Integers

Introduction to Integers

The topic of integers holds great relevance in mathematics and forms the foundation for many advanced topics. The word ‘integer’ is derived from the Latin word ‘integer’, which means whole or complete. So we can say that integers are whole numbers that may be less than, equal to, or greater than zero. 

Just like whole numbers, integers do not contain any fractional components. This means integers can be negative (e.g., –5), positive (e.g., 1, 2, 4), or zero (0). We can perform basic arithmetic operations such as addition, subtraction, multiplication, and division on integers, following specific rules. Common examples of integers include –10, –3, 0, 4, 11, and so on. The symbol used to represent integers is ‘Z’, derived from the German word ‘Zahlen’, meaning ‘numbers’. 

In this article, we will explore more about integers, their types, arithmetic operations, important rules, properties, and how to represent integers on a number line. Along with detailed solved examples, this step-by-step guide will help you to build a clear understanding of this topic. 

 

Table of Contents

• What are integers?
• Types of Integers
• Integers on Number Line
• Operations on Integers
• Properties of Integers
• Closure Property
• Commutative Property
• Associative Property
• Distributive Property
• Identity Property
• Additive Inverse Property
• Multiplicative Inverse Property
• Applications of Integers in Daily Life
• Solved Examples
• Practice Questions
• FAQs on Integers

 

What are integers?

The group of numbers that contains all positive numbers, negative numbers, and zero is called the integers. There are no decimals or fractional parts in these numbers. 

Examples of integers: –4, –6, 10, 21.

Types of Integers

Integers are categorised into 3 main types based on their value:

 

Integers on Number Line

We already know that integers include positive numbers, negative numbers, and zero. We can easily represent these integers on a number line.

On a number line:

• Zero (0) is always located in the centre.

• Positive integers (e.g., 1, 2, 3, 4, ...) appear to the right of zero.

• Negative integers (e.g., –1, –2, –3, –4, ...) appear to the left of zero.

Each number is located at the same distance from the next to the next. This helps us compare numbers and perform operations such as addition and subtraction.

Example: Here is a simple number line showing some integers:

• On the left of 0: –1, –2, –3, –4 → These are negative integers.

• On the right of 0: 1, 2, 3, 4 → These are positive integers on the right.

 

Operations on Integers 

In arithmetic operations, we can perform all basic mathematical operations (addition, subtraction, multiplication, and division) with integers. But we have to follow some simple rules, which depend on whether the number is positive or negative.

1. Addition of Integers

When adding two integers with the same sign, add the full value and write down the sum with the sign included with the numbers.

For example:

• (+5) +( +8) = +14

• (–6) + (–3) = –9

When adding two integers with different signs, you subtract the entire value and write the difference with the sign of the number with the largest absolute value.

2. Subtraction of Integers

When you subtract two integers, you change the sign of the second number that is subtracted, and also follow the rules.

For example:

• (–8) – (+4) = (–8) + (–4) = –12

• (+8) – (+6) = (+8) + (–6) = +2

3. Multiplication of Integers

When you multiply two integer numbers, the rule is simple.

• If both integers have the same sign, the result is positive.

• If there are different signs of integers, the result is negative.

For example:

• (+6) x (+3) = +18

• (+9) x (–4) = –36

4. Division of Integers

The rule for dividing integers is almost the same as multiplication.

• If both integers have the same sign, then the answer will be positive.

• If the integers have different signs, then the answer will be negative.

Examples:

• (+12) ÷ (+4) = +3

• (−15) ÷ (−3) = +5

• (+18) ÷ (−6) = −3

 

Properties of Integers 

Here are some important rules, also known as properties of integers that help us solve problems quickly:

1. Closure Property

2. Commutative Property

3. Associative Property

4. Distributive Property

5. Identity Property

6. Additive Inverse Property

7. Multiplicative Inverse Property

 

Closure Property

The closure property states that when we perform an operation (like addition, subtraction, or multiplication) on 2 numbers from a set, the answer will also be in the same set.

• For whole numbers (0, 1, 2, ...): If we add any 2 whole numbers, we will always get another whole number. For example: 4 + 4 = 8

• For Integers (..., −3, −2, −1, 0, 1, 2, …): If you multiply any 2 integers, the answer will also be an integer. For example: −8 × 3 = −24 

Commutative Property

The commutative property states that changing the order of numbers in addition or multiplication does not change the answer.

• Addition: A + B = B + A. For example: 8 + 7 = 15

• Multiplication: A × B = B × A. For example: 6 + (−2) = 4 = (−2) + 6

Associative Property

This property states that when we combine three integers differently, the answer remains the same, but it only works for addition and multiplication of integers.

