Integers: Definition, Types, Rules, Properties and Examples

What Are Integers?

Integers are numbers that include positive whole numbers, negative whole numbers, and zero. They do not include fractions or decimals.

The set of integers is written as:

Z = {..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...}

For example:

-10, -3, -1, 0, 4, 8 and 25 are integers.

Numbers such as 1/2, 3.5 and -2.75 are not integers because they are fractions or decimals.

Integers are used to represent quantities above and below a reference point, such as temperature, profit and loss, elevation, and changes in scores.

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Integer Meaning and Definition

An integer is a number with no fractional or decimal part. Integers can be:

• Positive

• Negative

• Zero

Examples of integers:

-8, -2, 0, 5, 12, 100

Examples that are not integers:

2.5, -3.7, 1/4, 7/9

A simple way to identify an integer is to check whether the number can be written without a fractional or decimal part.

Check: Consecutive Numbers: Meaning, Types, Formulas and Examples

Types of Integers

Integers can be grouped into three categories:

1. Positive integers

2. Negative integers

3. Zero

Positive Integers

Positive integers are integers greater than zero.

Examples:

1, 2, 3, 4, 5, 6, ...

They are written on the right side of zero on a number line.

Positive integers are commonly used for quantities such as the number of books, scores, floors above ground level, or money received.

Negative Integers

Negative integers are integers less than zero.

Examples:

-1, -2, -3, -4, -5, ...

They are written on the left side of zero on a number line.

Negative integers can represent situations such as temperatures below zero, losses, debts, or positions below a reference level.

Zero

0 is an integer.

Zero is neither positive nor negative.

It lies between the positive and negative integers on a number line.

For example:

-2 < -1 < 0 < 1 < 2

Zero is also the additive identity, because adding zero to any integer leaves the number unchanged:

7 + 0 = 7

How Many Types of Integers Are There?

At the school level, integers are commonly grouped into three types:

• Positive integers

• Negative integers

• Zero

For example:

Type	Examples
Positive integers	1, 2, 3, 4, 5
Zero	0
Negative integers	-1, -2, -3, -4, -5

Together, these numbers form the set of integers.

Integers on a Number Line

A number line is one of the easiest ways to understand integers.

• Positive integers are to the right of 0.

• Negative integers are to the left of 0.

• 0 is in the middle.

• Moving right means the values increase.

• Moving left means the values decrease.

For example:

5 > 2

because 5 is to the right of 2.

Similarly:

-6 < -2

because -6 is to the left of -2.

Comparing Negative Integers

Negative integers can sometimes be confusing.

Consider:

-3 and -8

Although 8 is greater than 3 as a positive number, -3 is greater than -8.

On a number line, -3 is to the right of -8.

Therefore:

-3 > -8

A useful rule is:

Among negative integers, the number closer to zero is greater.

Also, check: Successor and Predecessor in Maths: Meaning, Definition, Examples and Rules

Opposite Integers

Two integers are called opposites if they have the same distance from zero but lie on opposite sides of zero.

Examples:

• 5 and -5

• 8 and -8

• 12 and -12

Their sum is zero:

5 + (-5) = 0

-8 + 8 = 0

The opposite of zero is zero itself.

Rules of Integers

Integer rules help us determine the sign and value of a result when adding, subtracting, multiplying, or dividing integers.

Rules for Adding Integers

When adding integers with the same sign:

• Add their absolute values.

• Keep the common sign.

Examples:

5 + 3 = 8

-5 + (-3) = -8

When adding integers with different signs:

• Find the difference between their absolute values.

• Keep the sign of the number with the greater absolute value.

Example:

8 + (-3) = 5

Example:

-8 + 3 = -5

Rules for Subtracting Integers

A useful rule is:

Subtracting an integer is the same as adding its opposite.

So:

a - b = a + (-b)

Examples:

7 - 3 = 7 + (-3) = 4

7 - (-3) = 7 + 3 = 10

-5 - 2 = -5 + (-2) = -7

-5 - (-2) = -5 + 2 = -3

Rules for Multiplying Integers

When multiplying two integers:

Signs	Result
Positive × Positive	Positive
Positive × Negative	Negative
Negative × Positive	Negative
Negative × Negative	Positive

Examples:

4 × 3 = 12

4 × (-3) = -12

(-4) × 3 = -12

(-4) × (-3) = 12

Rules for Dividing Integers

The same sign rule applies to division:

Signs	Result
Positive ÷ Positive	Positive
Positive ÷ Negative	Negative
Negative ÷ Positive	Negative
Negative ÷ Negative	Positive

Examples:

12 ÷ 3 = 4

12 ÷ (-3) = -4

(-12) ÷ 3 = -4

(-12) ÷ (-3) = 4

Remember that division by zero is not defined.

