Question

Prove that √2 +√3 is irrational(Explain this step by step)​

Answer 1

Explanation:

let us assume that root 2+ root 3 is rational.

then there exists co prime positive integers p and q such that

root2+root3=p/q

p/q-root3=root2

squaring on both sides, we get

(p/q-root3)^2=(root 2)^2

p^2/q^2 - 2p/q root 3 +3=2

p^2/q^2 +3-2=2 p/q root 3

p^2/q^2 +1=2 root3 p/q

(p^2+q^2/q^2)×q/2p=root 3

(p^2+q^2/2pq)=root 3

root 3 is a rational number

( pq are integers

p^2+q^2/2pq) is rational)

This contradicts the fact that root 3 is irrational.

so assumption was incorrect. here root 2+root3 is irrational