Properties of Rational Numbers
Rational numbers are the collection of integers, whole numbers, and natural numbers. Rational numbers are numbers that can be represented in fraction form. They all can be represented as rational numbers of the form p/q or as terminating decimal numbers, or as nonterminating but repeating decimal numbers. Properties of rational numbers are the general properties such as associative, commutative, distributive, and closure properties. Let us read about all the properties of rational numbers.
What Are The Properties of Rational Numbers?
When numbers expressed in the form of p/q, then they are considered rational numbers, here both p and q are integers and q ≠ 0. There are six properties of rational numbers, which are listed below:
 Closure Property
 Commutative Property
 Associative Property
 Distributive Property
 Multiplicative Property
 Additive Property
Let us explore these properties on the four arithmetic operations (Addition, subtraction, multiplication, and division) in mathematics.
Closure Property of Rational Numbers
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 read about how the closure property of rational numbers works on all the basic arithmetic operations. We will understand this property on each operation using various illustrations.
Let us take two rational numbers 1/3 and 1/4, and perform basic arithmetic operations on them.
For Addition: 1/3 + 1/4 = (4 + 3)/12 = 7/12. Here, the result is 7/12, which is a rational number. We say that rational numbers are closed under addition. That is, for any two rational numbers a and b, (a + b) is also a rational number.
For Subtraction: 1/3  1/4 = (4  3)/12 = 1/12. Here, the result is 1/12, which is a rational number. We say that rational numbers are closed under subtraction. That is, for any two rational numbers a and b, (a  b) is also a rational number.
For Multiplication: 1/3 × 1/4 = 1/12. Here, the result is 1/12, which is a rational number. We say that rational numbers are closed under multiplication. That is, for any two rational numbers a and b, (a × b) is also a rational number.
For Division: 1/3 ÷ 1/4 = 4/3. Here, the result is 4/3, which is a rational number. But we find that for any rational number a, a ÷ 0 is not defined. So 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 of Rational Numbers
The commutative property of rational numbers states that when any two rational numbers are added or multiplied in any order it does not change the result. But in the case of subtraction and division if the order of the numbers is changed then the result will also change. We will understand this property on each operation using various illustrations.
Let us again take two rational numbers 1/3 and 1/4, and perform basic arithmetic operations on them.
For Addition: 1/3 + 1/4 = 1/4 + 1/3 = 7/12. We say that addition is commutative for rational numbers. That is, for any two rational numbers a and b, a + b = b + a.
For Subtraction: 1/3  1/4 ≠ 1/4  1/3 = 1/12 ≠ 1/12. You will find that subtraction is not commutative for rational numbers. That is, for any two rational numbers a and b, a  b ≠ b  a.
For Multiplication: 1/3 × 1/4 = 1/4 × 1/3 = 1/12. You will find that multiplication is commutative for rational numbers. In general, a × b = b × a for any two rational numbers a and b.
For Division: 1/3 ÷ 1/4 ≠ 1/4 ÷ 1/3 = 4/3 ≠ 3/4. You will find that expressions on both sides are not equal. In general, a ÷ b ≠ b ÷ a for any two rational numbers a and b. So division is not commutative for rational numbers.
Associative Property of Rational Numbers
The associative property of rational numbers states that when any three rational numbers are added or multiplied the result remains the same irrespective of the way numbers are grouped. But in the case of subtraction and division if the order of the numbers is changed then the result will also change. We will understand this property on each operation using various illustrations.
For Addition: For any three rational numbers associative property for addition is given as A, B, and C, (A + B) + C = A + (B + C). For example, (1/3 + 1/4) + 1/2 = 1/4 + (1/3 + 1/2) = 13/12. We say that addition is associative for rational numbers.
For Subtraction: For any three rational numbers associative property for subtraction is given as A, B, and C, (A  B)  C ≠ A  (B  C). For example, (1/3  1/4)  1/2 ≠ 1/3  (1/4  1/2) = 1/24 ≠ 1/12. You will find that subtraction is not associative for rational numbers.
For Multiplication: For any three rational numbers associative property for multiplication is given as A, B, and C, (A × B) × C = A × (B × C). For example, (1/3 × 1/4) × 1/2 = 1/4 × (1/3 × 1/2) = 1/24 = 1/24. You will find that multiplication is associative for rational numbers.
For Division: For any three rational numbers associative property for division is given as A, B, and C, (A ÷ B) ÷ C ≠ A ÷ (B ÷ C). For example, (1/3 ÷ 1/4) ÷ 1/2 ≠ 1/4 ÷ (1/3 ÷ 1/2) = 8/3 ≠ 2/3. You will find that expressions on both sides are not equal. So division is not associative for rational numbers.
Distributive Property of Rational Numbers
The distributive property of rational numbers states that any expression with three rational numbers A, B, and C, given in form A (B + C) then it is resolved as A × (B + C) = AB + AC or A (B – C) = AB – AC. This means operand A is distributed among the other two operands i.e., B and C. This property is also known as the distributivity of multiplication over addition or subtraction. Let us learn how the distributive property of rational numbers works. We will understand this property using the illustration given below.
Solve: 1/2(1/6 + 1/5)
Solution: The given expression is of the form A (B + C) = A × (B + C) = AB + AC
1/2(1/6 + 1/5) = 1/2 × 1/6 + 1/2 × 1/5 = 11/60
Let us solve the same expression with subtraction.
Solve: 1/2(1/6  1/5)
Solution: The given expression is of the form A (B  C) = A × (B  C) = AB  AC
1/2(1/6  1/5) = 1/2 × 1/6  1/2 × 1/5 = 1/60
Additive Property of Rational Numbers
There are two basic additive properties of rational numbers, additive identity, and additive inverse.
