Sums and products of irrational numbers  Summary and Q&A
TL;DR
The sums and products of two irrational numbers can be either rational or irrational, depending on the specific values of the irrational numbers involved.
Questions & Answers
Q: Can the sum of two irrational numbers be rational?
Yes, the sum of two irrational numbers can be rational. For example, if one irrational number is pi and the other is 1 minus pi, their sum is 1, which is a rational number.
Q: Can the sum of two irrational numbers always be irrational?
No, the sum of two irrational numbers can also be irrational. For example, if both irrational numbers are pi, their sum is 2pi, which is still an irrational number.
Q: Can the product of two irrational numbers be rational?
Yes, the product of two irrational numbers can be rational. For example, if one irrational number is 1 over pi and the other is pi, their product is 1.
Q: Is it always true that squaring an irrational number results in another irrational number?
No, it is not always true. For example, squaring the square root of 2 results in 2, which is a rational number.
Summary & Key Takeaways

The sum of two irrational numbers can be rational or irrational, depending on the specific values of the irrational numbers.

Examples of irrational number pairs can be chosen in such a way that their sum is rational, or their sum is irrational.

Similarly, the product of two irrational numbers can also be rational or irrational, depending on the specific values of the irrational numbers.

It is not always the case that the product of the same irrational number or the square of an irrational number is always irrational.