# Properties of division of rational numbers

In this post we will learn important properties of rational number with examples.

These properties are important as they are helpful to solve questions related to rational number in fast and easy manner.

## Division Property of rational numbers

Here we will discuss following properties;

(a) Closure property

(b) Commutative property

(c) Associative property

(d) Division with 1

(e) Division by itself

### Closure property of division of rational numbers

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

For example;
Let 2/9 and 16/27 are two rational numbers.

Its division is given as;

\mathtt{\Longrightarrow \frac{2}{9} \ \div \ \frac{16}{27}}\\\ \\ \mathtt{\Longrightarrow \frac{2}{9} \times \frac{27}{16}}\\\ \\ \mathtt{\Longrightarrow \frac{\cancel{2}}{\cancel{9}} \times \frac{\cancel{27} \ \mathbf{3}}{\cancel{16} \ \mathbf{8} \ }}\\\ \\ \mathtt{\Longrightarrow \ \frac{3}{8}}

We get 3/8 as the solution.

Note that 3/8 is also a rational number. Hence, division of two rational number results in solution which is also a rational number.

### Commutative property of division of rational number

The division of rational number is non – commutative.

It means that changing the order of number in division will produce different results.

If a/b and c/d are the two rational numbers, we can say that;

\mathtt{\frac{a}{b} \ \div \ \frac{c}{d} \ \ \cancel{=} \ \ \frac{c}{d} \ \div \ \frac{a}{b}}

Let’s prove the concept with the help of example.

Suppose 1/5 and 2/3 are given rational numbers.

(i) Divide 1/5 with 2/3

\mathtt{\Longrightarrow \ \frac{1}{5} \ \div \ \frac{2}{3}}\\\ \\ \mathtt{\Longrightarrow \frac{1}{5} \times \frac{3}{2}}\\\ \\ \mathtt{\Longrightarrow \frac{3}{10}}

(ii) Now divide 2/3 with 1/5

\mathtt{\Longrightarrow \ \frac{2}{3} \ \div \ \frac{1}{5}}\\\ \\ \mathtt{\Longrightarrow \frac{2}{3} \times \frac{5}{1}}\\\ \\ \mathtt{\Longrightarrow \frac{10}{3}}

You can see that changing the position of numbers produce different result.

Hence, division of rational numbers is non commutative in nature.

### Associative Property of division of rational numbers

Division of rational numbers is non – associative in nature.

It means that forming different groups in division will produce different results.

If a/b, c/d & e/f are the given rational numbers, then;

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

Let us verify this with the help of example.

Suppose 1/5, 2/5 and 7/5 are the given rational numbers.

\mathtt{\Longrightarrow \ \frac{1}{5} \div \left(\frac{2}{5} \div \frac{7}{5}\right)}\\\ \\ \mathtt{\Longrightarrow \frac{1}{5} \div \frac{2}{7}}\\\ \\ \mathtt{\Longrightarrow \frac{1}{5} \times \frac{7}{2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7}{10}}

7/10 is the solution.

Now change the group of number in above division.

\mathtt{\Longrightarrow \ \left(\frac{1}{5} \div \frac{2}{5}\right) \div \frac{7}{5}}\\\ \\ \mathtt{\Longrightarrow \frac{1}{2} \div \frac{7}{5}}\\\ \\ \mathtt{\Longrightarrow \frac{1}{2} \times \frac{5}{7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5}{14}}

5/14 is the solution.

Note that changing the group of number in division will change the end result.

Hence, division of rational number is non associative.

### Division of Rational number with 1

The division of any rational number with 1 will result in same rational number.

If a/b is the given rational number, then;

\mathtt{\frac{a}{b} \div 1=1}

Let us understand the concept with example.

Divide rational number 2/5 by 1.

\mathtt{\Longrightarrow \frac{2}{5} \div \frac{1}{1}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2}{5} \times \frac{1}{1}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2}{5}}

Hence, division of rational number by 1 will produce the same number.

### Division of rational number by itself

Any rational number divided by itself will result in number 1.

If a/b is the given rational number, then;

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