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.


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
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.


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