• Addition: (a + b) + c = a + (b + c).For example: (−1 + 2) + 3 = −1 + (2 + 3) = 4

• Multiplication: (a × b) × c = a × (b × c). For example: (2 × 3) × 4 = 2 × (3 × 4) = 24

Distributive Property

This property combines multiplication and addition. It says that if we multiply a number by a group of numbers, it is similar to multiplying separately and then adding.

• A × (B + C) = (A × B) + (A × C). For example: 3 × (4 + 2) = 3 × 6 = 18 → 3 × 4 + 3 × 2 = 12 + 6 = 18

Identity Property

This property tells us about special numbers (identity) that keep the original number equal after an operation.

• Additive Identity: 0 → A + 0 = A and 0 + A = A. For example: 7 + 0 = 7

• Multiplicative Identity: 1 → A × 1 = A and 1 × A = A. For example: (–5) × 1 = –5

Additive Inverse Property

This property states that each number has a negative version, and when you add them together, the answer is zero.

• For any integer a: A + (–A) = 0. For example: 8 + (–8) = 0, (–6) + 6 = 0

Multiplicative Inverse Property

This property states that every number has a reciprocal, and when we multiply them, the answer will be 1.

• For any integer a≠0: A × 1/A = 1. For example: 8 × 1/8 = 1, (–9) × (–1/9) = 1

 

Applications of Integers in Daily Life

The integer is used in many real-life situations. They help show the opposite things, such as profits and losses or high and low values.

Some examples:

• Temperatures: Shown with positive numbers above 0°C; temperatures below 0°C appear with negative numbers.

• Sport: Points in sports such as football or cricket can be positive or negative.

• Banking: The extra money is shown as positive; the withdrawn money is shown as negative.

• Ranking: Movies or songs can be considered using positive and negative numbers.
  
  

Solved Examples

By solving integer questions, you can easily build the conceptual fluency required for exams and competitive tests.

Example 1:
Plot the following integers on a number line:
–35, –28, –15, –6, 0, 4, 9, 17, 23, 39, 48, 66, 85, 99.

Solution:

We should place those numbers in the correct positions on the number line. & Negative numbers should be left of zero, and positive numbers to the right. 

Example 2:
Solve the following: Addition & Subtraction. 

1. 9 + (–5) = ?

2. (–10) + 9 = ?

3. (–8) – (–4) = ?

4. (–7) – 6 = ?

Answer:
1. 9 + (–5) = 4

2. (–10) + 9 = –1

3. (–8) – (–4) = –4 

4. (–7) – 6 = –13

Example 3:
Solve the following: Multiplication 

1. (–6) × (+8) 

2. (–11) × (–9)

3. 7 × (–13) 

4. (–4) × 0

Answer:

1. (–6) × (+8) = –48

2. (–11) × (–9) = 99

3. 7 × (–13) = –91

4. (–4) × 0 = 0

Example 4:
Solve the following: Division

1. (+94) ÷ (–7)

2. (–48) ÷ (–8)

3. (–64) ÷ (8)

4. (+49 ÷ (–8)

Answer:

1. (+94) ÷ (–7) = –13.4

2. (–48) ÷ (–8) = 6

3. (–64) ÷ (8) = –8

4. (+49) ÷ (–8) = –6.1
   
   

Practice Questions 

1. Is the sum of two positive integers always positive? 

2. What is the sum of the first four positive integers?

3. What is the product of the first 6 positive odd integers?

4. Draw and mark integers from –10 to +10 on a number line.
   
   

FAQs on Integers

1. What are integers and examples

Ans: Integers are whole numbers (no decimals or fractions) that can be positive, negative, or zero. They are a fundamental set of numbers used in mathematics and various real-world applications.

2. Is 1.5 an integer, yes or no?

Ans: An integer, also called a "round number" or "whole number", is any positive or negative number that does not include decimal parts or fractions. For example, 3, –10, and 1,025 are all integers, but 2.76 (decimal), 1.5 (decimal), and 3 ½ (fraction) are not.

3. Is 7 an integer?

Ans: Integers are numbers that cannot be decimals or fractions. They are either whole numbers or negative numbers. Some examples are 2, 7, 0, –9, –12, etc.

4. What are the 7 rules of integers?

Ans: Rules of Integers

• The sum of two positive integers is an integer.

• The sum of two negative integers is an integer.

• The product of two positive integers is an integer.

• The product of two negative integers is an integer.

• The sum of an integer and its inverse is equal to zero.

• The product of an integer and its reciprocal is equal to 1.

5. Is root 2 an integer?

Ans: It is not, as 2 is not a perfect square or irrational. The rational root theorem (or integer root theorem) may be used to show that any square root of any natural number that is not a perfect square is irrational.