Sum of a rational number and an irrational number will be 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 : Sum of a rational number and an irrational number will be an irrational number.

Proof :

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

Let "n" be a rational number. ------------(2)

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

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

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

So, m + n = p/q

Subtract "n" on both sides.

Then, m = (p/q) - n = difference of two rational numbers

Difference of two rational numbers will be a rational number.

Therefore, it implies "m" should have been a rational number.

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 + n = p/q i.e. m + n as a rational number. Therefore, our consideration should be false.

It implies m + n should be irrational.

Therefore, sum of a rational number and an irrational number will be an irrational number.

Question to think : Use the above result and show that the solution of the form rational number, irrational number does not exist for the below set of equations.

 x^(1/2) + y = 7 and  y^(1/2) + x = 11

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