Question

Show that 5 – √3 is irrational.

Answer 1

Let us assume, to the contrary, that 5 – √3 is rational.

That is, we can find coprime a and b (b≠0) such that 5-√3 =a/ b .

Therefore, 5-a/b=√3.

Rearranging this equation, we get √3=5-a/b=(5b-a)/b

Since a and b are integers, we get 5 –a/b is rational, and so √3 is rational.

But this contradicts the fact that √3 is irrational.

This contradiction has arisen because of our incorrect assumption that 5 – √3 is rational.

So, we conclude that 5-√3 is irrational.