Number used for counting
This article is about "positive integers" and "non-negative integers". For all the numbers ..., ?2, ?1, 0, 1, 2, ..., see
Integer
.
In
mathematics
, the
natural numbers
are the
numbers
0, 1, 2, 3, etc., possibly excluding 0.
[1]
Some define the natural numbers as the
non-negative integers
0, 1, 2, 3, ...
, while others define them as the
positive integers
1, 2, 3, ...
.
[a]
Some authors acknowledge both definitions whenever convenient.
[2]
Some texts define the
whole numbers
as the natural numbers together with zero, excluding zero from the natural numbers, while in other writings, the
whole numbers
refer to all of the
integers
(including negative integers).
[3]
The
counting numbers
refer to the natural numbers in common language, particularly in primary school education, and are similarly ambiguous although typically exclude zero.
[4]
The natural numbers can be used for counting (as in "there are
six
coins on the table"), in which case they serve as
cardinal numbers
. They may also be used for ordering (as in "this is the
third
largest city in the country"), in which case they serve as
ordinal numbers
. Natural numbers are sometimes used as labels?also known as
nominal numbers
, (e.g.
jersey numbers
in sports)?which do not have the properties of numbers in a mathematical sense.
[2]
[5]
The natural numbers form a
set
, commonly symbolized as a bold
N
or
blackboard bold
. Many other
number sets
are built by successively extending the set of natural numbers: the
integers
, by including an
additive identity
0 (if not yet in) and an
additive inverse
?
n
for each nonzero natural number
n
; the
rational numbers
, by including a
multiplicative inverse
for each nonzero integer
n
(and also the product of these inverses by integers); the
real numbers
by including the
limits
of
Cauchy sequences
[b]
of rationals; the
complex numbers
, by adjoining to the real numbers a
square root of
?1
(and also the sums and products thereof); and so on.
[c]
[d]
This chain of extensions canonically
embeds
the natural numbers in the other number systems.
Properties of the natural numbers, such as
divisibility
and the distribution of
prime numbers
, are studied in
number theory
. Problems concerning counting and ordering, such as
partitioning
and
enumerations
, are studied in
combinatorics
.
History
[
edit
]
Ancient roots
[
edit
]
The most primitive method of representing a natural number is to use one's fingers, as in
finger counting
. Putting down a
tally mark
for each object is another primitive method. Later, a set of objects could be tested for equality, excess or shortage?by striking out a mark and removing an object from the set.
The first major advance in abstraction was the use of
numerals
to represent numbers. This allowed systems to be developed for recording large numbers. The ancient
Egyptians
developed a powerful system of numerals with distinct
hieroglyphs
for 1, 10, and all powers of 10 up to over 1 million. A stone carving from
Karnak
, dating back from around 1500 BCE and now at the
Louvre
in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The
Babylonians
had a
place-value
system based essentially on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one?its value being determined from context.
[9]
A much later advance was the development of the idea that
0
can be considered as a number, with its own numeral. The use of a 0
digit
in place-value notation (within other numbers) dates back as early as 700 BCE by the Babylonians, who omitted such a digit when it would have been the last symbol in the number.
[e]
The
Olmec
and
Maya civilizations
used 0 as a separate number as early as the
1st century BCE
, but this usage did not spread beyond
Mesoamerica
.
[11]
[12]
The use of a numeral 0 in modern times originated with the Indian mathematician
Brahmagupta
in 628 CE. However, 0 had been used as a number in the medieval
computus
(the calculation of the date of Easter), beginning with
Dionysius Exiguus
in 525 CE, without being denoted by a numeral. Standard
Roman numerals
do not have a symbol for 0; instead,
nulla
(or the genitive form
nullae
) from
nullus
, the Latin word for "none", was employed to denote a 0 value.
[13]
The first systematic study of numbers as
abstractions
is usually credited to the
Greek
philosophers
Pythagoras
and
Archimedes
. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.
[f]
Euclid
, for example, defined a unit first and then a number as a multitude of units, thus by his definition, a unit is not a number and there are no unique numbers (e.g., any two units from indefinitely many units is a 2).
[15]
However, in the definition of
perfect number
which comes shortly afterward, Euclid treats 1 as a number like any other.
[16]
Independent studies on numbers also occurred at around the same time in
India
, China, and
Mesoamerica
.
[17]
Emergence as a term
[
edit
]
Nicolas Chuquet
used the term
progression naturelle
(natural progression) in 1484.
