Question

prove that√7 is an irrational number​

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   =7b 2

Therefore, a  2  is divisible by 7 and hence, a is also divisible by7  

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

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

This means, b  2  is 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.