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.
Common examples of irrational numbers are roots of numbers. Miscellaneous examples include numbers that are also transcendental such as pi and e.
|square root of 2||1.41421356|
|square root of 3||1.73205081|
|square root of 5||2.23606798|
|The Golden ratio||1.61803399|
Sample proof: is an irrational number:
|Theorem. Square root of 2 is irrational|
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.