Question

Is the product of a rational and irrational number always irrational? Give an example.

Answer 1

Find the product of rational and irrational numbers.

Both rational and irrational numbers are real numbers, but they have different characteristics. A rational number is a number that can be expressed in the form of the ratio$\left(\frac{P}{Q}\&Q\ne 0\right)$ and an irrational number cannot be expressed in the ratio $\left(\frac{P}{Q}\&Q\ne 0\right)$. However, both numbers are real and can be represented by a number line.

Example $1$: $a=\frac{2}{3},b=\sqrt{3}$

Product $ab=2\sqrt{3}$

$2\sqrt{3}$ is an irrational number.

Example $2$: $a=0,b=\sqrt{3}$

Product $ab=0$

$0$ is a rational number.

Example $3$: $a=4,b=\sqrt{3}$

Product $ab=4\sqrt{3}$

$4\sqrt{3}$ is an irrational number.

Using the above example, we can conclude that if the rational number is $0$, the product of the rational number and the non-rational number is always a rational number, but if the rational number is non-zero, the product of the rational number and irrational numbers are always irrational numbers.

Hence, the given statement is only true when the rational number is no zero.