Question

The product of a non-zero rational number with an irrational number is always
${\text{A}}{\text{.}}$ Irrational number
${\text{B}}{\text{.}}$ Rational number
${\text{C}}{\text{.}}$ Whole number
${\text{D}}{\text{.}}$ Natural number

Answer 1

Hint- Here, we will proceed by defining the rational and irrational numbers from which we will get to know which number is repeating or terminating after the decimal place and which number is non-repeating non-terminating number.

Complete step-by-step solution -

According to the definition of rational number, we can say that the rational numbers are the numbers which can be expressed as a fraction where that fraction can be some positive number, negative number and zero. These numbers either terminate or repeat after the decimal point when written in decimal form.
Any rational number can be expressed in the form $\dfrac{p}{q}$ where p is the numerator of the fractional number and q is the denominator of the fractional number. It is important to note that here q should never be equal to zero.
According to the definition of irrational numbers, we can say that the irrational numbers are the numbers which are not rational numbers (i.e., those numbers which can never be expressed in the form of $\dfrac{p}{q}$). These numbers have endless non-repeating digits after the decimal point where written in decimal form.
Now, the product of a non-zero rational number (i.e., repeating or terminating number) with an irrational number (i.e., non-repeating non-terminating number) will always result in an irrational number (i.e., non-repeating non- terminating number).
Hence, option A is correct.

Note- We can easily see this concept i.e., when a repeating or terminating number like $\dfrac{3}{2} = 1.5$ is multiplied with a non-repeating non-terminating number like $\sqrt 2 = 1.41421....$, we will always get a non-repeating non-terminating number i.e., for this example the number after multiplication will be $\left( {\dfrac{3}{2}} \right)\left( {\sqrt 2 } \right) = \left( {1.5} \right)\left( {1.41421....} \right) = 2.12132.....$.