# Properties of Rational Numbers – Closure, Commutative, Associative and Distributive

** Properties of Rational Numbers **

**Rational numbers**

Rational numbers are numbers which can be represented in the form of p/q, where p and q are any two integers and q is not equal to zero(q ≠ 0).

**Denominator and Numerator of a Rational Number**

In a rational number p/q, the integer p is the numerator and the integer q is the denominator (q ≠ 0).

**Properties of Rational numbers**

The properties of rational numbers are the associative property, the commutative property, the distributive property and the closure property.

Let us explore these properties on the four arithmetic operations (Addition, Subtraction, Multiplication, and Division).

When any two rational numbers are added, subtracted or multiplied the result of all three cases will also be a rational number.

We will understand this properties on each operation using various examples.

Let know about how the closure property of rational numbers works on all the basic arithmetic operations.

**Closure Property**

The closure property of rational numbers states that when any two rational numbers are added, subtracted or multiplied the result of all three cases will also be a rational number.

Let us understand the closure property on each operation with examples.

**Closure Property for Addition**

Closure property of addition of rational numbers states that the sum of two rational numbers is also a rational number, i.e.,

**if a and b are rational numbers, a + b is also a rational number.**

Example: Let us take two rational numbers 1/2 and 1/3

= 1/2 + 1/3

= (3 + 2)/6

= 5/6

Here result is 5/6, which is also a rational number.

We find that the sum of two rational numbers is again a rational number.

Check it for a few more pairs of rational numbers.

Example: Let us take two rational numbers 1/4 and 1/5

1/4 + 1/5

= (5 + 4)/20

= 9/20

We can say that the rational numbers are closed under addition.

That is for any two rational numbers a and b, a + b is also a rational number.

**Closure Property for Subtraction**

Closure property of subtraction of rational numbers states that the difference of two rational numbers is also a rational number, i.e.,

**if a and b are rational numbers then, a – b is also a rational number.**

Example: 1/2 – 1/3

= (3 – 2)/6

= 1/6

Here result is 1/6, which is also a rational number.

That is for any two rational numbers a and b, a – b is also a rational number.

We can say that the rational numbers are closed under subtraction.

**Closure Property for Multiplication**

The closure property of multiplication of rational numbers states that when any two rational numbers are multiplied the result will also be a rational number.

**For any two rational numbers a and b, a x b is also a rational number.**

Example: 1/2 x 1/3

= (1 x 1)/(2 x 3)

= 1/6

Here result is 1/6, which is also a rational number.

We can say that the rational numbers are closed under multiplication.

That is for any two rational numbers a and b, a x b is also a rational number.

**Closure Property for Division**

Example: 1/2 ÷ 1/3

= 1/2 x 3/1

= 3/2

Here result is 3/2, which is also a rational number.

We find that for any rational number a, a ÷ 0 is not defined. So, we can say rational numbers are not closed under division.

However, if we exclude zero, then the collection of all other rational numbers are closed under division.

**Commutative Property **

The commutative property of rational numbers states that when any two rational numbers are added or multiplied in any order the result does not change.

In case of subtraction and division, if order of numbers is change the result will also change.

We will understand these properties on each operation with examples.

**Commutative Property for Addition**

Commutative property for addition of rational numbers states that any two numbers are added in any order the result does not change.

i.e., if a and b are rational numbers,

**a + b = b + a** is also a rational number.

Example: 1/3 + 1/5 = (5 + 3)/15 = 8/15

= 1/5 + 1/3 = (3 + 5)/15 = 8/15

8/15 = 8/15

We can say that The sum of two rational numbers does not depend on the order in which they are added, i.e., a and b are rational numbers,

**a + b = b + a **

Here result is 8/15, which is also a rational number.

That is for any two rational numbers a and b, a + b is also a rational number.

We can say that the rational numbers are commutative under addition.

