Question

Prove that root 3 be a irrational no solving

Answer 1

Answer:

Let us assume on the contrary that

√3 is a rational number. Then, there exist positive integers a and b such that

√3 = a/b

where, a and b, are co-prime i.e. their HCF is 1

Now

√3 = a/b

3 = a2/b2

3b2= a2

3∣a2. [∵3∣3b2]

3∣a...(i)

a=3c for some integer c

a2 =9c2

3b2 =9c2 [∵a2 =3b2 ]

b2 =3c2

3∣b2 [∵3∣3c2 ]

3∣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,

√3 is an irrational number.

hope it will help you.........