Question

Prove that √5 is an irrational number.​

Answer 1

Step-by-step explanation:

let us assume to the contrary that √5 is a rational number.

so we can find integers r and s (nonzero) such that √5=r/s.

suppose r and s have a common factor other than 1

then we divide them by the common factor to get

=>√5=a/b, where a and b are coprime.

so , b√5=a

on squaring both sides we get,

5b^2=a^2 ...…….................(I)

therefore 5 divides a^2

now by theorem 1.3 it follows that 5 divides a

so we can write a=5c , for some integer c

substituting a in equation (I) we get,

5b^2=25

=>b^2=5

this means 5divides b^2 , 5 divides b , again by theorem 1.3

But this contradicts that a and b have no common factor other than 1.

this contradiction has arised due to our wrong assumption.

hence √5 is not a rational number.

proved