Question

Examine, whether the following number are rational or irrational:

`2+sqrt3`

Answer 1

Let `x=2+sqrt3` be the rational

squaring on both sides

`rArrx^2=(2+sqer3)^2`

`rArrx^2=4+3+4sqrt3`

`rArrx^2=7+4sqrt3`

`rArrx^2-7=4sqrt3`

`rArr(x^2-7)/4=sqrt3`

Since, x is rational

⇒ x² is rational

⇒ x² - 7 is rational

`rArr(x^2-7)/4` is rational

`rArr sqrt3` is rational

But, `sqrt3` is irrational

So, we arrive at a contradiction.

Hence, `2+sqrt3` is an irrational number.