What Are Whole Numbers?

Whole numbers are the numbers:

0, 1, 2, 3, 4, 5, 6, 7, ...

They include:

• zero
• positive counting numbers

Whole numbers do not include:

• negative numbers
• fractions
• decimals that are not whole values

Examples of whole numbers:

0, 7, 15, 102, 4000

Examples that are not whole numbers:

−3, 1.5, 2/3, −10.7

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Whole Numbers on a Number Line

Whole numbers can be shown on a number line.

They begin at:

0

and continue indefinitely to the right.

There is no largest whole number.

No matter how large a whole number is, we can always add:

1

to produce a larger whole number.

For example:

999 + 1 = 1000

1,000,000 + 1 = 1,000,001

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Whole Numbers and Natural Numbers

The terms whole numbers and natural numbers are closely related.

Whole numbers are normally defined as:

0, 1, 2, 3, 4, ...

Definitions of natural numbers can vary.

Some texts define natural numbers as:

1, 2, 3, 4, ...

while others include zero:

0, 1, 2, 3, 4, ...

For this topic, the important idea is that:

whole numbers include 0

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Properties of Whole Numbers

Numbers follow predictable rules when we perform operations.

These rules are called properties.

Important properties include:

• commutative property
• associative property
• distributive property
• identity property
• closure property

Understanding these properties helps us:

• simplify calculations
• rearrange expressions
• perform mental mathematics
• understand algebra
• check calculations

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The Commutative Property

The word commutative refers to changing the order.

For addition:

a + b = b + a

For multiplication:

a × b = b × a

This means changing the order of the numbers does not change the result.

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Commutative Property of Addition

Consider:

4 + 7 = 11

Reverse the order:

7 + 4 = 11

Therefore:

4 + 7 = 7 + 4

The order changed, but the answer did not.

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Another Addition Example

Consider:

18 + 25

We could calculate:

18 + 25 = 43

or:

25 + 18 = 43

Therefore:

18 + 25 = 25 + 18

Addition is commutative.

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Why Commutative Addition Is Useful

Suppose we need to calculate:

7 + 38 + 3

Instead of calculating from left to right:

7 + 38 = 45

then:

45 + 3 = 48

we can rearrange:

7 + 3 + 38

Now:

7 + 3 = 10

so:

10 + 38 = 48

The commutative property allows us to choose a more convenient order.

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Commutative Property of Multiplication

Multiplication is also commutative.

For example:

4 × 6 = 24

and:

6 × 4 = 24

Therefore:

4 × 6 = 6 × 4

An array helps us see why.

A:

4 × 6

array can be rotated to form a:

6 × 4

array.

The number of objects remains the same.

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Using Commutative Multiplication

Suppose:

25 × 8 × 4

Rearrange the factors:

25 × 4 × 8

Now:

25 × 4 = 100

so:

100 × 8 = 800

The rearrangement makes the calculation much easier.

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Does Commutative Property Work for Subtraction?

No.

Consider:

8 − 3 = 5

but:

3 − 8 = −5

Therefore:

8 − 3 ≠ 3 − 8

Subtraction is not commutative.

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Does Commutative Property Work for Division?

No.

Consider:

12 ÷ 4 = 3

but:

4 ÷ 12 = 1/3

Therefore:

12 ÷ 4 ≠ 4 ÷ 12

Division is not commutative.

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The Associative Property

The associative property deals with grouping.

For addition:

(a + b) + c = a + (b + c)

For multiplication:

(a × b) × c = a × (b × c)

Changing the grouping does not change the result.

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Commutative vs Associative

These properties are easy to confuse.

Commutative property:

changes the order

Associative property:

changes the grouping

A useful memory aid is:

Commutative → commute → move around

Associative → associate → group together

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Associative Property of Addition

Consider:

(2 + 5) + 8

First:

2 + 5 = 7

Then:

7 + 8 = 15

Now regroup:

2 + (5 + 8)

First:

5 + 8 = 13

Then:

2 + 13 = 15

Therefore:

(2 + 5) + 8 = 2 + (5 + 8)

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Using Associative Addition

Consider:

27 + 13 + 7

We can group:

27 + (13 + 7)

Since:

13 + 7 = 20

we get:

27 + 20 = 47

The associative property lets us group numbers in a convenient way.

