Prove that 3- √5 is irrational number.

Question

Prove that 3- (minus)√(root)5 is an irrational number.

Answer 1

If we are known with √5 is irrational than it can be proved as:

Let 3 - √5 be a rational number

 3 - √5 = p/q [ where p and q are integer , q ≠ 0 and q and p are co-prime number ]

=> √5 =  3 - p/q

=> √5 = (3q - p)/q

We know that number of form p/q is a rational number.

So, √5 is also a rational number.

But we know that √5 is irrational number. This contradicts our assumption.

Therefore, 3 - √5 is an irrational number.