Natural number

The natural numbers are the numbers (0), 1,2,3,etc. used for counting, and for enumerating an ordered sequence. As such they are the basis of all numbers used in everyday life for calculating and measuring. They are also used to indicate the number of equal parts into which a unit of measure is divided, and how many of such parts are needed for a measurement, thus being the basis for fractions and rational numbers.

Because of their importance every culture has developed a numeral system for representing and manipulating natural numbers, both in oral and in written language. Now the decimal system is almost universally used to write natural numbers while -- depending on history and the context -- other methods (e.g., Roman numerals) still coexist.

Moreover, since ancient times the natural numbers have been of interest not only for practical reasons. On the one hand, their properties have been studied out of (theoretical or mathematical) curiosity, and, on the other hand, some numbers have been assigned symbolic value.

In modern mathematics, the natural numbers are either defined axiomatically by the Peano axioms, i.e., they are characterized by their properties or, in set theory, as a specific set that serves as a concrete object (model) which can be shown to have the desired properties, i.e., to satisfy the Peano axioms.

Is zero a natural number? Whether 0 is a natural number or not is not a mathematical question but the matter of an essentially arbitrary definition, a decision which depends on the context and on personal taste. Historically, 0 was not considered as a "number" because it means that there is "nothing to count". In modern mathematics, in particular because of set theory and the concept of cardinality, 0 is usually included into the natural numbers.

Decimal system
In principle, a natural number could be represented by the corresponding number of dots, strokes, or similar. But this soon becomes impractical if the numbers get large.

Therefore, decimal numerals are used as a sort of shorthand: They are written with ten digits &mdash; 0,1,2,3,4,5,6,7,8,9 &mdash; which represent the numbers zero, one, two, three, four, five, six, seven, eight, nine. Larger numbers are represented by a sequence of digits, e.g., 325. Such a numeral is read starting from the right. The first (rightmost) digit represents the corresponding number of dots (in the example: five); the next (second-right) represents the corresponding number of groups of ten dots (in the example: two groups of ten dots each), the next digit indicates the corresponding number of "groups of ten groups of ten dots" (in the example, three groups of ten times ten dots), and so on.

Arithmetic
Elementary arithmetic with natural numbers is based on addition: Adding two natural numbers is equivalent to counting two sets in sequence. (This is equivalent to the mathematical notion of adding cardinal numbers.)

Addition of two numbers is written with a plus sign "+", and the result is called the sum of the two numbers.
 * e.g., 2 + 3 = 5, read "two plus three equals (or makes) five"

If an equation like 2 + n = 5 can be solved, i.e., if there is a number (denoted by n) which added to 2 makes 5, then its solution is unique and called the difference of these two numbers and is written with a minus sign "-":
 * e.g., 5 - 2 = 3, read "five minus two equals (or makes) three".

Addition of two or more numbers does not depend on the order in which they are added.

Multiplication can be considered as multiple addition, and the result is called product:
 * e.g., "3 times 5" means "5 plus 5 plus 5" (three summands of five).

It is written with a times sign "&times;" (or a central dot "•").
 * e.g., 3 • 5 = 15, read "three times five equals (or makes) fifteen".

The times sign is only used for elementary examples, in mathematics only the dot is usual – the times sign is used for other operations – amd usually is omitted when variables (denoted by letters) are involved:
 * 2 • 3 and n • 2, but 2n (only rarely 2 • n) and nm (only very rarely n • m).

If an equation like 2 • n = 6 can be solved, i.e., if there is a number (denoted by n) which multiplied by 2 makes 6, then its solution is unique and called the quotient of these two numbers and is written with a division sign ":" or as fraction with "/":
 * e.g., 6 : 2 = 6 / 2 = 3, read "six divided by two equals (or makes) three".

Multiplication of two or more numbers also does not depend on the order in which they are multiplied: 3 • 2 = 2+2+2 = 6 = 3+3 = 2 • 3.

Exponentiation can be considered as multiple multiplication.

Special properties
The natural numbers have been used since thousands of years. It is no surprise that many properties and curious facts have been observed and investigated, some for practical reasons, some out of curiosity, some because they turned out to have connections with seemingly unrelated problems.

The first properties that come to mind are the distinction between even and odd numbers, between prime and composite numbers, and the perfect squares.


 * A number is even, if it is a multiple of 2, and odd if not.
 * A number is composite, if it is the product of two other numbers, and prime it not.
 * A number is a (perfect) square if it is the product of a number with itself.

Figurate numbers
All these properties can be viewed as simple cases of figurate numbers. Figurate numbers arise if one tries to arrange a certain number of dots, coins, etc. into a nice geometrical shape.

