Just like properties of whole numbers and integers, properties of rational numbers can be classified under Closer property, Commutative property, Associative property, Distributive property, Role of 1, Role of 0.

## Properties of Rational Numbers | Closure Property

*If the result obtained after performing a specific mathematical operation between any two rational numbers is always a rational number, then rational numbers are said to be closed for that operation.*

**↬ **** Addition**: Rational numbers are closed under addition.

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

**a + b = c , c ∈ Q**

e.g.,

2/7 + 5/6 = (12+35)/42 = **47/42**** **(a rational number)

**↬ **** Subtraction**: Rational numbers are closed under subtraction.

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

**a − b = c ,**

**c ∈ Q**e.g.,

2/7 − 5/6 = (12−35)/42 = **−23/42** (a rational number)

**↬ **** Multiplication**: Rational numbers are closed under multiplication.

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

**a × b = c , c ∈ Q**e.g.,

2/7 × 5/6 = 10/42 = **5/21** (a rational number)

**↬ **** Division**: Rational numbers are not closed under division.

* For any two rational numbers a and b, c may not be a rational number.*

**a ÷ b = c , c ∉ Q for****b = 0**

e.g.,

2/7 ÷ 0/6 = 2/7 × 6/0 = **12/0** (* not defined, not a rational number*)

## Properties of Rational Numbers | Commutative Property

*If the result obtained before and after changing the orders of any two rational numbers for a particular mathematical operation is same, then rational numbers are said to be commutative for that operation.*

↬ ** Addition**: Rational numbers are commutative for addition.

* for any two rational numbers a and b,*

**a + b = b + a**e.g.,

2/7 + 5/6 = (12+35)/42 = **47/42**** **

5/6 + 2/7 = (35+12)/42 = **47/42*** *

↬ ** Subtraction**: Rational numbers are not commutative for subtraction.

* for any two rational numbers a and b, *

**a − b ≠ b − a**e.g.,

2/7 − 5/6 = (12−35)/42 = **−23/42**

5/6 − 2/7 = (35−12)/42 = **23/42**** (results are not same but opposite).**

↬ ** Multiplication**: Rrational numbers are commutative for multiplication.

* for any two rational numbers a and b, *

**a × b = b × a**e.g.,

2/7 × 5/6 = 10/42 = **5/21**

5/6 × 2/7 = 10/42 = **5/21**

↬ ** Division**: Rational numbers are not commutative for division.

* for any two rational numbers a and b, *

**a ÷ b ≠ b ÷ a**

**6/7**

1/3 ÷ 2/7 = 1/3 × 7/2 = **7/6 ***(results are not same but inverse).*

## Properties of Rational Numbers | Associative Property

*If the result obtained before and after changing the grouping of rational numbers for a particular mathematical operation is same, then rational numbers are said to be commutative for that operation.*

↬ ** Addition**: Rational numbers are associative for addition.

* for any two rational numbers a and b,*

**(a + b) + c = a + (b + c)**e.g.,

(2/7 + 5/6) + 4/3 = 47/42 + 4/3 = (47+56)/42 = **103/42**

2/7 + (5/6 + 4/3) = 2/7 + 13/6 = (12+91)/42 = **103/42**

↬ ** Subtraction**: Rational numbers are not associative for subtraction.

* for any two rational numbers a and b, *

**(a − b) − c ≠**

**a − (b − c)**e.g.,

(2/7 − 5/6) − 4/3 = −23/42 − 4/3 = (−23−56)/42 = **−79/42**

2/7 − (5/6 − 4/3) = 2/7 − (−3)/6 = {12−(−21)}/42 = **33/42**

↬ ** Multiplication**: Rational numbers are associative for multiplication.

* for any two rational numbers a and b, *

**(a****×**

**b)****×**

**c = a****×**

**(b****×**

**c)**e.g.,

(2/7 × 5/6) × 3/2 = 5/21 × 3/2 = **5/14**

2/7 × (5/6 × 3/2) = 2/7 × 5/4 = **5/14**

↬ ** Division**: Rational numbers are not associative for division.

* for any two rational numbers a and b, *

**(a ÷ b) ÷ c ≠ a ÷( b÷ c)**e.g.,

(2/7 ÷ 1/3) ÷ 4/5 = (2/7 × 3/1) ÷ 4/5 = 6/7 × 5/4 = **15/14 **,

2/7 ÷ (1/3 ÷ 4/5) = 2/7 ÷ (1/3×5/4) = 2/7 ÷ 5/12 = 2/7 × 12/5 = **24/35**

## Properties of Rational Numbers | Distributive Property

Multiplying a rational number with sum of two rational numbers will give the same result if the number is first multiplied with each addend and products are then added in next step.

↬ *For all rational numbers a, b **and c,*

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

**a (b – c) = ab – ac**## Properties of Rational Numbers | Role of 0

### Additive Identity

**Zero ***is called the identity for the addition of rational numbers because when we add 0 to any rational number we get the same number i.e. the number retain its identity. It is the additive identity for integers and whole numbers as well.*

e.g.,

2/3 + 0 = 2/3,

−2 + 0 = −2,

21 +0 = 21

### Additive Inverse (Negative) :

*When we add a rational number with its negative (opposite) we get 0. Negative of a rational number is called its additive inverse.*

e.g.,

Additive inverse of 2/3 is −2/3

because 2/3 + (−2/3) = 0

Additive inverse of −4/3 is 4/3

because −4/3 + 4/3 = 0

∴ *Additive inverse of any rational number .*

## Properties of Rational Numbers | Role of 1

### Multiplicative Identity

*One*** ***is called the identity for the multiplication of rational numbers because when we multiply 1 to any rational number we get the same number i.e. the number retain its identity. It is the multiplicative identity for integers and whole numbers as well.*

e.g.,

2/3 × 1 = 2/3,

−2 × 1 = −2,

21 × 1 = 21

### Multiplicative Inverse (Reciprocal)

*When we multiply a rational number with its reciprocal (inverse) we get 1. Reciprocal of a rational number is called its multiplicative inverse.*

e.g.,

Multiplicative inverse of 2/3 is 3/2

because, 2/3 × 3/2 = 6/6 =1

Multiplicative inverse of −4/3 is −3/4

because, −2/3 × −3/4 =12/12 =1

*Multiplicative inverse of any rational number .*