# 8th Class Mathematics Rational Numbers Properties of Multiplication

## Properties of Multiplication

Category : 8th Class

### Properties of Multiplication

Closure Property of Multiplication

For any two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ we have,

$\frac{a}{b}\times \frac{c}{d}=\frac{ac}{db}$,

which is again a rational number. Hence, multiplication of two rational number is again a rational. Therefore, multiplication is closed w.r.t. multiplication. Look at the following examples:

$\frac{1}{3}\times \frac{7}{8}=\frac{7}{24}\in Q;$

$\frac{-7}{3}\times \frac{2}{9}=-\frac{14}{27}\in Q;$

Thus we can say that multiplication of two rational numbers is closed w.r.t. multiplication.

Commutative Property of Multiplication

For any two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$,

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

Thus multiplication of two rational number is commutative. Look at the following example:

$-\frac{2}{5}\times \frac{1}{4}=\frac{1}{4}\times -\frac{2}{5}=-\frac{1}{10};$

$-\frac{11}{9}\times \left( -\frac{5}{6} \right)=-\frac{5}{6}\times \left( -\frac{11}{9} \right)=\frac{55}{54}$

Associative Property of Multiplication

For any three rational numbers $\frac{a}{b},\frac{c}{d}$ and $\frac{e}{f}\in Q$.

$\Rightarrow$$\frac{a}{b}\times \left( \frac{c}{d}\times \frac{e}{f} \right)=\left( \frac{a}{b}\times \frac{c}{d} \right)\times \frac{e}{f}$

This is called the Associative Property of Multiplication. Look at the following example:

$\left( \frac{4}{5}\times -\frac{2}{7} \right)\times \frac{3}{2}=\frac{4}{5}\times \left( -\frac{2}{7}\times \frac{3}{2} \right)=-\frac{12}{35}$

Multiplicative Identity

For every rational number $\frac{a}{b},$

$\Rightarrow$$\frac{a}{b}\times 1=1\times \frac{a}{b}=\frac{a}{b}$

Thus 1 is the Multiplicative Identity because if we multiply any rational number by 1, the result is the same. Look at the following example:

$-\frac{98}{75}\times 1=1\times -\frac{38}{75}=-\frac{98}{75}$

Zero Property of Multiplication

If we multiply any rational number with 0 the result is again 0. This property is called as the zero property of the rational number. For any rational number$\frac{a}{b}$,

$\Rightarrow$ $\frac{a}{b}\times 0=0\times \frac{a}{b}=0$

Look at the following example:

$-\frac{23}{4}\times 0=0\times \frac{a}{b}=0$

Remember for any two rational numbers$\frac{a}{b}$ and $\frac{c}{d}$, if$\frac{a}{b}\times \frac{c}{d}=0$ then either a = 0 or $c=0\,or\,a=c=0$

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