8th Class Mathematics Rational Numbers Properties of Multiplication

         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 \[c=0\,or\,a=c=0\]