Question

Prove that 3 + 2`sqrt5` is irrational.

Answer 1

If possible let a = 3 + 2`sqrt5` be a rational number.

We can find two co-prime integers a and b such that `3 + 2sqrt5 = a/b,` where b ≠ 0

`(a - 3b)/b`

= `2sqrt5`

= `(a - 3b)/(2b)`

= `sqrt5`

∵ a and b are integers,

∴ `(a - 3b)/(2b)`

= `"Integer - 3 (interger)"/"2 interger"`

= `(a - 3b)/ (2b)` is rational.

= From (1), `sqrt 5` is rational.

= But this contradicts the fact that `sqrt5` is rational.

∴ Our supposition is wrong.

Hence, `3 + 2sqrt5` is irrational.