Is the product of two irrational numbers always an irrational
number?
not neccessarily
When you have a statement like "...always...", it is sometimes (not always) easier to disprove with a counter example than to prove the truth of the statement.
For example,
√2 is irrational.
√2 * √2 = 2 is rational.
To determine whether the product of two irrational numbers is always irrational, we can consider a counterexample.
Let's take the irrational number √2 and its negative counterpart -√2. Both of these numbers are irrational because they cannot be expressed as a fraction (or ratio) of two integers.
When we multiply √2 with -√2, we get (-√2) * √2 = -2. As you can see, the result of this multiplication is the rational number -2, which can be expressed as -2/1.
Since we found an example where the product of two irrational numbers (√2 and -√2) is indeed rational (-2), we can conclude that the statement "the product of two irrational numbers is always an irrational number" is false.
To summarize, not all products of two irrational numbers are irrational. In fact, it is possible for the product to be rational, as demonstrated by the example above.