Integers
2026-02-28 10:34 Diff
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Last updated on December 4, 2025

Numbers are arithmetic values used to represent quantities. Integers are a type of number that includes all the positive numbers, negative numbers, and zero. In this topic, we will learn about integers, their types, operations, etc.

What are Integers?

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Integers are a type of number that includes whole numbers and negative numbers. This means they do not include fractions or decimal numbers. For example,  -6, -5, -3, 0, 1, 2, and 9. Integers are represented using the letter Z.
 

  • The positive numbers are the numbers greater than zero, i.e., 1, 2, 3, 4, …
     
  • The numbers that are less than zero are negative numbers. For example, -1, -2, -3, -4, …
     
  • Zero is a whole number and is neither positive nor negative.

What are the Types of Integers

Based on the type of numbers, integers can be classified into three categories 
 

  • Zero
  • Positive Integers
  • Negative Integers

Zero: Zero is a neutral number, as it is neither positive nor negative; it is represented as 0. 


Positive Integers: Positive integers are numbers that are greater than 0 and are represented as Z+. In a number line, it is represented to the right of zero. For example, 1, 2, 3, 4, 5, … 

Negative Integers: Negative integers are natural numbers with opposite signs, so they are represented as Z-. In a number line, negative integers are listed to the left of zero. For example, -1, -2, -3, -4, -5, …  

Difference Between Integers vs Non-integers

Integers are whole numbers without any fractional or decimal parts, whereas non-integers include all numbers that contain fractions, decimals, or irrational values. Here are the differences between integers and non-integers. 

Integers Non-integers Integers are whole numbers with no decimal or fractional part.        Non-integers are the fraction, decimal, or irrational part.   Integers include positive, negative, and zero.
    Non-integers include fractions, decimals, irrational numbers, and mixed numbers.   Examples: -8, -5, 0, 5, 6   Example: 5.2, -3, √4, 8(1/2)  Used for counting and whole quantities  Used for measurements like height, weight, temperature, and money.  On a number line, it can be represented as a fixed point.  On a number line, it can be represented between two integers.  

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Representation of Integers on Number Line

A number line visually represents integers as a straight line. Integers include all the positive numbers, negative numbers, and zero. In a number line, when representing the integers, zero is marked in the center, representing a neutral point. The numbers on the right are the positive integers, as the values keep increasing. So, the numbers on the left side are negative integers, and the values of the numbers decrease toward the left.   

Operations on Integers

There are four basic arithmetic operations on integers:

  • Addition of Integers
  • Subtraction of Integers
  • Multiplication of Integers
  • Division of Integers

Now, learn them in detail. 

Addition of Integers - Adding two or more integers is called integer addition, and the result is called the sum. The sum may increase or decrease depending on the signs of the numbers involved. When adding the integers, make sure to follow these rules. 
 

  • When adding integers with the same sign, add their absolute values and give the sum the same sign.


           For example, 5 + 5= 10, - 5 + (-5) = -10.
 

  • When adding the integers of different signs, that is, one positive and the other negative, find the difference of the numbers and give the sign of the larger number to the sum.


            For example, 5 + (-3) = 2, 9 + (-16) = -7.

Subtraction of Integers


We subtract numbers to find the difference between them. Here, the result depends on the integers being positive or negative. Follow these rules when subtracting two integers,
 

  • Convert the subtraction to addition by taking the opposite of the second number.
     
  • Apply the rules of addition of integers. For example, 5 – 3 = 25 – (3) = 5 + 3 = 8

Multiplication of Integers


Multiplication of integers is the process of finding the product of two or more numbers. To multiply integers, follow these steps;
 

Product of Signs Result Example (+) × (+)  + 5 × 6 = 30 (–) × (–)   + (-5) × (-6) = 30 (–) × (+)  - (-5) × (6) = -30 (+) × (–)  - (5) × (-6) = -30

Division of Integers


The division of an integer is splitting a number into groups. There are different rules to be followed, such as;
 

Division of Signs Result Example

(+) ÷ (+) 

+ 30 ÷ 6 = 5

(–) ÷ (–) 

+ (-30) ÷ (-6) = 5

(–) ÷ (+) 

- (-30) ÷ (6) = -5

(+) ÷ (–)

- (30) ÷ (-6) = -5

Properties of Integers

Integers follow several important properties, such as : 

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

Closure Property: The set is closed under an operation if performing that operation on any two elements of the set results in another element within the set.
 

For any two integers, p and q
 

  • p + q ∈ Z 
  • p – q ∈ Z
  • p × q ∈ Z
     

Example 1: 5 + 7 = 12

Example 2: 9 - 14 = -5

Example 3: -3 × 6 = -18

Associative Property: The associative property states that grouping of integers in both addition and multiplication does not affect the result.

That is, p + (q + r) = (p + q) + r and p × (q × r) = (p × q) × r. 

