Question

Prove √3 irrational number​

Answer 1

Answer:

Let us assume on the contrary that root 3 is a rational number.  

Then, there exist positive integers a and b such that

root 3= a/b

3=a^2/b^2

3b^2=a^2

3 divides a^2 [∵3 divides 3b^2]  

3 divides a  

a=3c for some integer c

a^2=9c^2

⇒3 divides b^2 [∵3 divides 3c^2 ]  

⇒3 divides b...(ii)  

From (i) and (ii), we observe that a and b have at least 3 as a common factor. But, this contradicts the fact that a and b are co-prime. This means that our assumption is not correct.

Hence, root 3 is an irrational number.

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