Explain why multiplying two irrational numbers could result in either an irrational number or a rational number. Give examples to support your explanation.

Answer 1

Multiplying two irrational numbers can result in either an irrational number or a rational number because the product of two irrational numbers may or may not have a pattern or repetition in its decimal expansion.

1. Result is an irrational number:
When multiplying two irrational numbers, if the decimal expansion of their product does not have a pattern or repetition, the result will be an irrational number. For example, let's consider the square root of 2 (√2) and the square root of 3 (√3), both of which are irrational. Their product (√2 * √3) is equal to √6, which is also irrational as its decimal expansion does not repeat or have a specific pattern.

2. Result is a rational number:
However, there are cases where multiplying two irrational numbers results in a rational number. This occurs when the decimal expansion of their product exhibits a pattern or repetition. Let's take the example of √2 and 2/√2. Both are irrational numbers. When we multiply them (√2 * 2/√2), the product simplifies to 2. Hence, in this case, the result is a rational number as the decimal expansion of 2 is 2.0000... with a repeating pattern of zeros.

In summary, multiplying two irrational numbers can result in either an irrational number or a rational number, depending on whether the product's decimal expansion exhibits a pattern or repetition.