Properties of Rational Numbers

Properties of Rational Numbers as the name suggests are the properties of the rational number that help us to distinguish rational numbers from the other types of numbers. rational numbers are the superset of the numbers such as natural numbers, whole numbers, even numbers, etc. So these properties are applicable to all these numbers. Properties of Rational numbers are very important for class 8.

Rational numbers are the numbers that can be represented in the form p/q where p and q are integers and q is never equal to zero. All fractions, terminating decimals, recurring decimals, etc. come under rational numbers. There are various properties of rational numbers such as associative property, commutative property, distributive property, and closure property.

In this article, you are going to learn about the properties of rational numbers with examples and solved problems.

What are Rational Numbers

Rational numbers are of the form p/q, where p and q are integers and q ≠ 0. Because of the underlying structure of numbers, p/q form, most individuals find it difficult to distinguish between fractions and rational numbers. The rational number is represented by Q.

When a rational number is divided, the output is in decimal form, which can be either ending or repeating. 3, 4, 5, and so on are some examples of rational numbers as they can be expressed in fraction form as 3/1, 4/1, and 5/1.

What are the Properties of Rational numbers?

There are various properties of rational numbers that differentiate it from other numbers. The main six properties of rational numbers are:

Properties of Rational Numbers

• Closure Property
• Commutative Property
• Associative Property
• Distributive Property
• Additive Property of Rational Numbers
• Multiplicative Property of Rational Numbers

Let’s learn about each in detail below for all operations addition, subtraction, division and multiplication.

Closure Property of Rational Numbers

The closure property of the rational number state that performing an operation on the rational number always results in a rational number, i.e. the operations are closed on the rational number. The operations allowed in this property are Addition, Subtraction, and Multiplication. The division operation is not allowed in the closure property of the rational number as dividing two rational numbers may or may not result in the rational number.

Now let’s learn about this property in detail below,

Suppose we take two rational numbers, X and Y then,

For Addition: The sum of any two rational numbers always results in the rational number this can be represented as,

X + Y = Rational Number

For example, 2/3 + 4/3 = 6/3 = 2 (Rational Number)

For Subtraction: The difference between any two rational numbers always results in the rational number this can be represented as,

X – Y = Rational Number

For example, 2/3 – 4/3 = -2/3 (Rational Number)

For Multiplication: The multiplication of any two rational numbers always results in the rational number this can be represented as,

X × Y = Rational Number

For example, 2/3 × 4/3 = 8/9 (Rational Number)

For Division: The division of any two rational numbers may or may not always results in the rational number this can be represented as,

X/Y = May or may not Rational Number

For example, 

• 2/3 /4/3 = 1/2 (Rational Number)
• 2/3/0 = undefined (Not a rational Number)

Commutative Property of Rational Numbers

The commutative property of the rational number state that changing the order of the multiplication and the addition of the rational number does not change the result.

Now let’s learn about this property in detail below,

Suppose we take two rational numbers, X and Y then,

For Addition: The sum of any two rational numbers is always commutative i.e. it does not change with the change of the order. This can be represented as, 

X + Y = Y + X

For example, 2/3 + 4/3 = 4/3 + 2/3 = 6/3

For Subtraction: The difference between any two rational numbers is not commutative, i.e. the difference if the order is changed is not equal. This can be represented as, 

X – Y ≠ Y – X

For example, 2/3 – 4/3 = -2/3 

4/3 – 2/3 = 2/3

2/3 – 4/3 ≠ 4/3 – 2/3

For Multiplication: The multiplication of any two rational numbers is always commutative i.e. it does not change with the change of the order. This can be represented as, 

X × Y = Y × X

For example, 2/3 × 4/3 = 4/3 × 2/3 = 8/9

For Division: The division of two rational numbers is not commutative, i.e. if the order of division is changed the result is not equal. This can be represented as, 

X ÷ Y ≠ Y ÷ X

For example, 2/3 ÷ 4/3 = 1/2

4/3 ÷ 2/3 = 2

2/3 ÷ 4/3 ≠ 4/3 ÷ 2/3

Associative Property of Rational Numbers

The associative property of the rational number state that if we take three or more rational number then adding or multiplying them in any order does not change the result.

Now let’s learn about this property in detail below,

Suppose we take three rational numbers, X, Y, and Z then,

For Addition: The sum of three rational numbers is always associative i.e. it does not change with the change of the order. This can be represented as, 

(X + Y) + Z = X + (Y + Z)

For example, (2/3 + 4/3) + 6/3 = 2/3 + (4/3 + 6/3)

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

= 6/3 + 6/3

= 12/3 = 4

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

= 2/3 + 10/3

= 12/3 = 4

Hence, LHS = RHS proved

Thus, the property is verified.

For Subtraction: The difference of three or more rational numbers is not commutative, i.e. the difference of three or more rational numbers if the order is changed is not equal. This can be represented as, 

(X – Y) – Z ≠ X – (Y – Z)

For Multiplication: The product of three rational numbers is always associative i.e. it does not change with the change of the order. This can be represented as, 

(X × Y) × Z = X × (Y × Z)

For example, (2/3 × 4/3) × 6/3 = 2/3 × (4/3 × 6/3)

LHS = (2/3 × 4/3) × 6/3

= 8/3 × 6/3

= 48/9 = 16/3

RHS = 2/3 × (4/3 × 6/3)

= 2/3 × 24/3

= 48/9 = 16/3

Hence, LHS = RHS proved

Thus, the property is verified.

