Properties of Rational Numbers

The properties of rational numbers help us to distinguish them from the other types of numbers. Rational numbers consist of integers, whole numbers, and natural numbers. They can be represented in the form of a fraction p/q or as terminating decimal numbers, or as non-terminating but repeating decimal numbers. The properties of rational numbers include the associative property, the commutative property, the distributive property, and the closure property. Let us read about all the properties of rational numbers on this page.

When numbers can be expressed in the form of p/q, then they are considered to be 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.

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 examples.

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.

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. We can see 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. We can see that multiplication is commutative for rational numbers. This means, a × b = b × a for any two rational numbers a and b.
• For Division: 1/3 ÷ 1/4 ≠ 1/4 ÷ 1/3 because 4/3 ≠ 3/4. We can see that the expressions 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.

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. However, 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, the associative property for addition is expressed 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, the associative property for subtraction is expressed 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). We can see that subtraction is not associative for rational numbers.
• For Multiplication: For any three rational numbers, the associative property for multiplication is expressed 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. We can see that multiplication is associative for rational numbers.
• For Division: For any three rational numbers, the associative property for division is expressed 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) because 8/3 ≠ 3/8. We can see that the expressions on both sides are not equal. So division is not associative for rational numbers.

The distributive property of rational numbers states that if any expression with three rational numbers A, B, and C is given in form A (B + C), then it can be solved as A × (B + C) = AB + AC. This applies to subtraction also which means A (B - C) = AB - AC. This means operand A is distributed between the other two operands, i.e., B and C. This property is also known as the distributive property 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.

Example: 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.

Example: 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

There are two basic additive properties of rational numbers, the additive identity property and the additive inverse property. For any rational number a/b, where b ≠ 0 these two properties are illustrated below.

Let us understand the additive identity property and the additive inverse property with the help of examples.

Additive Identity Property

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 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.

For example, the additive inverse of 3/7 is (-3/7).

(3/7) + (-3/7) = (-3/7) + 3/7 = 0.

There are two basic multiplicative properties of rational numbers, the multiplicative identity property, and the multiplicative inverse property. Let us understand these properties with examples.

Multiplicative Identity Property

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. 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 Property

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, a 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 Topics

• Properties of Natural Numbers
• Properties of Whole Numbers
• Properties of Integers
• Commutative Property Formula
• Distributive Property Calculator

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 applicable to subtraction which says p × (q - r) = (p × q) - (p × r). For example, 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, and 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 the multiplicative identity property and the multiplicative inverse property. 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 q/p. For example, the multiplicative inverse of 7/4 is 4/7. (7/4 × 4/7 = 1).

What is the Difference Between the Associative Property and the Commutative Property of Rational Numbers?

The commutative property of rational numbers says that 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).