Commutative property — lesson. Mathematics CBSE, Class 8.

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A set of numbers is said to be commutative for a specific mathematical operation if the result obtained when changing the order of the operands does not change the result.

Whole Numbers:

i) Addition: Changing the order of operands in addition to whole numbers does not change the result. Hence, whole numbers under addition are commutative.

\begin{array}{l} 2 + 3 = 3 + 2 \\ 0 + 6 = 6 + 0 \end{array}

ii) Subtraction: Changing the order of operands in the subtraction of whole numbers changes the result. Hence, whole numbers under subtraction are not commutative.

\begin{array}{l} 5 - 3 \neq 3 - 5 \\ 0 - 6 \neq 6 - 0 \end{array}

iii) Multiplication: Changing the order of operands in the multiplication of whole numbers does not change the result. Hence, whole numbers under multiplication are commutative.

\begin{array}{l} 5 \times 3 = 3 \times 5 \\ 2 \times 0 = 0 \times 2 \end{array}

iv) Division: Changing the order of operands in the division of whole numbers changes the result. Hence, whole numbers under division are not commutative.

\begin{array}{l} 4 \div 2 \neq 2 \div 4 \\ 10 \div 4 \neq 4 \div 10 \end{array}

Integers:

i) Addition: Changing the order of operands in addition to integers, does not change the result. Hence integers under addition are commutative.

\begin{array}{l} 2 + (- 3) = (- 3) + 2 \\ (- 1) + 6 = 6 + (- 1) \end{array}

ii) Subtraction: Changing the order of operands in the subtraction of integers changes the result. Hence, integers under subtraction are not commutative.

\begin{array}{l} 5 -(- 3) \neq (- 3) - 5 \\ (- 1) - 6 \neq 6 - (- 1) \end{array}

iii) Multiplication: Changing the order of operands in the multiplication of integers does not change the result. Hence, integers under multiplication are commutative.

\begin{array}{l} 5 \times (- 3) = (- 3) \times 5 \\ (- 2) \times 0 = 0 \times (- 2) \end{array}

iv) Division: Changing the order of operands in the division of integers changes the result. Hence, integers under division are not commutative.

\begin{array}{l} 4 \div (- 2) \neq (- 2) \div 4 \\ (- 10) \div 4 \neq 4 \div (- 10) \end{array}

Rational Numbers:

i) Addition: Changing the order of operands in addition to rational numbers, does not change the result. Hence, rational numbers under addition are commutative.

\frac{a}{b} + \frac{c}{d} = \frac{c}{d} + \frac{a}{b}

\frac{8}{5} + \frac{(- 2)}{5} = \frac{(- 2)}{5} + \frac{8}{5}

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ii) Subtraction: Changing the order of operands in the subtraction of rational numbers changes the result. Hence, rational numbers under subtraction are not commutative.

\frac{a}{b} - \frac{c}{d} \neq \frac{c}{d} - \frac{a}{b}

\frac{8}{5} - \frac{(- 2)}{5} \neq \frac{(- 2)}{5} - \frac{8}{5}

iii) Multiplication: Changing the order of operands in the multiplication of rational numbers does not change the result. Hence, rational numbers under multiplication are commutative.

\frac{a}{b} \times \frac{c}{d} = \frac{c}{d} \times \frac{a}{b}

\frac{8}{5} \times \frac{(- 2)}{5} = \frac{(- 2)}{5} \times \frac{8}{5}

iv) Division: Changing the order of operands in the division of rational numbers changes the result. Hence, rational numbers under division are not commutative.

\frac{a}{b} \div \frac{c}{d} \neq \frac{c}{d} \div \frac{a}{b}

\frac{4}{3} \div \frac{(- 2)}{7} \neq \frac{(- 2)}{7} \div \frac{4}{3}

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3. Commutative property

Theory: