal notation as .
     
   • He gives some interesting conditions for the decimal representation of a number n to be a square: .
     
   • 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.
     

7. Vlacq biography
   
   • This contained three items: a Dutch translation of Napier's Rabdologiae which described a method of multiplication using "numbering rods"; a paper by de Decker written to help businessmen undertake arithmetical calculations; and Stevin's La Theinde which gave an account of decimal fractions.
     
   • The 'Tweede deel' of 1627, actually the first complete table of decimal logarithms, was long forgotten until a copy was rediscovered in 1920.
     
   • In 1628 Vlacq republished the 10 decimal place logarithm tables as Arithmetica logarithma sive logarithmorum chiliades tentum, pro numeris naturali serie crescentibus ab unitate ad 100000 ..
     
   • These tables, carried to seven decimal places, were a great success and were often reprinted and reedited ..
     

8. Xiahou Yang biography
   
   • One significant idea which appears in the text concerns representation of numbers in the decimal notation.
     
   • Xiahou Yang notes that to multiply a number by 10, 100, 1000, or 10000 all that needs to be done is that the rods on the counting board are moved forwards by 1, 2, 3, or 4 decimal places.
     
   • Similarly to divide by 10, 100, 1000, or 10000 the rods are moved backwards by 1, 2, 3, or 4 decimal places.
     
   • What is significant here is that Xiahou Yang seems to understand not only positive powers of 10 but also decimal fractions as negative powers of 10.
     

9. Zu Chongzhi biography
   
   • He gave the rational approximation 355/113 to in his text Zhui shu (Method of Interpolation), which is correct to 6 decimal places.
     
   • To compute this accuracy for π, Zu must have used an inscribed regular 24,576-gon and undertaken the extremely lengthy calculations, involving hundereds of square roots, all to 9 decimal place accuracy.
     
   • Since his book is lost we will never know exactly how he found the rational approximation 355/113 from the decimal approximation.
     

10. 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.
      

11. Mouton biography
    
    • In this work Mouton became the first to propose the decimal system of measurement based on the size of the earth.
      
    • He also suggested a standard linear measurement, which he called the mille, based on the length of the arc of one degree of longitude on the Earth's surface and divided decimally.
      

12. 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.
      
    • It also gave tables of natural sine functions to 15 decimal places, and the tan and sec functions to 10 decimal places.
      

13. Roomen biography
    
    • One of Roomen's most impressive results was finding π to 16 decimal places.
      
    • Roomen was critical of the accuracy of the tables and wrote to Clavius at the Collegio Romano in Rome pointing out that, to calculate tangent and secant tables correctly to ten decimal places, it was necessary to work to 20 decimal places for small values of sine, see [Amphora : Festschrift for Hans Wussing on the occasion of his 65th birthday (Basel- Boston- Berlin, 1992), 55-66.',2)">2].
      

14. Fibonacci biography
    
    • The book, which went on to be widely copied and imitated, introduced the Hindu-Arabic place-valued decimal system and the use of Arabic numerals into Europe.
      
    • This converts to the decimal 1.3688081075 which is correct to nine decimal places, a remarkable achievement.
      

15. Aryabhata II biography
    
    • Aryabhata II constructed a sine table correct up to five decimal places when measured in decimal parts of the radius, see [J.
      

16. Napier biography
    
    • Napier also found exponential expressions for trigonometric functions, and introduced the decimal notation for fractions.
      
    • Napier chose the length AB to be 107, based on the fact that the best tables of sines available to him were given to seven decimal places and he thought of the argument x as being of the form 102.sin X.
      

17. Lemaitre biography
    
    • Instead of using the Arabic numerals 0 - 9 he proposes that the four letters i, j, k, l which lie within easy reach of the right hand be used to give a peculiar binary coded decimal representation.
      
    • In Pourquoi de nouveaux chiffres? (1955) he discusses the shortcomings of the decimal system and even the Arabic digits.
      

18. 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.
      
    • In this last mentioned paper Rychlik looked at the calculation of e to 225 decimal places carried out by the Czech mathematician Bohumir Tichanek in 1890 using a continued fraction method.
      

