Question

Why is the product of a rational number and an irrational number Irrational?

Answer 1

Because the product is always non-termination,non-repeating decimal.

Explanation:

If we have
`a` is irrational;
`b` is rational such that
`b \\eq 0` , then
`a \cdot b` is irrational.

A way to represent this is:

`a\in\mathbb{R}\setminus\mathbb{Q}, b\in\mathbb{Q},ab\in\mathbb{Q}\implies a\in\mathbb{Q}\implies\text{Contradiction}\therefore ab\\ot\in\mathbb{Q}.`

Note that we have a contradiction, because
`a` is not a rational number, as I stated in the beginning. Therefore, ab is irrational.

Answer 2

Answer: A

Explanation:

Multiplying an irrational number by a rational number will always be an irrational number because irrational numbers do not repeat and terminate.

So for example multiplying pi by a rational like two you will have an irrational number because pi is an irrational number.

`\pi * 2` = 6.28318530718 as you could see that is an irrational number because there is no repetition or termination.