Question

Which of the following is an example of why irrational numbers are not closed under addition?

Answer 1

Option C: $\sqrt{10} +(-\sqrt{10})=0$ is an example of why irrational numbers are not closed under addition.

Explanation:

For a irrational number to be closed under addition, the sum of two numbers of an irrational number must also be an irrational.

Option A : $\sqrt{4} +\sqrt{4} =2+2=4$ and 4 is not irrational.

From the expression, we can see that $\sqrt{4}$ is a rational number because it results in a rational number. That is, $\sqrt{4}=2$

Thus, Option A is not the correct answer.

Option B : $\frac{1}{2} +\frac{1}{2} =1$ and 1 is not irrational.

From the expression, we can see that $\frac{1}{2}$ is a rational number.

Hence, the addition of two rational numbers results in a rational number.

Thus, Option B is not the correct answer.

Option C : $\sqrt{10} +(-\sqrt{10})=0$ and 0 is not irrational.

From the expression, we can see that $\sqrt{10}$ is an irrational number because it is a non - terminating decimal number.

Hence, the addition of two irrational number is a rational number.

Therefore, the irrational numbers are not closed under addition because the addition of irrational numbers does not result in a irrational number.

Thus, Option C is the correct answer.

Option D : $-3+3=0$ and 0 is not irrational.

From the expression, we can see that 3 is a rational number.

Hence, the addition of two rational numbers results in a rational number.

Thus, Option D is not the correct answer.

Answer 2

Answer C:

Explanation:  For a irrational number to be closed under addition, the sum of two numbers of an irrational number must also be an irrational.