Produced by Charles Wells Revised 2016-10-23 Introduction to this website website TOC website index blog

The natural numbers are the positive integers (whole numbers): the numbers $1, 2, 3, 4,$ and so on. You have known about them since kindergarten.

Many authors include $0$ in the natural numbers, especially in computing science.

In nineteenth century mathematical writing, “natural number” may mean any integer.

This is a short list of some of the properties that make the natural numbers important in math.

If $m$ and $n$ are integers, then so are $m+n$ and $mn$. This is described by saying that the natural numbers are **closed under addition and multiplication.** This is intrinsic to the use of natural numbers as representing quantity.

The natural numbers are *not* closed under subtraction or division. For example, $3$ and $5$ are natural numbers but $3-5$ and $3/5$ are not natural numbers.

Every natural number greater than $1$ has a **unique factorization**: it can be expressed as a product of primes (counting multiplicity) in exactly one way. For example, $12=2\times2\times3$ and there is no other such factorization for $12$ (rearranging the primes doesn't count). This fact is the source of some of the deepest pure mathematics of all. The subject is called number theory, which in spite of its name means specifically the study of *natural* numbers.

The natural numbers are **well-ordered**. This allows **proof by mathematical induction.** Inductive proofs are a basic tool in mathematical logic and in theoretical computer science. Other number systems – integers, rational, real and complex numbers –
do not allow proof by induction.

This section is about useful ways to think about the natural numbers for the purpose of doing math.

It is not about what the natural numbers "**really are**".

In contrast to most objects that occur in abstract math, you have been thinking about the natural numbers for most of your life. Here I will point out several important aspects of natural numbers, making explicit some things you probably already know implicitly.

Each natural number corresponds to a position in a sequence.

For example, the letter ‘d’ is the fourth letter of the alphabet, which is a sequence of $26$ letters. This is the familiar use of integers as **ordinal numbers.** The natural numbers themselves are ordered in an infinite **sequence**
\[1\;2\;3\;4\;5\;\ldots\]
that starts at 1 *but has no end:* **There is no “last” natural number.**

Computer people sometimes start sequences at 0, so that for example the element $a_3$ is the *fourth* entry in the sequence $a_0, a_1, a_2, a_3,\ldots$

Each natural number corresponds to a quantity of distinct individual things. For example the set of letters $S:=\{\text{a},\text{c},\text{e},\text{r},\text{x}\}$ contains five letters. This is the use of integers as **cardinal numbers.** In this case, you can say "The set $S$ contains **five elements,"** or "The set $S$ has **cardinality $5$."**

I referred to the set $\{\text{a},\text{c},\text{e},\text{r},\text{x}\}$ as a **set of letters**. If these five symbols were five *variables,* then the set might contain *fewer than*
five elements. Example: Let $a=e=r=13$, $c=4$ and $x=7$. Then the set $\{a,c,e,r,x\}$ has *three* elements. It is the same set as $\{4,7,13\}$.

Notice that I use upright forms for letters and numbers, and italics for variables. Not everyone does this.

**Order** and **Quantity** are two *genuinely different ideas.* One aspect of the difference is that ordinal numbers should start at $1$ (for the first thing in a sequence) but cardinal numbers should start at $0$, since it is possible to discover that you don’t have *any* instances of some kind of thing. (See empty set.)

Order and quantity become **radically different** when you consider infinite sets. Compare the Wikipedia articles on cardinal numbers and ordinal numbers.

The set of natural numbers may be denoted by "$\mathbb{N}$", but be careful because some authors include $0$ in $\mathbb{N}$ and others do not. People sometimes informally write $\{1,2,3,4,\ldots\}$ or $\{0,1,2,3,4,\ldots\}$ to refer to $\mathbb{N}$.

Don't let the informal notation $\{1,2,3,4,\ldots\}$ for the set of natural numbers mislead you.

$\mathbb{N}$ has every natural number as an element, all at once.

It is plain wrong to think that the natural numbers "go on forever". The reason it is wrong to say that is that there is no sense in which a natural numbers pop into existence at a certain point in time, one after another. You should think of a number as an **unchanging mathematical object**.

I claim that that is the only useful way to think about natural numbers. I am not claiming anything about what they "really are".

One aspect of natural numbers that causes difficulty for people new to abstract math is this:

A number is not the same thing as its representation.

A natural number is a mathematical object. The number of states in the United States of America is a natural number. In the usual notation, that natural number is written '$50$'.

The expression '50' is a **sequence of typographical characters.** It is not itself the natural number it represents. The notation ‘50’ is not the number $50$, although it represents the number $50$.

That integer can be represented in many ways:

- in decimal notation as ‘50’.
- in hexadecimal as ‘32’.
- in binary as ‘110010’.
- as a Roman numeral `L'.
- as a product of powers of primes as $2\cdot5^2$ (see Fundamental Theorem of Arithmetic).
- by the English word "fifty".
- by the phrase "the number of states in the United States of America".

Some more examples of representations of natural numbers are given in the article Representations and models.

The first three items are examples of the representation of natural numbers to different bases. Decimal notation is what we normally use, but from the point of view of abstract mathematics no representation to a particular base is more or less valid than any other.

All base representations are equally valid, but one may be more useful than another in a given situation. For example, binary notation takes too long to write but provides a useful direct representation of computer memory.

You need to distinguish between **properties of natural numbers** and** properties of their representations**.

- Being even is a property of the number, not of its representation.
- On the other hand, “ending in an even digit” is a property of the representation. The number $16$ (in decimal) is even, but its representation in base $3$ is "$121$", which does not end in an even digit. Nevertheless, $121_{3}$ is even.
- If you are asked, "Is $24$ divisible by $3$?", don’t ask, "In what base?", because being divisible by $3$ (or by any other natural number) is a property of the number, not of its representation.

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