Last Modified on Saturday, August 10, 2002.A Chronology of the Development of the Concept of Irrational Number
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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 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 360BC
Eudoxus writes about irrational magnitudes and proportions.
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.
250
Diophantus of Alexandria writes Arithmetica, a study of number theory problems in which only rational numbers are allowed as solutions.
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 1010
Al-Biruni writes on many scientific topics, including irrational numbers.
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.
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.
1696
Wallis identifies the rational as periodic decimal numbers.
1707
Newton states that irrationals are indeed numbers.
1744
Euler distinguishes between algebraic and transcendental numbers.
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.
1844
Liouville finds the first transcendental numbers - numbers that cannot be expressed as the roots of an algebraic equation with rational coefficients.
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.
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.
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.
1874
Cantor publishes his first paper on set theory. He proves the controversial result that almost all numbers are transcendental.
1879
Fermat gives his method of infinite descent, which can be used to prove the irrationality of R(2).
1888
Dedekind publishes Was Sind Und Was Sollen Die Zahlen? (The Nature and Meaning of Numbers).
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.
1966
Alan Baker proves "Gelfond's Conjecture" about the linear independence of algebraic numbers over the rational numbers.
1975
Feigenbaum discovers a new constant, approximately 4.669201660910..., which is related to period-doubling bifurcations and plays an important part in chaos theory.
Last Modified on Saturday, August 10, 2002.