Concept
Natural number
Overview
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.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
Mathematicians have distinguished different degrees of continuity, and have confined the word "continuous," for technical purposes, to series having a certain high degree of continuity. But for philosophical purposes, all that is important in continuity is introduced by the lowest degree of continuity, which is called "compactness." A series is called "compact" when no two terms are consecutive, but between any two…
Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy
States what Natural number is; the author is tied to Natural number in the knowledge graph.
All types of series are capable of formal definition, and their properties can be deduced from the principles of symbolic logic by means of the Algebra of Relatives. The notion of a limit, which is fundamental in the greater part of higher mathematics, used to be defined by means of quantity, as a term to which the terms of some series approximate as nearly as we please. But nowadays the limit is defined quite…
Bertrand Russell, Mysticism and Logic and Other Essays
States what Natural number is; the author is tied to Natural number in the knowledge graph.
All numbers must be numbers of something: there are no such things as numbers in the abstract. _Ten_ must mean ten bodies, or ten sounds, or ten beatings of the pulse. But though numbers must be numbers of something, they may be numbers of anything. Propositions, therefore, concerning numbers, have the remarkable peculiarity that they are propositions concerning all things whatever; all objects, all existences of…
John Stuart Mill, A System of Logic
States what Natural number is.
Philosophers who discussed Natural number
Related ideas
- Finite set
- Giuseppe Peano
- Springer Science+Business Media
- Integer
- John von Neumann
- Foundations of mathematics
- Mathematics
- Axiomatic system
- Group (mathematics)
- Rational number
- Real number
- Constructivism (philosophy of mathematics)
- Numeral system
- Richard Dedekind
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Place in the tradition
Part of First-order logic.