Question

Prove that √2+√3 is an irrational number.

Answer 1

let √2 + √3 = r 
for some rational number r. 
By squaring both sides, we get: 
5 + 2√6 = r^2 
so that ,√6 = (r^2 - 5) / 2 is rational. 
Now √6 = m/n for some integers m, n, such that m and n are relatively prime. 
Squaring both sides, we get 
6 = m^2 / n^2 
Now, 2 divides the left-hand side, so it divides m^2, but then 2 divides m since 2 is prime. We may write m = 2k for some integer k. 
6n^2 = (2k)^2 = 4k^2 
3n^2 = 2k^2 
Now 2 divides the right-hand side, so 2 divides 3n^2. However, 2 and 3 are relatively prime, so 2 divides n^2, forcing n to be divisible by 2. 
However, this is a contradiction to the assumption that m and n are relatively prime.