The product of non zero rational and irrational number is

By Rebecca Published November 11, 2022

The product of non zero rational and irrational number is

A) always irrational

B) always rational

C) rational or irrational

D) one

Answer:

A) always irrational

Explanation:

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

Let us assume that $a$ is rational.

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

Let $x = n m$ , where $m$ and $n$ are integers.
Now, $x y = a$

n m y = q p

.
On cross multiplying we get,

$\Rightarrow y = q m p n$ .

Now,

p n

and

q m

are integers.

Hence,

q m p n

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.

The product of non zero rational and irrational number is