Question

Prove that √3 is an irrational number.

Answer 1

Let √3 be a rational number. 

Then √3 = \(\frac{q}{p}\)

HCF (p,q) =1 

Squaring both sides 

(√3)² = (\(\frac{q}{p}\))² 

3 = \(\frac{p^2}{q^2}\) 

3q² = p² 

3 divides p² » 3 divides p 

3 is a factor of p 

Take p = 3C 

3q² = (3c)² 

3q² = 9C² 

3 divides q² » 3 divides q 

3 is a factor of q 

Therefore 3 is a common factor of p and q 

It is a contradiction to our assumption that \(\frac{q}{p}\) is rational. 

Hence √3 is an irrational number.