Maths

Proof that 2 is irrational

Let x be 2 . Then

x 2 + = 2

If x were a rational number it would be expressible in the form

x = m n

Where m and n are rational numbers with no common divisor (other than 1). It follows that

m 2 = 2 n 2

So m 2 is even. This implies that m is even. So we can write

m = 2 k

Hence

4 k 2 = 2 n 2

So

2 k 2 = n 2

Thus n is even. We have therefore shown that m and n are divisible by 2. This is a contradiction. It therefore follows that x cannot be rational, i.e. 2 must be an irrational real number.