# Natural Number

Natural Number

In mathematics, the natural numbers are the ordinary whole numbers used for counting ("there are 6 coins on the table") and ordering ("this is the 3rd largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively (see English numerals). A later notion is that of a nominal number, which is used only for naming.

Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partition enumeration, are studied in combinatorics.

There is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the positive integers {1, 2, 3, ...}, while for others the term designates the non-negative integers {0, 1, 2, 3, ...}. The former definition is the traditional one, with the latter definition having first appeared in the 19th century. Some authors use the term "natural number" to exclude zero and "whole number" to include it; others use "whole number" in a way that excludes zero, or in a way that includes both zero and the negative integers.

Read more about Natural Number:  History of Natural Numbers and The Status of Zero, Notation, Algebraic Properties, Properties, Generalizations, Formal Definitions

### Other articles related to "natural numbers, numbers, natural number, number":

Proof Sketch For Gödel's First Incompleteness Theorem - Gödel Numbering
... formulas of the theory, and finite lists of these formulas, as natural numbers ... These numbers are called the Gödel numbers of the formulas ... Begin by assigning a natural number to each symbol of the language of arithmetic, similar to the manner in which the ASCII code assigns a unique binary number ...
Unary Coding
... sometimes called thermometer code, is an entropy encoding that represents a natural number, n, with n ones followed by a zero (if natural number is understood as non-negative ...
Kleene's O - Kleene's
... The natural number 0 belongs to Kleene's and ... If is total, with range contained in, and for every natural number, we have, then belongs to Kleene's, for each and, i.e ... is a notation for the limit of the ordinals where for every natural number ...
Examples - Example #3: Definition of The Unbounded μ Operator in Terms of An Abstract Machine
... IR), (ii) a few "registers" each of which can contain only a single natural number, and (iii) an instruction set of four "commands" described in the following table In the following, the symbolism ... of the function φ( x, y ) as the value of parameter y (a natural number) increases the process will continue (see Note † below) until a match occurs between the output of function φ( x, y ) and ... must be unbounded -- it must be able to "hold" a number of arbitrary size ...
Natural Number - Formal Definitions - Constructions Based On Set Theory - Other Constructions
... Each natural number is then equal to the set of the natural number preceding it ... The oldest and most "classical" set-theoretic definition of the natural numbers is the definition commonly ascribed to Frege and Russell under which each concrete natural number n is defined as ... The set of all natural numbers can be defined as the intersection of all sets containing 0 as an element and closed under S (that is, if the set ...

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