Associativity property — lesson. Mathematics State Board, 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.

Rational Numbers:

i) Addition:

Changing the group of operands in addition of 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})

Example:

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

ii) Subtraction:

Changing the group 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})

Example:

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

iii) Multiplication:

Changing the group 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})

Example:

(\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 group 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})

Example:

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

Important!

Therefore for rational number addition and multiplication operations only satisfy the associative property, not the subtraction and division operations.

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4. Associativity property

Theory: