Question

prove that √3 is irrational​

Answer 1

Answer:

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

√3 = p/q

√3 = p^2/q^2 ( squaring on both side )

3q^2 = p^2 ..............( 1 )

It means that 3 divides p^2 and also 3 divided p because each factor should appear two times for the square to exist.

So we have p = 3 x

Where x is some integer.

p^2 = q^2 ...........( 2 )

from equation ( 1 ) and ( 2 )

3q^2 = 9x^2

q^2 = 3x^2

where q^2 is multiply of 3 and also q is multiple of 3 .

Then p, q have a common factor of 3 . The run contrary to their being Co-primes. Consequently p/q is not a rational number. This demonstrates that √3 is an irrational number.