# Whole numbers

Here you will learn about whole numbers, including how to identify whole numbers, whole numbers on a number line, and the properties of whole numbers.

Students will first learn about whole numbers as part of counting and cardinality in Kindergarten and will expand their knowledge of whole numbers throughout elementary and middle school when learning about the properties of whole numbers and performing the four operations with whole numbers.

## What are whole numbers?

Whole numbers are a set of numbers starting at zero and increasing by one each time.

Whole numbers do not include fractions, decimals, or negative numbers. They are  positive integers.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10….

All whole numbers are integers, but not all integers are whole numbers since integers also include negative numbers. Both whole numbers and integers are rational numbers.

For example,

Examples of Whole NumbersExamples of Non-Whole Numbers

0 \quad \quad \quad \quad 100 \quad \quad \quad \quad 857 \quad

1,524 \quad \quad 125,031 \quad \quad 1,000,000

-5 \quad \quad \quad \quad \cfrac{1}{2} \quad \quad \quad \quad 0.75 \quad

0.\overline{88} \quad \quad \quad \;\, 2\cfrac{1}{4} \,\;\; \quad \quad \quad \quad \pi \quad \quad

### Properties of whole numbers

Commutative property of whole numbers

• The commutative property of whole numbers states that the order of two numbers being added or multiplied together does not matter and that changing the order of the numbers will still give the same result.

For example,

Commutative property of additionCommutative property of multiplication

a+b=b+a

4+5=5+4
I know this is true because
4+5=9
and
5+4=9

a \times b=b \times a

6 \times 3=3 \times 6
I know this is true because
6 \times 3=18
and
3 \times 6=18

See also: Commutative property

Associative property of whole numbers

• The associative property of whole numbers states that, when adding or multiplying three numbers, the grouping of two numbers within the expression can change and still give the same result.

For example,

Associative property of additionAssociative property of multiplication

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

(8+4)+6=8+(4+6)
I know this is true because
(8+4)+6=
12+6=18
and
8+(4+6)=
8+10=18

(a \times b) \times c=a \times(b \times c)

(2 \times 5) \times 7=2 \times(5 \times 7)
I know this is true because
(2 \times 5) \times 7=
10 \times 7=70
and
2 \times(5 \times 7)=
2 \times 35=70

See also: Associative property

Distributive property

• The distributive property of whole numbers says that multiplication is distributive over addition or subtraction. This means that when multiplying a number by a sum or difference of 2 numbers, you can multiply by each number separately and then add or subtract the products.

For example,

Distributive property of multiplication
over addition
Distributive property of multiplication
over subtraction

a(b+c)=(a \times b)+(a \times c)

\begin{aligned} 5(3+9) & =(5 \times 3)+(5 \times 9) \\ & =15+45 \\ & =60 \end{aligned}

a(b-c)=(a \times b)-(a \times c)

\begin{aligned} 8(10-1) & =(8 \times 10)-(8 \times 1) \\ & =80-8 \\ & =72 \end{aligned}

See also: Distributive property

Closure property

• The closure property of whole numbers says that the sum or product of two whole numbers will always be a whole number.

For example,

Closure property of additionClosure property of multiplication

a+b=c
If a and b are whole numbers, c will be
a whole number.

9+6=15
Since 9 and 6 are whole numbers, the
sum, 15, is also a whole number.

a \times b=c
If a and b are whole numbers, c will be
a whole number.

8 \times 4=32
Since 8 and 4 are whole numbers, the
product, 32, is also a whole number.

## Common Core State Standards

How does this relate to Kindergarten math through 6th grade math?

• Kindergarten – Counting and Cardinality (K.CC.1, K.CC.2, K.CC.3)
Count to 100 by ones and by tens; Count forward beginning from a given number within the known sequence (instead of having to begin at 1 ); Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).

• Grade 1 – Operations and Algebraic Thinking (1.0A.B.3 )
Apply properties of operations as strategies to add and subtract.
Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition).
To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition).

• Grade 2 – Operations and Algebraic Thinking (2.OA.C.3)
Determine whether a group of objects (up to 20 ) has an odd or even number of members, for example, by pairing objects or counting them by 2 s; write an equation to express an even number as a sum of two equal addends.

• Grade 3 – Operations and Algebraic Thinking (3.OA.B.5)
Apply properties of operations as strategies to multiply and divide.
Examples: If 6 \times 4 = 24 is known, then 4 \times 6 = 24 is also known. (Commutative property of multiplication).
3 \times 5 \times 2 can be found by 3 \times 5 = 15, then 15 \times 2 = 30, or by 5 \times 2 = 10, then 3 \times 10 = 30. (Associative property of multiplication).
Knowing that 8 \times 5 = 40 and 8 \times 2 = 16, one can find 8 \times 7 as 8 \times (5 + 2) = (8 \times 5) + (8 \times 2) = 40 + 16 = 56. (Distributive property).

• Grade 4 – Number and Operations Base Ten (4.NBT.B.5)
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

• Grade 6 – Number Systems (6.NS.B.4)
Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 \, (9 + 2).

## How to use whole numbers

In order to identify whole numbers:

1. Recall the definition of the type of number needed.
2. Show whether the number fits or does not fit the definition.

In order to apply a property of whole numbers:

1. Recall the property.
2. Use the property to get an answer.

## Identifying whole numbers examples

### Example 1: identifying whole numbers

Which of the following are whole numbers?

