 UPSKILL MATH PLUS

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### Theory:

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:

Changing the group of operands in addition of rational numbers does not change the result. Hence, rational numbers under addition are associative.

$\left(\frac{a}{b}+\frac{c}{d}\right)+\frac{e}{f}=\frac{a}{b}+\left(\frac{c}{d}+\frac{e}{f}\right)$
Example:
ii) Subtraction:
Changing the group of operands in the subtraction of rational numbers changes the result. Hence, rational numbers under subtraction are not associative.

$\left(\frac{a}{b}-\frac{c}{d}\right)-\frac{e}{f}=\frac{a}{b}-\left(\frac{c}{d}-\frac{e}{f}\right)$
Example:
$\left(\frac{2}{3}-\phantom{\rule{0.147em}{0ex}}\frac{3}{2}\phantom{\rule{0.147em}{0ex}}\right)-\phantom{\rule{0.147em}{0ex}}\frac{\left(-6\right)}{7}\ne \phantom{\rule{0.147em}{0ex}}\frac{2}{3}-\left(\phantom{\rule{0.147em}{0ex}}\frac{3}{2}\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}\frac{\left(-6\right)}{7}\right)\phantom{\rule{0.147em}{0ex}}$
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.

$\left(\frac{a}{b}×\frac{c}{d}\right)×\frac{e}{f}=\frac{a}{b}×\left(\frac{c}{d}+\frac{e}{f}\right)$
Example:
$\left(\frac{2}{3}×\phantom{\rule{0.147em}{0ex}}\frac{3}{2}\phantom{\rule{0.147em}{0ex}}\right)×\phantom{\rule{0.147em}{0ex}}\frac{\left(-6\right)}{7}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\frac{2}{3}×\left(\phantom{\rule{0.147em}{0ex}}\frac{3}{2}\phantom{\rule{0.147em}{0ex}}×\phantom{\rule{0.147em}{0ex}}\frac{\left(-6\right)}{7}\right)\phantom{\rule{0.147em}{0ex}}$
iv) Division:
Changing the group of operands in the division of rational numbers changes the result. Hence, rational numbers under division are not associative.

$\left(\frac{a}{b}÷\frac{c}{d}\right)÷\frac{e}{f}=\frac{a}{b}÷\left(\frac{c}{d}÷\frac{e}{f}\right)$
Example:
$\left(\frac{2}{3}÷\phantom{\rule{0.147em}{0ex}}\frac{3}{2}\phantom{\rule{0.147em}{0ex}}\right)÷\phantom{\rule{0.147em}{0ex}}\frac{\left(-6\right)}{7}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\frac{2}{3}÷\left(\phantom{\rule{0.147em}{0ex}}\frac{3}{2}\phantom{\rule{0.147em}{0ex}}÷\phantom{\rule{0.147em}{0ex}}\frac{\left(-6\right)}{7}\right)\phantom{\rule{0.147em}{0ex}}$
Important!
Therefore for rational number addition and multiplication operations only satisfy the associative property, not the subtraction and division operations.