Integers: Types, Rules, Properties, and Examples

Integers are numbers we use every day, whether counting, measuring, or solving problems. They include positive numbers, negative numbers, and zero, but never fractions or decimals.

From tracking temperatures to handling money, integers help in many real-life situations. They follow specific rules that make calculations easier. Let’s learn about integers in detail in this blog.

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What Are Integers?

Integers are groups of numbers that include whole numbers and their negative counterparts but do not include fractions or decimals. In simple terms, integers can be positive, negative, or zero.

On a number line, integers extend infinitely in both directions:

... -4, -3, -2, -1, 0, 1, 2, 3, 4 …

• Positive integers (1,2,3,4,...) are numbers greater than zero.

• Negative integers (−1,−2,−3,−4) are numbers less than zero.

• Zero (0) is a neutral integer—it is neither positive nor negative.

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Integers on a Number Line

Integers can be shown on a number line to help us understand their order and size.

On the number line, zero is the middle point. 

Positive integers are placed to the right of zero, starting with 1 and increasing as you move further right (1, 2, 3, 4, ...).

Negative integers are placed to the left of zero, starting with -1 and decreasing as you move further left (-1, -2, -3, -4, ...).

The size of integers increases as you move further away from zero. 

For example, on the right side of zero, 5 is larger than 3, and on the left side of zero, -6 is smaller than -2. 

The order follows a simple rule: numbers on the right are always larger than numbers on the left.

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Types of Integers

Integers are classified into three main types based on their position on the number line: zero, positive, and negative. Each category has unique properties and plays a distinct role in mathematics.

Zero (0)

Zero is a unique number that does not fall into positive or negative integers. It is a neutral point on the number line and is represented simply as "0", without a positive or negative sign. 

Zero is important in mathematical operations, acting as the identity element in addition and as a reference point between positive and negative values.

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Positive Integers (Natural Numbers)

Positive integers are also called natural or counting numbers and are represented by Z+. These numbers are always greater than zero and appear to the right of zero on the number line. 

They follow a sequential order and are commonly used in counting, ordering, and quantifying objects.

Z+ → 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, …

Negative Integers (Additive Inverses of Natural Numbers)

Negative integers are the opposite of positive integers and are represented by Z–. These numbers are always less than zero and are positioned to the left of zero on the number line. 

Each negative integer is the additive inverse of a positive integer, meaning they result in zero when added together. 

Negative integers are often used to represent losses, temperatures below freezing, and depths below sea level.

Z– → -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, …

Rules of Integers

Integers follow specific rules when performing mathematical operations like addition and multiplication. Here are the key rules that govern their behavior:

• Adding Positive Integers: The sum of two positive integers is always a positive integer.

• Adding Negative Integers: The sum of two negative integers is always a negative integer.

• Multiplying Positive Integers: The product of two positive integers is always a positive integer.

• Multiplying Negative Integers: The product of two negative integers is always a positive integer.

• Sum of an Integer and Its Inverse: Adding an integer to its inverse always results in zero.

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Properties of Integers

Integers follow several mathematical properties that help in simplifying calculations and understanding number relationships. These properties define how integers behave under different arithmetic operations.

Closure Property

The closure property means that when you add, subtract, or multiply two integers, the result will always be an integer. However, division does not always give an integer.

• Addition: Adding two integers always gives an integer.

• Subtraction: Subtracting one integer from another always results in an integer.

• Multiplication: Multiplying two integers always gives an integer.

• Division: Dividing one integer by another may or may not result in an integer.

Examples:

• (-8) + 4 = -4 (The sum is an integer.)

• 7 − 12 = −5 (The result is an integer.)

• −15 −(−5) = −15 + 5 = −10 (The result is an integer.)

• (-10) × 3 = -30 (The product is an integer.)

• 15 ÷ 4 = 3.75 (This is not an integer, so division does not always follow the closure property.)

Associative  Property

The associative property states that changing the way numbers are grouped in addition or multiplication does not affect the result. This property does not apply to subtraction or division.

For addition: The sum remains the same regardless of how the numbers are grouped. Example:  

3 + (4 + 6) = (3 + 4) + 6 = 13 

For multiplication: The product remains unchanged no matter how the numbers are grouped. 

Example: (−2) × (5 × 3) = ((−2) × 5) × 3 = −30

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Commutative Property

The commutative property states that changing the order of numbers in addition or multiplication does not change the result. However, this property does not apply to subtraction or division.

For addition: Swapping the numbers does not affect the sum.

Example: (−5) + 8 = 8 + (−5) = 3

For multiplication: Changing the order of the numbers does not affect the product.

Example: 6 × ( −3) = (−3) × 6 = −18

Distributive Property

The distributive property states that multiplication distributes over both addition and subtraction, meaning a number multiplied by a sum or difference is the same as multiplying each term separately and then adding or subtracting the results.

Example

• 5 × (3 + 7) = (5 × 3) + (5 × 7) = 15 + 35 = 50

• (−4) × (8 − 2) = (−4 × 8) − (− 4 × 2) = −32 + 8 = −24

Additive Inverse Property

The additive inverse property states that adding an integer to its opposite (negative counterpart) always results in zero.

For any integer x: x + ( −x) = 0

Example:

• 9 + ( − 9) = 0

• −15 + 15 = 0

Multiplicative Inverse Property

The multiplicative inverse property states that when an integer is multiplied by its reciprocal, the result is always 1. This property applies to all integers except zero, as zero has no reciprocal.

For any integer y:  y × 1/y ​= 1

Example: 

• 7 × 1/ 7 = 1

• (−5) × (−1/ 5) = 1

Real-Life Applications of Integers

Integers are useful in many real-life situations, helping us understand and calculate different aspects of everyday life.

1. Temperature: Positive integers represent temperatures above zero, and negative integers represent temperatures below zero.

2. Banking: Deposits are represented by positive integers, while withdrawals or losses are represented by negative integers.

3. Sports Scores: Positive integers are used for scores, and negative integers can be used for penalties.

4. Elevation: Positive integers are used for heights above sea level, while negative integers show depths below sea level.

5. GPS Coordinates: Positive integers represent locations north or east, and negative integers represent locations south or west.

Integers Solved examples 

1. John owes his friend $8, but he receives $15 from another friend. After repaying his debt, how much money does he have left?

Solution:
Since owing money is considered negative and receiving money is positive, the equation is:

(−8) + 15 = 7 

Answer: John has $7 left.

2. The temperature in a city is 12°C in the afternoon, but a cold wave causes it to drop by -5°C. What is the new temperature?

Solution:
Subtracting a negative number means adding its positive counterpart:

12 − (−5) = 12 + 5 = 17 

Answer: The new temperature is 17°C.

3.  A submarine is descending at a rate of 6 meters per second. How deep will it be after 4 seconds if the initial depth is zero?

Solution:
Since going downward is represented as negative, the equation is:

(−6) × 4 = −24

Answer: The submarine will be 24 meters below sea level.

4. A company lost $48,000 over 6 months. If the loss was evenly distributed, what was the company’s loss per month?

Solution:
Since loss is negative, we divide:

(−48,000) ÷ 6 = −8,000

Answer: The company lost $8,000 per month.

5.  A factory produces 5 batches of products daily, with each batch containing (7 defective + 3 non-defective) items. How many total defective and non-defective items are produced in 5 days?

Solution:
Using the distributive property:
5 × (7 + 3) = (5 × 7) + (5 × 3) = 35 + 15 = 50

Answer: The factory produces 50 items in 5 days.