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Some mathematicians (especially number theorists) prefer not to regard zero as a natural number, while some others (especially set theorists, logicians and computer scientists) take the opposite stance. In this encyclopedia, zero is considered to be a natural number.
Though even a small child will understand what we mean by natural numbers, their definition has not been easy. The Peano postulates essentially uniquely describe the set of natural numbers, which is denoted by N or <math>\mathbb{N}</math> (an N in blackboard bold).
The last postulate ensures that the proof technique of mathematical induction is valid.
A standard construction in set theory is to define each natural number as the set of natural numbers less than it, so that 0 = {}, 1 = {0}, 2 = {0,1}, 3 = {0,1,2}... when you see a natural number used as a set this is typically what is meant.
One can inductively define an addition on the natural numbers by requiring a + (b + 1) = (a + b) + 1. This turns the natural numbers (N, +) into a commutative monoid with neutral element 0, the socalled free monoid with one generator. This monoid satisfies the cancellation property and can therefore be embedded in a group. The smallest group containing the natural numbers is the integers.
Analogously, a multiplication * can be defined via a * (b + 1) = ab + a. This turns (N, *) into a commutative monoid; addition and multiplication are compatible which is expressed in the distribution law: a * (b + c) = ab + ac.
Furthermore, one defines a total order on the natural numbers by writing a <= b if and only if there exists another natural number c with 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 wellordered: every nonempty set of natural numbers has a smallest element.
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 is available as a substitute: For any two natural numbers a and b with b <> 0 we can find natural numbers q and r such that
The number q is called the quotient and r is called the remainder of division of a by b. The numbers q and r are uniquely determined by a and b.
The deeper properties of the natural numbers, such as the distribution of prime numbers for example, are studied in number theory.
Natural numbers can be used for two purposes: to describe the position of an element in an ordered sequence, which is generalized by the concept of ordinal number, and to specify the size of a finite set, which is generalized by the concept of cardinal number. In the finite world, these two concepts coincide: the finite ordinals are equal to N as are the finite cardinals. When moving beyond the finite, however, the two concepts diverge.
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