Integers
The term “integer” was adapted in Mathematics from the Latin word “integer”, which means intact or whole. Integers are very much like whole numbers, but they also include negative numbers among them.
What is an Integer?
An integer is a number with no decimal or fractional part, from the set of negative and positive numbers, including zero. Examples of integers are: 5, 0, 1, 5, 8, 97, and 3,043.
A set of integers, which is represented as Z, includes:
 Positive Integers: An integer is positive if it is greater than zero. Example: 1, 2, 3 . . .
 Negative Integers: An integer is negative if it is less than zero. Example: 1, 2, 3 . . .
 Zero is defined as neither negative nor positive integer.
Z = {... 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, ...}
Integers on a Number Line
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.
Graphing Integers on a Number Line
 The number on the right horizontal side is always greater than the left side number.
 Positive numbers are placed on the right side of 0, as they are greater than “0”.
 Negative numbers are placed on the left side of “0”, as they are smaller than “0”.
 Zero, which isn't positive or negative, is kept at the center.
Integer Operations
The four basic arithmetic operations associated with integers are:
 Addition of Integers
 Subtraction of Integer
 Multiplication of Integers
 Division of Integers
There are some rules for doing these operations.
Before we start learning these methods of integer operations, we need to remember a few things.
If there is no sign in front of a number, it means that the number is positive. For example, 5 means +5.
The absolute value of an integer is a positive number, i.e., −2 = 2 and 2 = 2.
Addition of Integers
While adding two integers, we come across the following cases:
 Both integers have the same signs: Add the absolute values of integers, and give the same sign as that of the given integers to the result.
 One integer is positive and the other is negative: Find the difference of the absolute values of the numbers and then give the original sign of the larger of these numbers to the result.
Example 1:
Adding two integers: Calculate the value of 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:
Adding two integers: Calculate the value of 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:
 always start from "0".
 move to the right side, if the number is positive.
 move to the left side, if the number is negative.
Let's find the value of 5 + (10) using a number line.
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.
The next number in the given problem is 10, which is negative.
We move (from the fifth unit) 10 units to the left side.
The number we have moved to finally is 5.
Subtraction of Integers
To carry out the subtraction of two integers:
 Convert the operation into an addition problem by changing the sign of the subtrahend.
 Apply the same rules of addition of integers and solve the problem thus obtained in the above step.
Example:
Subtracting two integers: Calculate the value of 7  10.
Solution:
Converting the given expression into an addition problem, we get: 7 + (10)
Now, the rules for this operation will be the same as for the addition of two integers.
Here, the absolute values of 7 and (10) are 7 and 10 respectively.
Their difference (larger number  smaller number) is 10  7 = 3
Now, among 7 and 10, 10 is the larger number and its original sign “”.
Hence, the result gets a negative sign, "”.
Therefore, 7  10 = 3
Multiplication of Integers
To carry out the multiplication of two integers:
 Multiply their signs and get the resultant sign.
 Multiply the numbers and add the resultant sign to the answer.
The different possible cases for multiplication of two signs can be observed in the following table:
Product of Signs  Result  Example 

+ × +  +  2 × 3 = 6 
+ ×     2 × (3) = 6 
 × +    (2) × 3 = 6 
 ×   +  2 × 3 = 6 
Example:
Multiplying integers on a number line: Calculate the value of 2 × 3 and 2 × 3 using a number line.
Solution:
We read 2 × 3 as “2 times 3”. We have to represent 3 on the number line 2 times. To do so, we will start from and move left by 3 units twice.
Thus, 2 × 3 = 6.
Also, 2 × 3 is similar to 2 × 3, but 2 is replaced by 2. Hence, we follow the same number line process as above but in the opposite direction (i.e., to the right side).
The number line will be represented in this way:
Therefore, 2 × 3 = 6
Division of Integers
To carry out the division operation between two integers:
 Divide the signs of the two operands and get the resultant sign.
 Divide the numbers and add the resultant sign to the quotient.
The different possible cases for the division of two signs can be observed in the following table:
Division of Signs  Result  Example 