Example 1: 2 + (3 + 4) = (2 + 3) + 4 = 9

Example 2: 3 ×( 4 × 5) = (3 × 4) × 5 = 60

Example 3: 5 + (3 + 8) = (5 + 3) + 8 = 16

Commutative Property: The Commutative property states that the order of the integers does not alter both addition and multiplication. For example, p + q = q + p and p × q = q × p. 

Example 1: 8 + 2 = 2 + 8 = 10

Example 2: 12 + 5 = 5 + 12 = 17

Example 3: 5 × 2 = 2 × 5 = 10

Distributive Property: The product of a number with two addends will be the same as the sum of the products of the number with the addends is the distributive property.

That is: p × (q + r) = (p × q) + (p × r)

Example 1: 4 × (6 + 2) = (4 × 6) + (4 × 2) = 32

Example 2: 5 × (8 - 2) = (5 × 8) - (5 × 2) = 30

Example 3: 2 × (5 + 9) = (2 × 5) + (2 × 9) = 28

Additive Inverse Property: The sum of any positive integer and its negative integer is zero, which means p + (-p) = 0.

Example 1: 5 + (-5) = 0

Example 2: (-9) + 9 = 0

Example 3: (-18) + 18 = 0

Multiplicative Inverse Property: Except for 1 and -1, most integers do not have multiplicative inverses within the set of integers. 

Example 1:  1 × 1 = 1

Example 2: -1 × -1 = 1

Example 3: 5 × 1 = 5

Identity Property: Identity property states that the sum of any number with 0 is equal to the number itself. That is, a + 0 = a. Same as the product of an integer with 1 is equal to the integer itself.

That is, a × 1 = a

Example 1: 12 + 0 = 12

Example 2: 15 × 1 = 15

Example 3: -5 × 1 = -5

Tips and Tricks to Master Integers

Integers can be challenging for learners because they involve both positive and negative values and different rules for each operation. Here are tips that can help students master the concept of integers.  

  • Students can use a number line to practice placing and moving numbers. This helps students to see how positive numbers move right, and negative numbers move left. 
     
  • Use an integer rule chart to show how signs work, as it helps them to solve operations with integers easily. 
     
  • Teachers can introduce integers to students through familiar scenarios like weather changes, bank balances, or game scores. This builds strong conceptual understanding from the beginning. 
     
  • Parents can ask students comparison questions like, "Which is greater: -1 or -5?"
     
  • Students should always pay close attention to the + and - signs before solving the problems.

Common Mistakes and How to Avoid Them in Integers

When working with integers, students usually tend to confuse them, which leads to errors. Now let’s learn a few common mistakes and ways to avoid them in integers. 
 

Real-world Applications of Integers

In the real world, we use integers in different fields, from balancing budgets to measuring temperature. Now, let’s learn some applications of integers. 

  • Integers are used to represent gains and losses in temperature, elevation, navigation, and so on.
  • In population studies, we use integers to represent the number of individuals in a region.
  • In computer science and programming, we use integers for efficient computation and logical control of algorithms, software applications, and digital systems.
  • Integers are used to record credits and debits in financial transactions.

Download Worksheets

Problem 1

Multiply -4 by 6.

Okay, lets begin

 (-4) × 6 = -24
 

Explanation

The product of a negative and a positive integer is a negative integer. So, (-4) × 6 = -24
 

Well explained 👍

Problem 2

Subtract -3 from -8.

Okay, lets begin

(-8) – (-3) = -5

Explanation

(-8) – (-3) = (-8) + 3 = -5.

Well explained 👍

Problem 3

What is the sum of -7 and 4?

Okay, lets begin

 (-7) + 4 = -3
 

Explanation

 The sum of a negative number and a positive number is the difference between the numbers, that is (-7) + 4 = -3
 

Well explained 👍

Problem 4

Divide -18 by 6.

Okay, lets begin

 (-18) ÷ 6 = -3
 

Explanation

A negative divided by a positive gives a negative result, so (-18) ÷ 6 = -3
 

Well explained 👍

Problem 5

Find the product of -9 and -3.

Okay, lets begin

(-9) × (-3) = 27
 

Explanation

The product of two negative integers is positive, so (-9) × (-3) = 27

Well explained 👍

FAQs on Integers

1.What are integers?

Integers are whole numbers including positive, negative, and zero. For example, -2, 0, 1.
 

2.Is zero an integer?

Yes, zero is an integer, as it is a whole number.
 

3.What is the difference between whole numbers and integers?

The integers include all the positive numbers, negative numbers, and zero, that is, -2, -1, 0, 1, 2. Whereas the whole numbers are the non-negative numbers from 0, 1, 2, 3. 
 

4.What are the rules for multiplying integers?

When multiplying two integers, we first multiply the numbers and then the signs. The product of two same signs is always positive, whereas the product of different signs is negative. 
 

5.Are fractions and decimals integers?

Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

Fun Fact

: She loves to read number jokes and games.