A number is even if the dots can be arranged pairwise, it is composite if they can be arranged in the form of a rectangle, and it is a square if this rectangle can be a square.

The simplest figurate numbers are the triangular numbers
 * 1, 3 = 1+.2, 6 = 1+2+3, 10 = 1+2+3+4

which are of the form k(k+1)/2.

Sum of divisors
Another type of property, also already investigated in Ancient Greece, concerns the sum of the divisors (excluding the number itself) of a number: A number is called perfect if it is equal to the sum of its divisors:
 * 6 = 1+2+3, 28 = 1+2+4+7+14, etc.

Two numbers are amicable, if each is the sum of the divisors of the other, as in the following pair which was already known to Euclid:
 * 220 and 284.

Properties in general
There is no limit to the various properties which can be considered. It can even be argued that every natural number is "interesting": Indeed, if there were a number which is not interesting then the set of such numbers were not empty, and thus it had a minimal element (by axiom (5a)). But the smallest uninteresting number certainly would be interesting, wouldn't it?

For more properties see the corresponding catalog.

Symbolic meaning
Many natural numbers had - and still have - a symbolic meaning, some are considered as "lucky" or even "holy" numbers, others are considered to indicatie bad luck, some are "nice", others are not. Some of these meanings are (almost) universal, while others vary from culture to culture. Numbers derived from words have significance in numerology. Numbers play a – sometimes secret – role in (secret) orders like the freemasons. But all such interpretations are not part of mathematics, even in cases where they are motivated by mathematical properties and arguments &mdash; they are determined by tradition, by mythology, or by esoteric beliefs.

For examples of symbolic meanings see the corresponding catalog.

Peano axioms
During the 19th century the foundations of mathematics which, of course, include the concept of number became a major topic of discussion and research. In 1889 Giuseppe Peano published a system of axioms that characterizes the natural numbers. The axioms, essentially, state that eventually every natural number will be reached if one starts to count at 0 (or 1, if that is preferred) and proceeds from that by stepping from one number to the next. The axioms are usually given as follows:


 * (1) 0 is a natural number.
 * (2) Every natural number has a unique successor.
 * (3) 0 is not the successor of a natural number.
 * (4) Different natural numbers have different successors.
 * (5) If a property of natural numbers is such that:
 * 0 has the property, and
 * if a natural number has the property then its successor has it as well.
 * Then every natural number has this property

The last axiom is equivalent to the following property:
 * (5a) Any non-empty set of natural numbers has a least element.

In these axioms, the first (least) element is taken to be 0, but this is arbitrary. It can be replaced by 1 (or any other number).

Axiom (5) (or (5a)) is the basis for proofs by induction, and for definition by recursion.

Set theoretic model
In modern mathematics, sets are used as a basis on which all other theories are built. In this context it is therefore necessary to construct a model which incorporates the natural numbers as sets such that the axioms can be verified (using the axioms of set theory). from the properties of these sets. Such constructions indeed exist of which one (due to John von Neumann) has particularly "nice" properties, and which is consequently usually used.

The construction starts with the number 0. It is represented by the empty set which has no – or 0 – elements. The number 1 is represented by a set which has one single element, namely the number 0 (i.e., the empty set). Next, 2 is represented by a set which has precisely 2 elements, the numbers 0 and 1. 3 is represented by a set with 3 elements (0,1,2), etc. In general, an arbitrary number n is represented by a set with precisely n elements, the numbers 0,1,2,..,n-1, i.e., the (previously constructed) numbers smaller than n.

This construction can be extended (in a natural way) to both the infinite cardinal numbers and the infinite ordinal numbers.

In all the formulas below $$ k,m,n \in \mathbb N $$ is assumed.

Formal construction
The formal construction of the natural numbers starts with the definition of an operator which can be applied to any set:


 * $$ S(A) := A \cup \{ A \} $$

This operator S can be interpreted as defining a successor for every set.

A set A is called closed under S if it contains the successor of all its elements:


 * $$ a \in A \Rightarrow S(a) \in A $$

The next step needs the smallest set that contains the empty set and is closed under S. It can be constructed by taking the intersection of such sets:


 * $$ \mathbb N := \bigcap \{ A \mid \emptyset \in A, A \text{ closed under } S \} $$

This construction is immediate in naive set theory and can be justified in axiomatic set theory.

$$ \mathbb N $$ is (by definition) a countably infinite set and, in the usual model that extends the von Neumann construction to infinite sets, is taken both as the smallest infinite cardinal number aleph-0 and the smallest infinite ordinal number omega:
 * $$ \aleph_0 := \mathbb N \textrm{ \ and \ } \omega := \mathbb N $$.

