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Produced by Charles Wells     Revised 2014 1109
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# NATURAL NUMBERS

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

Many authors include $0$ in the natural numbers, especially in computing science.  In nineteenth century mathematical writing, “natural number” may mean any integer.

## Properties of the natural numbers

### Closure

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.

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.

### Well-ordered

The natural numbers are well-ordered. This allows proof by induction.  Other number systems – integers, rational, real and complex numbers – do not allow proof by induction.

### Notation

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 $\{1,2,3,4,\ldots\}$ for $\mathbb{N}$.

## Images and metaphors for the natural numbers

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 already know implicitly.

### Thinking about $\mathbb{N}$

This section is about the right way to think about the natural numbers for the purpose of doing math. It is not about what the natural numbers "really are".

• 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.
• There is no sense in which you adjoin the numbers to $\mathbb{N}$ one at a time.
• It is plain wrong to think that the natural numbers "go on forever".

### Order

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

### Quantity

Each natural number corresponds to a quantity of distinct individual things.  For example the set of letters $\{\text{a},\text{c},\text{e},\text{r},\text{x}\}$ contains five letters. This is the use of integers as cardinal numbers.

#### Fine point about sets of letters:

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.

## Representation of natural numbers

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

That integer can be represented in many ways:

• in decimal notation as ‘50’.
• 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).
•  All base representations are equally valid, but one may be more useful in a given situa­tion. For example, binary notation takes too long to write but provides a very direct repre­senta­tion of computer memory.
• by the English word "fifty".
• by the phrase "the number of states in the United States of America".

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.

### Terminology

Some texts in computer science or foundations may distinguish typographically between

• The number, for example $50$, and
• The decimal representation of the number, for example writing '50'.

Most of the time no distinction is made. It is common for mathematicians to say, for example, “If a number ends in 0 then it is divisible by 10”, instead of the more precise “If the decimal representation of a number ends in ‘0’ then the number is divisible by 10.”

There are also various notations for representations to different bases.  One way is to use the base as a subscript.  For example, $110010_2=50_{10}=32_{16}$.

### Properties and representations

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

##### Examples
• 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 "12"’, 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.