Question

Prove that of the numbers `2 -3 sqrt(5)` is irrational:

Answer 1

Let `2 -3 sqrt(5)`  be rational.
Hence 2 and `2 -3 sqrt(5)` are rational.
∴ 2 – ( `2 -3 sqrt(5) )` = 2 – `2  + 3 sqrt(5)` = ` 3 sqrt(5)` = rational    [∵Difference of two rational is rational]
∴ `1/3 × 3sqrt(5) = sqrt(5)` = rational   [∵Product of two rational is rational]
This contradicts the fact that `sqrt(5)` is irrational.
The contradiction arises by assuming `2 -3 sqrt(5)` is rational.
Hence, `2 -3 sqrt(5)`  is irrational.