**Commutative Property for Subtraction**

** a – b ≠ b – a**

Example: 1/2 – 1/3 = (3 – 2)/6 = 1/6 whereas

1/3 – 1/2 = (2 – 3)/6 = -1/6

1/2 – 1/3 ≠ 1/3 – 1/2

That is for any two rational numbers a and b, a – b ≠ b – a subtraction is not commutative for rational number.

This means subtraction is not commutative for rational numbers.

**Commutative Property for Multiplication**

Commutative property for multiplication of rational numbers states that when any two numbers are multiplied in any order it does not change the result.

** a x b = b x a **

Example: 1/2 x 1/3 = 1/6

1/3 x 1/2 = 1/6

1/2 x 1/3 = 1/3 x 1/2 = 1/6

Here result is 1/6, which is also a rational number.

i.e., if a and b are rational numbers, **a x b = b x a** is also a rational number.

We can say that the rational numbers are commutative under multiplication.

**Commutative Property for Division**

**a ÷ b ≠ b ÷ a**

Example: 1/2 ÷ 1/3 = 3/2 and

1/3 ÷ 1/2 = 2/3 whereas

1/2 ÷ 1/3 ≠ 1/3 ÷ 1/2

We can see that the expression on both sides are not equal.

This means **a ÷ b ≠ b ÷ a**, for any two rational numbers a and b. So division is not commutative for rational numbers.

That is for any two rational numbers a and b, a ÷ b ≠ b ÷ a. division is not commutative for rational number.

**Associative Property**

Associative property of rational numbers states that three rational numbers are added or multiplied the result remains same.

we will understand this property on each operation using examples.

** Associative Property for Addition**

Associative property for addition of rational numbers states that sum of three or more rational numbers does not depend on the way the rational numbers are grouped, i.e.,

if a, b, c are three rational numbers then the sum of rational numbers is also a rational number.

For any three rational numbers, associative property for addition is expressed as

**(a + b) + c = a + (b + c) **

Example: (1/2 + 1/3) + 1/4

= (3 + 2)/6 + 1/4

= 5/6 + 1/4

= (20 + 6)/24

= 26 /24

= 13/12

1/2 + (1/3 + 1/4)

= 1/2 + (4 + 3)/12

= 1/2 + 7/12

= (6 + 7)/12

= 13 /12

13/12 = 13/12

We say that addition is associative for rational numbers.

**Associative Property for Subtraction**

For any three rational numbers, associative property for subtraction is expressed as

**(a – b) – c ≠ a – (b – c) **

Example: (1/2 – 1/3) – 1/4 ≠ 1/2 – (1/3 – 1/4)

We say that subtraction is not associative for rational numbers.

**Associative Property for Multiplication**

For any three rational numbers, associative property for multiplication is expressed as, i.e., if a, b, c are three rational numbers then

**(a x b) x c = a x (b x c) **

Example: (1/2 x 1/3) x 1/4 = 1/24

1/2 x (1/3 x 1/4) = 1/24

(1/2 x 1/3) x 1/4 = 1/2 x (1/3 x 1/4) = 1/24

When any three rational numbers are added or multiplied the result remains the same.

We can say that multiplication is associative for rational numbers.

**Associative Property for Division**

For any three rational numbers a b and c, associative property for division is expressed as

**(a ÷ b) ÷ c ≠ a ÷ (b ÷ c)**

Example: (1/2 ÷ 1/3) ÷ 1/4 ≠ 1/2 ÷ (1/3 ÷ 1/4)

We say that division is not associative for rational numbers.

**Distributive Property**

**Distributive Property of Multiplication over Addition**

Distributive property for multiplication over addition of rational numbers states that any expression of three rational numbers a, b, c in form of a (b + c), then it can be solved as,

**a x (b + c) = a x b + a x c **

Let us learn with examples.