19. Borda biography
    
    • One of his instruments, the Borda repeating circle, was used during the time of the French Revolution to measure an arc of a meridian as part of a project to introduce the decimal system.
      
    • After his death in 1804 Delambre published Decimal trigonometrical tables which extended Borda's work.
      

20. Farey biography
    
    • In the first paragraph Farey says that he noted the "curious property" while examining the tables of Complete decimal quotients produced by Henry Goodwin.
      
    • Haros, in 1802, wrote a paper on the approximation of decimal fractions by common fractions.
      

21. Ulugh Beg biography
    
    • This excellent book records the main achievements which include the following: methods for giving accurate approximate solutions of cubic equations; work with the binomial theorem; Ulugh Beg's accurate tables of sines and tangents correct to eight decimal places; formulae of spherical trigonometry; and of particular importance, Ulugh Beg's Catalogue of the stars, the first comprehensive stellar catalogue since that of Ptolemy.
      
    • These tables display a high degree of accuracy, being correct to at least 8 decimal places.
      

22. Ezra biography
    
    • 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.
      
    • A second work is the Book of the Number which describes the decimal system for integers with place values from left to right.
      

23. Laplace biography
    
    • This committee worked on the metric system and advocated a decimal base.
      
    • History Topics: Decimal time and angles .
      

24. Abu'l-Wafa biography
    
    • His trigonometric tables are accurate to 8 decimal places (converted to decimal notation) while Ptolemy's were only accurate to 3 places.
      

25. Turing biography
    
    • Some of the symbols written down will form the sequences of figures which is the decimal of the real number which is being computed.
      
    • He defined a computable number as real number whose decimal expansion could be produced by a Turing machine starting with a blank tape.
      

26. Narayana biography
    
    • He then finds the solutions x = 6, y = 19 which give the approximation 19/6 = 3.1666666666666666667, which is correct to 2 decimal places.
      
    • Finally Narayana gives the pair of solutions x = 8658, y = 227379 which give the approximation 227379/8658 = 3.1622776622776622777, correct to eight decimal places.
      

27. Deparcieux biography
    
    • In 1741 Deparcieux published Nouveaux traites de trigonometrie rectiligne et spherique which consisted of tables of sins, tans, secs (calculated to seven decimal places), and log sins and log tans (calculated to eight decimal places).
      

28. Oughtred biography
    
    • Oughtred's most important work, Clavis Mathematicae (1631), included a description of Hindu-Arabic notation and decimal fractions and a considerable section on algebra.
      
    • Page from Clavis mathematicae (1652) showing + and - and the notation for decimal fractions .
      

29. Lorenz Edward biography
    
    • Eventually he discovered that the data he had input to begin the second run had not been printed out to the same number of decimal places as the machine had stored, so the initial data was slightly different for the second run (differing in the fourth decimal place).
      

30. Viete biography
    
    • wrote decimal fractions with the fractional part printed in smaller type than the integral and separated from the latter by a vertical line.
      
    • Extract from Works (printed 1609) showing the use of decimal fractions.
      

31. Mahler biography
    
    • Mahler regretted that, apart from his own work, little interest had been shown by 20th century mathematicians in the study of arithmetical properties of decimal expansions.
      

32. Lalla biography
    
    • π = 3.1416 which is a value correct to the fourth decimal place.
      

33. Apastamba biography
    
    • which gives an answer correct to five decimal places.
      

34. Aitken biography
    
    • In 1962 he published an article very dear to his heart, namely The case against decimalisation.
      

35. Madhava biography
    
    • We know that Madhava obtained an approximation for π correct to 11 decimal places when he gave .
      

36. Li Zhi biography
    
    • Notice that negatives were allowed and so were decimal fractions.
      

37. Ramanujan biography
    
    • He investigated the series ∑(1/n) and calculated Euler's constant to 15 decimal places.
      

38. Lalande biography
    
    • History Topics: Decimal time and angles .
      

39. Rittenhouse biography
    
    • He published A method of finding the sum of several powers of the sines in 1793 and in a paper of 12 August 1795 he gave an expansion of log10 n, where n a positive number, as a simple continued fraction and then computed log10 99 to nine decimal places.
      

