The associative property of addition is [katex] (a + b) + c = a + (b + c) [/katex]

The commutative property of multiplication is [katex] a \times b = b \times c [/katex]

The distributive property is [katex] a \, (b + c) = ab + ac [/katex]

Natural Numbers - Math Steps, Examples & Questions

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Natural numbers

Here you will learn about natural numbers, including the types of natural numbers and properties of natural numbers.

Students will first learn about natural numbers in Kindergarten with counting and cardinality and extend that knowledge through elementary school into middle school with properties of addition and multiplication.

What are natural numbers?

Natural numbers help you count and represent objects or quantities. They are also called “counting numbers”. The set of natural numbers, usually represented by $N,$ can be defined as positive whole numbers beginning with $1.$

N = \{1, 2, 3, …\}

This number line shows the first $7$ natural numbers.

Let’s look at some types of natural numbers and properties of natural numbers.

Prime and Composite Natural Numbers $\hspace{1cm}$


A prime number is a number with exactly two factors: itself and $1.$ $2$ and $7$ are examples of prime numbers. $2$ has two factors: $\hspace{1cm}$ $7$ has two factors: $1, 2 \hspace{3cm} 1, 7$ $1 \times 2 = 2 \hspace{2.2cm} 1 \times 7 = 7$	A composite number is a number with more than two factors. $4$ and $12$ are examples of composite numbers. $4$ has three factors: $\hspace{1cm}$ $12$ has six factors: $1, 2, 4 \hspace{2.9cm} 1, 2, 3, 4, 6, 12$ $1 \times 4 = 4 \hspace{2.5cm} 1 \times 12 = 12$ $2 \times 2 = 4 \hspace{2.7cm} 2 \times 6 = 12$ $\hspace{4cm} 3 \times 4 = 12$

A prime number is a number with exactly
two factors: itself and $1.$

$2$ and $7$ are examples of prime numbers.

$2$ has two factors: $\hspace{1cm}$ $7$ has two factors:
$1, 2 \hspace{3cm} 1, 7$
$1 \times 2 = 2 \hspace{2.2cm} 1 \times 7 = 7$

A composite number is a number with more
than two factors.

$4$ and $12$ are examples of composite numbers.

$4$ has three factors: $\hspace{1cm}$ $12$ has six factors:
$1, 2, 4 \hspace{2.9cm} 1, 2, 3, 4, 6, 12$
$1 \times 4 = 4 \hspace{2.5cm} 1 \times 12 = 12$
$2 \times 2 = 4 \hspace{2.7cm} 2 \times 6 = 12$
$\hspace{4cm} 3 \times 4 = 12$

Step-by-step guide: Prime number

Even and Odd Natural Numbers $\hspace{1cm}$


Even numbers are divisible by $2$ without remainders; they end in $0, 2, 4, 6,$ or $8.$ Every other number on the number line is even.	Odd numbers are not divisible by $2$ without remainders and end in $1, 3, 5, 7,$ or $9.$ Every other number on the number line is odd.

Even numbers are divisible by $2$ without
remainders; they end in $0, 2, 4, 6,$ or $8.$

Every other number on the number line
is even.

Odd numbers are not divisible by $2$ without
remainders and end in $1, 3, 5, 7,$ or $9.$

Every other number on the number line
is odd.

There are four properties of natural numbers.

$\bf{1}$ . Closure Property of Natural Numbers
The sum and product of two natural numbers are also natural numbers. For example, $6 + 18 = 24 \;$ $7 \times 12 = 84 \;$ The closure property does not work all the time for subtraction and division. For example, $6-18 = -12 \;$ $7 \div 12 = \cfrac{7}{12} \;$

$\bf{1}$ . Closure Property of Natural Numbers

The sum and product of two natural numbers are also natural numbers.

For example,

6 + 18 = 24 \;

7 \times 12 = 84 \;

The closure property does not work all the time for subtraction and division.

