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In mathematics, the natural numbers are those numbers used for counting and ordering.
Numbers used for counting are called cardinal numbers, and numbers used for ordering are called ordinal numbers. Natural numbers are sometimes used as labels, known as nominal numbers, having none of the properties of numbers in a mathematical sense.
Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3,..., whereas others start with 1, corresponding to the positive integers 1, 2, 3,... Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers.
The natural numbers are a basis from which many other number sets may be built by extension: the integers, by including the identity element 0 and an additive inverse for each nonzero natural number n; the rational numbers, by including a multiplicative inverse for each nonzero integer n ; the real numbers by including with the rationals the limits of Cauchy sequences of rationals; the complex numbers, by including with the real numbers the unresolved square root of minus one ; and so on. This chain of extensions make the natural numbers canonically embedded in the other number systems.