Question

Prove that 5 - √3 is irrational, given that √3 is irrational.

Answer 1

Let us assume that 5 - √3 is a rational 

 We can find co prime a & b ( b≠ 0 )such that 

5 - √3  = a/b

Therefore 5 - a/b = √3

So we get 5b -a/b = √3

Since a & b are integers, we get 5b -a/b  is rational, and 

so √3 is rational. But √3 is an irrational number

Let us assume that 5 - √3 is a rational We can find co prime a & b ( b≠ 0 )such that 

∴ 5 - √3 = √3 = a/b

Therefore 5 - a/b = √3

So we get 5b -a/b = √3

Since a & b are integers, we get 5b -a/b  is rational, and so √3 is rational. But √3 is an irrational number

Which contradicts our statement 

∴ 5 - √3 is irrational