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An irrational number is any real number which is not rational. More systematically, it is the set of numbers which cannot be represented as the quotient of two integers $p$ and $q$, where $q \ne 0$, thus having a non-repeating, non-terminating decimal representation.

## Examples

Common examples of irrational numbers are roots of numbers. Miscellaneous examples include numbers that are also transcendental such as pi and e.

Name Representation Value
square root of 2 $\sqrt{2}$ 1.41421356
square root of 3 $\sqrt{3}$ 1.73205081
square root of 5 $\sqrt{5}$ 2.23606798
pi $\pi$ 3.14159265
Euler's number e 2.71828183
The Golden ratio $\varphi$ 1.61803399

Sample proof: $\sqrt{2}$ is an irrational number:

Theorem. Square root of 2 is irrational
Prerequisites:
Rules of exponents

Proof. Proof by contradiction: Assume $\sqrt{2}$ is rational. It can then be represented as an irreducible fraction of two integers, p and q. Therefore,
$\frac{p}{q} = \sqrt{2}$
$p = q\sqrt{2}$
$p^2 = 2q^2$

Since q is an integer, then 2q2 is even, and so is p2. Since p2 is even, then p must be even. If p is even, there exists an integer a such that p = 2a. Substituting,

$(2a)^2 = 2q^2$
$4a^2 = 2q^2$
$2a^2 = q^2$

Therefore, q2 must be even, and it follows that q must be even. $\frac{p}{q}$ can then be reduced (by 2) which contradicts the earlier statement (that it is irreducible).

Therefore, $\sqrt{2}$ is irrational.