Examples of irrational numbers are 21/2 (the square root of 2) and 31/3 (the cubic root of 3).
The first proof of the irrationality of 21/2 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 21/2 is a rational number.
- (2) Then 21/2 can be written as an irreducible fraction a / b such that (a / b)2 = 2.
- (3) It follows that a2 / b2 = 2 and a2 = 2 b2.
- (4) Therefore a2 is even.
- (5) It follows that a must be even.
- (6) Therefore a2 is divisible by 4.
- (7) So a2 / 2 is even.
- (8) From (3) it follows that a2 / 2 = b2.
- (9) From (7) and (8) it follows that b2 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 21/2 is a rational number must be false.
This proof is an example of Reductio ad absurdum.