Give one example each to show that the rational numbers are closed under addition, subtraction and multiplication. Are rational numbers closed under division? Give two examples in support of your answer.

When two rational numbers are added, their sum is always a rational number. For example,

, which is also a rational number.

Therefore, rational numbers are closed under addition.

When two rational numbers are subtracted, the result is always a rational number. For example, , which is also a rational number.

Therefore, rational numbers are closed under subtraction.

When two rational numbers are multiplied, their product is always a rational number. For example, , which is also a rational number.

Therefore, rational number are closed under multiplication.

As for any rational number a, a ÷ 0 is not defined, therefore not all rational numbers are closed under division. We can say that except zero, all rational numbers are closed under division. For example,

(i) , which is a rational number.

(ii) , which is a rational number.