In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used. The set of the natural numbers is commonly denoted by a bold N or a blackboard bold N {\displaystyle \mathbb {N} } . The natural numbers are used for counting, and for labeling the result of a count, such as: "there are seven days in a week", in which case they are called cardinal numbers. They are also used to label places in an ordered series, such as: "the third day of the month", in which case they are called ordinal numbers. Natural numbers are commonly expressed in writing using ten symbols called numerals ("0 1 2 3 4 5 6 7 8 9"). These numerals can also be used as unique identifiers or labels (like the jersey numbers of a sports team) that are referred to as nominal numbers, which resemble natural numbers but have no specific mathematical properties. Natural numbers can be compared by magnitude, with larger numbers coming after smaller ones in the list 1, 2, 3, .... Two basic arithmetical operations are defined on natural numbers: addition and multiplication. However, the inverse operations, subtraction and division, only sometimes give natural-number results: subtracting a larger natural number from a smaller one results in a negative number and dividing one natural number by another commonly leaves a remainder. The most common number systems used throughout mathematics – the integers, rational numbers, real numbers, and complex numbers – contain the natural numbers, and can be formally defined in terms of natural numbers. Arithmetic is the study of the ways to perform basic operations on these number systems. Number theory is the study of the properties of these operations and their generalizations. Much of combinatorics involves counting mathematical objects, patterns and structures that are defined using natural numbers.
Intuitive concept An intuitive and implicit understanding of natural numbers is developed naturally through using numbers for counting, ordering and basic arithmetic. Within this are two closely related aspects of what a natural number is: the size of a collection; and a position in a sequence.
Size of a collection Natural numbers can be used to answer questions like "how many apples are on the table?" A natural number used in this way describes a characteristic of a finite collection of objects. This characteristic, the size of a collection, is called cardinality and a natural number used to describe or measure it is a cardinal number.
Two finite collections have the same size or cardinality if they have a one-to-one correspondence, meaning the objects can be arranged in pairs (one from each collection), with every object in exactly one pair. In the adjacent image every apple is paired with exactly one orange and every orange is paired with exactly one apple. As such, the group of apples has the same cardinality as the group of oranges, or put more simply the number of apples is the same as the number of oranges. Because this equality can be established without counting or using any prior notion of number, it can form the definition of a cardinal number. In this case, the number of apples, oranges – and of any other collection that could be paired off to either group – is 3. If two collections do not have the same cardinality, pairing will leave one of the collections with objects that are unpaired and this can be used to define a size relationship between them. The collection in which all objects are paired is said to be "smaller" and the one left with unpaired objects "larger", than the other.
Position in a sequence A sequence is a list of objects in a specific order. More precisely, a sequence is a function that assigns an object to each position in that list. The positions themselves are labeled using a well-ordered set; every element always has a clear next element. Every well-ordered set has an order type, which is the ordinal number that describes its shape of ordering. The position labels here are not counts or size like with the cardinal numbers, just ordered elements. The natural numbers are the most common choice for labeling infinite sequences because they form the simplest infinite well-ordered set, with order type ω. They start at either 0 or 1 and continue in their familiar fixed order – 1, 2, 3, and so on – with no end point. Each natural number labels a specific position in the sequence based on where it falls relative to all other positions. For example, 1 is the first position, 2 is the position right after 1, and 3 is the position after both 1 and 2 and before 4, 5, and so on. This ordering matches the usual ordering, smaller numbers before larger ones. But the natural numbers are simply the most familiar example; any well-ordered set would work equally well for indexing a sequence, for example the set of letters a, b, c, and so on.
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