Is the sum of two irrational numbers always an irrational number?

No, the sum of two irrational numbers is not always irrational. There are cases where the sum of two irrational numbers can be a rational number. For example:

Consider the irrational numbers \(\sqrt{2}\) and \(-\sqrt{2}\). Both are irrational, but their sum is:

\[
\sqrt{2} + (-\sqrt{2}) = 0
\]

In this case, the sum of two irrational numbers is 0, which is a rational number.

Thus, while the sum of two irrational numbers **can** be irrational, it is not always guaranteed to be so.

Would you like more examples or a deeper explanation on this? Here are five related questions:

1. Can the product of two irrational numbers be rational?
2. Why is \(\sqrt{2}\) considered an irrational number?
3. Are there any conditions under which the sum of two irrational numbers is guaranteed to be irrational?
4. What is the difference between rational and irrational numbers?
5. How can we prove that certain numbers are irrational?

**Tip**: When adding or multiplying irrational numbers, pay attention to possible cancellation effects that may lead to rational outcomes.

Discover whether the sum of two irrational numbers is always irrational. This problem dives into key concepts of number theory and irrational numbers. Learn through examples like √2 + (-√2) = 0, where irrational numbers sum to a rational number. Suitable for Grades 8-10.

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