40. Bernoulli Johann(III) biography
    
    • In the field of mathematics he worked on probability, recurring decimals and the theory of equations.
      

41. Euler biography
    
    • He calculated the constant γ to 16 decimal places.
      

42. Qadi Zada biography
    
    • Qadi Zada computed sin 1° to an accuracy of 10-12 (if expressed in decimals), as did al-Kashi.
      

43. Adams biography
    
    • He also gave tables of the positions of the moons of Jupiter, spent much time on a catalogue of Newton's papers, and calculated Euler's constant to 236 decimal places.
      

44. Zhu Shijie biography
    
    • The book contains examples of computations with fractions and decimals giving results such as 1/16 = 0.0625 and 2/16 = 0.125.
      

45. Al-Nasawi biography
    
    • There were three different types of arithmetic used in Arab countries around this period: (i) a system derived from counting on the fingers with the numerals written entirely in words; this finger-reckoning arithmetic was the system used by the business community, (ii) the sexagesimal system with numerals denoted by letters of the Arabic alphabet, and (iii) the arithmetic of the Indian numerals and fractions with the decimal place-value system.
      

46. Condorcet biography
    
    • History Topics: Decimal time and angles .
      

47. Baudhayana biography
    
    • This gives √2 correct to five decimal places.
      

48. Bhaskara II biography
    
    • Let an n-digit number be represented in the usual decimal form as .
      

49. Lagrange biography
    
    • They worked on the metric system and advocated a decimal base.
      

50. Wishart biography
    
    • He instructed me to do three hours computing a day on a table of the 1% level of z to 7 decimal places ..
      

51. Haselgrove biography
    
    • For some cases 15 figure logarithms were needed as input data to obtain barely 6 decimal accuracy.
      

52. Al-Biruni biography
    
    • We should record that in the Chronology al-Biruni refers to seven earlier works which he had written: one on the decimal system, one on the astrolabe, one on astronomical observations, three on astrology, and two on history.
      

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

54. Al-Samawal biography
    
    • The work here is carried out using the sexagesimal system, showing that, although mathematicians of this period favoured the decimal system, commercial use still favoured the sexagesimal system.
      

55. Mascheroni biography
    
    • In Adnotationes ad calculum integrale Euleri (1790) Mascheroni calculated Euler's constant to 32 decimal places.
      

56. Glaisher biography
    
    • His historical interests were on the early development of numerical computation, Stevin and the beginnings of the decimal system, Napier, Briggs and the beginnings of logarithms as well as the mathematical notation + and -.
      

57. Collins biography
    
    • His major works were An introduction to merchant's accounts (1652), The sector on a quadrant (1658), Geometrical dialling (1659), The mariner's plain scale new plained (1659) and, in 1664, he published Doctrine of Decimal Arithmetick.
      

58. Clavius biography
    
    • He was the first, however, to use the decimal point.
      

59. Brouncker biography
    
    • However after Brouncker correctly computed the first 10 places in the decimal expansion of π using his continued fraction expansion, Huygens accepted the result.
      

60. Cotes biography
    
    • Newton gave the following rational approximations (we add decimal values to see their accuracy) .
      

61. Mechain biography
    
    • There he was less than cooperative in presenting his observations to the commissioners charged with setting up the decimal metric system.
      

62. Legendre biography
    
    • Following the work of the committee on the decimal system on which Legendre had served, de Prony in 1792 began a major task of producing logarithmic and trigonometric tables, the Cadastre.
      

63. Speiser biography
    
    • The machine is to have a magnetic drum memory of 1200 word capacity, each word being 12 decimal digits.
      

64. Duarte biography
    
    • In 1908 he published an article where he calculated π to 200 decimal places.
      

65. Al-Khwarizmi biography
    
    • the decimal place-value system was a fairly recent arrival from India and ..
      

66. Poincare biography
    
    • History Topics: Decimal time and angles .
      

67. Brouwer biography
    
    • His 1920 lecture Does Every Real Number Have a Decimal Expansion? was published in the following year.
      