For any rational number a/b, b ≠ 0 the relationship between additive identity and the additive inverse is given as:
This property is very useful in solving complicated calculations. Let us understand the additive identity and additive inverse with the help of examples.
Additive Identity
The additive identity property 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 the additive identity for rational numbers. Let us understand this with an example:
3/7 + 0 = 0 + 3/7 = 3/7
Additive Inverse
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.
For example, the additive inverse of 3/7 is (3/7).
(3/7) + (3/7) = (3/7) + 3/7 = 0.
Multiplicative Property of Rational Numbers
There are two basic multiplicative properties of rational numbers, multiplicative identity, and multiplicative inverse. This property is also very useful in solving complicated calculations. Let us understand the two with examples.
Multiplicative Identity
The additive identity property of rational numbers states that the product of any rational number and 1 is the rational number itself. Here, 1 is the multiplicative identity for rational numbers that expressed in a/b form. If a/b is any rational number, then a/b × 1 = 1 × a/b = a/b. For example: 5/3 × 1 = 1 × 5/3 = 5/3.
Multiplicative Inverse
The 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 × b/a = 1. In this case, rational number b/a is the multiplicative inverse of a rational number a/b. For example, the multiplicative inverse of 7/3 is 3/7. (7/3 × 3/7 = 1).
Note: Every rational number multiplied with 0 gives 0. If a/b is any rational number, then a/b × 0 = 0 × a/b = 0. For example, 7/2 × 0 = 0 × 7/2 = 0.
Related Articles on Properties of Rational Numbers
Check out these interesting articles to learn more about the properties of rational numbers.
Properties of Rational Numbers Examples

Example 1: Using the commutative property of rational numbers, justify whether the given situations are commutative or not:
a) You initially had 2/3 juice in your bottle, and you added 1/6 more
b) You initially had 1/6 juice in your bottle, and you added 2/3 moreSolution:
By addition, we can calculate the total quantity of juice in your bottle.
Total quantity of Juice = Initial quantity of Juice + Added quantity = 2/3 + 1/6 = 5/6.
Let's do the same calculation for the second case,
Total quantity of Juice = Initial quantity of Juice + Added quantity = 1/6 + 2/3 = 5/6.
Here, in both cases, the total quantity of juice is the same.
We know that the commutative property of rational numbers on addition operation states that, for any two rational numbers a and b, a + b = b + a. Here also 2/3 + 1/6 = 1/6 + 2/3 = 5/6. 
Example 2: Help Jack in solving 7/2(1/6 + 3/7) by using the distributive property of rational numbers.
Solution:
Using the distributive property of rational numbers let us write the given expression in the form A (B + C) = A × (B + C) = AB + AC
= 7/2(1/6 + 3/7)
= 7/2 × (1/6 + 3/7)
= (7/2 × 1/6) + (7/2 × 3/7)
= 25/12 
Example 3: If 8/3 × (7/6 × 5/4) = 35/9, then find (8/3 × 7/6) × 5/4.
Solution:
The associative property of rational numbers says that for any three rational numbers (A, B, and C) expression can be expressed as (A × B) × C = A × (B × C)
Given = 8/3 × (7/6 × 5/4) = 35/9Using the associative property of rational numbers, we can evaluate (8/3 × 7/6) × 5/4 is also equal to 35/9.
To verify: (8/3 × 7/6) × 5/4. First, solve the terms inside parentheses.
= 56/18 × 5/4
= 35/9
Hence, 8/3 × (7/6 × 5/4) = (8/3 × 7/6) × 5/4 = 35/9.
FAQ's on Properties of Rational Numbers
What are the Six Important Properties of Rational Numbers?
The six major properties of rational numbers are listed below:
 Closure Property
 Commutative Property
 Associative Property
 Distributive Property
 Multiplicative Property
 Additive Property
What is the Distributive Property of Rational Numbers?
The distributive property states, if p, q, and r are three rational numbers, then the relation between the three is given as, p × (q + r) = (p × q) + (p × r). For example, 1/3(1/2 + 1/5) = 1/3 × 1/2 + 1/3 × 1/5 = 7/30. This property is also known as the distributivity of multiplication over addition. This property is also known as the distributivity of multiplication over subtraction p × (q  r) = (p × q)  (p × r). 1/3(1/2  1/5) = 1/3 × 1/2  1/3 × 1/5 = 1/10.
The Commutative Property of Rational Numbers is Applicable on Which Two Operations?
The commutative property of rational numbers is applicable for addition and multiplication. Example, for addition 1/6 + 1/4 = 1/4 + 1/6 = 5/12, for multiplication 1/3 × 1/7 = 1/7 × 1/3 = 1/21.
What are the Two Multiplicative Properties of Rational Numbers?
The two basic multiplicative properties of rational numbers are multiplicative identity and multiplicative inverse. Let us understand the two with examples.
 Multiplicative identity for rational numbers is expressed as, p/q × 1 = 1 × p/q = p/q. For example: 5/4 × 1 = 1 × 5/4 = 5/4.
 Multiplicative Inverse for rational numbers is expressed as p/q × q/p = 1 such that p/q is the multiplicative inverse of a q/p. For example, the multiplicative inverse of 7/4 is 4/7. (7/4 × 4/7 = 1).
What is the Difference Between Associative Property and Commutative Property of Rational Numbers?
The commutative property of rational numbers holds the expression as A + B = B + A (here, A and B are rational numbers in a form of p/q), and on the other hand, the associative property of rational numbers states that (A + B) + C = A + (B + C) (here, A, B, and C are rational numbers in a form of p/q).
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