Question

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

Answer 1

To prove `(2+sqrt3)/5` is irrational, let us assume that `(2+sqrt3)/5` is rational.

`(2+sqrt3)/5 = "a"/"b"; "b" ≠ 0` and a and b are integers.

`=> 2"b" + sqrt3 "b" = 5"a"`

`=> sqrt 3 "b" = 5"a" - 2 "b"`

`=> sqrt 3 = (5"a" - 2"b")/"b"`

Since a and b are integers so, 5a – 2b will also be an integer.

So, `(5"a" - 2"b")/"b"` will be rational which means `sqrt 3` is also rational.

But we know `sqrt 3` is irrational (given).

Thus, a contradiction has risen because of incorrect assumptions. 

Thus, `(2+sqrt3)/5` is irrational.