Question

The sum of the difference between a rational number and an irrational number is always an irrational number.

• A. True
• B. False

Answer 1

The correct option is A True

Rational number:

The numbers that can be written in the form $\frac{p}{q}$, where $p,q$are integers and coprime and $q$≠$0$ are called rational numbers.

Irrational numbers:

The numbers that cannot be written in the form $\frac{p}{q}$ where $p,q$are integers and coprime and $q$≠$0$ and have a non-terminating and a non-recurring decimal expansion are called irrational numbers.

Explanation:

Let, $a$ be a rational number and $b$ be an irrational number.

Let us take the addition of them $a+b$ is a rational number.

Then, $a$ and $a+b$ can be written in the form

$a=\frac{c}{d},a+b=\frac{m}{n}$

Substituting $a=\frac{c}{d}$ in $a+b=\frac{m}{n}$ we get,

$$\begin{gathered} \frac{c}{d}+b=\frac{m}{n} \\ \Rightarrow b=\frac{m}{n}-\frac{c}{d} \\ \Rightarrow b=\frac{m}{n}+(-\frac{c}{d}) \end{gathered}$$

Since the rational numbers are closed under addition, $b=\frac{m}{n}+(-\frac{c}{d})$ is a rational number.

But we have taken $b$ as an irrational number.

Which is a contradiction.

Hence, $a+b$ is an irrational number.

So, the sum or the difference between a rational number and an irrational number is always an irrational number.

Hence, the given statement is true.