In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, and also to contrast the discreteness of counting to the continuity of measurement, established by the real numbers. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems. The natural numbers are a basis from which many other number sets may be built by extension: the integers ( Grothendieck group), by including (if not yet in) the neutral element 0 and an additive inverse (− n) for each nonzero natural number n the rational numbers, by including a multiplicative inverse (1/ n) for each nonzero integer n (and also the product of these inverses by integers) the real numbers by including with the rationals the limits of (converging) Cauchy sequences of rationals the complex numbers, by including with the real numbers the unresolved square root of minus one (and also the sums and products thereof) and so on. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers (including negative integers). 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, …. In common mathematical terminology, words colloquially used for counting are " cardinal numbers" and words connected to ordering represent " ordinal numbers". In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). Natural numbers can be used for counting (one apple, two apples, three apples, …)
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