Question

Prove that root 3 is irrational



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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 = ba

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

Now,

3 = ba

⇒3= b 2a 2

⇒3b 2

=a 2

⇒3 divides a 2

[∵3 divides 3b 2 ]

⇒3 divides a...(i)

⇒a=3c for some integer c

⇒a 2

=9c 2

⇒3b 2

=9c 2

[∵a 2 =3b 2 ]

⇒b 2 =3c 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, 3 is an irrational number.

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Answer 2

Answer:

hope that it will be helpful

Step-by-step explanation:

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