Question

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

Answer 1

Yes, if $x$ and $y$ are rational and irrational numbers, respectively, then $x+y$ is an irrational number.

For example,

Let $x=5$ and $y=\surd 2.$

Then, $x+y=5+\surd 2=5+1.41421\dots =\mathrm{6.41421...}$

Here, $\mathrm{6.41421......}$ is a non-terminating and non-recurring decimal and therefore is an irrational number.

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