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 fraction a / b with a and b integers and b not zero. The irrational numbers are precisely those numbers whose decimal expansion never ends and never enters a periodic pattern. "Almost all" real numbers are irrational, in a sense which is defined more precisely below.

Some irrational numbers are algebraic numbers such as 2^1/2 (the square root of two) and 3^1/3 (the cube root of 3); others are transcendental numbers such as π and e.

History of the theory of irrational numbers

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 De Moivre, and especially of Euler. For the nineteenth century it remained to complete the theory of complex numbers, to separate irrationals into algebraic and transcendent, to prove the existence of transcendent 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 Weierstrass (by his pupil Kossak), Heine (Crelle, 74), G. Cantor (Annalen, 5), and 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 indorsement 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 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.

Transcendent numbers were first distinguished from algebraic irrationals by Kronecker. Lambert proved (1761) that $\pi$ cannot be rational, and that $e^{n}$ ($n$ being a rational number) is irrational, a proof, however, which left much to be desired. Legendre (1794) completed Lambert's proof, and showed that $\pi$ is not the square root of a rational number. Liouville (1840) showed that neither $e$ nor $e^{2}$ can be a root of an integral quadratic equation. But the existence of transcendent numbers was first established by Liouville (1844, 1851), the proof being subsequently displaced by G. Cantor's (1873). Hermite (1873) first proved $e$ transcendent, and Lindemann (1882), starting from Hermite's conclusions, showed the same for $\pi$. Lindemann's proof was much simplified by Weierstrass (1885), still further by Hilbert (1893), and has finally been made elementary by Hurwitz and Gordan.

Irrationality of certain logarithms

Perhaps the numbers most easily proved to be irrational are logarithms like log₂3. The argument by reductio ad absurdum is as follows:

• Suppose log₂3 is rational. Then for some positive integers m and n, we have log₂3 = m/n.
• Consequently 2^m/n = 3.
• So 2^m = 3ⁿ.
• But 2^m is even (because at least one of its prime factors is two) and 3ⁿ is odd (because none of its prime factors is two (they're all three)) so that is impossible.

Irrationality of the square root of 2

The discovery of irrational number is usually attributed attributed to Pythagoras or one of his followers, who produced a (most likely geometrical) proof of the irrationality of the square root of 2.

One proof of this irrationality is the following reductio ad absurdum. The proposition is proved by assuming the opposite and showing that that is false, which in mathematics means that the proposition must be true.

1. Assume that √2 is a rational number. Meaning that there exists an integer a and b so that a / b = √2.
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.
3. It follows that a² / b² = 2 and a² = 2 b².
4. Therefore a² is even because it is equal to 2 b² which is obviously even.
5. It follows that a must be even. (Odd numbers have odd squares and even numbers have even squares.)
6. Because a is even, there exists a k that fullfills: a = 2k.
7. We insert the last equation of (3) in (6): 2b² = (2k)² is equivalent to 2b² = 4k² is equivalent to b² = 2k².
8. Because 2k² is even it follows that b² is also even which means that b is even because only even numbers have even squares.
9. 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.

Other irrational numbers

All transcendental numbers are irrational, and the article on transcendental numbers lists several examples. e^r is irrational if r ≠ 0 is rational; πⁿ is irrational for positive integers n.

Another way to construct irrational numbers is as zeros of polynomials: start with a polynomial equation

p(x) = a_n xⁿ + a_n-1 xⁿ⁻¹ + ... + a₁ x + a₀ = 0

where the coefficients a_i are integers. Suppose you know that there exists some real number x with p(x) = 0 (for instance 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 a_n; 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 = (2^1/2 + 1)^1/3 is irrational: we have (x³ − 1)² = 2 and hence x⁶ − 2x³ − 1 = 0, and this latter polynomial doesn't 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).

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 repeating decimal, for example:

$A=0.7\,162\,162\,162\,\dots$

Since the length of the repitend is 3, multiply by 10³:

$1000A=7\,16.2\,162\,162\,\dots$

and then subtract A from both sides:

$999A=715.5\,.$

Then

$A={\frac {999}{715.5}}={\frac {9990}{7155}}={\frac {135\times 53}{135\times 74}}={\frac {53}{74}}.$

(The "135" above can be found quickly via the Euclid's algorithm.)

Numbers not known to be irrational

It is not known whether π + e or 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 2^e, π^e, π^√2 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). 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.