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# Natural number

zh-cn:自然数 zh-tw:自然數

A natural number is a non-negative integer: 0, 1, 2, 3, 4, ... These are the first numbers learned by children, and the easiest to understand. Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table), or they can be used for ordering ("this is the 3rd largest city in the state"). The deeper properties of the natural numbers, such as the distribution of prime numbers, are studied in number theory.

 Table of contents 1 History of natural numbers and the status of zero 2 Notation 3 Formal definitons 4 Properties 5 Generalizations

## History of natural numbers and the status of zero

Natural numbers were originally invented to count physical objects. Their first systematic study as things in themselves (separated from physical objects) is usually credited to the Greek philosophers Pythagoras and Archimedes. However, independent studies occurred at around the same time in India, China, and Mesoamerica.

Zero is relatively newborn. A zero digit was used in place-value notation as early as 400 BC by the Babylonians. The Olmec and Maya civilization used zero as a separate number as early as 1st century BC, apparently developed independently, but they did not pass it along to anyone outside of Mesoamerica. The modern concept dates to the Indian mathematician Brahmagupta in 628 AD. It took more than five centuries for European mathematicians to accept zero as a number, and even when they did, it was not counted as a natural number.

In the nineteenth century, a set-theoretical definition of the natural numbers was developed. With this definition, it was more convenient to include zero (corresponding to the empty set) in the naturals. Wikipedia follows this convention, as do set theorists, logicians, and computer scientists. Some other mathematicians, mainly number theorists, prefer to follow the old tradition and exclude zero from the natural numbers.

The term whole number is used by different authors for the set of integers, the set of non-negative integers, or the set of positive integers. It is best avoided in Wikipedia.

## Notation

Mathematicians use N or (an N in blackboard bold) to refer to the set of all natural numbers. This set is infinite but countable by definition.

W or is sometimes used to refer to the set of whole numbers, by authors who do not identify it with the integers.

## Formal definitons

The precise mathematical definition of the natural numbers has not been easy. The Peano postulates state conditions that any successful definition must satisfy:

• There is a natural number 0.
• Every natural number a has a successor, denoted by a + 1.
• There is no natural number whose successor is 0.
• Distinct natural numbers have distinct successors: if ab, then a + 1 ≠ b + 1
• If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers. (This postulate ensures that the proof technique of mathematical induction is valid.)

If zero is excluded from the natural numbers, every 0 in the Peano postulates should be replaced by a 1.

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. Under this definition, there are exactly n elements in the set n and if m is bigger than n, then n is a subset of m.

## Properties

One can inductively define an addition on the natural numbers by requiring a + 0 = a and a + (b + 1) = (a + b) + 1. This turns the natural numbers (N, +) into a commutative monoid with neutral element 0, the so-called 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 * 0 = 0 and 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 ab 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 + cb + c and acbc. An important property of the natural numbers is that they are well-ordered: every non-empty 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

a = bq + r and r < b

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. This, the quotient-remainder theorem, is key to several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory.

## Generalizations

Two generalizations of natural numbers arise from the two uses: ordinal numbers are used to describe the position of an element in a ordered sequence and cardinal numbers are used to specify the size of a given set.

For finite sequences or finite sets, both of these are of course the same as the natural numbers.

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