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| {{other uses|24 (disambiguation)}}
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| {{Infobox number
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| | number = 24
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| | divisor = 1, 2, 3, 4, 6, 8, 12, 24
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| }}
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| '''24''' ('''twenty-four''') is the [[natural number]] following [[23 (number)|23]] and preceding [[25 (number)|25]].
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| The [[SI prefix]] for 10<sup>24</sup> is [[yotta]] (Y), and for 10<sup>−24</sup> (i.e., the reciprocal of 10<sup>24</sup>) [[Yocto-|yocto]] (y). These numbers are the largest and smallest number to receive an SI prefix to date.
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| ==In mathematics==
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| *24 is the [[factorial]] of 4 and a [[composite number]], being the first number of the form <math>2^3q</math>, where <math>q</math> is an [[odd prime]].
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| *It is the smallest number with exactly eight [[divisor]]s: [[1 (number)|1]], [[2 (number)|2]], [[3 (number)|3]], [[4 (number)|4]], [[6 (number)|6]], [[8 (number)|8]], [[12 (number)|12]], and 24.
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| *It is a [[highly composite number]], having more divisors than any smaller number.
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| *24 is a [[semiperfect number]], since adding up all the proper divisors of 24 except 4 and 8 gives 24.
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| *Subtracting 1 from any of its divisors (except 1 and 2, but including itself) yields a [[prime number]]; 24 is the largest number with this property.
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| *24 has an [[aliquot sum]] of 36 and the [[aliquot sequence]] (24, 36, 55, 17, 1, 0).
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| *The [[aliquot sum]] of only one number, 529 = 23<sup>2</sup>, is 24.
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| *There are 10 solutions to the equation [[Euler's totient function|φ]](''x'') = 24, namely 35, 39, 45, 52, 56, 70, 72, 78, 84 and 90. This is more than any integer below 24, making 24 a [[highly totient number]].
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| *24 is a [[nonagonal number]].
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| *24 is the sum of the [[twin prime|prime twins]] 11 and 13.
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| *24 is a [[Harshad number]].
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| *24 is a [[semi-meandric number]].
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| *The product of any four consecutive numbers is divisible by 24. This is because among any four consecutive numbers there must be two even numbers, one of which is a multiple of four, and there must be a multiple of three.
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| *The [[tesseract]] has 24 two-dimensional faces (which are all [[square (geometry)|squares]]).
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| *24 is the only nontrivial solution to the [[cannonball problem]], that is: 1<sup>2</sup>+2<sup>2</sup>+3<sup>2</sup>+...+24<sup>2</sup> is a [[square number|perfect square]] (=70<sup>2</sup>). The trivial case is just 1<sup>2</sup> = 1<sup>2</sup>.
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| *In 24 dimensions there are 24 even positive definite [[unimodular lattice]]s, called the [[Niemeier lattice]]s. One of these is the exceptional [[Leech lattice]] which has many surprising properties; due to its existence, the answers to many problems such as the [[kissing number problem]] and [[sphere packing|densest lattice sphere-packing problem]] are known in 24 dimensions but not in many lower dimensions. The Leech lattice is closely related to the equally nice length-24 [[binary Golay code]] and the [[Steiner system]] S(5,8,24) and the [[Mathieu group]] M<sub>24</sub>. (One construction of the [[Leech lattice]] is possible because 1<sup>2</sup>+2<sup>2</sup>+3<sup>2</sup>+...+24<sup>2</sup> =70<sup>2</sup>.)
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| *The [[modular discriminant]] Δ(τ) is proportional to the 24th power of the [[Dedekind eta function]] η(τ): Δ(τ) = (2π)<sup>12</sup>η(τ)<sup>24</sup>.
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| *The [[Barnes-Wall lattice]] contains 24 [[Lattice (group)|lattice]]s.
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| *24 is the only number whose divisors — namely, {1, 2, 3, 4, 6, 8, 12, 24} — are exactly those numbers ''n'' for which every invertible element of the [[commutative ring]] [[modular arithmetic|'''Z'''/''n'''''Z''']] is a square root of 1. It follows that the multiplicative group ('''Z'''/24'''Z''')<sup>×</sup> = {±1, ±5, ±7, ±11} is [[group isomorphism|isomorphic]] to the additive group ('''Z'''/2'''Z''')<sup>3</sup>. This fact plays a role in [[monstrous moonshine]].