[18]
The earliest known use of "natural number" as a complete English phrase is in 1763.
[19]
[20]
The 1771 Encyclopaedia Britannica defines natural numbers in the logarithm article.
[20]
Starting at 0 or 1 has long been a matter of definition. In 1727,
Bernard Le Bovier de Fontenelle
wrote that his notions of distance and element led to defining the natural numbers as including or excluding 0.
[21]
In 1889,
Giuseppe Peano
used N for the positive integers and started at 1,
[22]
but he later changed to using N
0
and N
1
.
[23]
Historically, most definitions have excluded 0,
[20]
[24]
[25]
but many mathematicians such as
George A. Wentworth
,
Bertrand Russell
,
Nicolas Bourbaki
,
Paul Halmos
,
Stephen Cole Kleene
, and
John Horton Conway
have preferred to include 0.
[26]
[20]
Mathematicians have noted tendencies in which definition is used, such as algebra texts including 0,
[20]
[g]
number theory and analysis texts excluding 0,
[20]
[27]
[28]
logic and set theory texts including 0,
[29]
[30]
[31]
dictionaries excluding 0,
[20]
[32]
school books (through high-school level) excluding 0, and upper-division college-level books including 0.
[1]
There are exceptions to each of these tendencies and as of 2023 no formal survey has been conducted. Arguments raised include
division by zero
[27]
and the size of the
empty set
.
Computer languages
often
start from zero
when enumerating items like
loop counters
and
string-
or
array-elements
.
[33]
[34]
Including 0 began to rise in popularity in the 1960s.
[20]
The
ISO 31-11
standard included 0 in the natural numbers in its first edition in 1978 and this has continued through its present edition as
ISO 80000-2
.
[35]
Formal construction
[
edit
]
In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers.
Henri Poincare
stated that axioms can only be demonstrated in their finite application, and concluded that it is "the power of the mind" which allows conceiving of the indefinite repetition of the same act.
[36]
Leopold Kronecker
summarized his belief as "God made the integers, all else is the work of man".
[h]
The
constructivists
saw a need to improve upon the logical rigor in the
foundations of mathematics
.
[i]
In the 1860s,
Hermann Grassmann
suggested a
recursive definition
for natural numbers, thus stating they were not really natural?but a consequence of definitions. Later, two classes of such formal definitions were constructed; later still, they were shown to be equivalent in most practical applications.
Set-theoretical definitions of natural numbers
were initiated by
Frege
. He initially defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set. However, this definition turned out to lead to paradoxes, including
Russell's paradox
. To avoid such paradoxes, the formalism was modified so that a natural number is defined as a particular set, and any set that can be put into one-to-one correspondence with that set is said to have that number of elements.
[39]
In 1881,
Charles Sanders Peirce
provided the first
axiomatization
of natural-number arithmetic within this second class of definitions.
[40]
[41]
In 1888,
Richard Dedekind
proposed another axiomatization of natural-number arithmetic,
[42]
and in 1889, Peano published a simplified version of Dedekind's axioms in his book
The principles of arithmetic presented by a new method
(
Latin
:
Arithmetices principia, nova methodo exposita
). This approach is now called
Peano arithmetic
. It is based on an
axiomatization
of the properties of
ordinal numbers
: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is
equiconsistent
with several weak systems of
set theory
. One such system is
ZFC
with the
axiom of infinity
replaced by its negation.
[43]
Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include
Goodstein's theorem
.
[44]
Notation
[
edit
]
The
set
of all natural numbers is standardly denoted
N
or
[2]
[45]
Older texts have occasionally employed
J
as the symbol for this set.
[46]
Since natural numbers may contain
0
or not, it may be important to know which version is referred to. This is often specified by the context, but may also be done by using a subscript or a superscript in the notation, such as:
[35]
[47]
- Naturals without zero:
- Naturals with zero:
Alternatively, since the natural numbers naturally form a
subset
of the
integers
(often
denoted
),
they may be referred to as the positive, or the non-negative integers, respectively.
[48]
To be unambiguous about whether 0 is included or not, sometimes a superscript "
" or "+" is added in the former case, and a subscript (or superscript) "0" is added in the latter case:
[35]
Properties
[
edit
]
This section uses the convention
.
