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 |
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Prerequisites:
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,
Since q is an integer, then 2q^{2} is even, and so is p^{2}. Since p^{2} is even, then p must be even. If p is even, there exists an integer a such that p = 2a. Substituting,
Therefore, q^{2} 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. |