Question

Prove that root 3 is irrational.

Pls I’m in a hurry!

Answer 1

Answer:

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

Answer 2

Answer:

since, √3 is a irrational number

Step-by-step explanation:

Let us assume √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 1. Then, we divide by the common factor to get √3=a/b,where a and b are co-prime. So, b√3=a.

On squaring both sides and rearranging, we get 3b²=a². Therefore, 3 divides a².

Now,by Theorem 1.6, it follows that since 3 is dividing a², It is also divides a.

So, we can write a=3c for some integer c.

Substituting for a, we get 3b²=9c², that is, b²=3c².

This means that 3 divides b², and so 3 divides b.

Therefore, both a and b have 3 as a common factor.

But this contradicts the fact that a and b are

co-prime.

This contradiction has arisen because of our assumption that √3 is rational. Thus our assumption is false. So, we conclude that √3 is irrational.