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Properties of the natural numbers 1
Images and metaphors for the natural numbers
1
Representation of natural numbers 2
Addenda
3
Posted 30 April 2017
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.
Terminology: 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. The other number
systems treated here – integers, rational, real and complex numbers – do not allow proof by induction.
Notation
The set of natural numbers may be denoted by N , but be careful because some authors include 0 in
N and others do not. People sometimes write informally {1, 2, 3, 4, ...} for N (see below.)
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.
Order
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
1
2
3
4
5
…
that starts at 1 but has no ending: There is no “last” natural number.
Computer people sometimes
start sequences at 0, so that
for example the element a3 is
the fourth entry in the
sequence a0 , a1, a2 ,...
Tsk.
Bad metaphors for N
 Don’t let the notation {1, 2, 3, 4, ...} for N mislead you. N has every natural number as an
element, all at once. There is no sense in which you adjoin the numbers to N one by one.
 The words “last” and “ending” in the preceding section on order are misleading, because there is
no time involved. You must think of 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 N . (I am not making a metaphysical statement. I am telling you how
to think of N .)
Quantity
Each natural number corresponds to a quantity of distinct individual things. For example the set of
letters {a,c,e,r,x} contains five letters (see fine point). This is the use of integers as cardinal numbers
(MW, Wi).
About order and quantity
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.
Representation of natural numbers
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. 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 hexadecimal as ‘32’
 in binary as ‘110010’.
 as a Roman numeral `L' .
2
 as a product of powers of primes as 2 ×5 (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
All base representations are
equally valid but one may be
more useful in a given
situation, of course. For
example binary notation
takes too long to write but
provides a very direct
representation of computer
memory.
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’. (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, 1100102 = 5010 = 3216 .
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 ‘121’, which does not end in an even digit. Nevertheless, 1213 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.
Addenda
Fine point about sets of letters:
About the set {a,c,e,r,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.