Question

Rational numbers are closed under ……….
(a) addition
(b) subtraction
(c) multiplication
(d) all of the above

Answer 1

Hint: Here, we will check whether the rational numbers are closed under the given operations or not. A set is closed under an operation if performance of that operation on members of the set always produces a member of that set.

Complete step-by-step answer:

A rational number is a number that can be expressed as the quotient or fraction $\dfrac{p}{q}$ of two integers, a numerator p and a non-zero denominator q. Since, q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as ‘ the field of rationals’ is usually denoted by Q .

A set that is closed under an operation or collection of operations is said to satisfy a closure property. Often a closure property is introduced as an axiom, which is then usually called the axiom of closure.

For any two rational numbers, say x and y, the results of addition, subtraction and multiplication operations give a rational number. Division is not under closure property because division by zero is not defined.

For example, if we take two rational numbers $\dfrac{1}{2}\text{ and }\dfrac{3}{4}$, then:
$\dfrac{1}{2}\text{ +}\dfrac{3}{4}=\dfrac{1\times 2+3\times 1}{4}=\dfrac{2+3}{4}=\dfrac{5}{4}$, which is a rational number.

Also, $\dfrac{1}{2}\text{ -}\dfrac{3}{4}=\dfrac{1\times 2-3\times 1}{4}=\dfrac{2-3}{4}=\dfrac{-1}{4}$, which is a rational number.

And, $\dfrac{1}{2}\times \dfrac{3}{4}=\dfrac{3}{8}$, which is also a rational number.

Thus, we see that for addition, subtraction as well as multiplication, the result that we get is itself a rational number. This means that rational numbers are closed under addition, subtraction and multiplication.

Hence, option (d) is the correct answer.


Note: Students should remember the meaning of the closure property. One should also note that the denominator of a rational number can never be 0 otherwise it will not be defined.