Table of contents  
Introduction 
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Integers include positive numbers, negative numbers, and zero. 'Integer' is a Latin word which means 'whole' or 'intact'. This means integers do not include fractions or decimals.
Integers include all whole numbers and negative numbers. This means if we include negative numbers along with whole numbers, we form a set of integers.
An integer is a number with no decimal or fractional part and it includes negative and positive numbers, including zero. A few examples of integers are: 5, 0, 1, 5, 8, 97, and 3,043. A set of integers, which is represented as Z, includes:
Z = {... 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, ...}
Observe the figure given below to understand the definition of integers.
A number line is a visual representation of numbers on a straight line. This line is used for the comparison of numbers that are placed at equal intervals on an infinite line that extends on both sides, horizontally. Just like other numbers, the set of integers can also be represented on a number line.
Positive and negative integers can be visually represented on a number line. Integers on a number line help in performing arithmetic operations. The basic points to keep in mind while placing integers on a number line are as follows:
The four basic arithmetic operations associated with integers are:
There are some rules for performing these operations of integers. Before we start learning these methods of integer operations, we need to remember a few things.
Adding integers is the process of finding the sum of two or more integers where the value might increase or decrease depending on the integer being positive or negative. The different rules and the possible cases for the addition of integers are given in the following section.
While adding two integers, we use the following rules:
Example 1: Add the given integers: 2 + (5)
Solution:
Here, the absolute values of 2 and (5) are 2 and 5 respectively.
Their difference (larger number  smaller number) is 5  2 = 3
Now, among 2 and 5, 5 is the larger number and its original sign “”.
Hence, the result gets a negative sign, "”.
Therefore, 2 + (5) = 3
Example 2: Add the given integers: (2) + 5
Solution: Here, the absolute values of (2) and 5 are 2 and 5 respectively.
Their difference (larger number  smaller number) is 5  2 = 3.
Now, among 2 and 5, 5 is the larger number and its original sign “+”.
Hence, the result will be a positive value. Therefore,(2) + 5 = 3
We can also solve the above problem using a number line. The rules for the addition of integers on the number line are as follows.
Example 3: Find the value of 5 + (10) using a number line.
Solution: In the given problem, the first number is 5 which is positive. So, we start from 0 and move 5 units to the right side.
Addition using Number Line
The next number in the given problem is 10, which is negative. We move 10 units to the left side from 5.
Addition using Number Line
Finally, we reach at 5. Therefore, the value of 5 + (10) = 5
Subtracting integers is the process of finding the difference between two or more integers where the final value might increase or decrease depending on the integer being positive or negative. The different rules and the possible cases for the subtraction of integers are given in the following section.
In order to carry out the subtraction of two integers, we use the following rules:
Example: Subtract the given integers: 7  10
Solution: 7  10 can be written as (+ 7)  (+)10
For the multiplication of integers, we use the following rules given in the table. The different rules and the possible cases for the multiplication of integers are given in the following section.
Rules of Integers in Multiplication
In order to carry out the multiplication of two integers, we use the following rules:
Example: Multiply (6) × 3
Solution: Using the rules of multiplication of integers, when we multiply a positive and negative integer, the product has a negative sign.
Therefore, (6) × 3 = 18
Division of integers means equal grouping or dividing an integer into a specific number of groups. For the division of integers, we use the rules given in the following table. The different rules and the possible cases for the division of integers are given in the following section
Rules of Integers in Division
In order to carry out the division of two integers, we use the following rules.
Example: Divide (15) ÷ 3
Solution: Using the rules of division of integers, when we divide a negative integer by a positive integer, the quotient has a negative sign.
Therefore, (15) ÷ 3 = 5
The main properties of Integers are as follows:
1. Closure Property The closure property states that the set is closed for any particular mathematical operation. Z is closed under addition, subtraction, multiplication, and division of integers. For any two integers, a and b:
2. Associative Property
According to the associative property, changing the grouping of two integers does not change the result of the operation. The associative property applies to the addition and multiplication of two integers.
For any two integers, a and b:
3. Commutative Property
According to the commutative property, changing the position of the operands in an operation does not affect the result. The addition and multiplication of integers follow the commutative property.
For any two integers, a and b:
4. Distributive Property
The distributive property states that for any expression of the form a (b + c), which means a × (b + c), operand a can be distributed among the operands b and c as: (a × b) + (a × c) that is,
a × (b + c) = (a × b) + (a × c)
5. Additive Inverse Property
The additive inverse property states that the addition operation between any integer and its negative value will give the result as zero (0).
For any integer, a:
a + (a) = 0
6. Multiplicative Inverse Property
The multiplicative inverse property states that the multiplication operation between any integer and its reciprocal will give the result as one (1).
For any integer, a: a × 1/a = 1
7. Identity Property
Integers follow the identity property for addition and multiplication operations. The additive identity property states that when zero is added to an integer, it results in the integer itself. This means, a + 0 = a
Similarly, the multiplicative identity states that when 1 is multiplied to any integer, it results in the integer itself. This means, a × 1 = a
Example 1: Can you identify the property of the integer used in the following expression? 3 + (7 + 2) = (3 + 7) + 2
Solution: The given expression, 3 + (7 + 2) = (3 + 7) + 2, shows the associative property of integers, which says that changing the grouping of two integers does not change the result of the operation because 3 + (7 + 2) = (3 + 7) + 2 = 12.
Example 2: Add the following integers: (9) + (5)
Solution: According to the rules of integers in addition, when both the integers have the same signs, we add the absolute values of integers and give the same sign as that of the given integers to the result.
The absolute values of the given integers is 9 and 5.
So we will add 9 + 5 = 14 and the sign of the sum will be negative.
Therefore, (9) + (5) = 14
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