1. Prove that √5 is irrational.
Solutions:
Let us assume, that √5 is rational number. i.e. √5 = x/y (where, x and y are co-primes)
Y√5=X
Squaring both the sides, we get,
(y√5)² = x²
⇒5y² = x²..........(1)
Thus, x is divisible by 5, so x is also divisible by 5.
Let us say, x = 5k, for some value of k and substituting the value of x in equation (1), we get,
5y = (5k)
⇒y=5k
is divisible by 5 it means y is divisible by 5.
Clearly, x and y are not co-primes. Thus, our assumption about √5 is rational is incorrect.
Hence, √5 is an irrational number.