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| *:It follows that any number ''n'' relatively prime to 24, and in particular any prime ''n'' greater than 3, has the property that ''n''<sup>2</sup>-1 is divisible by 24.
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| *The [[24-cell]], with 24 octahedral cells and 24 vertices, is a [[dual polyhedron|self-dual]] [[convex regular 4-polytope]]; it has no analogue in any other dimension. Its 24 vertices can be expressed as the set {±1, ±i, ±j, ±k, (±1 ± i ± j ± k)/2} of unit quaternions, using all choices of signs. This set forms a group under quaternion multiplication, isomorphic to the [[binary tetrahedral group]]. The quotient of the unit quaternions '''S'''<sup>3</sup> by this subgroup is identical as a metric space to the [[configuration space]] of a regular tetrahedron centered at the origin in 3-space. The 24-cell tiles 4-dimensional space.
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| *24 is the [[kissing number]] in 4-dimensional space: the maximum number of unit spheres that can touch another one without overlapping. (The centers of such 24 spheres form the vertices of a 24-cell.)
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| *24 is the second [[Granville number]], the previous being [[6 (number)|6]] and the next being [[28 (number)|28]]. It is the first Granville number that is not also a conventional perfect number.
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| *24 is the largest [[integer]] that is evenly divisible by all [[natural numbers]] no larger than its square root.
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| ==In science==
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| * The [[atomic number]] of [[chromium]]
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| * The number of hours in a day
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| * [[Factorial|24!]] is an approximation (by 3%) of the [[Avogadro constant]].
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| ==In religion==
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| *The number of books in the [[Tanakh]]
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| *In Christian [[apocalyptic]] literature it represents the complete Church, being the sum of the 12 [[tribes of Israel]] and the 12 [[Apostles]] of the Lamb of God
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| *Number of [[Tirthankaras]]
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| *Number of spokes in the [[Ashok Chakra]]
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| ==In music==
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| *There are 24 major and minor keys in Western [[tonal music]], not counting [[enharmonic]] equivalents. Therefore, for collections of pieces written in each key, the number of pieces in such a collection; e.g., [[Frédéric Chopin|Chopin]]'s ''[[Preludes Op. 28 (Chopin)|24 Preludes]]''
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| ==In sports==
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| *The number of teams that participated in each [[FIFA World Cup]] finals tournament between 1982 and 1994. Also the number of teams that will participate in the finals of the [[FIFA Women's World Cup]] starting in [[2015 FIFA Women's World Cup|2015]].
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| * [[Four-and-Twenty]] was an American racehorse.
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| * Under the [[basketball]] rules of [[International Basketball Federation|FIBA]], the [[National Basketball Association|NBA]], and the [[Women's National Basketball Association|WNBA]], the offensive team has [[Shot clock|24 seconds]] to attempt a shot.
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| * There are 24 players on the field at any given time in [[Canadian football]] (12 on each team).
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| * The [[Squad number|jersey number]] 24 has been retired by several [[Major professional sports leagues in the United States and Canada|North American sports]] teams in honor of past playing greats or other key figures:
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| ** In [[Major League Baseball]]:
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| *** The [[Cincinnati Reds]], for [[National Baseball Hall of Fame and Museum|Hall of Famer]] [[Tony Pérez]].
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| *** The [[Los Angeles Dodgers]], for Hall of Fame [[manager (baseball)|manager]] [[Walter Alston]].
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| *** The [[St. Louis Cardinals]], for Hall of Fame manager [[Whitey Herzog]].
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| *** The [[San Francisco Giants]], for Hall of Famer [[Willie Mays]].
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| *** The [[Seattle Mariners]] have yet to retire any numbers, but took #24 out of circulation when [[Ken Griffey, Jr.]] was traded from the team after the 1999 season. The number was reissued to Griffey when he rejoined the team in 2009, and taken out of circulation again after his retirement during the 2010 season.
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| ** In the [[National Basketball Association|NBA]]:
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| *** The [[Boston Celtics]], for [[Naismith Memorial Basketball Hall of Fame|Hall of Famer]] [[Sam Jones (basketball)|Sam Jones]].
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| *** The [[Golden State Warriors]], for Hall of Famer [[Rick Barry]].
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| *** The [[Houston Rockets]], for Hall of Famer [[Moses Malone]].
