Question

Choose the correct statement:

A: The sum of any two rational numbers is not always a rational number.

B: The difference of any two rational numbers is always a rational number.

C: The difference of any two rational numbers is not always a rational number.

D: Subtraction of two rational numbers is commutative.

$\left[1\right]$

Answer 1

Answer: Option B

We know, rational numbers are closed under addition. That is why, for any two rational numbers $a$ and $b$, $a+b$ is also a rational number.
For example if $a=\frac{3}{8},b=\frac{-5}{7}$, then
$a+b=\frac{3}{8}+\frac{-5}{7}=\frac{-19}{56}$.
$\frac{-19}{56}$ is a rational number.

It is true that the difference between two rational numbers must be a rational number.
Let us consider two rational numbers, $\frac{4}{5}$ and $\frac{1}{5}$.
Difference $=\frac{4}{5}-\frac{1}{5}=\frac{3}{5}$.
$\frac{3}{5}$ is a rational number, therefore, the difference between two rational numbers must be a rational number.

Subtraction of two rational numbers is not commutative.
Consider the rational numbers $\frac{5}{8}$ and $\frac{2}{8}$ then:
$=\frac{5}{8}–\frac{2}{8}=\frac{3}{8}$
Now, $\frac{2}{8}–\frac{5}{8}=\frac{-3}{8}$
Since, $\frac{5}{8}-\frac{2}{8}\ne \frac{2}{8}-\frac{3}{8}$, we can say that subtraction of two rational numbers is not commutative.