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Associative Property of Multiplication

Consider:

(2 × 5) × 4

First:

2 × 5 = 10

Then:

10 × 4 = 40

Now regroup:

2 × (5 × 4)

First:

5 × 4 = 20

Then:

2 × 20 = 40

Therefore:

(2 × 5) × 4 = 2 × (5 × 4)

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Using Associative Multiplication

Consider:

5 × 17 × 2

Group:

(5 × 2) × 17

This uses both commutative and associative properties.

Now:

5 × 2 = 10

Then:

10 × 17 = 170

This is easier than calculating:

5 × 17 = 85

then:

85 × 2

although both methods give the same result.

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Associative Property Does Not Work for Subtraction

Consider:

(10 − 5) − 2

= 5 − 2

= 3

But:

10 − (5 − 2)

= 10 − 3

= 7

Therefore:

(10 − 5) − 2 ≠ 10 − (5 − 2)

Subtraction is not associative.

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Associative Property Does Not Work for Division

Consider:

(24 ÷ 6) ÷ 2

= 4 ÷ 2

= 2

But:

24 ÷ (6 ÷ 2)

= 24 ÷ 3

= 8

Therefore, division is not associative.

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The Distributive Property

The distributive property connects multiplication with addition or subtraction.

It states:

a(b + c) = ab + ac

and:

a(b − c) = ab − ac

The number outside the brackets is multiplied by every term inside the brackets.

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Example of the Distributive Property

Consider:

3(4 + 2)

Calculate inside the brackets:

3(6) = 18

Or distribute the 3:

3(4) + 3(2)

= 12 + 6

= 18

Therefore:

3(4 + 2) = 3(4) + 3(2)

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Why It Is Called Distributive

The multiplication is distributed across every term inside the brackets.

For:

5(8 + 3)

the 5 multiplies both:

8

and:

3

Therefore:

5(8 + 3)

becomes:

5 × 8 + 5 × 3

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Distributive Property with Subtraction

The property also works with subtraction.

For example:

6(10 − 2)

Distribute:

6 × 10 − 6 × 2

= 60 − 12

= 48

Check directly:

6(8) = 48

Both methods agree.

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Using Distribution for Mental Mathematics

The distributive property is extremely useful for mental calculations.

Consider:

7 × 19

Think of:

19 = 20 − 1

Therefore:

7 × 19

= 7(20 − 1)

= 140 − 7

= 133

This may be easier than using the standard multiplication algorithm.

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Another Mental Mathematics Example

Calculate:

8 × 27

Break 27 into:

20 + 7

Then:

8(20 + 7)

= 8 × 20 + 8 × 7

= 160 + 56

= 216

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Breaking Apart Large Numbers

Calculate:

6 × 103

Think:

103 = 100 + 3

Therefore:

6(100 + 3)

= 600 + 18

= 618

The distributive property lets us break difficult numbers into easier parts.

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Breaking Numbers Around Benchmarks

Numbers close to:

• 10
• 20
• 50
• 100
• 1000

can often be handled efficiently using distribution.

For example:

9 × 48

Think:

48 = 50 − 2

Then:

9(50 − 2)

= 450 − 18

= 432

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Area Model for the Distributive Property

Suppose a rectangle has:

height = 6

and:

width = 10 + 4

Its total area is:

6(10 + 4)

We can divide the rectangle into two smaller rectangles.

First rectangle:

6 × 10 = 60

Second rectangle:

6 × 4 = 24

Total:

60 + 24 = 84

Therefore:

6(10 + 4) = 84

This visual model explains why the distributive property works.

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Working Backwards: Factoring

The distributive property can also be used backwards.

Consider:

24 + 16

Both numbers are multiples of:

8

Therefore:

24 + 16

= 8(3 + 2)

= 8(5)

= 40

This process of taking out a common factor is called factoring.

It becomes very important in algebra.

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The Identity Property

Although the main focus is on commutative, associative, and distributive properties, two additional properties of whole numbers are useful.

The identity property describes numbers that leave another number unchanged.

For addition:

a + 0 = a

Therefore, 0 is the additive identity.

For example:

17 + 0 = 17

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Multiplicative Identity

For multiplication:

a × 1 = a

Therefore, 1 is the multiplicative identity.

For example:

42 × 1 = 42

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The Zero Property of Multiplication

Any whole number multiplied by zero equals zero.

a × 0 = 0

Examples:

5 × 0 = 0

93 × 0 = 0

1,000,000 × 0 = 0

This is called the zero property of multiplication.

