Question

Prove that √5-√3is not a rational number

Answer 1

Assume that √5 - √3 = p/q (it's rational).

Multiple both sides by (√5 + √3).

(√5 - √3) (√5 + √3) = 5-3 = 2 = p/q * (√5 + √3)

(√5 + √3) = 2q/p, therefore √5 +√3 is rational = 2q/p

√5 - √3 = p/q
√5 +√3 = 2q/p
√5 = [(p/q) + (2q/p)]/2, a rational number.

But we know that √5 is IRRATIONAL (easily provable, let me know if you need the proof).

Therefore the assumption is wrong and √5 - √3 is irrational.