Question

Prove that √p + √q is an irrational Fast.......,☺☺

Answer 1

Assume that

• $\sf{\sqrt p+\sqrt q$ is rational, where p and q are distinct primes.

• $\sf{\sqrt p+\sqrt q=x$ , where x is rational.

Rational numbers are closed under multiplication, so if we square both sides, we still get rational numbers on both sides.  

$\implies\sf{(\sqrt p+\sqrt q)^2=(x)^2$

$\implies\sf{p+2\sqrt{pq}+q=x^2$

$\implies\sf{2\sqrt{pq}=x^2-p-q$

$\implies\sf{\sqrt{pq}=\frac{(x^2-p-q)}{2}$

Now x, x², p, q and 2 are all rational, and rational numbers are closed under subtraction and division.

So (x² - p - q) / 2 is rational.  

But since p and q are both primes, then pq is not a perfect square and therefore √(pq) is not rational.

But this is a contradiction.

Original assumption must be wrong.  

So √p + √q is irrational, where p and q are distinct primes.