Posted 8 July 2011

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.

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

**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.

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

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 write informally $\left\{ {1,\,\,2,\,\,3,\,\,4,\,\,...} \right\}$ for ${\mathbb{N}}$ (see below.)

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.

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****
**(MW,Wi).** **The natural
numbers themselves are ordered in an infinite list

that starts at 1 **but has no ending: ** There is no “last” natural number.

¨ Don’t let the notation $\left\{
1,\,\,2,\,\,3,\,\,4,\,\,...\right\}$
for $ {\mathbb{N}}$ 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 ** ${\mathbb{N}}$

**¨ **The words “last” and “ending” in the preceding section on order are misleading,
because there is **no time involved**. You must think of ${\mathbb{N}}$
in the rigorous way; it is **unchanging
****and ****has
every natural number as an element.**

**¨ ****Statements such as “the
natural numbers go on forever” are similarly bad metaphors. ****All the natural numbers are already in ** ${\mathbb{N}}$**. ****(I am not making a metaphysical statement. I am telling you
how to think of ** ${\mathbb{N}}$**.)**

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 (see fine point). This is the use of integers
as **cardinal numbers**** **(MW, Wi).** **

**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.)
Another aspect is that both ideas can be extended to infinite sets, but then
they diverge quite radically: Infinite cardinals are
very different from infinite
ordinals.

One basic aspect of natural numbers that causes difficulty
for people new to abstract math is that they are *not the same thing as
their***representations.** This is also true for the
other kinds of numbers.

*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.

** **

The notation ‘50’ is not 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 $2 \cdot {5^2}$(see Fundamental Theorem of Arithmetic)

¨ 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**.

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’. (They almost always use single quotes for this.)

**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”, which could be more precisely reworded as, “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,${110010_2} = {50_{10}} = {32_{16}}$.

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.

About the set $\{
{\text{a,}}\,{\text{c,}}\,{\text{e,}}\,{\text{r,}}\,{\text{x}}\} $:
I referred to this set as a *set of letters. *If these five symbols were
five *variables,* then the set might contain *less 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. Return.