Number

: ''This article is about the concept in mathematics. For a discussion of numbers in linguistics, see grammatical number. For other uses, see #Disambiguation.'' A number is an abstract entity used originally to describe quantity. At least since the invention of complex numbers, this definition must be relaxed. Preserving the main ideas of "quantity" except for the total order, one can define numbers as elements of any integral domain. ===Examples=== The most familiar numbers are the natural numbers {0, 1, 2, ...} or {1, 2, 3, ...}, used for counting, and denoted by N. If the negative number whole numbers are included, one obtains the integers Z. ("Whole numbers" are sometimes denoted by W, but it depends on the author if this means positive, non-negative, or all integers.) Fraction (mathematics)s of integers are called rational numbers. The set of all rational numbers is denoted by Q. Rational numbers having a finite decimal are called decimal fractions or decimal numbers, sometimes denoted by D. The real numbers R, can have an infinite and non-repeating decimal expansion. Real numbers which are not rational are called irrational numbers. The real numbers can be extended to the complex numbers C, which leads to an algebraically closed field in which every polynomial with complex coefficients can be completely factored. The above symbols are often written in blackboard bold, thus: :

\mathbb{N}\sub\mathbb{Z}\sub\mathbb{D}\sub\mathbb{Q}\sub\mathbb{R}\sub\mathbb{C}

In another respect, roots of polynomials with rational coefficients lead to algebraic numbers, and those complex numbers which are not algebraic are called transcendental number. Iterating the adjunction of imaginary units allows to extend complex numbers to quaternions H, losing commutativity of multiplication, and then to octonions, losing associativity and thus leaving the category of associative division algebras. === Further generalizations === Elements of function fields of finite characteristic (algebra) behave in some ways like numbers and are often regarded as a kind of number by number theorists. == Numerals and numbering == Numbers should be distinguished from ''numerals'', which are (combinations of) symbols used to represent numbers. The notation of numbers as a series of digits is discussed in numeral systems. People like to assign numbers to objects in order to have unique names. There are various numbering schemes for doing so. == Extensions == Superreal number, hyperreal number and surreal numbers extend the real numbers by adding infinitesimal and infinitely large numbers. While (most) real numbers have infinitely long expansions to the right of the decimal point, one can also try to allow for infinitely long expansions to the left in base p, where p is a prime number, leading to the p-adic numbers. For dealing with infinite collections, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former give the ordering of the collection, the latter its size. (For the finite case, the ordinal and cardinal numbers are equivalent; they diverge in the infinite case.) The arithmetical operations of numbers, such as addition, subtraction, multiplication and division (mathematics), are generalized in the branch of mathematics called abstract algebra; one obtains the group (mathematics)s, ring (algebra)s and field (mathematics)s. ==See also== * Arabic numeral system * even and odd numbers * floating point numbers are used by computers to approximate real numbers * large numbers * list of numbers * mathematical constant * mythical number * negative and non-negative numbers * orders of magnitude (numbers) * physical constant * prime number * small number * subitizing and counting * numbers in various languages * number sign * numero sign ==External links== * [http://www.virtuescience.com/number.html Database of Number Correlations] * [http://wiktionary.org/wiki/Number Wiktionary article on ''number''] * [http://www.stetson.edu/~efriedma/numbers.html What's special about this number?] * [http://www.cut-the-knot.org/do_you_know/numbers.shtml What's a Number?] ==Disambiguation== * The Book of Numbers, the fourth book of the Bible * ''NUMB3RS'', a CBS television show Group theory Numbers __NOTOC__ bn:সংখ্যা br:Niver hi:अंक simple:Number su:Wilangan th:จำนวน