Addition
[
edit
]
Given the set
of natural numbers and the
successor function
sending each natural number to the next one, one can define
addition
of natural numbers recursively by setting
a
+ 0 =
a
and
a
+
S
(
b
) =
S
(
a
+
b
)
for all
a
,
b
. Thus,
a
+ 1 =
a
+ S(0) = S(
a
+0) = S(
a
)
,
a
+ 2 =
a
+ S(1) = S(
a
+1) = S(S(
a
))
, and so on. The
algebraic structure
is a
commutative
monoid
with
identity element
0. It is a
free monoid
on one generator. This commutative monoid satisfies the
cancellation property
, so it can be embedded in a
group
. The smallest group containing the natural numbers is the
integers
.
If 1 is defined as
S
(0)
, then
b
+ 1 =
b
+
S
(0) =
S
(
b
+ 0) =
S
(
b
)
. That is,
b
+ 1
is simply the successor of
b
.
Multiplication
[
edit
]
Analogously, given that addition has been defined, a
multiplication
operator
can be defined via
a
× 0 = 0
and
a
× S(
b
) = (
a
×
b
) +
a
. This turns
into a
free commutative monoid
with identity element 1; a generator set for this monoid is the set of
prime numbers
.
Relationship between addition and multiplication
[
edit
]
Addition and multiplication are compatible, which is expressed in the
distribution law
:
a
× (
b
+
c
) = (
a
×
b
) + (
a
×
c
)
. These properties of addition and multiplication make the natural numbers an instance of a
commutative
semiring
. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that
is not
closed
under subtraction (that is, subtracting one natural from another does not always result in another natural), means that
is
not
a
ring
; instead it is a
semiring
(also known as a
rig
).
If the natural numbers are taken as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that they begin with
a
+ 1 =
S
(
a
)
and
a
× 1 =
a
. Furthermore,
has no identity element.
Order
[
edit
]
In this section, juxtaposed variables such as
ab
indicate the product
a
×
b
,
[49]
and the standard
order of operations
is assumed.
A
total order
on the natural numbers is defined by letting
a
≤
b
if and only if there exists another natural number
c
where
a
+
c
=
b
. This order is compatible with the
arithmetical operations
in the following sense: if
a
,
b
and
c
are natural numbers and
a
≤
b
, then
a
+
c
≤
b
+
c
and
ac
≤
bc
.
An important property of the natural numbers is that they are
well-ordered
: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an
ordinal number
; for the natural numbers, this is denoted as
ω
(omega).
Division
[
edit
]
In this section, juxtaposed variables such as
ab
indicate the product
a
×
b
, and the standard
order of operations
is assumed.
While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of
division with remainder
or
Euclidean division
is available as a substitute: for any two natural numbers
a
and
b
with
b
≠ 0
there are natural numbers
q
and
r
such that
The number
q
is called the
quotient
and
r
is called the
remainder
of the division of
a
by
b
. The numbers
q
and
r
are uniquely determined by
a
and
b
. This Euclidean division is key to the several other properties (
divisibility
), algorithms (such as the
Euclidean algorithm
), and ideas in number theory.
Algebraic properties satisfied by the natural numbers
[
edit
]
The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties:
- Closure
under addition and multiplication: for all natural numbers
a
and
b
, both
a
+
b
and
a
×
b
are natural numbers.
[50]
- Associativity
: for all natural numbers
a
,
b
, and
c
,
a
+ (
b
+
c
) = (
a
+
b
) +
c
and
a
× (
b
×
c
) = (
a
×
b
) ×
c
.
[51]
- Commutativity
: for all natural numbers
a
and
b
,
a
+
b
=
b
+
a
and
a
×
b
=
b
×
a
.
[52]
- Existence of
identity elements
: for every natural number
a
,
a
+ 0 =
a
and
a
× 1 =
a
.
- If the natural numbers are taken as "excluding 0", and "starting at 1", then for every natural number
a
,
a
× 1 =
a
. However, the "existence of additive identity element" property is not satisfied
- Distributivity
of multiplication over addition for all natural numbers
a
,
b
, and
c
,
a
× (
b
+
c
) = (
a
×
b
) + (
a
×
c
)
.
- No nonzero
zero divisors
: if
a
and
b
are natural numbers such that
a
×
b
= 0
, then
a
= 0
or
b
= 0
(or both).
Generalizations
[
edit
]
Two important generalizations of natural numbers arise from the two uses of counting and ordering:
cardinal numbers
and
ordinal numbers
.
