Question

If p,q are prime positive integers, prove that √p+ √q is an irrational number.

My doubt: how to find whether √p is an irrational number???

Answer 1

Answer to your doubt
If possible,let √p be a rational number.
also a and b is rational.
then,√p = a/b
on squaring both sides,we get,
(√p)²= a²/b²
→p = a²/b²
→b² = a²/p [p divides a² so,p divides a]
Let a= pr for some integer r
→b² = (pr)²/p
→b² = p²r²/p
→b² = pr²
→r² = b²/p [p divides b² so, p divides b]
Thus p is a common factor of a and b.
But this is a contradiction, since a and b have no common factor.
This contradiction arises by assuming √p a rational number.
Hence,√p is irrational

Now proof to the question given

Assume √p + √q is rational.
√p + √q = x, where x is rational
√p + √p = x
2√p = x
√p = x/2

Since both x and 2 are rational, and rational numbers are closed under division, then x/2 is rational. But since p is not a perfect square, then √p is not rational. But this is a contradiction. Original assumption must be wrong.

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