Question

Question 1
Let x and y be rational and irrational numbers respectively. Is x + y necessarily an irrational number? Give an example in support of your answer.

Answer 1

Yes. ( x + y) is necessarily an irrational number.

Example:
Let us assume x = 2 , y = $\sqrt{3}$
Then, $x+y=2+\sqrt{3}$
If possible, let's consider $x+y=2+\sqrt{3}$ be a rational number.
Consider. $a=2+\sqrt{3}$
On squaring both sides, we get
$a^2=(2+\sqrt{3}{)}^2$ [Using identity $(a+b{)}^2=a^2+b^2+2ab]$
$\Rightarrow a^2=2^2+(\sqrt{3}{)}^2+2(2\left)\right(\sqrt{3})\Rightarrow a^2=4+3+4\sqrt{3}\Rightarrow \frac{a^2-7}{4}=\sqrt{3}$

According to our assumption, if a is rational, then $\frac{a^2-7}{4}$ is rational. And, if $\frac{a^2-7}{4}$ is rational, then $\sqrt{3}$ is rational.

But, this contradicts the fact that $\sqrt{3}$ is an irrational number. Thus, our assumption x + y is a rational number, is wrong.

Hence, x + y is an irrational number.

This example clearly explains that the addition of a rational and an irrational number leads to an irrational number.