For Division: The division of three or more rational numbers is not commutative, i.e. the division of three or more rational numbers if the order is changed is not equal. This can be represented as, 

(X ÷ Y) ÷ Z ≠ X ÷ (Y ÷ Z)

Distributive Property of Rational Numbers

The distributive property of the rational number distributes multiplication over addition and subtraction. Suppose we have three rational numbers,

X, Y, and Z then the distributive property of the rational number states that,

X × ( Y + Z) = X × Y + Y × Z

This is called the distributive property of rational numbers over addition.

X × ( Y – Z) = X × Y – Y × Z

This is called the distributive property of rational numbers over subtraction.

Example 1: Simplify 3 × (11 + 4)

Solution:

= 3 × (11 + 4)

=  3×11 + 3×4

= 33 + 12

= 45

Example 2: Simplify 3 × (11 – 4)

Solution:

= 3 × (11 – 4)

=  3×11 – 3×4

= 33 – 12

= 21

Additive Property of Rational Numbers

The additive property of the rational number is classified into categories that include,

• Additive Identity Property
• Additive Inverse Property

Let’s learn about them in detail.

Additive Identity Property

The additive identity property states that among rational numbers we have an identity element such that adding it to any other rational number results in the same rational number. The additive identity element of the rational number is 0. Thus, for any rational number A

A + 0 = A

Additive Inverse Property

Additive inverse property states that among rational numbers we have an inverse element of all the elements such that adding these elements results in the identity element(0). The inverse element of any rational number A is (-A). Thus,

A + (-A) = 0

Identity and Inverse Properties of Rational Numbers

The additive property of the rational number is classified into categories that include,

• Multiplicative Identity Property
• Multiplicative Inverse Property

Let’s learn about them in detail.

Multiplicative Identity Property

The multiplicative identity property states that among rational numbers we have an identity element such that multiplying it by any other rational number results in the same rational number. The multiplicative identity element of the rational number is 1. Thus, for any rational number A

A × 1 = A

Multiplicative Inverse Property

Multiplicative inverses property states that among rational numbers we have an inverse element of all the elements such that multiplying these elements results in the identity element(1). The inverse element of any rational number A is (1/A) which is also called the reciprocal of the number. Thus,

A × (1/A) = 1

The image added below shows all the properties of the rational numbers.

Conclusion

We have discussed all the properties of rational numbers in detail including examples of each property. These properties are:

• Closure Property
• Commutative Property
• Associative Property
• Distributive Property
• Identity Property
• Inverse Property

Each property is equally important to explain various abstract concepts in mathematics.

Read More,

• Natural Numbers
• Whole Numbers
• Prime Numbers

Solved Examples on Properties of Rational Numbers

Example 1: Verify the associative property of a rational number if a = 1/2 and b = 3/4 and c = 2/3.

Solution:

Given,

• a = 1/2
• b = 3/4
• c = 2/3

For Associative Property of Addition,

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

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

5/4 + 2/3 = 1/2 + 17/12

23/12 =  23/12

Proved.

For Associative Property of Addition,

(a.b).c = a.(b.c)

(1/2 . 3/4 ). 2/3 = 1/2 . (3/4 . 2/3)

2/8 = 2/8

3/8 . 2/3 = 1/2 . 2/4

1/4 =  1/4

Proved

Example 2: Verify the distributive property of a rational number if  a = 1/2 and b = 3/4 and c = 2/3

Solution: 

Given,

• a = 1/2
• b = 3/4
• c = 2/3

Distributive Property of Multiplication over Addition 

a × (b + c) = a × b + a × c 

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

1/2 × 17/12 =  3/8 + 2/6

17/24 = 17/24

Proved

Distributive Property of Multiplication over Subtraction

a × (b – c) = a × b – a × c

1/2 × (3/4 – 2/3) = 1/2 × 3/4 – 1/2 × 2/3

1/2 × 1/12 = 3/8 – 2/6

1/24 = 1/24

Proved

FAQs on Properties of Rational Numbers

1. What are Rational Numbers? 

A rational number is a number that is in the form of p/q, where p and q are integers, and q is not equal to 0. Some examples of rational numbers include 1/3, 2/4, 1/5, 9/3, and so on.

2. What are the Properties of the Rational Number?

The basic six properties of the rational number are,

• Closure Property
• Commutative Property
• Associative Property
• Distributive Property
• Additive Property of Rational Numbers
• Multiplicative Property of Rational Numbers

3. Is 0 a Rational Number?

Yes, 0 is a rational number because it is an integer that can be written in any form such as 0/1, 0/2, where b is a non-zero integer. It can be written in the form: p/q = 0/1. Hence, we conclude that 0 is a rational number.

4. Is Pi(π) a Rational Number?

No, Pi (π) is not a rational number. It is an irrational number and its value equals 3.142857…

5. What is the Distributive Property of Rational Numbers?

The distributive property states, if a, b and c are three rational numbers, then;

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

6. What are the Two Multiplicative Properties of Rational Numbers?

Multiplicative Identity and Multiplicative Inverse are the two Multiplicative properties of rational numbers.

• Multiplicative identity for rational numbers is expressed as, p/q × 1 = 1 × p/q = p/q.
• Multiplicative Inverse for rational numbers is expressed as p/q × q/p = 1 such that p/q is the multiplicative inverse of q/p.

7. How important are Properties of Rational Numbers for class 8?

Properties of rational numbers are quiet important for class 8. As it is a topic that help students to build a strong base for rational numbers and after that they can easily study real numbers, complex numbers etc.