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

(e) Additive Property

(f) Additive Inverse 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**

Add \mathtt{\frac{1}{5} \ +\ \frac{2}{7}}

**Solution**

Adding the given rational numbers.

\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}}

Adding the numbers we get;

\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.

### Additive Property

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

### Additive Inverse Property

**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**

Find additive inverse of -6/7

The additive inverse is;

⟹ – (-6/7)

⟹ 6/7

If we add this inverse with original number, we will get 0.

-6/7 + 6/7 = 0