Irrational number

In mathematics, an irrational number is any real number that is not a rational number, i.e., one that cannot be written as a ratio of two integers, i.e., it is not of the form

History of the theory of irrational numbers

The discovery of irrational numbers is usually attributed to Pythagoras, more specifically to the Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the square root of 2. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction (proof below). However Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but his beliefs would not accept the existence of irrational numbers and so he sentenced Hippasus to death by drowning.

Related Topics:
Pythagoras - Pythagorean - Hippasus of Metapontum - Square root of 2

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The sixteenth century saw the final acceptance of negative, integral and fractional numbers. The seventeenth century saw decimal fractions with the modern notation quite generally used by mathematicians. The next hundred years saw the imaginary become a powerful tool in the hands of Abraham de Moivre, and especially of Leonhard Euler. For the nineteenth century it remained to complete the theory of complex numbers, to separate irrationals into algebraic and transcendental, to prove the existence of transcendental numbers, and to make a scientific study of a subject which had remained almost dormant since Euclid, the theory of irrationals. The year 1872 saw the publication of the theories of Karl Weierstrass (by his pupil Kossak), Heine (Crelle, 74), Georg Cantor (Annalen, 5), and Richard Dedekind. Méray had taken in 1869 the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method has been completely set forth by Pincherle (1880), and Dedekind's has received additional prominence through the author's later work (1888) and the recent endorsement by Tannery (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of real numbers, separating all rational numbers into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, Kronecker (Crelle, 101), and Méray.

Related Topics:
Negative - Fractional - Abraham de Moivre - Leonhard Euler - Complex number - Transcendental number - Euclid - Karl Weierstrass - Kossak - Heine - Crelle - Georg Cantor - Richard Dedekind - Méray - Pincherle - Tannery - Cut (Schnitt) - Real number - Rational number - Kronecker

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Continued fractions, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler, and at the opening of the nineteenth century were brought into prominence through the writings of Joseph Louis Lagrange. Other noteworthy contributions have been made by Druckenmüller (1837), Kunze (1857), Lemke (1870), and Günther (1872). Ramus (1855) first connected the subject with determinants, resulting, with the subsequent contributions of Heine, Möbius, and Günther, in the theory of Kettenbruchdeterminanten. Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.

Related Topics:
Continued fraction - Euler - Joseph Louis Lagrange - Druckenmüller - Kunze - Lemke - Günther - Ramus - Determinant - Möbius

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Transcendental numbers were first distinguished from algebraic irrationals by Kronecker. Lambert proved (1761) that π cannot be rational, and that en is irrational if n is rational (unless n = 0), a proof, however, which left much to be desired. Legendre (1794) completed Lambert's proof, and showed that π is not the square root of a rational number. Joseph Liouville (1840) showed that neither e nor e2 can be a root of an integral quadratic equation. But the existence of transcendental numbers was first established by Liouville (1844, 1851), the proof being subsequently displaced by Georg Cantor (1873). Charles Hermite (1873) first proved e transcendental, and Ferdinand von Lindemann (1882), starting from Hermite's conclusions, showed the same for π. Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and has finally been made elementary by Hurwitz and Paul Albert Gordan.

Related Topics:
Transcendental number - Lambert - Legendre - Joseph Liouville - Quadratic equation - Charles Hermite - Ferdinand von Lindemann - David Hilbert - Hurwitz - Paul Albert Gordan

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