An **irrational number** is any real number that is not a rational number, i.e., it cannot be written as a fraction *a* / *b* with *a* and *b* integers and *b* not zero.

Examples of irrational numbers are 2^{1/2} (the square root of 2) and 3^{1/3} (the cubic root of 3).

The first proof of the irrationality of 2^{1/2} is usually ascribed to Pythagoras or one of his followers and seen as the discovery of the irrational numbers. This proof proceeds as follows.

- (1) Assume that 2
^{1/2}is a rational number. - (2) Then 2
^{1/2}can be written as an irreducible fraction*a*/*b*such that (*a*/*b*)^{2}= 2. - (3) It follows that
*a*^{2}/*b*^{2}= 2 and*a*^{2}= 2*b*^{2}. - (4) Therefore
*a*^{2}is even. - (5) It follows that
*a*must be even. - (6) Therefore
*a*^{2}is divisible by 4. - (7) So
*a*^{2}/ 2 is even. - (8) From (3) it follows that
*a*^{2}/ 2 =*b*^{2}. - (9) From (7) and (8) it follows that
*b*^{2}is even. - (10) It follows that
*b*must be even. - (11) By (5) and (10)
*a*and*b*are both even wich contradicts that*a*/*b*is irreducible as stated in (2). - (12) Since we have found a contradiction the assumption (1) that 2
^{1/2}is a rational number must be false.

This proof is an example of Reductio ad absurdum.