Question

Prove that the (square root of 3)/3 is irrational.

Answer 1

(√3)/3 is irrational

proof:-

suppose

(√3)/3 is rational then

(√3)/3 is rational

then it's of the form a/b where a and b are in lowest form that is

gcd(a,b)=1

And a,b are integers and b≠0

that is (√3)/3 =a/b

implies 3/9=a²/b²

impies 1/3=a²/b²

=> b²=3a²,

so b² is a multiple of 3 and hence b must be multiple of 3

let b=3k where k is some integer

this implies

9k²=3a²

=> 3k²=a²

this implies a² is a multiple of 3

and hence a must b multiple of 3

So finally we got 'b' as a multiple of 3 and 'a' as a multiple of

which is a big contradiction as we have a and b in lowest form that is gcd(a,b)=1

So our supposition that √3/3 is rational is wrong and

hence this is an irrational number

this completes the proof