Question

Prove that root 5 is irrational ?

Answer 1

The value of √5 is 2.23.....Hence,it is a irrational number.

Answer 2

Let assume to the countrary , that √5 is rational

Now  

Let $\bf\huge \sqrt{5} \bf\huge\frac{a}{b}$ where a and b are co prime  

Squaring on both sides  

5 = a^2 / b^2

5b^2 = a^2

This shows that a^2 is divisible by 5

It also shows that a is also divisible by 5

Let a = 5m for some integer m

Put a = 5m in 5b^2 = a^2

5b^2 = (5m)^2

5b^2 = 25m^2

b^2 = 5m^2

Hence

b^2 is divisible by 5  

So b is also divisible by 5

From above we can say that 5 is a common factor for both a and b

But this contradicts our assumption that a and b are co prime

Hence √5 is an irrational number . Proved