Square root of an irrational number is an irrational number

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Hello Everyone. Before going to the proof, let us go into the basics of Rational and Irrational numbers.

Rational Number : A number that can be expressed as the fraction p/q of two integers p and q, where q not equal to zero.

Irrational Number : A number that cannot be expressed as the fraction p/q of two integers p and q,  where q not equal to zero.

What is required to prove : Square root of an irrational number will be an irrational number.

Proof :

Let m be an irrational number. --------- (1)

Let us think that we don't know whether √m is rational or irrational number.

So, let us assume √m as a rational number initially.

Then, √m = p/q where p and q are some integers and q is not equal to zero. (According to the definition)

So, √m = p/q

Square on both sides

Then, m = (p^2)/(q^2)

It implies that m is a rational number. (According to the definition)

But m is an irrational number [From (1)]

A number cannot be both rational and irrational at a time.

This condition is developed due to our consideration of √m = p/q i.e. √m as a rational number. Therefore our consideration should be false.

It implies √m should be irrational.

Therefore, square root of an irrational number is an irrational number.

Question to think : Where can we see Irrational numbers in our life ?

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