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| *** The [[New York Knicks]], for Hall of Famer [[Bill Bradley]].
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| *** The [[Philadelphia 76ers]], for [[Bobby Jones (basketball, born 1951)|Bobby Jones]].
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| *** The [[Phoenix Suns]], for [[Tom Chambers (basketball)|Tom Chambers]].
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| *** The [[Seattle SuperSonics]], for [[Spencer Haywood]]. The team has since [[Seattle SuperSonics relocation to Oklahoma City|relocated]] to become the [[Oklahoma City Thunder]], but the Thunder has yet to issue any of the numbers retired by the franchise in Seattle.
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| ** In the [[National Hockey League|NHL]]:
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| *** The [[Boston Bruins]], for [[Terry O'Reilly]].
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| *** The [[St. Louis Blues]], for [[Hockey Hall of Fame|Hall of Famer]] [[Bernie Federko]].
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| ** In the [[National Football League|NFL]]:
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| *** The [[History of the Baltimore Colts|Baltimore Colts]], for [[Pro Football Hall of Fame|Hall of Famer]] [[Lenny Moore]]. The current [[Indianapolis Colts]] have yet to retire any numbers, but continue to honor all numbers retired by the franchise in Baltimore.
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| *The number of accepted runners in the [[Melbourne Cup]]
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| *The number of a car in the [[NASCAR]] [[Sprint Cup Series]] owned by [[Hendrick Motorsports]]. The car has been driven by [[Jeff Gordon]] since its introduction in 1992.
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| ==In other fields==
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| [[Image:Prague Apr04 015.jpg|200px|thumb|Astronomical clock in Prague]]
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| '''24''' is also:
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| * The number of [[bit]]s a computer needs to represent [[TrueColor]] images (for a maximum of 16,777,216 colours). (But greater numbers of bits provide more accurate colors. "TrueColor" is one of many possible representations of colors.)
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| * The number of [[carat (purity)|carat]]s representing 100% pure [[gold]].
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| * The number of cycles in the [[Chinese calendar|Chinese solar year]].
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| * The number of frames per second at which motion picture film is usually projected.
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| * The number of hours in a day.
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| * The number of letters in both the modern and classical [[Greek alphabet]]. For the latter reason, also the number of chapters or "books" into which [[Homer]]'s ''[[Odyssey]]'' and ''[[Iliad]]'' came to be divided.
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| {| style="clear: right" align="right"
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| | [[Image:Seven-segment 2.svg]] [[Image:Seven-segment 4.svg]]
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| * ''[[24 (TV series)|24]]'', television series starring [[Kiefer Sutherland]]. Each episode covers one hour, with 24 episodes making up one entire "day".
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| * The number of points on a [[backgammon]] board.
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| * When pronounced "two-four", a 24-pack of beer ([[Canadian English|Canadianism]]).
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| * A children's mathematical game involving the use of any of the four standard operations on four numbers on a card to get 24 (see [[Math 24]])
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| * The maximum number of Knight Companions in the [[Order of the Garter]]
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| * 24 is considered an unlucky number in Cantonese culture because its pronunciation is similar to that of "easy to die".{{citation needed|date=November 2010}} For this reason, many buildings skip this floor number. The same goes for 13, 34, 44, etc.
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| * [[X]] is the 24th letter of the Latin alphabet
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| * In [[Brazil]], the number is associated with [[Homosexuality|homosexuals]] due to the number representing the [[deer]] in the gambling game [[Jogo do Bicho]] and the word ''viado'' (a misspelling of ''veado'', deer in [[Portuguese language|Portuguese]]) being a slang for homosexuals.
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| * The number of the French department [[Dordogne]].
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| * In a [[Spongebob Squarepants]] episode called "New Student Starfish", when Patrick is asked for his name in driving school, he struggles and says 24, causing everyone to burst out laughing.
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| * Four and twenty is the number of blackbirds baked in a pie in the traditional English nursery rhyme ''[[Sing a Song of Sixpence]]''.
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| ==References==
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| {{reflist}}
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| ==External links==
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| {{Commons category|24 (number)}}
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| * [http://math.ucr.edu/home/baez/numbers/index.html#24 My Favorite Numbers: 24], [[John C. Baez]]
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| {{Integers|zero}}
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| {{DEFAULTSORT:24 (Number)}}
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| [[Category:Integers|2 4]]
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