Let x and y be rational and irrational numbers,


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.


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

e.g., Let $\quad x=2, y=\sqrt{3}$

Then, $x+y=2+\sqrt{3}$

If possible, let $x+y=2+\sqrt{3}$ be a rational number.

Consider $a=2+\sqrt{3}$

On squaring both sides, we get

$a^{2}=(2+\sqrt{3})^{2}$           [using identity $(a+b)^{2}=a^{2}+b^{2}+2 a b$ ]

$\Rightarrow$            $a^{2}=2^{2}+(\sqrt{3})^{2}+2(2)(\sqrt{3})$

$\Rightarrow$            $a^{2}=4+3+4 \sqrt{3} \Rightarrow \frac{a^{2}-7}{4}=\sqrt{3}$

So, $a$ is rational $\Rightarrow \frac{a^{2}-7}{4}$ is rational $\Rightarrow \sqrt{3}$ is rational.


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