Famous Theorems of Mathematics/Proof style

This is an example on how to design proofs. Another one is needed for definitions and axioms.

Irrationality of the square root of 2

This result uses the following:

The square root of 2 is irrational, \sqrt{2} \notin \mathbb{Q}

Proof

This is a proof by contradiction, so we assume that \sqrt{2} \in \mathbb{Q} and hence \sqrt{2} = \frac{a}{b} for some a, b that are coprime.

This implies that 2 = \frac{a^2}{b^2}. Rewriting this gives 2b^2 = a^2 \!\,.

Since the left-hand side of the equation is divisible by 2, then so must the right-hand side, i.e., 2 | a^2 . Since 2 is prime, we must have that 2 | a .

So we may substitute a with 2a', and we have that 2b^2 = 4a^2 \!\,.

Dividing both sides with 2 yields b^2 = 2a^2 \!\,, and using similar arguments as above, we conclude that 2 | b .

Here we have a contradiction; we assumed that a and b were coprime, but we have that 2 | a and 2 | b .

Hence, the assumption were false, and \sqrt{2} cannot be written as a rational number. Hence, it is irrational.

Notes

  • As a generalization one can show that the square root of every prime number is irrational.
  • Another way to prove the same result is to show that x^2-2 is an irreducible polynomial in the field of rationals using Eisenstein's criterion.
Last modified on 22 November 2009, at 06:03