68. Stirling biography
    
    • He calculated T3/2 to ten decimal places.
      

69. Sharp biography
    
    • All the dimensions are given with great accuracy, far more than makes any sense from a practical point of view since he gives figures to between 15 and 20 decimal place accuracy.
      

70. 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.
      

71. Khayyam biography
    
    • We know now that the length of the year is changing in the sixth decimal place over a person's lifetime.
      

72. Arbogast biography
    
    • was responsible for the law introducing the decimal metric system in the whole of the French Republic.
      

73. De Prony biography
    
    • vast, with values calculated to between fourteen and twenty-nine decimal places.
      

74. Murnaghan biography
    
    • Tables of this factor to 63 decimal places are included for n ranging from 2 to 64.
      

75. Knuth biography
    
    • For example he computed Euler's constant to 1271 decimal places and published the result in 1962.
      

History Topics

1. Decimal time
   
   • Decimal time and angles .
     
   • The proposals regarding the decimalisation of time and angle did not catch on at all, although the calendar continued to be used until 1 January 1806 when the French calendar reverted to the old style.
     
   • In the end, despite a strong showing by the French delegation, the near unanimous choice was for the zero through Greenwich but, at least partly to soften the blow for the French, the Washington Conference passed a rather vague resolution which hoped that studies on the decimalisation of time and angle would be resumed.
     
   • Bouquet de la Grye, an engineer, expressed clearly the problems encountered by the earlier attempts at decimalisation described above:- .
     
   • The metric system succeeded because it was the simplest and it put an end to a veritable incoherence in local measures; the decimalisation of time and circumference failed because the whole world employed the same measures and the proposals sinned precisely because of their lack of unity.
     
   • There was also support for dividing the circle into 360 parts (the traditional degree) but then decimalising subdivisions of a degree.
     
   • The push for decimal time and angle was at an end.
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/Decimal_time.html .
     

2. Mental arithmetic
   
   • In his spare time, between 1844 and 1847, he calculated the natural logarithms of the first 1005000 numbers to 7 decimal places.
     
   • In 1873, Shanks carried this to 707 decimals; but it was not until 1948 that it was discovered that the last 180 of these were wrong.
     
   • When π was calculated to 1000 and indeed more decimals, I re-memorised it.
     
   • You can see the 1000 digits of the decimal expansion of π as Aitken recited them.
     
   • For example if asked for the decimal expansion of 1/851 he would think of 851 as 23 cross 37, if asked for the square root of 851 then he thought of it as 292 + 10, if asked for the decimal expansion of 17/851 then he would think of it as almost 0.02.
     
   • Aitken showed an ability divide, and thus to calculate decimal expansions of rationals, which other calculators could not do.
     
   • On asked to compute the decimal expansion of 1/697 he explained his method.
     
   • You see you've got to run one decimal and divide it at the same time by something else.
     
   • He explained how many decimal expansions could be carried out by short division.
     
   • Here we have the decimal for 1/59, obtained by dividing 1 by 60; as we obtain each digit we merely enter it in the dividend, one place later, and continue with the division.
     
   • In fact 5/23 = 0.2173913043478260869565, a recurring decimal with a period of 22 digits.
     

3. Babylonian numerals
   
   • Now of course this comment is based on knowledge of our own decimal system which is a positional system with nine special symbols and a zero symbol to denote an empty place.
     
   • 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.
     
   • For example the decimal 12345 represents .
     
   • which, in decimal notation is 424000.
     
   • 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.
     
   • Although not a very serious comment, perhaps it is worth remarking that if we assume that all our decimal digits are equally likely in a number then there is a one in ten chance of an empty place while for the Babylonians with their sexagesimal system there was a one in sixty chance.
     
   • The Babylonians used a system of sexagesimal fractions similar to our decimal fractions.
     
   • Of course a fraction of the form a/b, in its lowest form, can be represented as a finite decimal fraction if and only if b has no prime divisors other than 2 or 5.
     
   • So 1/3 has no finite decimal fraction.
     
