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Prove that 3-root 3 is irrational number

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Raghunath answered 3 year(s) ago

Prove that root 3 is an irrational number.

Prove that  3 is an irrational number.

Find two irrational numbers lying between 2 and  3

Class-IX Maths

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Asked by Prashanth

Dec 1

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Raghunath , SubjectMatterExpert

Member since Apr 11 2014

Sol:
Let us assume that √3 is a rational number.

That is, we can find integers a and b (≠ 0) such that √3 = (a/b)

Suppose a and b have a common factor other than 1, then we can divide by the common factor, and assume that a and b are coprime.

√3b = a

⇒ 3b2=a2 (Squaring on both sides) → (1)

Therefore, a2 is divisible by 3

Hence ‘a’ is also divisible by 3.

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

Equation (1) becomes,

3b2 =(3c)2

⇒ 3b2 = 9c2

∴ b2 = 3c2

This means that b2 is divisible by 3, and so b is also divisible by 3.

Therefore, a and b have at least 3 as a common factor.

But this contradicts the fact that a and b are coprime.

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

Now
√2 = 1.4142...
√3 = 1.7321...

1.45, 1.5, 1.55, 1.6, 1.65, 1.7 lies between √2 and √3.

Hence the rational numbers between √2 and √3 are:

145/100, 15/10, 155/100, 16/10, 165/100 and 17/10