Question

__________ are not commutative for rational numbers.

Answer 1

Hint: Here we are going to use trial and error methods on basic operations like addition, subtraction, multiplication and division. 

Complete step-by-step answer:

“Commutative property: If “a” and “b” be two variables then by commutative property $a\times b=b\times a$” where $\times$ represents any of the binary operations.

Using the above property we are going to find the answer. 

For rational numbers, we know that addition is commutative,

Since, if “a” and “b” are two rational numbers then $a+b=b+a$ .

For example, ${\displaystyle \frac{1}{3}}+{\displaystyle \frac{2}{3}}={\displaystyle \frac{2}{3}}+{\displaystyle \frac{1}{3}}=1$ 

Also for rational numbers, we know that multiplication is also commutative,

Since, if “a” and “b” are two rational numbers then $a\times b=b\times a$ .

For example, ${\displaystyle \frac{1}{3}}\times {\displaystyle \frac{2}{3}}={\displaystyle \frac{2}{3}}\times {\displaystyle \frac{1}{3}}={\displaystyle \frac{2}{9}}$

Let us check whether the subtraction is commutative or not 

If a counter example is given then we can say that the operation is not commutative.

For example, 

${\displaystyle \frac{1}{3}}-{\displaystyle \frac{2}{3}}=-{\displaystyle \frac{1}{3}}$

${\displaystyle \frac{2}{3}}-{\displaystyle \frac{1}{3}}={\displaystyle \frac{1}{3}}$

Here $a-b\ne b-a$  therefore subtraction is not commutative.

Let us check whether the division is commutative or not.

If a counter example is given then we can say that the operation is not commutative.

For example, 

${\displaystyle \frac{1}{3}}÷{\displaystyle \frac{2}{3}}={\displaystyle \frac{1}{3}}\times {\displaystyle \frac{3}{2}}={\displaystyle \frac{1}{2}}$

${\displaystyle \frac{2}{3}}÷{\displaystyle \frac{1}{3}}={\displaystyle \frac{2}{3}}\times {\displaystyle \frac{3}{1}}=2$

Here $a÷b\ne b÷a$  therefore division is not commutative.

Hence, we can come to a conclusion that subtraction and division are not commutative for rational numbers.

Note:

A rational number is a number that can be written in the form p/q where “p” and “q” are integers and q is not equal to zero.

Subtraction and division are not commutative for rational numbers because while performing those operations, if the order of numbers is changed, then the result also changes.