   • Since 60 is divisible by the primes 2, 3 and 5 then a number of the form a/b, in its lowest form, can be represented as a finite decimal fraction if and only if b has no prime divisors other than 2, 3 or 5.
     
   • More fractions can therefore be represented as finite sexagesimal fractions than can as finite decimal fractions.
     
   • Some historians think that this observation has a direct bearing on why the Babylonians developed the sexagesimal system, rather than the decimal system, but this seems a little unlikely.
     
   • It is the "sexagesimal point" and plays an analogous role to a decimal point.
     
   • His idea basically is that a decimal counting system was modified to base 60 to allow for dividing weights and measures into thirds.
     
   • This version has the advantage that there is a natural unit for 10 in the Babylonian system which one could argue was a remnant of the earlier decimal system.
     

4. Decimal time references
   
   • References for: Decimal time and angles .
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Decimal_time.html .
     

5. Decimal time references
   
   • References for: Decimal time and angles .
     
   • [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Decimal_time.html] .
     

6. Bakhshali manuscript
   
   • Hence we see that the Bakhshali formula gives the result correct to four decimal places.
     
   • This time the Bakhshali formula gives the result correct to five decimal places.
     
   • Here 9 decimal places are correct .
     
   • Here 5 decimal places are correct .
     
   • Here 8 decimal places are correct] .
     
   • Here 11 decimal places are correct .
     
   • He finds in [Ganita Bharati 1 (3-4) (1979), 25-27.',7)" onmouseover="window.status='Click to see reference';return true">7] that it is 38% faster than Newton's method in giving √41 to ten places of decimals.
     

7. Real numbers 3
   
   • For example "one third" has 9 characters so will be decoded from c around 1018 digits after the decimal point.
     
   • This article is there, both with the misprints which inevitably occur and a corrected version is there (but one has to go rather a long way to the right of the decimal point to find it!).
     
   • First assume that we have a real number written in base 10, that is a decimal expansion.
     
   • Then if it is a "random" number the digit 1 should occur about 1/10 of the time so, if we denote by N(1,n) the number of times 1 occurs in the first n decimal digits, then N(1,n)/n should tend to 1/10 as n tends to infinity.
     
   • in turn to form the decimal expansion of a number .
     
   • In The Construction of Decimals Normal in the Scale of Ten published in the Journal of the London Mathematical Society in 1933, Champernowne proved that his number was normal in base 10.
     
   • We can describe rationals easily enough, for example either as, say, one-seventh or by specifying the repeating decimal expansion 142857.
     

8. Fair book
   
   • The decimal fraction is then multiplied by 4 to obtain roods (4 roods to 1 acre).
     
   • The decimal fraction is then multiplied by 40 to obtain square poles (40 sq poles to a rood).
     
   • Then the decimal fraction is multiplied by 301/4 to obtain square yards (301/4 sq yds to a sq pole).
     
   • He uses six figure logs obtaining the answer correctly to 2 decimal places.
     
   • The area is then 2/3 base times height which is computed without logs as 27303.33 (in fact 27305.06 with the error coming from taking the height to 2 decimal places).
     
   • Walker only computes the square root correct to two decimal places which is where the error arises.
     
   • Again error comes from stopping the square root computation after one decimal place without realising that the next figure would be 9.
     

9. Measurement
   
   • One exception, and the earliest known decimal system of weights and measures, is the Harappan system.
     
   • An analysis of the weights discovered in excavations suggests that they had two different series, both decimal in nature, with each decimal number multiplied and divided by two.
     
   • One was a decimal scale based on a unit of measurement of 1.32 inches (3.35 centimetres) which has been called the "Indus inch".
     
   • He proposed that decimal subdivisions should be used to determine the lengths of shorter units of length.
     
   • The system should have decimal subdivisions, all measures of area, volume, weight etc should be linked to the fundamental unit of length.
     
   • The decimal metric system was required to be used by law in the Low Countries in 1820.
     

10. The number e
    
    • Again out of this comes the logarithm to base 10 of e, which Huygens calculated to 17 decimal places.
      
    • Euler gave an approximation for e to 18 decimal places, .
      
