Natural number | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-03-26

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Natural_number


00:02:16 1 History
00:02:25 1.1 Ancient roots
00:04:49 1.2 Modern definitions
00:08:36 2 Notation
00:09:12 3 Properties
00:09:25 3.1 Infinity
00:09:34 3.2 Addition
00:10:24 3.3 Multiplication
00:10:33 3.4 Relationship between addition and multiplication
00:11:02 3.5 Order
00:11:15 3.6 Division
00:11:55 3.7 Algebraic properties satisfied by the natural numbers
00:12:06 4 Generalizations
00:12:13 5 Formal definitions
00:12:23 5.1 Peano axioms
00:12:41 5.2 Constructions based on set theory
00:12:54 5.2.1 Von Neumann ordinals
00:13:05 5.2.2 Zermelo ordinals
00:14:04 6 See also
00:14:16 7 Notes
00:15:07 8 References
00:15:40 9 External links
00:15:45 Algebraic properties satisfied by the natural numbers
00:16:20 (a + b) + c and a × (b × c)
00:16:40 b + a and a × b
00:16:58 a and a × 1
00:17:25 0, then a
00:17:36 0 (or both).
00:17:40 Generalizations
00:20:13 Formal definitions
00:20:22 Peano axioms
00:21:46 Constructions based on set theory
00:21:56 Von Neumann ordinals
00:22:52 0 ∪ {0}
00:23:04 1 ∪ {1}
00:23:16 2 ∪ {2}
00:23:29 n−1 ∪ {n−1}
00:24:19 Zermelo ordinals
00:24:45 {0}
00:24:54 {1}
00:25:03 {n−1}
00:25:24 See also



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SUMMARY
=======
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"). In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers". The natural numbers can, at times, appear as a convenient set of codes (label or "names"), that is, as what linguists call nominal numbers, foregoing many or all of the properties of being a number 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, but in other writings, that term is used instead for the integers (including negative integers).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. These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems.
Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics.
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.