For example,

6-18 = -12 \;

7 \div 12 = \cfrac{7}{12} \;

$\bf{2}$ . Associative Property of Natural Numbers
The sum and product of three natural numbers are the same even if the numbers are grouped differently. $→ (a + b) + c = a + (b + c) \; \& \; (a \times b) \times c = a \times (b \times c)$ For example, $(1 + 3) + 5 = 9$ $1 + (3 + 5) = 9 \;$ $(1 \times 3) \times 5 = 15$ $1 \times (3 \times 5) = 15 \;$ The associative property does not work all the time for subtraction and division. $(6 -1)-3$ ≠ $6- (1-3) \;$ $(6 \div 1) \div 3$ ≠ $6 \div (1 \div 3) \;$

$\bf{2}$ . Associative Property of Natural Numbers

The sum and product of three natural numbers are the same even if the numbers
are grouped differently. $→ (a + b) + c = a + (b + c) \; \& \; (a \times b) \times c = a \times (b \times c)$

For example,

$(1 + 3) + 5 = 9$

1 + (3 + 5) = 9 \;

$(1 \times 3) \times 5 = 15$

1 \times (3 \times 5) = 15 \;

The associative property does not work all the time for subtraction and division.

(6 -1)-3

≠

6- (1-3) \;

(6 \div 1) \div 3

≠

6 \div (1 \div 3) \;

Step by step guide: Associative Property

$\bf{3}$ . Commutative Property of Natural Numbers
The sum and product of two natural numbers is the same even if the order is different. $→ a + b = b + a \; \& \; a \times b = b \times a$ For example, $7 + 9 = 16$ and $9 + 7 = 16 \;$ $7\times 9 = 63$ and $9 \times 7 = 63 \;$ The commutative property does not work all the time for subtraction and division. $7-9$ ≠ $9-7$ $7 \div 9$ ≠ $9 \div 7 \;$

$\bf{3}$ . Commutative Property of Natural Numbers

The sum and product of two natural numbers is the same even if the order is
different. $→ a + b = b + a \; \& \; a \times b = b \times a$

For example,

7 + 9 = 16

and

9 + 7 = 16 \;

7\times 9 = 63

and

9 \times 7 = 63 \;

The commutative property does not work all the time for subtraction and division.

7-9

≠

9-7

7 \div 9

≠

9 \div 7 \;

Step-by-step guide: Commutative Property

$\bf{4}$ . Distributive Property of Natural Numbers
For three natural numbers, multiplication is distributive over addition and subtraction meaning that $a \times (b + c) = a \times b + a \times c \; \& \; a \times (b-c) = a \times b-a \times c$ For example,

Step-by-step guide: Distributive Property

What are natural numbers?

Common Core State Standards

How does this apply to kindergarten through 6th grade?

Kindergarten – Counting and Cardinality (K.CC.B.4)
Understand the relationship between numbers and quantities; connect counting to cardinality.

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, e.g., 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 natural numbers

In order to classify natural numbers:

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

In order to apply a property of natural numbers:

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

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Identifying natural numbers examples

Example 1: identifying natural numbers

Is the number identified on the number line a natural number?

Recall the definition of the type of number needed.

Natural numbers are whole positive numbers beginning with $1.$

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

The number identified on the number line is $\, \cfrac{2}{5} \,$ which is a fraction.

So, the number identified on the number line is not a natural number.

Example 2: identifying types of natural numbers

Is $13$ an odd natural number?

Recall the definition of the type of number needed.

An odd number is not divisible by $2$ without a remainder. Odd numbers end in $1, 3, 5, 7, 9.$

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

$13$ is not divisible by $2$ without a remainder and ends in a $3.$

So, $13$ is an odd number.

Example 3: identifying types of natural numbers

Is $19$ a prime number?

Recall the definition of the type of number needed.

A prime number is a number with exactly two factors: itself and $1.$

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

The factors of $19$ are $1$ and $19.$

So, $19$ is a prime number.

Example 4: apply a property of natural numbers

Find the missing number using the associative property of addition.

$(5 + 2) + \, \rule{0.5cm}{0.15mm} \, = 5 + (2 + 9)$

Recall the property.

The associative property of addition is $(a + b) + c = a + (b + c)$

Use the property to get an answer.

