Question

prove that √3 is irrational​

Answer 1

let, assume that √3 is a rational (where it is written in the form of p / q where p and q is integers or coprime numbers and q is not equal to zero)

=√3=a/b

=b√3=a

by squaring both the sides

=(b)^2 (√3)^2=(a)^2

=3b^2=a^2______equation 1

therefore ,

3 divides a

3 divides a^2

now,

= √3b=√3c

by squaring both the sides

=(b)^2=(√3)^2 (c)^2

=b^2=9c^2__________equation 2

therefore,

3 divides b

therefore,

from equation 1 and 2 , 3 divides p/q where p/q are coprime numbers this contridicts arises due to our wrong assumption that √3 is a rational number

hence, √3 is irrational number

proved