Integer Sign Rules

A quick way to remember the multiplication and division sign rules is:

Same signs → Positive

Different signs → Negative

For example:

This rule applies to both multiplication and division.

Properties of Integers

Integers follow several important properties under arithmetic operations.

The main properties commonly studied are:

1. Closure property

2. Commutative property

3. Associative property

4. Distributive property

5. Additive identity

6. Additive inverse

Closure Property

A set is closed under an operation when performing that operation on members of the set gives another member of the same set.

For integers:

• Addition: closed

• Subtraction: closed

• Multiplication: closed

• Division: not closed

Addition

(-8) + 4 = -4

The result is an integer.

Subtraction

7 - 12 = -5

The result is an integer.

Multiplication

(-10) × 3 = -30

The result is an integer.

Division

15 ÷ 4 = 3.75

The result is not an integer.

Therefore, integers are not closed under division.

Commutative Property

The commutative property means that changing the order of numbers does not change the result.

It applies to addition and multiplication.

Addition

(-5) + 8 = 8 + (-5) = 3

Multiplication

6 × (-3) = (-3) × 6 = -18

The property does not apply to subtraction or division.

For example:

8 - 3 = 5

but

3 - 8 = -5

Therefore, subtraction is not commutative.

Associative Property

The associative property means that changing the grouping of numbers does not change the result.

It applies to addition and multiplication.

Addition

3 + (4 + 6) = (3 + 4) + 6

Both sides equal:

13

Multiplication

(-2) × (5 × 3) = ((-2) × 5) × 3

Both sides equal:

-30

The associative property does not apply to subtraction or division.

Distributive Property

The distributive property connects multiplication with addition or subtraction.

For addition:

a × (b + c) = (a × b) + (a × c)

For subtraction:

a × (b - c) = (a × b) - (a × c)

Example:

5 × (3 + 7)

= (5 × 3) + (5 × 7)

= 15 + 35

= 50

Another example:

(-4) × (8 - 2)

= (-4 × 8) - (-4 × 2)

= -32 - (-8)

= -24

Additive Identity Property

Zero is the additive identity for integers.

Adding zero to any integer gives the same integer.

a + 0 = a

Examples:

9 + 0 = 9

-15 + 0 = -15

Additive Inverse Property

Every integer has an additive inverse.

The additive inverse of a number is the number that gives zero when added to it.

For any integer a:

a + (-a) = 0

Examples:

9 + (-9) = 0

-15 + 15 = 0

The additive inverse of:

• 7 is -7

• -12 is 12

• 0 is 0

Multiplicative Identity

The multiplicative identity is 1 because multiplying an integer by 1 gives the same integer.

a × 1 = a

Examples:

8 × 1 = 8

-6 × 1 = -6

Is Multiplicative Inverse a Property of Integers?

The reciprocal relationship is important in mathematics, but it should not be treated as a closure property of integers.

For example:

5 × 1/5 = 1

The reciprocal of 5 is 1/5, which is not an integer.

Also, 0 has no multiplicative inverse because division by zero is not defined.

Therefore, unlike addition, subtraction, and multiplication, integers are not closed under taking multiplicative inverses.

Integer Properties at a Glance

Property	Addition	Subtraction	Multiplication	Division
Closure	Yes	Yes	Yes	No
Commutative	Yes	No	Yes	No
Associative	Yes	No	Yes	No
Distributive	Not applicable in this form	Not applicable in this form	Yes, over addition/subtraction	No

Students can practise integers, number lines, and arithmetic concepts through interactive maths learning resources on CuriousJr.

Real-Life Uses of Integers

Integers are useful whenever quantities can be represented as whole-number values above or below a reference point.

1. Temperature

Temperatures can be represented using positive and negative integers.

For example:

5°C is above zero, while -5°C is below zero.

2. Money and Debt

Positive and negative integers can represent financial changes.

For example:

• Receiving ₹500 can be represented as +500.

• Owing ₹200 can be represented as -200.

The exact representation depends on the situation and the convention being used.

3. Elevation

Integers can represent locations above or below a reference level.

For example:

• +300 m can represent a location 300 m above sea level.

• -20 m can represent a location 20 m below sea level.

4. Sports

Integers can represent scores, point changes, or penalties when the scoring system allows negative values.

For example:

A team with a score change of -2 has lost two points under that scoring system.

5. Floors of a Building

Integers can be used to label floors above and below ground level.

For example:

• Floor 3 = +3

• Basement 1 = -1

10 Examples of Integers in Daily Life

Here are simple examples of situations where integers can be used:

1. +10°C temperature above zero

2. -5°C temperature below zero

3. +₹500 received

4. -₹200 debt or loss

5. +3 floors above ground

6. -2 basement levels

7. +8 points gained

8. -3 points lost

9. +100 m above a reference level

10. -50 m below a reference level

The important point is that the integer represents a whole-number value with a sign.

