Question

Prove that `3 - 2sqrt(5)` is an irrational number, given that `sqrt(5)` is an irrational number.

Answer 1

To Prove: `3 - 2sqrt(5)` is an irrational number

Given `sqrt(5)` is an irrational number

Let `3 - 2sqrt(5)` is a rational number

∴ `3 - 2sqrt(5) = p/q`  ...(Where q ≠ 0)

`3 - 2sqrt(5)q` = p

3q – p = `2sqrt(5)q`

`(3q - p)/(2q) = sqrt(5)`

p and q of are integers

∴ `(3p - q)/(2q)` is a rational number but `sqrt(5)` is an irrational number

Hence rational number ≠ irrational number

So our assumption is wrong by contradiction fact

∴ `3 - 2sqrt(5)` is an irrational number.

Hence Proved.