prove that√p+√q is an irrational,wherep,q are primes
Answers
Step-by-step explanation:
What would you like to ask?
MATHS
Show that
p
+
q
is an irrational number, where p,q are primers.
Share
Study later
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.