Integers Examples

Example 1: Identify the Integers

Which of these numbers are integers?

-7, 0, 3.5, 8, 1/2, -12

The integers are:

-7, 0, 8 and -12

The numbers 3.5 and 1/2 are not integers.

Example 2: Add Two Integers

Find:

-8 + 13

Since the signs are different, find the difference:

13 - 8 = 5

The larger absolute value is 13, which is positive.

Answer: 5

Example 3: Subtract Two Integers

Find:

6 - (-4)

Change subtraction into addition:

6 + 4 = 10

Answer: 10

Example 4: Multiply Two Negative Integers

Find:

(-7) × (-3)

Two negative signs give a positive result:

7 × 3 = 21

Answer: 21

Example 5: Divide Integers

Find:

(-24) ÷ 6

Different signs give a negative result:

24 ÷ 6 = 4

Therefore:

Answer: -4

Example 6: Use the Additive Inverse

Find the additive inverse of -15.

The opposite of -15 is 15.

Check:

-15 + 15 = 0

Answer: 15

Example 7: Compare Two Negative Integers

Which is greater: -4 or -9?

On the number line, -4 is to the right of -9.

Therefore:

-4 > -9

Example 8: Temperature Change

The temperature is 12°C and falls by 5°C.

Represent the change as -5:

12 + (-5) = 7

Answer: 7°C

Example 9: Submarine Depth

A submarine starts at sea level and descends 6 metres per second. How far below sea level is it after 4 seconds?

Represent downward movement as negative:

(-6) × 4 = -24

Answer: The submarine is 24 metres below sea level.

Example 10: Loss Over Several Months

A company records a total loss of ₹48,000 over 6 months. If the loss is evenly distributed:

(-48,000) ÷ 6 = -8,000

Answer: The loss is ₹8,000 per month.

Consecutive Integers

Consecutive integers are integers that follow one another without gaps.

Examples:

3, 4, 5, 6

and

-4, -3, -2, -1

are sets of consecutive integers.

The difference between consecutive integers is always 1.

For example:

7, 8, 9

are consecutive integers.

Non-Consecutive Integers

Non-consecutive integers have one or more integers between them.

For example:

2, 5, 8

are non-consecutive because there are other integers between the numbers.

Integers vs Whole Numbers

Feature	Integers	Whole Numbers
Positive numbers	Yes	Yes
Zero	Yes	Yes
Negative numbers	Yes	No
Fractions	No	No
Decimals	No	No
Examples	-3, -1, 0, 2, 5	0, 1, 2, 3, 5

Every whole number is an integer, but every integer is not a whole number.

For example:

-4 is an integer but not a whole number.

Integers vs Natural Numbers

Natural numbers are commonly used for counting. Depending on the textbook or convention, they may begin with 1 or include 0.

Integers include both positive and negative whole numbers as well as zero.

For example:

-3 is an integer but is not a natural number.

Because definitions of natural numbers can vary, students should follow the convention used in their textbook or syllabus.

Integers vs Fractions and Decimals

Integers do not contain a fractional or decimal part.

Examples of integers:

-5, -1, 0, 7, 20

Examples that are not integers:

1/2, 2.5, -3.75

A decimal such as 5.0 represents the same value as 5, so its value is an integer even though it is written in decimal notation.

Common Mistakes While Working With Integers

Mistake 1: Thinking every negative number is smaller in absolute value

Compare:

-2 and -8

The correct relationship is:

-2 > -8

because -2 is closer to zero.

Mistake 2: Forgetting the sign during subtraction

For:

5 - (-3)

do not subtract 3.

Instead:

5 + 3 = 8

Mistake 3: Assuming two negative numbers always give a negative product

For multiplication:

(-4) × (-2) = +8

Two negative signs give a positive result.

Mistake 4: Including fractions as integers

3/4 is not an integer.

Mistake 5: Treating zero as positive or negative

Zero is neither positive nor negative.

Mistake 6: Assuming division is closed for integers

7 ÷ 2 = 3.5

Since 3.5 is not an integer, integers are not closed under division.

Quick Revision of Integers

• Integers include positive numbers, negative numbers, and zero.

• Integers do not include fractions or non-integer decimals.

• The integer set is represented by Z.

• Zero is neither positive nor negative.

• Positive integers lie to the right of zero.

• Negative integers lie to the left of zero.

• Integers are closed under addition, subtraction, and multiplication.

• Integers are not closed under division.

• Addition and multiplication are commutative and associative.

• Multiplication is distributive over addition and subtraction.

• Every integer has an additive inverse.

• Zero is the additive identity.

• One is the multiplicative identity.

• For multiplication and division, same signs give positive results and different signs give negative results.