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.

Table of Contents

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, ...}

Definition of integers


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.

Integers on a number line


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.

Addition of Integers: Adding integers using the number line

The next number in the given problem is -10, which is negative.

We move (from the fifth unit) 10 units to the left side.

Addition of Integers: Adding integers using the number line

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.

Multiplication of Integers: Multiplying integers using the number line

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:

Multiplication of Integers: Multiplying integers using the number line

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


Important Topics

Given below is the list of topics that are closely connected to integers. These topics will also give you a glimpse of how such concepts are covered in Cuemath.

FAQs on Integers

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, 34, etc.

What are the Different Types of Integers?

There are generally three types of integers: Positive Integers (example 1, 2, 3 . . .), Negative Integers (example -1, -2, -3 . . .), and Zero (neither negative nor positive integer)

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.

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.

What is the Rule for Adding a Positive and Negative Integer?

The 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.

What are Some Rational Numbers that are not Integers?

Rational numbers are those numbers that can be represented in the p/q form. While some rational numbers that are either unit fractions or can be simplified to the form of unit fractions are termed as Integers. Whereas, those rational numbers that cannot be simplified to unit fractions and have a fractional part, are non-integers.

Therefore, rational numbers like 2/3, -3/4, -1/4, etc are not integers.

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 are:

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

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 real-life 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 the amount.

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?

Integers problem example

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

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 
Download Free Grade 10 Worksheets
Arithmetic Integers
grade 10 | Questions Set 2
Arithmetic Integers
grade 10 | Answers Set 2
Arithmetic Integers
grade 10 | Questions Set 1
Arithmetic Integers
grade 10 | Answers Set 1
Download Free Grade 6 Worksheets
An Introduction to Integers
Grade6 | Worksheet 1
An Introduction to Integers
Grade6 | Worksheet 2
Download Free Grade 7 Worksheets
Integers-A Consolidation
Grade7 | Worksheet 2
Integers-A Consolidation
Grade7 | Worksheet 1
Download Free Grade 9 Worksheets
Numbers and Number Systems
grade 9 | Questions Set 2
Numbers and Number Systems
grade 9 | Answers Set 2
Numbers and Number Systems
grade 9 | Answers Set 1
Numbers and Number Systems
grade 9 | Questions Set 1
Download Numbers and Number Systems Worksheets
Numbers and Number Systems
grade 9 | Answers Set 2
Numbers and Number Systems
grade 9 | Questions Set 1
  
More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus