Irrational number

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 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 2^1/2 is a rational number must be false.

This proof is an example of Reductio ad absurdum.