# Properties of addition of rational number

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

The following properties are discussed in this chapter;

(a) Closure Property
(b) Commutative Property
(c) Associative Property
(d) Distributive Property

We will discuss all these property one by one.

## Addition Property of rational numbers

Important properties of addition of rational number has been discussed below with examples.

### Closure Property

The property states that ” addition of two rational numbers result in another rational number“.

For example;
Let 2/3 and 6/3 are the rational numbers.

The addition of these numbers is expressed as;

\mathtt{\Longrightarrow \ \frac{2}{3} \ +\ \frac{6}{3}}\\\ \\ \mathtt{\Longrightarrow \frac{2+6}{3} \ }\\\ \\ \mathtt{\Longrightarrow \frac{8}{3}}

The final result is 8/3 which is also a rational number.

Example 02

Solution

\mathtt{\Longrightarrow \ \frac{1}{5} \ +\ \frac{2}{7}}\\\ \\ \mathtt{\Longrightarrow \frac{7+10}{35} \ }\\\ \\ \mathtt{\Longrightarrow \frac{17}{35}}

The final result is 17 / 35 which is also a rational number.

Conclusion
Hence, the sum of two rational numbers is always a rational number.

### Commutative Property

The Commutative property of rational number states that changing the position of rational number in addition will not change the final result.

So if (a/b) & (c/d) are two rational numbers then;

(a/b) + (c/d) = (c/d) + (a/b)

For example;
Consider the rational numbers \mathtt{\frac{13}{11} \ \&\ \ \frac{5}{11}}

\mathtt{\frac{13}{11} \ +\ \frac{5}{11} \ \Longrightarrow \ \frac{18}{11}}

Now interchanging the position of numbers.

\mathtt{\frac{5}{11} \ +\ \frac{13}{11} \ \Longrightarrow \ \frac{18}{11}}

Hence, interchanging the position of rational number in addition will not change the final result.

### Associative property

It states that during addition, changing the group of rational number will not affect the end result.

If a/b, c/d & d/e are the three rational numbers. Then according to the additive property.

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

For Example;
Let 2/3, 5/3 and 7/3 are the given rational numbers.

Then according to associative property.

(2/3 + 5/3) + 7/3 = 2/3 + (5/3 + 7/3)

7/3 + 7/3 = 2/3 + 12/3

14/3 = 14/3

Hence Proved.

### Distributive Property

The multiplication of rational number with sum of two rational number is equal to sum of product of rational numbers.

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

Then according to distributive property.

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

Let’s prove the distributive property with the help of example.

Given are three rational numbers 1/5, 4/5 and 3/5.

According to distributive property;

\mathtt{\frac{1}{5} \ \left(\frac{4}{5} +\frac{3}{5}\right) =\ \frac{1}{5} .\frac{4}{5} \ +\ \frac{1}{5} .\frac{3}{5} \ }\\\ \\ \mathtt{\frac{1}{5} \ .\frac{7}{5} \ \ =\frac{4}{25} +\frac{3}{25}}\\\ \\ \mathtt{\frac{7}{25} \ =\ \frac{7}{25}}

Since, LHS = RHS.

We have proved the distributive property of addition of rational number.

The property states that ” when we add any rational number with 0 we get the same rational number

Hence by adding with number 0, the identity of rational number will be preserved.

a/b + 0 = a/b

The number which addition with rational number produce 0 result is called additive inverse.

If a/b is the rational number then -a/b is the additive inverse.

a/b + (-a/b) = 0

Hence, for any rational number if we insert negative sign in the front it becomes its additive inverse.

Example 01
Find the additive inverse of 2/3

-2/3 is the additive inverse as it adds up with 2/3 to produce 0.

⟹ 2/3 + (-2/3)

⟹ 0

Example 02