Question

use division method to show that √3 is an irrational number

Answer 1

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Answer 2

assume that √3 is rational 

if it is rational then there must exist  two integers r and s (S≠0) such that √3=r/s

if r and s have a common factor other than one  then we divide by the common factor to get √3=a and b are co- prime.  so,b√3=a
sqaurring on both side we get 

(b√3)²=a²
3b²=a²
b²=a²/3   
3 divides a² so it also divides a 

so, we can write a=3c
substituting a , we get 3b²=(3c)²
3b²=9c²
b²=9c²/3
b²=3c²
c²=b²/3
3 divides b² so it also divides b 

therefore both a and b have 3 as common factor 
but this contradicts the fact that a and b  are co- primes.
this contradicts has arisen because of our assumption that√3 is rational 
so, our assumption is wrong 
so, we conclude √3 is a irrational