Question

Prove that each of the following number is irrational ( 2 - 3 √5 )​

Answer 1

Answer:

Let us assume that 2-3√5 is a rational.

If it is rational then there exists two coprimes a and b (b≠0).

Such that,

2-3√5=a/b

-3√5=a/b-2

-3√5=a-2b/b

√5=a-2b/-3b

since a and b are integers,

RHS=a-2b/-3b is rational and so LHS=√5 also become rational. But this contradicts the fact that√5 is irrational. This contradiction has arisen because our assumption that 2-3√5 is rational.

So, we conclude that 2-3√5 is an irrational.