UPSKILL MATH PLUS

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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.

**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.

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

Example:

$\frac{8}{5}+\phantom{\rule{0.147em}{0ex}}\frac{(-2)}{5}=\frac{(-2)}{5}+\frac{8}{5}$.

**ii) Subtraction:**

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

$\frac{a}{b}-\frac{c}{d}\phantom{\rule{0.147em}{0ex}}\ne \phantom{\rule{0.147em}{0ex}}\frac{c}{d}-\frac{a}{b}$.

Example:

$\frac{8}{5}-\frac{(-2)}{5}\ne \phantom{\rule{0.147em}{0ex}}\frac{\phantom{\rule{0.147em}{0ex}}(-2)}{5}-\frac{8}{5}$.

**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.

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

Example:

$\frac{8}{5}\phantom{\rule{0.147em}{0ex}}\times \frac{(-2)}{5}=\frac{(-2)}{5}\times \frac{8}{5}$

**iv) Division:**

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

$\frac{a}{b}\xf7\frac{c}{d}\phantom{\rule{0.147em}{0ex}}\ne \phantom{\rule{0.147em}{0ex}}\frac{c}{d}\xf7\frac{a}{b}$.

Example:

$\frac{4}{3}\xf7\frac{(-2)\phantom{\rule{0.147em}{0ex}}}{7}\phantom{\rule{0.147em}{0ex}}\ne \phantom{\rule{0.147em}{0ex}}\frac{(-2)\phantom{\rule{0.147em}{0ex}}}{7}\xf7\frac{4}{3}$.

Important!

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