Question

Find the √7 irrational numbers​

Answer 1

Answer:

Let us assume that 7

is rational. Then, there exist co-prime positive integers a and b such that

7 =b

a

⟹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=7p

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, 7is irrational.