Properties of multiplication of rational numbers

In this chapter we will discuss important properties of multiplication of rational numbers with solved examples.

These properties are important as they will help you to solve different algebraic expression faster.

Multiplication property of rational numbers

Here we have described following properties;

(a) Closure properties

(b) Commutative property

(c) Associative property

(d) Distributive property

(e) Multiplication with 0

(f) Multiplicative inverse

Closure property of Multiplication of Rational numbers

The property states that ” multiplication of two rational number always results in another rational number “

For example;
Let 1/5 and 2/7 are two rational numbers.

It’s multiplication is given as;

\mathtt{\Longrightarrow \ \frac{1}{5} \times \frac{2}{7}}\\\ \\ \mathtt{\Longrightarrow \frac{2}{35}}

Here the final result 2/35 is also a rational number.

Commutative property of multiplication of rational number

The property states that “ in multiplication, changing the order of numbers will not affect the final result “.

Let a/b and c/d be the given rational numbers.

Then according to commutative property.

\mathtt{\Longrightarrow \frac{a}{b} \times \frac{c}{d} \ =\ \frac{c}{d} \times \frac{a}{b}}

For example;
Let 3/5 and 11/7 are the two rational numbers.

Multiplying the numbers we get;

\mathtt{\Longrightarrow \frac{3}{5} \times \frac{11}{7} \ =\ \frac{33}{35}}

Now interchange the position of numbers and multiply.

\mathtt{\Longrightarrow \frac{11}{7} \times \frac{3}{5} \ =\ \frac{33}{35}}

Note that even after interchanging the position of numbers we get the same result.

Hence, commutative property applies in multiplication of rational numbers.

Associative Property of Multiplication of Rational number

If three of more rational numbers are present in multiplication then changing the association (or group) will not change the end result.

If a/b, c/d & e/f are the three rational numbers.

Then according to associative property.

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

For example;
Let 1/2 , 4/5 and 13/9 are the rational numbers.

Multiplying the numbers we get;

\mathtt{\Longrightarrow \left(\frac{1}{2} \times \frac{4}{5}\right) \times \frac{13}{9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4}{10} \times \frac{13}{9}}\\\ \\ \mathtt{\Longrightarrow \frac{52}{90}}

Now multiplying the number by forming different group.

\mathtt{\Longrightarrow \frac{1}{2} \times \left(\frac{4}{5} \times \frac{13}{9}\right)}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{2} \times \frac{52}{45}}\\\ \\ \mathtt{\Longrightarrow \frac{52}{90}}

Note that changing the group of multiplication results in same numbers.

Hence, associative property works in multiplication of rational numbers.

Distributive property of Multiplication of Rational number

According to the property the multiplication with sum of two rational number is equal to the sum of multiplication of rational numbers.

The distributive property can be represented as;

a/b ( c/d + e/f ) = a/b . c/d + a/b .e/f

Let us understand the property with yeh help of example.

Let 1/2 , 3/5 and 7/5 are the three rational numbers.

According to distributive property of multiplication.

\mathtt{\frac{1}{2} .\left(\frac{3}{5} +\frac{7}{5}\right) \ =\frac{1}{2} .\frac{3}{5} +\frac{1}{2} .\frac{7}{5}}\\\ \\ \mathtt{\frac{1}{2} .\frac{10}{5} =\frac{3}{10} +\frac{7}{10}}\\\ \\ \mathtt{\frac{10}{10} =\frac{10}{10}}\\\ \\ \mathtt{1=1}

Since, LHS = RHS.

The distributive property is verified.

Multiplication property of rational number with 0

Any number multiplied with 0 results in 0.

If a/b be the given rational number. The multiplication with number 0 will result in 0.

\mathtt{\frac{a}{b} \times 0=0}

Multiplicative inverse of rational number

Multiplicative inverse is a number which when multiplied with original numbers gives result 1.

The multiplicative inverse of rational number can be found by taking its reciprocal.

If a/b is the given rational number.

Then its multiplicative inverse will be b/a.

If we multiply the original number with its inverse, we will get 1 as a solution.

\mathtt{\Longrightarrow \ \frac{a}{b} \times \frac{b}{a}}\\\ \\ \mathtt{\Longrightarrow \ 1}

For example;
Let the given rational number is -4/5

The multiplicative inverse is -5/4.

Multiplying the original number & its multiplicative inverse, we get;

\mathtt{\Longrightarrow \ \frac{-4}{5} \times \frac{-5}{4}}\\\ \\ \mathtt{\Longrightarrow \ 1}