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

Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. So it is basically a fraction with non-zero denominator.
Irrational numbers are real numbers which cannot be represented as simple fractions.
Given that x and y be rational and irrational number respectively.
So, $(x + y)$  is necessarily an irrational number.
For example, let $x=2,y=\sqrt{3}$
Then, $x + y = 2+\sqrt{3}$
If possible, let us assume $x + y = 2+\sqrt{3}$ be a rational number.

Consider $a = 2 + \sqrt{3}$
On squaring both sides, we get
$a^{2}=(2+\sqrt{3})^{2}$
(using identity $(a + b)^{2} = a^{2} + b^{2} + 2ab$)

$\Rightarrow a^{2}=2^{2}+(\sqrt{3})^{2}+2(2)(\sqrt{3})$
$\Rightarrow a^{2}=4+3+4\sqrt{3}$
$\Rightarrow \frac{a^{2}-7}{4}=\sqrt{3}$
But we have assumed a is rational
$\Rightarrow \frac{a^{2}-7}{4}$ is rational
$\Rightarrow \sqrt{3}$ is rational which is not true.
Hence our assumption was incorrect, so $2+ \sqrt{3}$ is irrational.
Hence proved