Question

if p and q are prime positive integers then prove that a√p -b√q are irrational numbers​

Answer 1

Answer:

First, we'll assume that

p

and

q

is rational , where p and q are distinct primes

p

+

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.

(

p

+

q

)

2

=x

2

p+2

pq

+q=x

2

2

pq

=x

2

−p−q

pq

=

2

(x

2

−p−q)

Now, x, x

2

, p, q, & 2 are all rational, and rational numbers are closed under subtraction and division.

So,

2

(x

2

−p−q)

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 contradiction. Original assumption must be wrong.

So,

p

and

q

is irrational, where p and q are distinct primes.