Table of Contents
Natural numbers are usually the first kind of numbers we learn to work with. They are used for counting and simple ordering: how many objects there are, or which comes first, second, third, and so on.
In this chapter, we look at what natural numbers are, how they are written, and some of their most basic properties.
What are natural numbers?
Natural numbers are the numbers used for counting whole objects.
Some textbooks define the set of natural numbers as
$$\{1, 2, 3, 4, 5, \dots\}$$
Others include $0$ as a natural number and write
$$\{0, 1, 2, 3, 4, 5, \dots\}$$
Both conventions appear in mathematics. When it matters, people usually say clearly whether $0$ is included or not. In this course, whenever the inclusion of $0$ is important, it will be stated explicitly.
We often use the symbol $\mathbb{N}$ to denote the set of natural numbers. For example, we might write:
- $\mathbb{N} = \{1, 2, 3, \dots\}$ (if $0$ is not included), or
- $\mathbb{N}_0 = \{0, 1, 2, 3, \dots\}$ to emphasize that $0$ is included.
A number that is in $\mathbb{N}$ is called a natural number.
Counting and ordering
Natural numbers answer questions like:
- “How many apples are there?” (counting)
- “Which person is first in line?” (ordering)
If you place all natural numbers in order from smallest to largest, you get:
$$1, 2, 3, 4, 5, 6, \dots$$
This list never ends. We say that the natural numbers go on without bound or that they are infinite. No matter how large a natural number you pick, you can always add $1$ to get a larger one.
We can compare two natural numbers to see which is larger or smaller:
- $3 < 7$ (“3 is less than 7”)
- $10 > 4$ (“10 is greater than 4”)
- $5 = 5$ (“5 is equal to 5”)
These comparison symbols ($<$, $>$, $=$) help us describe the order of natural numbers.
Numerals and place value
The number itself is an abstract idea (for example, the idea of “seven”). A numeral is how we write that number using symbols (for example, “7”).
In everyday life we use the decimal (base‑10) system to write natural numbers. It uses ten digits:
$$0, 1, 2, 3, 4, 5, 6, 7, 8, 9$$
The position of each digit tells you its value: ones, tens, hundreds, thousands, and so on. For example, in the number $3\,482$:
- $2$ is in the ones place,
- $8$ is in the tens place,
- $4$ is in the hundreds place,
- $3$ is in the thousands place.
So
$$3\,482 = 3 \times 1000 + 4 \times 100 + 8 \times 10 + 2.$$
This idea is called place value. It is how we can write any natural number using only ten digits.
Basic operations on natural numbers
Four basic operations will be treated more fully in the Arithmetic chapters, but it helps to know how they behave on natural numbers.
Addition stays within the natural numbers
If you add two natural numbers, the result is again a natural number. For example:
- $2 + 3 = 5$
- $10 + 7 = 17$
We say that $\mathbb{N}$ is closed under addition:
- For all natural numbers $a$ and $b$, the sum $a + b$ is a natural number.
Addition has some important features for natural numbers:
- You can add in any order: $a + b = b + a$.
- You can group them in any way: $(a + b) + c = a + (b + c)$.
These facts are examples of general laws that hold for many kinds of numbers.
Multiplication stays within the natural numbers
Multiplication can be viewed as repeated addition. For example:
- $3 \times 4$ means $4 + 4 + 4 = 12$.
If you multiply two natural numbers, the result is again a natural number:
- $5 \times 6 = 30$,
- $7 \times 1 = 7$.
We say that $\mathbb{N}$ is closed under multiplication:
- For all natural numbers $a$ and $b$, the product $a \times b$ is a natural number.
Multiplication also has order and grouping properties similar to addition:
- $a \times b = b \times a$,
- $(a \times b) \times c = a \times (b \times c)$.
Subtraction and division may leave the natural numbers
Subtraction and division do not always stay within the natural numbers:
- $7 - 3 = 4$ is a natural number,
- but $3 - 7$ is not a natural number if we only look at $\mathbb{N}$.
Similarly:
- $12 \div 3 = 4$ is a natural number,
- but $5 \div 2 = 2.5$ is not a natural number.
This limitation is one reason we introduce larger number systems later (integers, rational numbers, real numbers).
Zero and one
Two natural numbers have especially important roles: $0$ (if included) and $1$.
The role of zero (if included)
When $0$ is considered a natural number, it has special properties:
- For any natural number $a$, $a + 0 = a$.
We say $ is the additive identity: adding $ does not change a number.
- Multiplying by $0$ gives $0$: $a \times 0 = 0$.
Zero is also used to show “no objects” when counting, for example “0 apples”.
The role of one
The number $1$ plays a special role in multiplication:
- For any natural number $a$, $a \times 1 = a$.
We say $ is the multiplicative identity: multiplying by $ does not change a number.
Also, $1$ is the smallest positive natural number. When we talk about “first”, “single”, or “one of something”, we are using this basic idea.
Even and odd natural numbers
Natural numbers can be divided into two types: even and odd.
- A natural number is even if it is a multiple of $2$:
$$\text{even } n \quad\text{means}\quad n = 2k \text{ for some natural number } k.$$
Examples: , 4, 6, 8, 10, \dots$ - A natural number is odd if it is one more than an even number:
$$\text{odd } n \quad\text{means}\quad n = 2k + 1 \text{ for some natural number } k.$$
Examples: , 3, 5, 7, 9, \dots$
Every natural number is either even or odd, but not both.
Even and odd numbers behave in simple ways under addition and multiplication. For example:
- even $+$ even $=$ even
- even $+$ odd $=$ odd
- odd $\times$ odd $=$ odd
- (any) $\times$ even $=$ even
These patterns are often used in reasoning and proofs.
Natural numbers and size: no largest natural number
Natural numbers keep going forever:
$$1, 2, 3, 4, 5, \dots$$
There is no “last” or “largest” natural number. If you think you have found the largest natural number, say $N$, you can always form $N + 1$, which is larger. This idea is sometimes summarized as:
- For every natural number $n$, $n + 1$ is also a natural number, and $n + 1 > n$.
This is a very simple statement, but it lies at the heart of mathematical ideas such as counting without end and building arguments step by step.
The successor idea
Closely related is the idea of the successor of a natural number. For a natural number $n$:
- The successor of $n$ is $n + 1$.
So, the successor of $3$ is $4$, the successor of $4$ is $5$, and so on. Every natural number (except perhaps $0$, depending on convention) has a unique predecessor and successor within $\mathbb{N}$:
- $4$ is the successor of $3$ and the predecessor of $5$.
This “next number” idea expresses the step‑by‑step nature of the natural numbers.
Natural numbers as building blocks
Natural numbers are the foundation for many later concepts:
- Larger number systems (integers, rational numbers, real numbers) extend and modify the idea of natural numbers.
- Concepts like divisibility and prime numbers start with natural numbers.
- Many formulas in mathematics count objects, and their results are natural numbers.
Whenever you see counting, listing, or “$1^{\text{st}}$, $2^{\text{nd}}$, $3^{\text{rd}}$” positions, natural numbers are in the background.
Understanding natural numbers as precise mathematical objects—rather than just familiar “everyday numbers”—sets the stage for exploring other number systems in later chapters.