Associativity property — lesson. Mathematics CBSE, Class 8.

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A set of numbers is said to be associative for a specific mathematical operation if the result obtained when changing grouping (parenthesizing) of the operands does not change the result.

Whole Numbers:

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

\begin{array}{l} 2 + (3 + 6) = (2 + 3) + 6 \\ (0 + 6)+ 8 = 0 + (6 + 8) \end{array}

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

\begin{array}{l} 5 - (3 - 4) \neq (5 - 3) - 4 \\ (2 - 0) - 6 \neq 2 - (0 - 6) \end{array}

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

\begin{array}{l} 5 \times (3 \times 6) = (5 \times 3) \times 6 \\ (2 \times 0) \times 9 = 2 \times (0 \times 9) \end{array}

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

\begin{array}{l} 4 \div (2 \div 6) \neq (4 \div 2) \div 6 \\ (10 \div 4) \div 7 \neq 10 \div (4 \div 7) \end{array}

Integers:

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

\begin{array}{l} 2 + (3 + (- 6)) = (2 + 3) + (- 6) \\ (0 + (- 6)) + (- 8) = 0 + ((- 6) + (- 8)) \end{array}

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

\begin{array}{l} 5 - (3 - (- 4)) \neq (5 - 3) - (- 4) \\ ((- 2) - 0) - 6 \neq (- 2) - (0 - 6) \end{array}

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

\begin{array}{l} 5 \times ((- 3) \times 6) = (5 \times (- 3)) \times 6 \\ ((- 2) \times 0) \times (- 9)= (- 2) \times (0 \times (- 9)) \end{array}

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

\begin{array}{l} 4 \div (2 \div (- 6)) \neq(4 \div 2) \div (- 6) \\ ((- 10) \div 4) \div 7 \neq(- 10) \div (4 \div 7) \end{array}

Rational Numbers:

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

(\frac{a}{b} + \frac{c}{d}) + \frac{e}{f} = \frac{a}{b} + (\frac{c}{d} + \frac{e}{f})

(\frac{2}{3} + \frac{3}{2}) + \frac{(- 6)}{7} = \frac{2}{3} + (\frac{3}{2} + \frac{(- 6)}{7})

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

(\frac{a}{b} - \frac{c}{d}) - \frac{e}{f} = \frac{a}{b} - (\frac{c}{d} - \frac{e}{f})

(\frac{2}{3} - \frac{3}{2}) - \frac{(- 6)}{7} \neq \frac{2}{3} - (\frac{3}{2} - \frac{(- 6)}{7})

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

(\frac{a}{b} \times \frac{c}{d}) \times \frac{e}{f} = \frac{a}{b} \times (\frac{c}{d} + \frac{e}{f})

(\frac{2}{3} \times \frac{3}{2}) \times \frac{(- 6)}{7} = \frac{2}{3} \times (\frac{3}{2} \times \frac{(- 6)}{7})

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

(\frac{a}{b} \div \frac{c}{d}) \div \frac{e}{f} = \frac{a}{b} \div (\frac{c}{d} \div \frac{e}{f})

(\frac{2}{3} \div \frac{3}{2}) \div \frac{(- 6)}{7} = \frac{2}{3} \div (\frac{3}{2} \div \frac{(- 6)}{7})

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4. Associativity property

Theory: