Is the sum of two irrational numbers rational or irrational

Is the sum of two irrational numbers rational or irrational? It depends. Since Squareroot 2 is irrational (that is, not rational), 1 - Squareroot 2 is irrational (otherwise it would be easy to get an expression for Squareroot 2 as a fraction of integers). Adding these: Squareroot 2 + (1 - Squareroot 2) = 1. Thus, the sum of two irrational numbers may be rational. On the other hand: Claim: (n Squareroot 2 + Squareroot 3) Q.

Solution

Assume it is rational so, there is some rational number p so that

p=sqrt{2}+sqrt{3}

Squaring gives

p^2=2+3+2sqrt{6}

p^2-5=2sqrt{6}

sqrt{6} is irrational hence, 2sqrt{6} is irrational

, p is rational hence, p^2 is rational , hence, p^2-5 is rational

So a contradiction

Hence, sqrt{2}+sqrt{3} is irrational