Prove  is irrational

Wednesday, 20 March 2019

10:30 AM

Proof by Contradiction

Suppose that  is rational.

Then  where  and  are integers with no common factor.

Squaring both sides

 

Hence  is even, so  is even. Hence we can write  for some integer

Substituting into the previous equation gives

And so

 

Therefore  is even, and hence  is even

 

 and  have a common factor of

This contradicts our assumption that  and  have no common factor

 

Therefore our original assumption is false