$(5 + 2) + {\color{blue} 9} = 5 + (2 + 9)$

$9$ is the missing number because $(5 + 2) + 9 = 5 + (2 + 9) = 16$

Example 5: apply a property of natural numbers

Find the missing number using the commutative property of multiplication.

$19 \times 12 = 12 \times \, \rule{0.5cm}{0.15mm}$

Recall the property.

The commutative property of multiplication is $a \times b = b \times c$

Use the property to get an answer.

$19 \times 12 = 12 \times {\color{blue} 19}$

$19$ is the missing number because $19 \times 12 = 228$ and $12 \times 19 = 228$

Example 6: properties of natural numbers

Find the missing number in the distributive property equation.

$8 \times (5 + 3) = \, \rule{0.5cm}{0.15mm} \, + 24$

Recall the property.

The distributive property is $a \, (b + c) = ab + ac$

Use the property to get an answer.

$8 \times (5 + 3) = {\color{blue} 40} + 24$

$\color{blue} 40$ is the missing number.

Teaching tips for natural numbers lessons

Natural numbers are students’ first experience with numbers because they are the counting numbers. When first teaching the concept of numbers to young mathematicians, associate the actual numeral with the correct number of dots so that students can associate the numeral with a quantity.

Use a number line so students can visually see a representation of the numbers.

To build number sense, expose students to number patterns.

Use a Venn diagram when teaching students in middle school the real number system. Using the diagram will help students visualize how whole numbers, integers, and rational numbers (all the number sets) build upon the natural numbers.

Easy mistakes to make

Thinking $\bf{0}$ is a natural number
Natural numbers are counting numbers and begin with the number $1.$
$0$ is not part of the natural number set.

Thinking all integers are natural numbers
Only the non-negative integers are natural numbers not including $0.$

Thinking that natural numbers are not part of integers
Natural numbers are a subset of the integers because they represent the positive integers.

This natural 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 natural numbers

0

1.03

15

\cfrac{4}{7}

Natural numbers are the set of whole positive numbers that start at $1.$

$15$ is a positive whole number so it is a natural number.

0, 1, 2

1, 2, 3

-3, -2, -1

-2, -1, 0

Natural numbers are the set of whole positive numbers that start at $1.$

So, $1, 2, 3$ are natural numbers.

2, 3, 7

3, 7, 9

1, 5, 9

-2, -3, -9

A prime number is a number that has exactly two factors, $1$ and itself.

$2, 3, 7$ represent prime numbers because:

The factors of $2$ are $1$ and $2.$

The factors of $3$ are $1$ and $3.$

The factors of $7$ are $1$ and $7.$

21

16

10

5

The associative property is $(a + b) + c = a + (b + c).$

So, $(6 + 5) + 10 = 6 + (5 + 10)$

$(6 + 5) + 10 = 21$

$6 + (5 + 10) = 21$

12

96

12

8

The commutative property is $a \times b = b \times a.$

So, $12 \times 8 = 8 \times 12.$

$12 \times 8 = 96$

$8 \times 12 = 96$

9

36

13

4

The distributive property is $a \times (b + c) = a \times b + a \times c.$

So, $4 \times (2 + 9) = 8 + 36 = 44$

Natural numbers FAQs

Is $\bf{0}$ a natural number?

No, natural numbers are positive whole numbers that start with $1.$

Are all positive numbers natural numbers?

No, only positive whole numbers are natural numbers. Positive fractions and decimals are not natural numbers.

Are all integers natural numbers?

No, only the non-negative integers starting with $1$ are natural numbers.

What are cardinal numbers?

Cardinal numbers are natural numbers used for counting. They are countable numbers.

What are ordinal numbers?

Ordinal numbers are natural numbers used for ordering objects such as $1$ st, $2$ nd, $3$ rd, etc…

Are rational numbers natural numbers?

Only the positive integers are rational numbers, not including $0.$ Negative numbers are not natural numbers, and fractions and decimals are not natural numbers. Natural numbers are a subset of rational numbers.

Why is set theory important?

Set theory serves as a foundation for everything that is done in mathematics because it builds concepts of numbers.

The next lessons are

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