Last Modified on Friday, August 9, 2002.A Chronology of Irrational Number Development
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(Includes Special Information on e, phi, pi, R(2), R(3), etc.)
Note: This page uses R(n) to indicate the square root of n (this notation was commonly used before the modern day notation was adopted). This page also uses pi for the irrational number 3.14159..., phi for the golden mean (which is the irrational number 1.618...), and ^ to indicate exponents. About 2000BC
The Babylonians use 3 1/8 for pi.
About 2000BC
The Egyptians use (16/9)^2 = 3.1605 for pi.
About 1750BC
The Babylonians compile tables of square and cube roots. They have a very accurate value for R(2).
About 1200BC
The Chinese use 3 for pi.
About 550BC
The Bible implies that pi=3.
About 450BC
Zeno of Elea presents his paradoxes.
About 430BC
It is thought that around this time incommensurable lengths were first realized.
About 425BC
Theodorus of Cyrene shows that the square roots of 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15 and 17 are irrational. Theodorus approximates R(3) as 7/4 since 49/16 is almost equal to 3.
About 385 BC
Theaetetus investigates and classifies the various types of incommensurable lengths that can be generated with a compass and a straight edge.
About 375BC
Archytas gives a method for approximating R(n). This method was known to the Babylonians.
About 360BC
Eudoxus writes about irrational magnitudes and proportions.
About 350BC
Plato approximates R(2) as 7/5 since 49/25 is almost 2.
About 325BC
Aristotle gives a proof that R(2) is irrational. This proof was known long before him.
About 300BC
Euclid, in The Elements, says that the line AB is divided in "extreme and mean ratio" (phi) by C if AB:AC = AC:CB. The Elements also deals with the idea of incommensurability, although the proof that R(2) is irrational was probably added in later editions of the work.
About 250BC
In Measurement of the Circle, Archimedes gives an approximation of the value of pi with a method which will allow improved approximations. He declares that 3 10/71 < pi < 3 1/7 and approximates pi as 211875/67441 = 3.14163. He needs to approximate R(3) to make these calculations, so he bounded R(3) by 1351/780 > R(3) > 265/153, which he probably finds by what will later be called Heron's Method.
About 225BC
Appolonius improves the Archimedean value of pi, but it is unknown to what extent.
About 150BC
Hypsicles writes on regular polyhedra, and the golden ratio enters into his constructions.
About 20BC
Vitruvius estimates pi as 25/8 = 3.125.
About 50
Heron begins to compute approximate ratios, and in his work he gives approximate values for the ratio of the area of the pentagon to the area of the square of one side. He also gives a method for approximating the square root of a number (this method was probably known to the Babylonians).
About 120
Ptolemy calculates the side of a regular pentagon in terms of the radius of the circumscribed circle. He also gives a value for R(3) that is correct to 6 decimal places.
About 130
Chung Hing uses pi=R(10) = 3.16....
About 150
Ptolemy uses pi=377/120 = 3.1416666....
About 200
Wang Fau uses pi=142/45 = 3.15555....
250
Diophantus of Alexandria writes Arithmetica, a study of number theory problems in which only rational numbers are allowed as solutions.
263
By using a regular polygon with 192 sides Liu Hui calculates the value of pi as 3.14159 which is correct to five decimal places.
About 450
Proclus uses the term "section," although some historians later dispute that his reference to "section" refers to the golden ratio.
About 480
Tsu Ch'ung Chi gives the approximation 355/113 to pi which is correct to 6 decimal places and establishes that 3.1415926 < pi < 3.1415927.
499
Aryabhata I calculates pi to be 3.1416. He produces his Aryabhatiya, a treatise on quadratic equations, the value of pi, and other scientific problems.
About 600
The Hindus operate correctly with irrationals.
628
Brahmagupta writes Brahmasphutasiddanta (The Opening of the Universe), a work on astronomy and mathematics. He uses zero and negative numbers, gives methods to solve quadratic equations, sum series, and compute square roots. Brahmagupta uses pi=R(10) = 3.16....
