Question

If p is a prime number then prove that√p is irrational.

Answer 1

Hi there!


Let, √p be a rational number.

Also a and b is rational.

Then,
√p = a/b

On squaring both sides,

(√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.

Cheers!

Answer 2

As prime no consists of examples like 2,3,5,7......

Take root 2 as example.
Then prove root 2 is irrational.

Root 2=a/b
2=a^2/b^2
So a=2b
So refer these similar steps in Ed sharma......