- A natural number can be used to express the size of a finite set; more precisely, a cardinal number is a measure for the size of a set, which is even suitable for infinite sets. The numbering of cardinals usually begins at zero, to accommodate the
empty set
. This concept of "size" relies on maps between sets, such that two sets have
the same size
, exactly if there exists a
bijection
between them. The set of natural numbers itself, and any bijective image of it, is said to be
countably infinite
and to have
cardinality
aleph-null
(
?
0
).
- Natural numbers are also used as
linguistic ordinal numbers
: "first", "second", "third", and so forth. The numbering of ordinals usually begins at zero, to accommodate the order type of the
empty set
. This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any
well-ordered
countably infinite set without
limit points
. This assignment can be generalized to general well-orderings with a cardinality beyond countability, to yield the ordinal numbers. An ordinal number may also be used to describe the notion of "size" for a well-ordered set, in a sense different from cardinality: if there is an
order isomorphism
(more than a bijection) between two well-ordered sets, they have the same ordinal number. The first ordinal number that is not a natural number is expressed as
ω
; this is also the ordinal number of the set of natural numbers itself.
The least ordinal of cardinality
?
0
(that is, the
initial ordinal
of
?
0
) is
ω
but many well-ordered sets with cardinal number
?
0
have an ordinal number greater than
ω
.
For
finite
well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite,
sequence
.
A countable
non-standard model of arithmetic
satisfying the Peano Arithmetic (that is, the first-order Peano axioms) was developed by
Skolem
in 1933. The
hypernatural
numbers are an uncountable model that can be constructed from the ordinary natural numbers via the
ultrapower construction
. Other generalizations are discussed in
Number § Extensions of the concept
.
Georges Reeb
used to claim provocatively that "The naive integers don't fill up
".
[53]
Formal definitions
[
edit
]
There are two standard methods for formally defining natural numbers. The first one, named for
Giuseppe Peano
, consists of an autonomous
axiomatic theory
called
Peano arithmetic
, based on few axioms called
Peano axioms
.
The second definition is based on
set theory
. It defines the natural numbers as specific
sets
. More precisely, each natural number
n
is defined as an explicitly defined set, whose elements allow counting the elements of other sets, in the sense that the sentence "a set
S
has
n
elements" means that there exists a
one to one correspondence
between the two sets
n
and
S
.
The sets used to define natural numbers satisfy Peano axioms. It follows that every
theorem
that can be stated and proved in Peano arithmetic can also be proved in set theory. However, the two definitions are not equivalent, as there are theorems that can be stated in terms of Peano arithmetic and proved in set theory, which are not
provable
inside Peano arithmetic. A probable example is
Fermat's Last Theorem
.
The definition of the integers as sets satisfying Peano axioms provide a
model
of Peano arithmetic inside set theory. An important consequence is that, if set theory is
consistent
(as it is usually guessed), then Peano arithmetic is consistent. In other words, if a contradiction could be proved in Peano arithmetic, then set theory would be contradictory, and every theorem of set theory would be both true and wrong.
Peano axioms
[
edit
]
The five Peano axioms are the following:
[54]
[j]
- 0 is a natural number.
- Every natural number has a successor which is also a natural number.
- 0 is not the successor of any natural number.
- If the successor of
equals the successor of
, then
equals
.
- The
axiom of induction
: If a statement is true of 0, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number.
These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. In ordinary arithmetic, the successor of
is
.
Set-theoretic definition
[
edit
]
Intuitively, the natural number
n
is the common property of all
sets
that have
n
elements. So, it seems natural to define
n
as an
equivalence class
under the relation "can be made in
one to one correspondence
". This does not work in
set theory
, as such an equivalence class would not be a set (because of
Russell's paradox
). The standard solution is to define a particular set with
n
elements that will be called the natural number
n
.
The following definition was first published by
John von Neumann
,
[55]
although Levy attributes the idea to unpublished work of Zermelo in 1916.
[56]
As this definition extends to
infinite set
as a definition of
ordinal number
, the sets considered below are sometimes called
von Neumann ordinals
.
The definition proceeds as follows:
- Call
0 = { }
, the
empty set
.
- Define the
successor
S
(
a
)
of any set
a
by
S
(
a
) =
a
∪ {
a
}
.
- By the
axiom of infinity
, there exist sets which contain 0 and are
closed
under the successor function. Such sets are said to be
inductive
. The intersection of all inductive sets is still an inductive set.
- This intersection is the set of the
natural numbers
.
