Question

Show that $5-\sqrt{3}$ is an irrational number.

Answer 1

We have to prove is irrational.
Let us assume that is rational.. Hence can be written in the form , where a and b (b not equals to 0) are co prime (no common factor other than 1).
Hence,


Here is a rational number but is an irrational number. Since rational can never to irrational number so this is a contradiction.
But this contradiction arrises due to our assumption that is a rational number. And hence our assumption was wrong.
Hence is an irrational number.