Question

prove that √3 is irrational. ​

Answer 1

Let us assume to the contrary that √3 is a rational number.

It can be expressed in the form of p/q

where p and q are co-primes and q≠ 0.

⇒ √3 = p/q

⇒ 3 = p²/q² (Squaring on both the sides)

⇒ 3q² = p²______________(1)

It means that 3 divides p² and also 3 divides p because each factor should appear two times for the square to exist.

So we have p = 3r

where r is some integer.

⇒ p² = 9r²__________________(2)

from equation (1) and (2)

⇒ 3q² = 9r²

⇒ q² = 3r²

We have two cases to consider now.