Question

The difference of a rational number and an irrational number is an irrational number.

• A. True
• B. False

Answer 1

The correct option is A True

Difference of a rational and irrational number :

Assume that $a$ is rational, $b$ is irrational number and $a-b$ is rational.

Since $a$ and $a-b$ are rational, we can write them as fractions.

so assume that $a=\frac{d}{c}$ and $a-b=\frac{n}{m}$.

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

$\frac{d}{c}-b=\frac{n}{m}$

Now, subtract $\frac{d}{c}$​ from both sides of the above equation.

$-b=\frac{n}{m}-\frac{d}{c}$​,

Since the rational numbers are closed under subtraction, $b=\frac{d}{c}-\frac{n}{m}$​ will be a rational number.

However, the assumption said that $b$ is irrational, it cannot be both rational and irrational simultaneously.

So, this is a contradiction to our assumption.

Hence, the given statement is True.