Question

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

Determine whether the sum of $x$ and $y$ is an irrational number:

As we know, the sum of a rational and irrational number is always an irrational number.

So, if $x$ and $y$ be rational and irrational numbers respectively.

Then $x+y$ necessarily an irrational number.

For example: assume that $x=1$and $y=\sqrt{7}$.

Then, $x+y=1+\sqrt{7}$ which is an irrational number.

Hence, $x+y$ is necessarily an irrational number.