    • The same passion that drove people to calculate to more and more decimal places of π never seemed to take hold in quite the same way for e.
      
    • There were those who did calculate its decimal expansion, however, and the first to give e to a large number of decimal places was Shanks in 1854.
      
    • It is worth noting that Shanks was an even more enthusiastic calculator of the decimal expansion of π.
      
    • Further calculations of decimal expansions followed.
      

11. 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.
      
    • In Gregory's series, for example, to get 4 decimal places correct we require the error to be less than 0.00005 = 1/20000, and so we need about 10000 terms of the series.
      
    • Brouwer's question: In the decimal expansion of π, is there a place where a thousand consecutive digits are all zero? .
      
    • Is π simply normal to base 10? That is does every digit appear equally often in its decimal expansion in an asymptotic sense? .
      
    • Is π normal to base 10? That is does every block of digits of a given length appear equally often in its decimal expansion in an asymptotic sense? .
      
    • As a postscript, here is a mnemonic for the decimal expansion of π.
      
    • J Borwein (The record for calculating π (200 Billion decimal places!) and some other details of its calculation) .
      

12. Arabic mathematics
    
    • Certainly there was an important influence which came from the Hindu mathematicians whose earlier development of the decimal system and numerals was important.
      Go directly to this paragraph
      
    • The third system was the arithmetic of the Indian numerals and fractions with the decimal place-value system.
      
    • Al-Baghdadi also contributed to improvements in the decimal system.
      
    • The discovery of the binomial theorem for integer exponents by al-Karaji (born 953) was a major factor in the development of numerical analysis based on the decimal system.
      Go directly to this paragraph
      
    • 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
      
    • His contribution to decimal fractions is so major that for many years he was considered as their inventor.
      Go directly to this paragraph
      

13. Real numbers 2
    
    • By the time Stevin proposed the use of decimal fractions in 1585, the concept of a number had developed little from that of Euclid's Elements.
      
    • If we move forward almost exactly 100 years to the publication of A treatise of Algebra by Wallis in 1684 we find that he accepts, without any great enthusiasm, the use of Stevin's decimals.
      
    • 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.
      

14. Trigonometric functions
    
    • Converted to decimals this is 0.0087268 which is correct to 6 decimal places, the answer to 7 decimal places being 0.0087265.
      

15. 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.
      
    • An analysis of the weights discovered suggests that they belong to two series both being decimal in nature with each decimal number multiplied and divided by two, giving for the main series ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500.
      
    • One was a decimal scale based on a unit of measurement of 1.32 inches (3.35 centimetres) which has been called the "Indus inch".
      

16. Nine chapters
    
    • Standard decimal units of length were established in China around 200 BC and later further subdivisions occurred.
      
    • Consider the fact that Britain changed to a decimal currency in 1970.
      
    • If you pick up a book with mathematics problems given in decimal currency then we could argue as above and say that the book was written after 1970.
      
    • However new editions of popular textbooks were brought out when the currency changed, so many older books appeared in decimal editions.
      

17. Maxwell's House
    
    • Other surprises which we noticed looking through the notebook were that e is given to 7 decimal places and π is given to 36 decimal places.
      Go directly to this paragraph
      
    • With π it looks as if Tait has only stopped writing down the decimal places when he has reached the edge of the page.
      Go directly to this paragraph
      

18. Real numbers 1
    
    • A major advance was made by Stevin in 1585 in La Theinde when he introduced decimal fractions.
      
    • Only finite decimals were allowed, so with his notation only certain rationals to be represented exactly.
      

19. Indian numerals
    
    • three essential features of our numeral notation system: (i) nine signs and the concept of zero, (ii) a place value system and (iii) a decimal base.
      
    • 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.
      

20. Fair book insert
    
    • The decimal fraction is then multiplied by 4 to obtain roods (4 roods to 1 acre).
      
    • The decimal fraction is then multiplied by 40 to obtain square poles (40 sq poles to a rood).
      
    • Then the decimal fraction is multiplied by 301/4 to obtain square yards (301/4 sq yds to a sq pole).
      

