Question

prove that √7 is rational​

Answer 1

Answer:

Let us assume that √7 is rational. Then, there exist co-prime positive integers a and b such that

√7 = a/b

⟹a=b √7

Squaring on both sides, we get

a 2 =7b2

Therefore, a

2

is divisible by 7 and hence, a is also divisible by 7

so, we can write a=7p, for some integer p.

Substituting for a, we get 49p2 =7b2 ⟹b 2 =7p 2 .

This means, b 2is also divisible by 7 and so, b is also divisible by 7.

Therefore, a and b have at least one common factor, i.e., 7.

But, this contradicts the fact that a and b are co-prime.

Thus, our supposition is wrong.

Hence, √7 is irrational.