Question

The product of a non-zero rational and an irrational number is

• A. always irrational
• B. always rational
• C. rational or irrational
• D. one

Answer 1

Answer: (A)

always irrational

Let $x$ be a rational number and $y$ be an irrational number.
Let $xy=a$.

Let us assume that $a$ is rational.

Since, $a$ is rational it can be expressed as $\frac{p}{q}$, where $p$ and $q$ are integers.

Let $x=\frac{m}{n}$ , where $m$ and $n$ are integers.
Now, $xy=a$

$\frac{my}{n}=\frac{p}{q}$.
On cross multipliying we get,

$\Rightarrow y=\frac{pn}{qm}$.

Now, $pn$ and $qm$ are integers.

Hence, $\frac{pn}{qm}$ is a rational number.

However, $y$ is irrational.

Hence, our assumption is incorrect.
Hence, the product of a non-zero rational and an irrational number is always an Irrational number.

So, option $A$.