Number

blackboard bold does not mesh well with a paragraph. User:Pizza Puzzle Eh, I'm used to seeing blackboard bold; plain bold reminds me of a variable, not a set. I agree that we should use it; however, the current font for it is too big and doesn't mesh with the rest of the page. Perhaps, if somebody submits a slightly smaller .png we can use that. User:Pizza Puzzle I plan to upload a whole bunch of <.png>s like this later this month. Of course, feel free to beat me to it. ^_^ Still we should still prefer markup that renders most directly in HTML, when such a thing works -- for now. -- User:Toby Bartels 05:57 12 Jun 2003 (UTC) ---- Could somebody produce a little Venn diagram picture showing the various number sets? I removed this verbal description of the Venn diagram. User:AxelBoldt 15:23, 29 Sep 2003 (UTC) ---- This statement is incorrect: "Ratios of integers are called rational numbers or fractions." In fact, ratios of integers are fractions but NOT rational numbers. The union of the set of integers and the set of fractions equals the set of rational numbers. The distinction between integers and fractions is that no less than two integers (by ratio) are required to define a fraction. I suspect this error has been carried over to a couple of other related pages. It must be corrected. User:OmegaMan :Rationals are usually defined as equivalence classes of ordered pairs of integers. Saying that they are ratios of integers is reasonable; this is only supposed to be an informal statement. A formal construction is given in the rational number article. --User:Zundark 08:17, 17 Nov 2003 (UTC) Yes but the formal construction you directed me to is for the rational numbers- NOT the fractions. The presentation of the various sets of numbers is more clearly understandable if it incorporates a brief summary of their methodical construction, one built onto the next. Where the integers have already been defined seperately, all trivial cases of fractions which equal integers should then be eliminated as redundant (i.e., those where ratios of two integers, converted to fractions, can be simplified such that the denominator is equal to one). Then, the rationals can be defined as the union of the two underlying subsets. Note that integers require only one integer (obviously) to define themselves reflexively which is not possible for fractions. You may think I am splitting trivial hairs. Still, the distinction I am making is reality-based and relevant. I am not just making this stuff up as I go along. It came directly from a "theory of arithmetic" textbook I own. User:OmegaMan -------------------------------------- This article is missing the ordinal/cardinal distinction for finite numbers. Although the finite ordinals are the same as the finite cardinals, the use to which they are put is different: "I have five beads" vs. "this is door number 5", and so they are conceptually different, even if mathematically equivalent. Can someone help put this distinction in the article? -- User:The Anome 13:54, 27 Jan 2004 (UTC) ===Umpteen=== I've added umpteen in the "see also" list, largely to de-orphan it. I'm not absolutely sure this is the right article to link to it, but I can't think of an alternative. Suggestions would be more than welcome. User:DavidWBrooks 20:01, 16 Feb 2004 (UTC) == Natural Numbers and Zero == I have never known zero to be included in the set of natural numbers (a.k.a. ''counting numbers'', hence the exclusion of zero, as one never counts the zeroth member of a set). Rather, it is the only non-natural member of the set of whole numbers. I hope someone will correct this, or at least address the question if I am in error. Arnold Karr :It's very common to include 0 as a natural number (so that "natural number" and "finite ordinal" mean the same thing). It's undesirable for some purposes, however, so not everyone does it. --User:Zundark 07:42, 23 May 2004 (UTC) :: I think it's very natural to have zero items of something (in contrast to having a negative amount of something). In fact, all of us own zero items of almost everything. It is at least as natural as the usual definition of (the number) zero as the empty set. :: Of course we do not count the zeroth member of a set, but when we count something, we start out with zero items counted, before adding the first to the inventory, if there is any. When somebody asks you to count the number of apples in your pocket, you would not protest saying "I cannot count them". You would maybe say "there are none", but this is just a synonym of zero. User:MFH 13:31, 7 Apr 2005 (UTC) == Mixing Numbers & Biology == I advocate the total removal of the speculative "biological basis" section, regardless of whether or not it may be wholly or partially correct. We should stick to provable information in an article involving mathematics in an online encyclopedia. User:OmegaMan == extensions and generalizations == The section "generalizations" should be merged into "extensions" (which could receive subsectioning). I suggest to put it after the nonstandard stuff and before the comment on abstract algebra. Please feel free to do so. User:MFH 18:07, 7 Apr 2005 (UTC)

Number

{| width="250" border="1" style="float: right; background: #feffff; border-collapse: collapse; margin: 0.5em" |- | colspan="2" | |- | Cardinal number | #
? |- | Ordinal number | #th
?th |- | Numeral system | ? |- | Factorization | prime number |

? \cdot ?

|- | Divisors | ? (number) |- | Roman numeral | ? |- | Prefixes | ? (Greek language)
? (Latin) |- | Binary numeral system | ? |- | Octal | ? |- | Duodecimal | ? |- | Hexadecimal | ? |- |}

See other meanings of words starting from letter:

N