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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 and , where , 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 1.41421356
square root of 3 1.73205081
square root of 5 2.23606798
pi 3.14159265
Euler's number e 2.71828183
The Golden ratio 1.61803399

Sample proof: is an irrational number:

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

Proof. Proof by contradiction: Assume is rational. It can then be represented as an irreducible fraction of two integers, p and q. Therefore,

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,

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

Therefore, is irrational.

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