+ ÷ +  +  12 ÷ 3 = 4 
+ ÷     12 ÷ 3 = 4 
 ÷ +    12 ÷ 3 = 4 
 ÷   +  12 ÷ 3 = 4 
Rules of Integers
Rules defined for integers are:
 Sum of two positive integers is an integer.
 Sum of two negative integers is an integer.
 Product of two positive integers is an integer.
 Product of two negative integers is an integer.
 Addition operation between any integer and its negative value will give the result as zero
 Multiplication operation between any integer and its reciprocal will give the result as one.
Integers Worksheets
Download integers worksheets, including addition and subtraction of integers, adding and subtracting multiple integers, and multiplication and division of integers.
 Integers Worksheets for Grade 7
 Integers Worksheets for Grade 6
 Adding and Subtracting Integers Worksheet Grade 5
 Adding and Subtracting Integers Worksheet Grade 8
Properties of Integers
The major Properties of Integers are:
 Closure Property
 Associative Property
 Commutative Property
 Distributive Property
 Additive Inverse Property
 Multiplicative Inverse Property
 Identity Property
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:
 a + b ∈ Z
 a  b ∈ Z
 a × b ∈ Z
 a/b ∈ Z
Associative Property:
According to the associative property, changing the grouping of two integers does not alter the result of the operation. The associative property applies to the addition and multiplication of two integers.
For any two integers, a and b:
 a + (b + c) = (a + b) + c
 a ×(b × c) = (a × b) × c
Commutative Property:
According to the commutative property, swapping the positions of 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:
 a + b = b + a
 a × b = b × a
Distributive Property:
Distributive property states that for any expression of the form a (b + c), which means a × (b + c), operand a can be distributed among operands b and c as: (a × b + a × c) i.e.,
a × (b + c) = a × b + a × c
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.
For any integer, a:
a + (a) = 0
Multiplicative Inverse Property:
The multiplicative inverse property states that the multiplication operation between any integer and it's reciprocal will give the result as one.
For any integer, a: a × 1/a = 1
Identity Property:
Integers follow the Identity property for addition and multiplication operations.
Additive identity property states that: a × 0 = a
Similarly, multiplicative identity states that: a × 1/a = 1
Solved Examples

Example 1: Can you identify the property of integer used in the following expression? 3 × 7 = 7 × 3
Solution:
The commutative property says that According to the commutative property, swapping the positions of operands in an operation does not affect the result, i.e., for the addition of two integers, we have: a × b = b × a Therefore, the property used in the given expression is commutative property.

Example 2: A plane is flying at the height of 3000 m above sea level. At some point, it is exactly above the submarine floating 700 m below the sea level. What is the vertical distance between them?
Solution:
The height at which the plane is flying = 3000 m
The depth of the submarine = 700 m (Negative as it is below the sea level)
To calculate the vertical distance between them, we will use the subtraction of two integers operation:
3000 (700) = 3000 + 700 = 3700 m
Therefore, Vertical distance between them = 3700 m
Practice Questions
FAQs on Integers
1.What is an Integer in Math?
An integer is a number with no decimal or fractional part from the set of negative and positive numbers, including zero. Examples of integers are: 5, 0, 1, 5, 8, 97, and 3,043.
2.What are the Different Types of Integers?
There are generally three types of integers:

Positive Integers: An integer is positive if it is greater than zero. Example 1, 2, 3 . . .

Negative Integers: An integer is negative if it is less than zero. Example 1, 2, 3 . . .

Zero is defined as neither negative nor positive integer.
3.Can a Negative Number be an Integer?
Yes, a negative number can also be an integer, given that it should not have a decimal or fractional part.
For example: Negative numbers: 2, 234, 71, etc are all integers.
4.What are Consecutive Integers?
The integers that follow each other in order are called consecutive integers.
For example: Numbers 2,3,4, and 5 are four consecutive integers.
5.What is the Rule for Adding a Positive and Negative Integer?
For rule for the addition of a positive and negative integer states that the difference between the two integers needs to be calculated in order to find their addition.
The sign of the result will be the same as that of the larger integer of the two.
6.What are Some Rational Numbers that are not Integers?
Rational numbers are those numbers that can be represented in the \(\frac{p}{q}\) form.
While some rational numbers that either have "1" in their denominator or can be simplified to this form, can be termed as Integers.
Whereas, those rational numbers that cannot be simplified to the form of fractions having "1" in their denominator, are nonintegers.
Rational numbers like \(\frac{2}{3}, \frac{3}{4}, \frac{1}{4}\),etc are not integers.
7.What are the Properties of Integers?
Various arithmetic operations can be performed on integers, like addition, subtraction, multiplication and division.
The major properties of integers associated with these different operations are:
 Closure Property
 Associative Property
 Commutative Property
 Distributive Property
 Additive Inverse Property
 Multiplicative Inverse Property
 Identity Property
8.What are the applications of integers?
The application of positive and negative numbers in the real world is different. They are generally used to represent two contradicting situations.
One common reallife application of integers is temperature measurement. The negative and positive numbers and zero in the scale denote different temperature readings.
Bank credit and debit statements also use integers to represent the negative or positive values of amount.