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Closure Property

Whole numbers are closed under addition and multiplication.

This means that when two whole numbers are added or multiplied, the result is also a whole number.

For example:

7 + 12 = 19

19 is a whole number.

And:

7 × 12 = 84

84 is also a whole number.

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Whole Numbers Are Not Closed Under Subtraction

Consider:

3 − 8 = −5

But:

−5

is not a whole number.

Therefore, whole numbers are not closed under subtraction.

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Whole Numbers Are Not Closed Under Division

Consider:

5 ÷ 2 = 2.5

But:

2.5

is not a whole number.

Therefore, whole numbers are not closed under division.

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Combining Number Properties

The greatest benefit comes from using several properties together.

Consider:

25 × 17 × 4

Use the commutative property:

25 × 4 × 17

Use the associative property:

(25 × 4) × 17

Calculate:

100 × 17

= 1700

The original calculation has become much easier.

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Worked Example 1

Calculate efficiently:

38 + 17 + 2 + 3

Rearrange using the commutative property:

38 + 2 + 17 + 3

Group using the associative property:

(38 + 2) + (17 + 3)

Calculate:

40 + 20

= 60

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Worked Example 2

Calculate:

4 × 23 × 25

Rearrange:

4 × 25 × 23

Group:

(4 × 25) × 23

= 100 × 23

= 2300

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Worked Example 3

Calculate:

9 × 102

Use the distributive property:

9(100 + 2)

= 900 + 18

= 918

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Worked Example 4

Calculate:

12 × 49

Think:

49 = 50 − 1

Then:

12(50 − 1)

= 600 − 12

= 588

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Worked Example 5

Calculate:

125 × 16

Break 16 into:

8 × 2

Then:

125 × 8 × 2

Group:

(125 × 8) × 2

= 1000 × 2

= 2000

Recognizing useful number combinations can greatly simplify calculations.

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Worked Example 6

Calculate:

37 + 46 + 63 + 54

Rearrange:

37 + 63 + 46 + 54

Group:

(37 + 63) + (46 + 54)

= 100 + 100

= 200

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Worked Example 7

Calculate:

15 × 98

Think:

98 = 100 − 2

Then:

15(100 − 2)

= 1500 − 30

= 1470

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Worked Example 8

Calculate:

32 × 11

Think:

11 = 10 + 1

Then:

32(10 + 1)

= 320 + 32

= 352

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Worked Example 9

Calculate:

7 × 48 + 7 × 52

Notice the common factor:

7

Factor:

7(48 + 52)

= 7(100)

= 700

This uses the distributive property backwards.

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Worked Example 10

Calculate:

18 × 25 + 2 × 25

Both terms contain:

25

Factor:

(18 + 2) × 25

= 20 × 25

= 500

This is much faster than calculating both products separately.

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Properties and Algebra

These number properties become extremely important when learning algebra.

For example:

3(x + 4)

can be expanded using the distributive property:

3x + 12

Similarly:

x + 5 + 3

can be regrouped as:

x + (5 + 3)

which becomes:

x + 8

The properties learned with whole numbers continue to work in much more advanced mathematics.

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Properties and Mental Mathematics

Strong mental mathematics often involves looking for useful patterns.

For addition, look for pairs that make:

10, 20, 50, 100, 1000

For multiplication, look for combinations such as:

2 × 5 = 10

4 × 25 = 100

8 × 125 = 1000

Then use the commutative and associative properties to put those numbers together.

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Real-World Example: Shopping

Suppose four items cost:

$27, $13, $18, and $12

Instead of adding in order:

27 + 13 + 18 + 12

group convenient pairs:

(27 + 13) + (18 + 12)

= 40 + 30

= $70

Number properties make the calculation easier.

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Real-World Example: Packing Boxes

Suppose there are:

24 boxes

with:

15 items per box

Calculate:

24 × 15

Break 15 into:

10 + 5

Then:

24(10 + 5)

= 240 + 120

= 360

There are:

360 items

in total.

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Real-World Example: Seating

A theatre has:

18 rows

with:

49 seats per row

Instead of calculating 18 × 49 directly:

18(50 − 1)

= 900 − 18

= 882

Therefore:

882 seats

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Recognizing the Property

Consider:

6 + 9 = 9 + 6

The order changes.