Example: 1/3 (1/4 + 1/5)

= (1/3 x 1/4) + (1/3 x 1/5)

= (1/12) + (1/15)

= (5 + 4)/60

= 9/60

**Distributive Property of Multiplication over Subtraction**

If a, b and c are three rational numbers then, this applies to subtraction as,

**a (b – c) = a x b – a x c**

Let us learn with examples.

Example: 1/3 (1/4 – 1/5)

= (1/3 x 1/4) – (1/3 x 1/5)

= (1/12) – (1/15)

= (5 – 4)/60

= 1/60

a (b – c) = a x b – a x c

**There are two basic additive properties of rational numbers. **

**(1). Additive Identity Property**

** (2). Additive Inverse Property **

**(1). Additive Identity Property:** Additive identity of rational numbers states that the sum of any rational number (a/b) and zero is the rational number itself. Suppose a/b is any rational number, then

**a/b + 0 = 0 + a/b = a/b**

Here 0 is called the additive identity of rational numbers.

We understand with example,

4/7 + 0 = 0 + 4/7

**(1). Additive Inverse Property:** The additive inverse property of rational numbers states that if a/b is a rational number, then there exists a rational number (-a/b) such that

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

Example: 2/3 + (-2/3) = (-2/3) + 2/3 = 0

Example: If a and b are rational numbers such that a + b = 0, then a and b are additive inverse of each other.

For a rational number x/y additive inverse is (-x/y).

**There are two basic multiplicative properties of rational numbers.**

**(1). Multiplicative Identity Property**

**(2). Multiplicative Inverse Property**

Let us understand these properties with examples.

**(1). Multiplicative Identity Property: **Multiplicative Identity Property of rational numbers states the product of for any rational number and 1 is the rational number itself.

Here, 1 is the multiplicative identity for rational numbers.

If (a/b) is any rational number, then

**a/b x 1 = 1 x a/b**

Example: 3/5 x 1 = 1 x 3/5

**(2). Multiplicative Inverse Property:** Multiplicative Inverse Property of rational numbers states that for every rational number a/b, b ≠ 0 there exists a rational number b/a such that

**a/b x b/a =1**

In this case a rational number b/a is multiplicative inverse of a rational number a/b.

Examples 3/5 x 5/3 = 1

The multiplicative inverse of 1/5 is 5.

Every rational number multiplied with 0 gives 0.

If a/b is any rational number then

a/b x 0 = 0 x a/b = 0

For example 2/3 x 0 = 0 x 2/3 = 0

Multiplicative inverse of 1/8 is 8.

1/8 x 8 = 1

For a rational number x/y multiplicative inverse is (y/x).

**There are few other properties of rational numbers that we need to know and they are explained below.**

- If a/b is a rational number and n is a non-zero integer then a/b = (an/bn).

In other words we can say that, if we multiply both numerator and denominator with same integer then the value of rational number is remains same.

Example: 3/2 = (3 x 2)/(2 x 2) = 6/4,

3/2 = (3 x 3)/(2 x 3) = 9/6,

3/2 = (3 x 4)/(2 x 4) = 12/8,

3/2 = 6/4 = 9/6 = 12/8,

- If a/b is a rational number and n is a common divisor then

a/b = (a ÷ n)/(b ÷ n)

- When we divide the numerator and denominator of a rational number with a common divisor the rational number remains unchanged.

24/36 = (24 ÷ 2)/(36 ÷ 2) = 12/18

**Solved Questions**

**What are the properties of rational numbers? **

Basic properties of rational numbers are

- Closure Property of rational numbers

- Commutative Property of rational numbers

- Associative Property of rational numbers

- Distributive Property of rational numbers

- Identity Property of rational numbers

- Inverse Property of rational numbers

**What are four types of rational numbers?**

Four types of rational numbers are

Natural Numbers

Whole Numbers

Integers

Fractions

**What is the commutative property of rational numbers?**

the commutative property of rational numbers states that adding or multiplying any two rational numbers in any order products the same result.

However, if the order of the numbers is changed in subtraction and division, the result will vary.