Question

Prove that √3 is irrational.​

Answer 1

Answer:

Therefore there exists no rational number r such that r2=3. Hence the root of 3 is an irrational number.

Answer 2

let us assume to the contrary that root 3 is a rational number.

now then √3 = a/b ( where a and b are co primes and b is not equal to zero)

√3 = a/b

squaring both the sides we get,

3 = a^2/ b^2

b^2 = a^2 / 3 ( mark this as 1)

if p^2 is divisible by a , p is also divisible by a.

now let a = 3m

√3 = a/b

squaring both the sides,

3 = 9m^2/b^2

b^2 = 9m^2/3

b^2= 3m^2

b^2/3 = m^2.( mark this as 2)

if p^2 is divisible by a, p is divisible by a..

from 1 and 2 we can see that 3 is a common factor of both and a and b , this contradicts our assumption that a and b are co primes. hence √3 is a irrational no.