Question

prove that√3 is an irrational number

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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= 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 divides a²

3 divides a. [.3 divides 3b²]... (1)

a = 3c for some integer c

a² = 9c²

3b² = 9c²

b² = 3c²

3 divides b²

3 divides b

From (1) and (ii), we observe that a and b have at least 3 as a common factor. But, this contradicts the fact that a and bare co-prime. This means that our assumption is not correct.

Hence, ✓3 is an irrational number.

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