Question

How are irrational numbers the result of adding rational numbers and irrational numbers or multiplying non zero rational numbers and irrational numbers

Answer 1

Rational numbers are numbers that can be expressed as a fraction (ratio). Irrational numbers can not be expressed like that (like sqrt(2)).

To prove your statement, assume the opposite until you have a contradiction.

If the result of adding them would be rational, then your irrational number can be expressed as the difference of two rational numbers, which itself is also a rational number. That cannot be, because it should be an irrational number. This contradiction forces that rational + irrational = irrational.

You can reason the same way for multiplication. Suppose rational * irrational = rational, you find that your irrational can be expressed as the fration of two rationals, which is a contradiction.