# irrational number

An irrational number is a real number that is not rational, that is, cannot be expressed as a fraction (or ratio ) of the form p / q , where p and q are integers.

## proofs that the square root of 2 is irrational

#### Pythagorean proof, as given by Euclid in his Elements

1. Assume that 2 is rational, that is, there exists integers p and q such that 2 = p / q ; take the irreducible form of this fraction, so that p and q have no factors in common
2. square both sides, to give 2 = p 2 / q 2
3. rearrange, to give 2 q 2 = p 2
4. hence p 2 is even
5. hence p is even (trivial proof left as an excercise for the reader); write p = 2 m
6. substitute for p in (3), to give 2 q 2 = (2 m ) 2 = 4 m 2
7. divide through by 2, to give q 2 = 2 m 2
8. hence q 2 is even
9. hence q is even

(1) assumes that p and q have no factors in common; (5) and (9) show they they both have 2 as a factor. This is a contradiction. Hence the assumption (1) is false, and 2 is not rational.

#### proof based on unique factorisation

[I first saw this rather elegant little proof in Chaitin's Meta Math! , p98. It uses the result of unique factorisation into powers of primes (the fundamental theorem of arthimetic ).]

1. Assume that 2 is rational, that is, there exists integers p and q such that 2 = p / q ; take the irreducible form of this fraction, so that p and q have no factors in common
2. square both sides, and rearrange, to give 2 q 2 = p 2
3. factorise p and q into their unique prime factoriastions, p = 2 a 3 b 5 c ..., q = 2 x 3 y 5 z ...
4. substitute into (2), and simplify, to give 2 2 x +1 3 2 y 5 2 z ... = 2 2 a 3 2 b 5 2 c ...

Since the LHS and the RHS refer to the same number, and since such factorisation is unique, we have 2 x +1 = 2 a (or odd = even ). This is a contradiction. Hence the assumption (1) is false, and 2 is not rational.