In mathematics, an irrational number is any real number that is not a rational number, i.e., 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.
There are two types of irrational numbers, the algebraic ones such as 2^{1/2} (the square root of two) and 3^{1/3} (the cube root of 3), and the transcendental numbers such as π and e.

Perhaps the numbers most easily proved to be irrational are logarithms like log_{2}3. The argument by reductio ad absurdum is as follows:
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.
This proof can be generalized to show that any root of any natural number is either a natural number or irrational.
Another reductio ad absurdum showing that √2 is irrational is less wellknown and has sufficient charm that it is worth including here. It proceeds by observing that if √2=m/n then √2=(2nm)/(mn), 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 straightedgeandcompass construction, a smaller isosceles right triangle whose leg and hypotenuse have respective lengths mn and 2nm. That construction proves the irrationality of √2 by the kind of method that was employed by ancient Greek geometers.
All transcendental numbers are irrational, and the article on transcendental numbers lists several examples. e^{r} is irrational if r ≠ 0 is rational; π^{n} is irrational for positive integers n.
Another way to construct irrational numbers is as zeros of polynomials: start with a polynomial equation
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).
It is not known whether π + e or π  e are irrational or not. In fact, there is no pair of nonzero 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 EulerMascheroni gamma constant γ are irrational.
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.
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