Question

Show that $2-\sqrt{3}$ is an irrational number.

Answer 1

Let us assume, to the contrary, that 2−√3 is rational.
That is, we can find co-prime a and b (b ≠ 0) such that 2−√3 =
Therefore, 2−=√3
Rearranging this equation, we get √3=2−=
Since a and b are integers, we get 2− 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 2−√3 is rational.
So, we conclude that 2
−√3 is irrational.