Division of Integers: Understanding the Properties

Properties of division of integers help us understand how division works with positive and negative numbers. In this topic, you will learn important rules and properties that explain whether division always gives an integer, how order affects the result, and how grouping changes the answer. These properties make it easier to solve problems correctly and avoid mistakes.

Table of Contents

• Definition of Division of Integers
• Properties of Division of Integers
• Closure Property
• Non-Commutative Property
• Non-Associative Property
• Identity Property

Definition of Division of Integers

Division of integers means splitting one integer by another integer. The result is called the quotient. While dividing integers, we follow sign rules:

• The same signs mean a positive result

• Different signs mean a negative result

Properties of Division of Integers

1. Closure Property (Not Always True)

If a and b are integers, a ÷ b may not always be an integer.

Example:

• 45 ÷ (−15) = −3 (integer)

• (−10) ÷ (−4) = 2.5 (not an integer)

2. Non-Commutative Property

Division of integers is not commutative, which means the following:
a ÷ b ≠ b ÷ a

Example:

• (−12) ÷ 3 = −4

• 3 ÷ (−12) = −1/4

So, both results are different.

3. Non-Associative Property

Division is not associative, meaning the following:
a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c

Example:

• (−8) ÷ (4 ÷ 2) = −8 ÷ 2 = −4

• [(−8) ÷ 4] ÷ 2 = −2 ÷ 2 = −1

4. Identity Property

Dividing any integer by 1 gives the same number.

a ÷ 1 = a

Example:

• (−25) ÷ 1 = −25