Question

Prove that  √7 is an irrational number.

Answer 1

Let x = √7 and assume it is rational. Then it can be written x = p/q where p and q are integers. Further assume that the fraction p/q is in simplest form. That is, GCF(p,q) = 1. Now: 

x^2 = (√7)^2 
x^2 = 7 
(a/b)^2 = 7 
a^2 = 7b^2 

So a must be a multiple of 7. Let a = 7k for some integer k. Then: 

(7k)^2 = 7b^2 
49k^2 = 7b^2 
7k^2 = b^2 

Now, b must also be a multiple of 7. But this contradicts the fact that a/b is in simplest form. So our assumptions that √7 is rational must be false.