About 800
Al-Khwarizmi estimates pi to be 3.1416.
About 825
Al-Khwarizmi gives a problem on dividing a line of length 10 into two parts and finds a quadratic equation for the length of the smaller part of the line of length 10 divided in the golden ratio. It is not known whether Al-Khwarizmi recognized the golden mean as a special number.
About 900
Abu Kamil gives equations which arise from dividing a line of length 10 in various ways. Two of these ways are related to the golden ratio but it is unclear whether Abu Kamil was aware of this.
About 1010
Al-Biruni writes on many scientific topics, including irrational numbers.
1040
Ahmad al-Nasawi writes Al-Muqni'fi Al-Hisab Al-Hindi which studies four different number systems. He explains the operations of arithmetic, particularly taking square and cube roots in each system.
About 1100
The Arabs work freely with irrationals.
1202
Fibonacci produces Liber Abaci, using many Arabic sources and one of them contains the problems of Abu Kamil. Fibonacci clearly indicates his awareness of the connection between Abu Kamil's two problems and the golden ratio. In Liber Abaci he gives the lengths of the segments of a line of length 10 divided in the golden ratio as R(125) -5 and 15 - R(125). He also shows that not all types of irrationals may be generated using a compass and straight edge.
1220
Fibonacci finds pi=3.141818.
About 1343
John of Meurs extracts R(2) as equal to (R(2000000))/1000, which is approximately equal to 1414/1000 or 1.414. Although it is clear that this is done using base ten computations, he uses mixed notations (which is very common during this time period) and expresses his result in sexagesimal (base 60) terms.
About 1375
Narayana gives a rule to calculate approximate values of a square root.
1400
Madhava of Sangamagramma proves a number of results about infinite sums giving Taylor expansions of trigonometric functions. He uses these to find an approximation for pi correct to 11 decimal places.
1424
Al-Kashi writes Treatise on the Circumference giving a remarkably good approximation to pi in both sexagesimal and decimal forms. Al-Kashi calculates pi to 14 decimal places. Later, in 1429, he calculates pi to 16 decimal places.
1486
Widman considers computation with irrational numbers and polynomials to be part of algebra.
1509
Pacioli writes Divina Proportione (Divine Proportion), which is his name for the golden ratio. The book contains little new on the topic, collecting results from Euclid, Fibonacci and other sources on the golden ratio. He states (without any attempt at a proof or a reference) that the golden ratio cannot be rational. Pacioli also gave a method for the approximation of square roots (using a special case of Newton's method).
1525
Rudolff introduces a symbol which resembles the modern day notation for square roots in his Die Coss, the first German algebra book. He calculates the solutions for polynomials with rational and irrational coefficients and was aware that ax^2 + b = cx has 2 roots.
1544
Stifel publishes Arithmetica Integra which contains binomial coefficients and the notation +, -, and the modern day square root symbol.
1573
Otho finds pi = 355/133 = 3.1415929.
1583
Duchesne finds pi=(39/22)^2 = 3.14256...
1590
Cataldi uses continued fractions in finding square roots.
1593
Van Roomen calculates pi to 16 decimal places. Romanus also calculates pi to 15 decimal places this year.
1593
Viete finds an infinite irrational product for pi.
1596
Ludolph van Ceulen calculates pi to 32 places, and later in 1610, he calculates it to 35 places. Some still call pi the Ludolphine Number.
1597
The first known calculation of the golden ratio as a decimal is given in a letter written by Michael Maestlin, at the University of Tübingen, to his former student Kepler. He gives "about 0.6180340" for the length of the longer segment of a line of length 1 divided in the golden ratio. The correct value is 0.61803398874989484821...
1609
Kepler knows that the ratio of adjacent terms of the Fibonacci sequence tends to the golden ratio.
1613
Cataldi publishes Trattato del Modo Brevissimo di Trovar la Radice Quadra Delli Numeri in which he finds square roots using continued fractions. His methods make precise some ideas which date back to Heron.