21. Babylonian Pythagoras
    
    • Assuming that the first number is 1; 24,51,10 then converting this to a decimal gives 1.414212963 while √2 = 1.414213562.
      
    • step decimal sexagesimal .
      

22. Zero
    
    • 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.
      
    • The Book of the Number describes the decimal system for integers with place values from left to right.
      

23. Chinese numerals
    
    • The number system which was used to express this numerical information was based on the decimal system and was both additive and multiplicative in nature.
      
    • What is significant here is that Xiahou Yang seems to understand not only positive powers of 10 but also decimal fractions as negative powers of 10.
      

24. Chinese overview
    
    • By the fourth century BC counting boards were used for calculating, which effectively meant that a decimal place valued number system was in use.
      
    • 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.
      

25. Arabic numerals
    
    • For example there were at least three different types of arithmetic used in Arab countries in the eleventh century: a system derived from counting on the fingers with the numerals written entirely in words, this finger-reckoning arithmetic was the system used for by the business community; the sexagesimal system with numerals denoted by letters of the Arabic alphabet; and the arithmetic of the Indian numerals and fractions with the decimal place-value system.
      
    • It is also historically important as it is the earliest known text offering a direct treatment of decimal fractions.
      

26. Prime numbers
    
    • The largest is M25964951 which has 7816230 decimal digits.
      
    • The largest known prime (found by GIMPS [Great Internet Mersenne Prime Search] in August 2008) was the 45th Mersenne prime: M43112609 which has 1209780189 decimal digits.
      

27. Squaring the circle
    
    • In the Journal of the Indian Mathematical Society in 1913 in a paper named Squaring the circle Ramanujan gave a construction which was equivalent to giving the approximate value of 355/113 for π, which differs from correct value only in the seventh decimal place.
      
    • which differs from π only in the ninth decimal place (π = 3.1415926535897932385..
      

28. Golden ratio
    
    • The first known calculation of the golden ratio as a decimal was given in a letter written in 1597 by Michael Maestlin, at the University of Tubingen, to his former student Kepler.
      

29. Greek numbers
    
    • Notice that this system of currency was not based on the decimal system although the number system had 10 as a base and 5 as a secondary base.
      

30. Pi chronology
    
    • Calculating π to many decimal places was used as a test for new computers in the early days.
      

31. 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.
      

32. 10000 digits of e
    
    • These are arranged 75 decimal places on each line.
      

33. Mathematics and Architecture
    
    • There are at least nine theories which claim to explain the shape of the Pyramid and at least half of these theories agree with the observed measurements to one decimal place.
      

34. Indian Sulbasutras
    
    • Compare the correct value √2 = 1.414213562 to see that the Apastamba Sulbasutra has the answer correct to five decimal places.
      

35. Babylonian mathematics
    
    • As a base 10 fraction the sexagesimal number 5; 25, 30 is 5 4/10 2/100 5/1000 which is written as 5.425 in decimal notation.
      

Famous Curves

Societies etc

1. BMC 2008
   
   • Bugeaud, YOn the decimal expansion of an algebraic number .
     

References

1. References for Al-Uqlidisi
   
   • H Tllasev and A T Umarov, Decimal fractions in 'The book of elements on Indian arithmetic' by al-Uqlidisi (Xth cent.) (Russian), in Mathematics in the East in the Middle Ages (Dubna, 1978), 191-193; 198.
     

Quotations

1. A quotation by Newcomb
   
   • Ten decimal places of π are sufficient to give the circumference of the earth to a fraction of an inch, and thirty decimal places would give the circumference of the visible universe to a quantity imperceptible to the most powerful microscope.
     

2. A quotation by Stevin
   
   • Introducing decimals in De Thiende .
     

3. Quotations by Maxwell
   
   • that, in a few years, all great physical constants will have been approximately estimated, and that the only occupation which will be left to men of science will be to carry these measurements to another place of decimals.
     

4. Quotations by Gauss
   
   • made that discovery! [= the decimal system of numeration or its equivalent (with some base other than 10)] .
     

5. Quotations by Cajori
   
   • The miraculous powers of modern calculation are due to three inventions: the Arabic Notation, Decimal Fractions and Logarithms.