Question

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

Answer 1

Answer:

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

Step-by-step explanation:

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

A way to represent this is:

$a\in\mathbb{R}\setminus\mathbb{Q}, b\in\mathbb{Q},ab\in\mathbb{Q}\implies a\in\mathbb{Q}\implies\text{Contradiction}\therefore ab\not\in\mathbb{Q}.$

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

Step-by-step 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.