1618
A table of natural logarithms appeared in Wright's English translation of Napier's work on logarithms, it was probably created by Oughtred, but he does not mention e. Napier's logarithms are actually logarithms with base 1/e, though he never thought of them in terms of a base.
1621
Snell refines the Archimedean Method for calculating digits of pi. Grienberger uses this refinement to calculate pi to 39 decimal places in 1630. Huygens, in 1654, proves the validity of this refinement.
1624
Briggs gives a numerical approximation for the base 10 logarithm of e, but does not mention e itself.
1632
Albert Girard discovers (independently of Kepler) that the ratio of adjacent terms of the Fibonacci sequence tends to the golden ratio. This is not published until 1634, two years after Albert Girard's death.
1655
Brouncker gives a continued fraction expansion of 4/pi based on the infinite rational product for pi that Wallis has just discovered.
1661
Huygens studies the area under the rectangular hyperbola xy=1. The area under this curve from 1 to e is 1, which is why e is the base of the natural logarithms, but Huygens does not realize this. Huygens also calculates the logarithm to the base 10 of e to 17 decimal places, but again he fails to realize its importance.
1663
Muramatsu Shigekiyo finds seven accurate digits for pi.
1665
Newton calculates pi to at least 16 decimal places using his own idea, but his results are not published until 1737, after his death. This same year, Newton also comes up with the MacLaurin series for e.
1668
Mercator publishes Logarithmotechnia, which uses the term "natural logarithm" for the first time for logarithms to base e. The number e itself again fails to appear as such and again remains elusively just around the corner.
1671
James Gregory discovers Taylor's Theorem and writes to Collins telling him of his discovery. His series expansion for arctan(x) later gives a series for pi/4 (Leibniz is credited with this additional idea in 1674).
1672
Mengoli publishes The Problem of Squaring the Circle which studies infinite series and gives an infinite product expansion for pi/2.
1673
Leibniz demonstrates his incomplete calculating machine to the Royal Society. It can multiply, divide and extract roots.
1683
Jacob Bernoulli investigates the limit of (1 + 1/n)^n as n approaches infinity. He uses the binomial theorem to show that this limits lies between 2 and 3 (the first approximation for e). This is the first time a number is defined by a limiting process.
1685
Kochanski gives an approximate method to find the length of the circumference of a circle.
1690
The number e appears in its own right in a letter from Leibniz to Huygens, although he calls it b.
1696
Wallis identifies the rational as periodic decimal numbers.
1700
Seki Kowa calculates pi to 10 decimal places.
1705
Sharp calculates pi to 72 decimal places.
1706
Jones introduces the Greek letter, pi, to represent the ratio of the circumference of a circle to its diameter in his Synopsis Palmariorum Matheseos (A New Introduction to Mathematics).
1706
Machin calculates pi to 100 places using his modification of the Gregory-Leibniz arctan(x) series.
1707
Newton states that irrationals are indeed numbers.
1713
Su-li Ching-yun is published, containing pi accurate to 19 digits.
1719
De Lagny calculates pi to 127 places, but only 112 are correct.
1722
Takebe Kenko finds 40 digits of pi.
1727
Euler introduces the symbol e for the base of natural logarithms in a manuscript entitled Meditation upon Experiments Made Recently on Firing of Cannon. The manuscript was not published until 1862.
1730
Kamata calculates 25 decimal places of pi.
1737
Euler is able to show that both e and e^2 are irrational and gives several continued fractions involving e. Euler popularizes the use of the symbol pi for the ratio of the circumference to the diameter of a circle. The year before, he had determined a simple series for (pi^2)/6.
1739
Matsunaga calculates 50 decimal places of pi.
1744
Euler distinguishes between algebraic and transcendental numbers.
1748
Euler publishes Analysis Infinitorum (Analysis of the Infinite) which is an introduction to mathematical analysis. The famous formula e^(i*pi) = -1 appears for the first time in this text, as well as many series for pi and pi^2. A series for e appears, as well as the fact that e is the limit as n approaches infinity of (1 + 1/n)^n. Euler also approximates e to 18 decimal places and gives several continued fractions involving e.