It follows that the natural numbers are defined iteratively as follows:
- 0 = { }
,
- 1 = 0 ∪ {0} = {0} = {{ }}
,
- 2 = 1 ∪ {1} = {0, 1} = {{ }, {{ }}}
,
- 3 = 2 ∪ {2} = {0, 1, 2} = {{ }, {{ }}, {{ }, {{ }}}}
,
- n
=
n
?1 ∪ {
n
?1} = {0, 1, ...,
n
?1} = {{ }, {{ }}, ..., {{ }, {{ }}, ...}}
,
- etc.
It can be checked that the natural numbers satisfy the
Peano axioms
.
With this definition, given a natural number
n
, the sentence "a set
S
has
n
elements" can be formally defined as "there exists a
bijection
from
n
to
S
. This formalizes the operation of
counting
the elements of
S
. Also,
n
≤
m
if and only if
n
is a
subset
of
m
. In other words, the
set inclusion
defines the usual
total order
on the natural numbers. This order is a
well-order
.
It follows from the definition that each natural number is equal to the set of all natural numbers less than it. This definition, can be extended to the
von Neumann definition of ordinals
for defining all
ordinal numbers
, including the infinite ones: "each ordinal is the well-ordered set of all smaller ordinals."
If one
does not accept the axiom of infinity
, the natural numbers may not form a set. Nevertheless, the natural numbers can still be individually defined as above, and they still satisfy the Peano axioms.
There are other set theoretical constructions. In particular,
Ernst Zermelo
provided a construction that is nowadays only of historical interest, and is sometimes referred to as
Zermelo ordinals
.
[56]
It consists in defining
0
as the empty set, and
S
(
a
) = {
a
}
.
With this definition each natural number is a
singleton set
. So, the property of the natural numbers to represent
cardinalities
is not directly accessible; only the ordinal property (being the
n
th element of a sequence) is immediate. Unlike von Neumann's construction, the Zermelo ordinals do not extend to infinite ordinals.
See also
[
edit
]
Notes
[
edit
]
- ^
See
§ Emergence as a term
- ^
Any Cauchy sequence in the Reals converges,
- ^
Mendelson (2008
, p. x) says: "The whole fantastic hierarchy of number systems is built up by purely set-theoretic means from a few simple assumptions about natural numbers."
- ^
Bluman (2010
, p. 1): "Numbers make up the foundation of mathematics."
- ^
A tablet found at Kish ... thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place.
[10]
- ^
This convention is used, for example, in
Euclid's Elements
, see D. Joyce's web edition of Book VII.
[14]
- ^
Mac Lane & Birkhoff (1999
, p. 15) include zero in the natural numbers: 'Intuitively, the set
of all
natural numbers
may be described as follows:
contains an "initial" number
0
; ...'. They follow that with their version of the
Peano's axioms
.
- ^
The English translation is from Gray. In a footnote, Gray attributes the German quote to: "Weber 1891?1892, 19, quoting from a lecture of Kronecker's of 1886."
[37]
[38]
- ^
"Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." (
Eves 1990
, p. 606)
- ^
Hamilton (1988
, pp. 117 ff) calls them "Peano's Postulates" and begins with "1.
0 is a natural number."
Halmos (1960
, p. 46) uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I)
0 ∈ ω
(where, of course,
0 = ?
" (
ω
is the set of all natural numbers).
Morash (1991)
gives "a two-part axiom" in which the natural numbers begin with 1. (Section 10.1:
An Axiomatization for the System of Positive Integers
)
References
[
edit
]
- ^
a
b
Enderton, Herbert B. (1977).
Elements of set theory
. New York: Academic Press. p. 66.
ISBN
0122384407
.
- ^
a
b
c
Weisstein, Eric W.
"Natural Number"
.
mathworld.wolfram.com
. Retrieved
11 August
2020
.
- ^
Ganssle, Jack G. & Barr, Michael (2003).
"integer"
.
Embedded Systems Dictionary
. Taylor & Francis. pp. 138 (integer), 247 (signed integer), & 276 (unsigned integer).
ISBN
978-1-57820-120-4
.
Archived
from the original on 29 March 2017
. Retrieved
28 March
2017
– via Google Books.
- ^
Weisstein, Eric W.
"Counting Number"
.
MathWorld
.
- ^
"Natural Numbers"
.
Brilliant Math & Science Wiki
. Retrieved
11 August
2020
.
- ^
"Introduction"
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