Question

prove that irrational number of√3​

Answer 1

Answer:

Conversely, let us assume that √3 is a rational number. That is, we can get 2 integers a and b such that

√3 = a/b.

If a and b have a common factor other than 1, we can divide it to make a and b coprime.

So b√3 = a

square on both sides,

3b^(2) = a^(2)

Hence a^(2) is divisible by 3.

Hence 3 will be divisible by a.

Thus a = 3c (where c is an integer)

3b^(2) = 9c^(2)

[i.e. b^(2) = 3c^(2)]

Hence, b^(2) is divisible by 3 and b will also be divisible by 3.

Hence a and b have at least one common factor of 3.

But this contradicts the fact that a and b are coprime.

So we conclude that √3 is an irrational number.