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with a and b integers, and b not zero. It can readily be shown that the irrational numbers are precisely those numbers whose expansion in any given base (decimal, binary, etc) never ends and never enters a periodic pattern, but no mathematician takes that to be a definition. "Almost all" real numbers are irrational, in a sense which is defined more precisely below.
When the ratio of lengths of two line segments is irrational, the line segments are also described as being incommensurable, meaning they share no measure in common. A measure of a line segment I in this sense is a line segment J that "measures" I in the sense that some whole number of copies of J laid end-to-end occupy the same length as I.
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.
The sixteenth century saw the final acceptance of negative numbers, integral and fractional. 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.
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.
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.
Irrationality of the square root of 2
One proof of the irrationality of the square root of 2 is the following reductio ad absurdum. The proposition is proved by assuming the negation and showing that that leads to a contradiction, which means that the proposition must be true.
- Assume that √2 is a rational number. Meaning that there exists an integer a and b so that a / b = √2.
- Then √2 can be written as an irreducible fraction (the fraction is shortened as much as possible) a / b such that a and b are coprime integers and (a / b)2 = 2.
- It follows that a2 / b2 = 2 and a2 = 2 b2.
- Therefore a2 is even because it is equal to 2 b2 which is obviously even.
- It follows that a must be even. (Odd numbers have odd squares and even numbers have even squares.)
- Because a is even, there exists a k that fulfills: a = 2k.
- We insert the last equation of (3) in (6): 2b2 = (2k)2 is equivalent to 2b2 = 4k2 is equivalent to b2 = 2k2.
- Because 2k2 is even it follows that b2 is also even which means that b is even because only even numbers have even squares.
- By (5) and (8) a and b are both even, which contradicts that a / b is irreducible as stated in (2).
Since we have found a contradiction the assumption (1) that √2 is a rational number must be false. The opposite is proven. √2 is irrational.
This proof can be generalized to show that any root of any natural number is either a natural number or irrational.
A different proof
Another reductio ad absurdum showing that √2 is irrational is less well-known and has sufficient charm that it is worth including here. It proceeds by observing that if √2 = m/n then √2 = (2n − m)/(m − n), so that a fraction in lowest terms is reduced to yet lower terms. That is a contradiction if n and m are positive integers, so the assumption that √2 is rational must be false. It is possible to construct from an isosceles right triangle whose leg and hypotenuse have respective lengths n and m, by a classic straightedge-and-compass construction, a smaller isosceles right triangle whose leg and hypotenuse have respective lengths m − n and 2n − m. That construction proves the irrationality of √2 by the kind of method that was employed by ancient Greek geometers.
A simpler proof of a similar result
When a line segment is divided into two disjoint subsegments in such a way that the ratio of the whole to the longer part equals the ratio of the longer part to the shorter part, then that ratio is the golden ratio, equal to
Assume this is a rational number n/m in lowest terms. Take n to be the length of the whole and m the length of the longer part. Then the length of the shorter part is n − m. Then we have
But this puts a fraction already in lowest terms into lower terms—a contradiction. Therefore the initial assumption that φ is rational is false.
Other irrational numbers
Almost all irrational numbers are transcendental and all transcendental numbers are irrational: the article on transcendental numbers lists several examples. er and πr are irrational if r ≠ 0 is rational; eπ is also irrational.
- p(x) = an xn + an-1 xn−1 + ... + a1 x + a0 = 0
where the coefficients ai are integers. Suppose you know that there exists some real number x with p(x) = 0 (for instance if n is odd and an is non-zero, then because of the intermediate value theorem). The only possible rational roots of this polynomial equation are of the form r/s where r is a divisor of a0 and s is a divisor of an; there are only finitely many such candidates which you can all check by hand. If neither of them is a root of p, then x must be irrational. For example, this technique can be used to show that x = (21/2 + 1)1/3 is irrational: we have (x3 − 1)2 = 2 and hence x6 − 2x3 − 1 = 0, and this latter polynomial does not have any rational roots (the only candidates to check are ±1).
Because the algebraic numbers form a field, many irrational numbers can be constructed by combining transcendental and algebraic numbers. For example 3π+2, π + √2 and e√3 are irrational (and even transcendental).
The simplest irrationality proofs: certain logarithms
Perhaps the numbers most easily proved to be irrational are certain logarithms. Here is a proof by reductio ad absurdum that log23 is irrational:
- Assume log23 is rational. For some positive integers m and n, we have log23 = m/n.
- Then 2m/n = 3.
- 2m = 3n.
- But 2 to any power greater than 0 is even (because at least one of its prime factors is 2) and 3 to any power greater than 0 is odd (because none of its prime factors is 2), so the original assumption is false.
Similar cases such as log102 can be treated similarly.
Irrational numbers and decimal expansions
It is often erroneously assumed that mathematicians define "irrational number" in terms of decimal expansions, calling a number irrational if its decimal expansion neither repeats nor terminates. No mathematician takes that to be the definition, since the choice of base 10 would be arbitrary and since the standard definition is simpler and more well-motivated. Nonetheless it is true that a number is of the form n/m where n and m are integers, if and only if its decimal expansion repeats or terminates. When the long division algorithm that everyone learns in grammar school is applied to the division of n by m, only m remainders are possible. If 0 appears as a remainder, the decimal expansion terminates. If 0 never occurs, then the algorithm can run at most m − 1 steps without using any remainder more than once. After that, a remainder must recur, and then the decimal expansion repeats! Conversely, suppose we are faced with a recurring decimal, for example:
Since the length of the repitend is 3, multiply by 103:
and then subtract A from both sides:
(The "135" above can be found quickly via Euclid's algorithm.)
Numbers not known to be irrational
It is not known whether π + e and π − e are irrational or not. In fact, there is no pair of non-zero integers m and n for which it is known whether mπ + ne is irrational or not. It is not known whether 2e, πe, or the Euler-Mascheroni gamma constant γ are irrational.
The set of all irrational numbers
The set of all irrational numbers is uncountable (since the rationals are countable and the reals are uncountable). The set of algebraic irrationals, that is, the non-transcendental irrationals, is countable. Using the absolute value to measure distances, the irrational numbers become a metric space which is not complete. However, this metric space is homeomorphic to the complete metric space of all sequences of positive integers; the homeomorphism is given by the infinite continued fraction expansion. This shows that the Baire category theorem applies to the space of irrational numbers.
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