Property:

Commutative

---

Consider:

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

The grouping changes.

Property:

Associative

---

Consider:

5(10 + 2) = 5(10) + 5(2)

Multiplication is spread across addition.

Property:

Distributive

---

Consider:

8 + 0 = 8

Property:

Additive identity

---

Consider:

8 × 1 = 8

Property:

Multiplicative identity

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A Quick Identification Strategy

Ask yourself:

Did the order change?

→ Commutative

Did the grouping change?

→ Associative

Was multiplication spread across terms?

→ Distributive

Was 0 added?

→ Additive identity

Was the number multiplied by 1?

→ Multiplicative identity

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Common Mistakes

Mistake 1: Thinking subtraction is commutative

Incorrect:

8 − 3 = 3 − 8

These expressions are not equal.

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Mistake 2: Thinking division is commutative

Incorrect:

12 ÷ 3 = 3 ÷ 12

Changing the order changes the answer.

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Mistake 3: Confusing commutative and associative properties

Commutative changes:

order

Associative changes:

grouping

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Mistake 4: Distributing to only one term

Incorrect:

5(3 + 7) = 15 + 7

Correct:

5(3 + 7) = 15 + 35

The 5 must multiply both terms.

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Mistake 5: Thinking whole numbers include negative numbers

Negative integers are not whole numbers.

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Mistake 6: Thinking all operations are closed for whole numbers

Whole numbers are closed under:

addition and multiplication

but not always under:

subtraction and division

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Did You Know?

The number properties are not just rules for elementary arithmetic.

They form part of the foundation of algebra.

When you:

• rearrange terms
• expand brackets
• factor expressions
• simplify equations
• manipulate formulas

you are often using the same commutative, associative, and distributive properties learned with whole numbers.

Understanding why these properties work is more powerful than simply memorizing their names.

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Key Terms

• Whole number: Zero or any positive counting number.
• Property: Rule describing how numbers behave under an operation.
• Operation: Mathematical process such as addition, subtraction, multiplication, or division.
• Commutative property: Changing the order does not change the result.
• Associative property: Changing the grouping does not change the result.
• Distributive property: Multiplication can be distributed across addition or subtraction.
• Additive identity: Zero, because adding zero leaves a number unchanged.
• Multiplicative identity: One, because multiplying by one leaves a number unchanged.
• Zero property of multiplication: Any number multiplied by zero equals zero.
• Closure: Property describing whether an operation on members of a set always produces another member of that set.
• Factor: Number multiplied by another number.
• Term: Part of an expression separated by addition or subtraction.
• Expression: Mathematical combination of numbers, operations, and sometimes variables.
• Factoring: Rewriting an expression by taking out a common factor.

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Key Rules

Whole numbers:

0, 1, 2, 3, 4, ...

Commutative addition:

a + b = b + a

Commutative multiplication:

ab = ba

Associative addition:

(a + b) + c = a + (b + c)

Associative multiplication:

(ab)c = a(bc)

Distributive property:

a(b + c) = ab + ac

a(b − c) = ab − ac

Additive identity:

a + 0 = a

Multiplicative identity:

a × 1 = a

Zero property:

a × 0 = 0

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Key Takeaways

• Whole numbers are 0, 1, 2, 3, 4, ...
• Whole numbers do not include negative numbers, fractions, or non-whole decimals.
• There is no largest whole number.
• Mathematical properties describe predictable ways that numbers behave.
• The commutative property allows the order of numbers to change in addition and multiplication.
• Addition and multiplication are commutative.
• Subtraction and division are not commutative.
• The associative property allows the grouping of numbers to change in addition and multiplication.
• Addition and multiplication are associative.
• Subtraction and division are not associative.
• The distributive property connects multiplication with addition and subtraction.
• Distribution requires multiplying every term inside the brackets.
• The distributive property can be used to break difficult calculations into easier ones.
• The distributive property can also be used backwards through factoring.
• Zero is the additive identity.
• One is the multiplicative identity.
• Any whole number multiplied by zero equals zero.
• Whole numbers are closed under addition and multiplication.
• Whole numbers are not always closed under subtraction or division.
• Several properties can be combined to simplify a single calculation.
• Useful mental-math strategies include rearranging numbers to make 10, 100, or 1000.
• These same number properties become essential when simplifying algebraic expressions and solving equations.