Peano axioms
Now it can be verified that the elements of this set satisfy the Peano axioms, and that it therefore can be taken as a model for the natural numbers:

Axioms (1) and (2) become definitions:
 * $$ (1) \quad 0 := \emptyset \in \mathbb N $$
 * $$ (2) \quad (\forall n \in \mathbb N) n' := S(n) \in \mathbb N $$

Axioms (3) and (4) state that S is an injective mapping:
 * $$ S : \mathbb N \rightarrow \mathbb N \setminus\{0\} $$

Axiom (5) states that a set which contains 0 and is closed under S is the set $$\mathbb N$$. This proposition is true because $$ \mathbb N $$ is the smallest set with this property.

The set theoretical order (by set inclusion) in $$\mathbb N$$ coincides – because of $$ a \subset S(a) $$ – with that needed for the natural numbers:
 * $$ n \le m :\Leftrightarrow n \subset m \Leftrightarrow  n \in m $$

The alternative axiom (5a) states that $$\mathbb N$$ is well-ordered. If A is a non-empty set, than its minimum coincides (because of the model) with the intersection of its elements:
 * $$ \emptyset \not= A \subset \mathbb N \Rightarrow \min A = \cap A \in \mathbb N $$

Once established, this construction is no longer needed. All further investigations can be founded on the Peano axioms.

Notation
The symbol $$ \mathbb N $$ – an N in the Blackboard Bold font – for the natural numbers has become usual (often even assumed as well-known standard which needs no further explanation) during the second half of the 20th century, replacing the previously customary bold N.

However, some authors do not include 0 into $$ \mathbb N $$. In order to explicitly indicate the inclusion or exclusion of 0, often $$ \mathbb N_0 $$ (with 0) and $$ \mathbb N^\ast $$ (without 0) are used. $$ {}^\ast \mathbb N $$ is used for the nonstandard extension of the natural numbers.

Addition
Addition is a binary operation $$\;+\;$$ in $$\mathbb N$$ which is defined recursively (using axiom (5)) as follows:


 * $$ n+0 := n,\; n+k' := (n+k)' \qquad (k' = k+1) $$

$$ (\mathbb N,+) $$ is a semigroup with neutral element 0. It is:


 * $$\quad n+m = m+n          \textrm{} $$ commutative
 * $$\quad k+(n+m) = (k+m)+n  \textrm{} $$ associative
 * $$\quad n+0 = 0+n = n      \textrm{} $$ neutral element

Moreover,
 * $$\quad n+k = m+k \Rightarrow n=m \textrm{} $$

Multiplication
Multiplication is a binary operation $$\;\cdot\;$$ in $$\mathbb N$$ which is defined recursively (using axiom (5) and addition) as follows:


 * $$ 0 \cdot n := 0 ,\quad k' \cdot n := ( k \cdot n ) + n ( 1 \cdot n = n ) $$

$$ (\mathbb N \setminus\{0\},\cdot) $$ is a semigroup with neutral element 1. It is:


 * $$\quad n \cdot m = m \cdot n                     \textrm{} $$ commutative
 * $$\quad k \cdot (n \cdot m) = (k \cdot n)\cdot m  \textrm{} $$ associative
 * $$\quad 1 \cdot n = n \cdot 1 = n                 \textrm{} $$ neutral element

Moreover,
 * $$\quad n \cdot k = m \cdot k \Rightarrow n=m \textrm{} $$

Order relation
An order relation $$\le$$ is defined using addition:


 * $$ n \le m :\Leftrightarrow (\exists k)\ n+k = m $$.

(Another order, a partial order, can be defined using divisibility.)

$$ (\mathbb N, \le) $$ is linearly ordered set:


 * $$\quad n \le n                               \textrm{} $$ reflexive
 * $$\quad n \le m, m \le n \Rightarrow n=m     \textrm{} $$ antisymmetric
 * $$\quad k \le n, n \le m \Rightarrow k \le m  \textrm{} $$ transitive

This order is a well-ordering (by axiom 5), i.e., all non-empty sets have a minimal element:
 * $$ \emptyset \not= A \subset \mathbb N \Leftrightarrow (\exist n) n \in A, (\forall m \in A)\ n \le m \qquad (n:=\min A)$$.

Relations between the structures
The binary operations $$+$$ and $$\;\cdot$$, and the order relation $$\le$$ are compatible with each other:


 * $$\quad k \cdot (n+m) = k \cdot n+k \cdot m          \textrm{}   $$
 * $$\quad n \le m \Rightarrow k + n \le k + m          \textrm{} $$
 * $$\quad n \le m \Rightarrow k \cdot n \le k \cdot m  \textrm{} $$

Exponentiation

 * $$ n^0 := 1, n^{k'} := n^k \cdot n $$