1753
Simson notes that in the Fibonacci sequence the ratio between adjacent numbers approaches the golden ratio.
1755
Euler derives a very rapidly converging arctangent series.
1757
Riccati first notices that the hyperbolic trigonometric functions behave very similarly to the trigonometric functions. He introduces a notation that is only slightly different from what we use today for the hyperbolic sine and cosine.
1761
Lambert proves that pi is irrational. He publishes a more general result in 1768. He also shows that the functions e^x and tanx cannot assume rational values if x is a non-zero rational number.
1775
Euler suggests that pi is transcendental.
1777
Buffon carries out his probability experiment to calculate pi by throwing sticks over his shoulder onto a tiled floor and counting the number of times the sticks fall across the lines between the tiles.
1794
Legendre proves the irrationality of pi and pi^2.
1794
Vega calculates pi to 140 decimal places.
1814
Barlow produces Barlow's Tables which give factors, squares, cubes, square roots, reciprocals and hyperbolic logs of all numbers from 1 to 10000.
1817
Bolzano publishes Rein Analytischer Beweis (Pure Analytical Proof), which defines continuous functions without the use of infinitesimals.
1821
Cauchy attempts to rigorously define irrational numbers, but his argument is circular and he later retracts it.
1824
Rutherford calculates 208 decimal places of pi, but only 152 are correct.
1835
The first known use of the term "golden section" appears in a footnote in a later edition of Die Reine Elementar-Matematik by Martin Ohm.
1844
Liouville finds the first transcendental numbers - numbers that cannot be expressed as the roots of an algebraic equation with rational coefficients.
1844
Strassnitsky and Dase calculate pi to 200 places.
1847
Clausen calculates 248 digits of pi.
1851
Liouville publishes a second work on the existence of specific transcendental numbers which are now known as "Liouville numbers." In particular, he gives the example 0.1100010000000000000000010000... where there is a 1 in place n! and 0 elsewhere.
1853
Lehmann correctly calculates 261 decimal places of pi.
1854
Shanks calculates e to 205 places, but in 1884, Boorman shows that only 187 were correct.
1855
Richter calculates pi to 500 decimal places.
1858
Dedekind discovers a rigorous method to define irrational numbers with "Dedekind cuts." The idea comes to him while he is thinking about teaching differential and integral calculus.
1864
Benjamin Pierce has his picture taken in front of a blackboard with the formula i^-i=R(e^pi) inscribed on it. In 1859, he had tried to introduce new symbols for e and pi, but they did not become popularly accepted.
About 1865
Stolz states that every irrational number can be represented as a non-periodic decimal (and this can then be used as a defining property).
1869
Méray is the first to publish an arithmetical theory of irrational numbers in his paper Remarques sur la Nature des Quantités Définies par la Condition de Servir de Limites à des Variables Données (the earliest coherent and rigorous theory of the irrational numbers to appear in print).
1872
Dedekind publishes his formal construction of real numbers and gives a rigorous definition of an integer. In this same year, Cantor, who is good friends with Dedekind, publishes his ideas about how to define irrational numbers. Cantor uses convergent sequences of rational numbers to define the irrationals.
1873
Hermite publishes Sur la Fonction Exponentielle (On the Exponential Function) in which he proves that e is a transcendental number.
1873
Shanks gives pi to 707 places (in 1944 it was discovered, by Ferguson, that Shanks was wrong from the 528th place on).
1874
Cantor publishes his first paper on set theory. He proves the controversial result that almost all numbers are transcendental.
1874
Tseng Chi-hung finds 100 digits of pi.
1879
Fermat gives his method of infinite descent, which can be used to prove the irrationality of R(2).
1882
Lindemann proves that pi is transcendental. This proves that it is impossible to construct a square with the same area as a given circle using only a ruler and compass. The classic mathematical problem of squaring the circle dates back to ancient Greece and had proved a driving force for mathematical ideas through many centuries.