0, \, 8.5, \, -1, \, 32, \, 6 \cfrac{1}{4} \, , \, 3.05, \, 927

1. Recall the definition of the type of number needed.

Since the set of whole numbers does not include decimals, fractions, and negative numbers, you can eliminate 8.5, -1, 6 \cfrac{1}{4} \, , and 3.05 from the list.

2Show whether the number fits or does not fit the definition.

The remaining numbers are 0, 32, and 927. All three fit the definition and are whole numbers.

Answer: 0, 32, and 927

### Example 2: identifying whole numbers

Maya says -4 is a whole number since it doesn’t have a decimal or fractional part. Is she correct?

Recall the definition of the type of number needed.

Show whether the number fits or does not fit the definition.

### Example 3: identifying whole numbers

Which point on the number line represents a whole number?

Recall the definition of the type of number needed.

Show whether the number fits or does not fit the definition.

### Example 4: identifying whole numbers

Which whole number fills in the blank in the sequence?

26, \, 27, \, 28, \, \rule{0.5cm}{0.15mm} \, , \, 30, \, 31

Recall the definition of the type of number needed.

Show whether the number fits or does not fit the definition.

### Example 5: apply a property of whole numbers

Fill in the blank using your knowledge of the commutative property of multiplication to make the equation true.

\rule{0.5cm}{0.15mm} \, \times 15=15 \times 3

Recall the property.

Use the property to get an answer.

### Example 6: apply a property of whole numbers

Fill in the blank using your knowledge of the distributive property to make the equation true.

3 \times(7 + 9)= \, \rule{0.5cm}{0.15mm} \, +27

Recall the property.

Use the property to get an answer.

### Teaching tips for whole numbers

• Allow students to use concrete manipulatives to explore whole numbers when first building number sense.

• Use a number line to give students a visual representation of whole numbers. As they progress to higher grades, the number line can be partitioned into fractional and decimal parts as well, so students can see the difference between whole numbers and fractions/decimals. Later, a number line can also be extended past the number zero to show negative numbers. Students will gain better number sense when they are able to see non-examples of whole numbers.

• Display a chart or poster in the classroom showing the different types of numbers – whole numbers, natural numbers, integers, real numbers, etc. New types of numbers can be added to these displays in higher grade levels. This will help students differentiate between the sets of numbers.

### Easy mistakes to make

• Thinking that zero is not a whole number
Zero is the first and smallest whole number. The set of whole numbers begins at zero and increases by one with each number.

• Thinking that whole numbers are the same as integers
Whole numbers are a subset of integers. Integers include all negative numbers, positive numbers, and zero, while whole numbers include only non-negative integers.

This whole numbers topic guide is part of our series on types of numbers. You may find it helpful to start with the main types of numbers topic guide for a summary of what to expect or use the step-by-step guides below for further detail on individual topics. Other topic guides in this series include:

### Practice identifying whole numbers questions

1. What is the smallest whole number?

1

-6

0

\cfrac{1}{2}

The set of whole numbers starts at zero. Whole numbers do not include negative numbers, fractions, or decimals. Therefore, the smallest whole number listed is zero.

2. Look at the number line. What is the missing whole number?

19 \cfrac{1}{2}

20

0

19.5

When counting whole numbers by ones, the number after 19 will be 20.

3. Colin wrote a set of whole numbers on the whiteboard using the numbers 0, 1, 3, and 9. What number should he not have included?

93,130

0

139

1.039

1.039 should not have been included because it is a decimal, not a whole number.

4. Select the group of numbers made up of only whole numbers.

856.1, \, 30,000, \, 0, \, 15

95 \cfrac{1}{2} \, , \, 65, \, 10, \, 100

101, \, 556, \, 8,000, \, 1

0.1, \, 18, \, 20, \, 200

101, \, 556, \, 18,000, \, 1 is the only group of numbers comprised of only whole numbers. The other groups include at least one fraction or decimal.

5. Which property is demonstrated by the following equation?

5(9+8)=(5 \times 9)+(5 \times 8)

Associative property

Commutative property

Distributive property

Closure property

This shows the distributive property because multiplication is being distributed over addition. The distributive property allows you to perform the multiplication separately, then add the products.

6. Fill in the blank to make the equation true.

8 \times\left(6 \times \, \rule{0.5cm}{0.15mm} \, \right)=(8 \times 6) \times 4

4

8

6

24

This equation shows the associative property of multiplication, which states that when multiplying three numbers, the grouping of two numbers within the expression can change and still give the same result.

Therefore, since the right side shows 8, 6, and 4 being multiplied, I know the same 3 numbers are being multiplied on the left side of the equation.

## Whole numbers FAQs

What are whole numbers?

Whole numbers are a set of numbers (also known as natural numbers or counting numbers) starting at the number zero and increasing by one each time. Whole numbers do not include fractions, decimals, or negative numbers.

Are whole numbers the same as natural numbers?

Whole numbers and natural numbers are very similar but not the same. The set of natural numbers starts at one instead of zero.

What is the difference between whole numbers and integers?

Whole numbers are a subset of integers. Integers include positive whole numbers, negative whole numbers, and zero, while whole numbers only include non-negative integers.

Can a fraction ever be a whole number?

If the fraction has the same numerator and denominator, or if its numerator is a multiple of its denominator, it can be written as a whole number. For example, the fraction \cfrac{4}{2} can be written as the whole number 2.

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