If p, q are prime positive integers, prove that (√p + √q) is an irrational number.
Answers
ANSWER
To prove :
p
+
q
is irrational
Proof :
We will proof by the method of contradiction,
Let's assume
p
+
q
is rational, where p & q are distinct primes
i.e.
p
+
q
=k, where k is rational
Rational numbers are closed under multiplication, So if we square both sides, we should still get rational number on both sides
(
p
+
q
)
2
=k
2
⇒p+q+2
pq
=k
2
⇒2
pq
=k
2
−p−q
⇒
pq
=
2
1
(k
2
−p−q)
Now, k
2
,p,q & 2 are all rational, and rational number being closed under subtraction and division, (k
2
−p−q)/2 will be rational.
But as p & q are both primes, then pq is not a perfect square and there by
pq
is not rational. Hence , we reach a contradiction. Therefore, our original assumption must be wrong.
So,
p
+
q
is irrational, where p & q are distinct primes.