1884
Boorman calculates 346 places of e.
1887
Adams calculates the base 10 log of e to 272 places.
1888
Dedekind publishes Was Sind Und Was Sollen Die Zahlen? (The Nature and Meaning of Numbers).
1896
Hadamard and Vallee-Poussin show that the number e is indirectly connected with the prime numbers.
About 1900
Mark Barr first uses the Greek letter, phi, for the golden ratio.
1934
Gelfond and Schneider solve "Hilbert's Seventh problem" independently. They prove that a^q is transcendental when a is algebraic (and not equal 0 or 1) and q is an irrational algebraic number. Gelfond proves that e^pi is transcendental.
1946
Ferguson publishes 620 decimal places of pi. Later, in 1947, he extends this to 808 places using a desk calculator.
1949
ENIAC is programmed to compute 2037 decimals of pi. This same year, Smith & Wrench use a desk calculator to compute 1120 places of pi.
1954
NORC is programmed to compute 3089 decimals of pi.
1957
Pegasus computer computes 7480 places of pi. The next year it computes 10021 digits of pi.
1959
IBM 704 computes 16167 decimal places of pi.
1961
Shanks and Wrench improve the IBM 7090 computer program for pi, and compute 100,000 decimal places for pi.
1966
Alan Baker proves "Gelfond's Conjecture" about the linear independence of algebraic numbers over the rational numbers.
1966
IBM 7030 computes 250,000 decimal places for pi.
1967
CDC 6600 computes 500,000 decimal places for pi.
1973
Guilloud and Bouyer compute 1 million decimal places of pi.
1975
Feigenbaum discovers a new constant, approximately 4.669201660910..., which is related to period-doubling bifurcations and plays an important part in chaos theory.
1976
Salamin and Brent find an arithmetic-geometric mean algorithm for pi.
1981
Miyoshi and Kanada compute over 2 million digits of pi. Kanada becomes a life-long "pi digit-hunter."
1983
Tamura and Kanada compute 16 million digits of pi.
1988
Kanada computes over 200 million digits of pi.
1989
The Chudnovsky brothers find 480 million digits of pi. In the same year, Kanada calculates 536 million digits of pi, and the Chudnovsky brothers reclaim the "most digits of pi" title by calculating 1 billion digits of pi.
1994
The Chudnovsky brothers put together a home-made parallel computer and use it to calculate over 4 million digits of pi.
1995
The Borwein brothers develop a method to find the nth hexadecimal digit of pi (without calculating the preceding digits). Also, in this year, Kanada computes 6 billion digits of pi.
1996
The Chudnovsky brothers compute over 8 billion digits of pi.
1997
Plouffe finds a method to calculate the nth digit of pi in any base. Also, Kanada and Takahashi calculate over 51 billion digits of pi.
1999
Kanada captures the current record for the number of digits of pi, over 206 billion.
References
Beckmann, P. (1971). {A history of} pi. New York: St. Martin's Press.
Blatner, D. (1997). The joy of pi. New York: Walker.
Boyer, C. (1991). A history of mathematics (2nd ed.). New York: Wiley & Sons, Inc.
Burton, D. (1997). The history of mathematics (3rd ed.). New York: McGraw-Hill.
Eves, H. (1992). An introduction to the history of mathematics (6th ed.). New York: Saunders College Publishing.
Katz, V. (1993). A history of mathematics: An introduction. New York: Harper Collins.
Kline, M. (1972). Mathematical thought from ancient to modern times. New York: Oxford.
Moar, E. (1994). e: The story of a number. Princeton, NJ: Princeton University Press.
Nahin, P. (1998). An imaginary tale: The story of R(-1). Princeton, NJ: Princeton University Press.
http://www-gap.dcs.st-and.ac.uk/~history/
http://www.mste.uiuc.edu/mathed/HumanResources/daleleibforth/timeline
Last Modified on Friday, August 9, 2002.