Question

The sum of a rational number and an irrational number is always an irrational number true or false
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Answer 1

true

Step-by-step explanation:

add 2 and 22/7 its irrational

Answer 2

★ Answer

The sum of a rational number and an irrational number is always an irrational number.

This statement is absolutely True.

Let's know the concept first.

➹ What is rational Number?

• The number which can be written in the form of ${\dfrac{p}{q}}$ where p and q are integers and q ≠ 0 then, it is said to be the rational number.

For Example - ${\dfrac{3}{4}}$ , ${\dfrac{7}{4}}$ , 0 , etc.

➹ What is irrational Number?

• The number which cannot be written in the form of ${\dfrac{p}{q}}$ where p and q are integers. They have non - negative root values.

For Example - √2 , √23, √17 , 4√3 etc

Let's prove the above Statement now :

Let us assume " The sum of a rational number and an irrational number is a rational number. "

Considering " x " be the rational number.

" y " be the irrational number.

So, now

x + y = Rational Number {According to our assumption}

As we know that rational numbers can be expressed or written in the form of p/q .

Then, we know x and ( x + y) are basically rational numbers.

Thus,

$:\implies\sf{x = \dfrac{p}{q}}$

$:\implies\sf{x + y = \dfrac{a}{b}}$

Putting up the value of " x " in it, we get :

$:\implies\sf{ \dfrac{p}{q} + y = \dfrac{a}{b}}$

$:\implies\sf{ y = \dfrac{a}{b} - \dfrac{p}{q}}$

That means y can be expressed in the p/q form. But as from our knowledge " y " can't be both rational and irrational.

This contradicts our fact that the sum of rational and irrational is rational.

Hence, Proved!!