Question

Give an example of two irrational numbers, whose sum is a rational number.

• A. $4+\sqrt{5},-\sqrt{5}$
• B. $4+\sqrt{5},\sqrt{5}$
• C. $4-\sqrt{5},-\sqrt{5}$
• D. $2+\sqrt{5},2+\sqrt{5}$

Answer 1

Answer: (A)

$4+\sqrt{5},-\sqrt{5}$

Consider option $A$.

The number are $4+\sqrt{5}$ and $-\sqrt{5}$.
Their sum $(4+\sqrt{5})+(-\sqrt{5})$
$=4+\sqrt{5}-\sqrt{5}=4$, which is a rational number.

Consider option $B$.

The number are $4+\sqrt{5}$ and $\sqrt{5}$.
Their sum $(4+\sqrt{5})+\left(\sqrt{5}\right)$
$=4+2\sqrt{5}$, which is an irrational number.

Consider option $C$.

The number are $4-\sqrt{5}$ and $-\sqrt{5}$.
Their sum $(4-\sqrt{5})+(-\sqrt{5})$
$=4-2\sqrt{5}$, which is an irrational number.

Consider option $D$.

The number are $2+\sqrt{5}$ and $2+\sqrt{5}$.
Their sum $=2+\sqrt{5}+2+\sqrt{5}=4+2\sqrt{}5$, which is an irrational number.

Therefore, option $A$ is correct.