24 (number)

← 23 25 → 24
← 20 21 22 23 24 25 26 27 28 29 → List of numbers — Integers 0 10 20 30 40 50 60 70 80 90 →
Cardinal	twenty-four
Ordinal	24th (twenty-fourth)
Factorization	$2^3 \cdot 3$
Divisors	1, 2, 3, 4, 6, 8, 12, 24
Roman numeral	XXIV
Binary	11000₂
Octal	30₈
Duodecimal	20₁₂
Hexadecimal	18₁₆

24 (twenty-four) is the natural number following 23 and preceding 25.

The SI prefix for 10²⁴ is yotta (Y), and for 10⁻²⁴ (i.e., the reciprocal of 10²⁴) yocto (y). These numbers are the largest and smallest number to receive an SI prefix to date.

1 In mathematics
2 In science
3 In religion
4 In music
5 In sports
6 In other fields
7 Historical years
8 References
9 External links

In mathematics

24 is the factorial of 4 and a composite number, being the first number of the form $2^3q$ , where $q$ is an odd prime.
It is the smallest number with exactly eight divisors: 1, 2, 3, 4, 6, 8, 12, and 24. It is a highly composite number, having more divisors than any smaller number. Adding up all the proper divisors of 24 except 4 and 8 gives 24, hence 24 is a semiperfect number.
Subtracting one from any of its divisors (except 1 and 2, but including itself) yields a prime number. 24 is the largest number with this property, for to have this property a number cannot be divisible by a prime greater than three, nor can it be divisible by 9 or 16.
24 has an aliquot sum of 36 and the aliquot sequence (24,36,55,17,1,0). Only one number has an aliquot sum 24; 529, the square of 23.
There are 10 solutions to the equation 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.
24 is a nonagonal number. This number is also the sum of a twin prime (11 + 13). It is a Harshad number and a semi-meandric number.
Together with the number one, 24 is one of the few numbers $n$ for which the sum of $\mu(d)d^2$ over the divisors $d$ of $n$ is equal to itself.^[1]
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.^[2]
In 24 dimensions there are 24 even positive definite unimodular lattices, called the Niemeier lattices. 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 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 M₂₄. One construction of the Leech lattice is possible because of the remarkable fact that 1²+2²+3²+...+24² =70² is a perfect square; 24 is the only integer greater than 1 with this property. These properties of 24 are related to the fact that the number 24 also appears in several places in the theory of modular forms; for example, the modular discriminant is the 24th power of the Dedekind eta function.
The Barnes-Wall lattice contains 24 lattices.
The divisors of 24 — namely, {1, 2, 3, 4, 6, 8, 12, 24} — are exactly those n for which every element x of the group of units (Z/nZ)^* of the commutative ring Z/nZ satisfies x² = 1. Thus the multiplicative group (Z/24Z)^* = {±1, ±5, ±7, ±11} is isomorphic to the additive group (Z/2Z)³. This fact plays a role in monstrous moonshine.
The 24-cell, with 24 octahedral cells and 24 vertices, is a self-dual convex regular 4-polytope; it has no analogue in any other dimension. Its 24 vertices can be expressed as the set of unit quaternions {±1, ±i, ±j, ±k, (±1±i±j±k)/2} using all choices of signs; this set forms a group under quaternion multiplication. (This is isomorphic to the binary tetrahedral group.)
For any prime $p$ greater than 3, $p^2-1$ is divisible by $24$ . Hence, any square of a prime $p$ > 3 expressed in radix 24 ends with an 1. For example, 2² = 4₂₄ and 3² = 9₂₄, but 5² = 11₂₄, 7² = 21₂₄, 11² = 51₂₄, 13² = 71₂₄, etc.
24 is the second Granville number, the previous being 6 and the next being 28. It is the first Granville number that is not also a conventional perfect number.
24 is the largest integer that is evenly divisible by all natural numbers smaller or equal to its square root.

In science

The atomic number of chromium
The number of hours in a day
24! is an approximation (by 3%) of the Avogadro constant.

In religion

The number of books in the Tanakh
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
Number of Tirthankaras

In music

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., Chopin's 24 Preludes

In sports

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.
Four-and-Twenty was an American racehorse.
Under the basketball rules of FIBA, the NBA, and the WNBA, the offensive team has 24 seconds to attempt a shot.
There are 24 players on the field at any given time in Canadian football (12 on each team).
The jersey number 24 has been retired by several North American sports teams in honor of past playing greats or other key figures:
- In Major League Baseball:
  - The Cincinnati Reds, for Hall of Famer Tony Pérez.
  - The Los Angeles Dodgers, for Hall of Fame manager Walter Alston.
  - The St. Louis Cardinals, for Hall of Fame manager Whitey Herzog.
  - The San Francisco Giants, for Hall of Famer Willie Mays.
  - 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.
- In the NBA:
  - The Boston Celtics, for Hall of Famer Sam Jones.
  - The Golden State Warriors, for Hall of Famer Rick Barry.
  - The Houston Rockets, for Hall of Famer Moses Malone.
  - The New York Knicks, for Hall of Famer Bill Bradley.
  - The Philadelphia 76ers, for Bobby Jones.
  - The Phoenix Suns, for Tom Chambers.
  - The Seattle SuperSonics, for Spencer Haywood. The team has since relocated to become the Oklahoma City Thunder, but the Thunder has yet to issue any of the numbers retired by the franchise in Seattle.
  - The Los Angeles Lakers, for Kobe Bryant.
- In the NHL:
  - The Boston Bruins, for Terry O'Reilly.
  - The St. Louis Blues, for Hall of Famer Bernie Federko.
- In the NFL:
  - The Baltimore Colts, for 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.
The number of accepted runners in the Melbourne Cup
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.

In other fields

24 is also:

The number of bits 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.)
The number of carats representing 100% pure gold.
The number of cycles in the Chinese solar year.
The number of frames per second at which motion picture film is usually projected.
The number of hours in a day.
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.

24, television series starring Kiefer Sutherland. Each episode covers one hour, with 24 episodes making up one entire "day".
The number of points on a backgammon board.
When pronounced "two-four", a 24-pack of beer (Canadianism).
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)
The maximum number of Knight Companions in the Order of the Garter
24 is considered an unlucky number in Cantonese culture because its pronunciation is similar to that of "easy to die". For this reason, many buildings skip this floor number. The same goes for 13, 34, 44, etc.
A Straight Edge symbol, for X is the 24th letter of the alphabet.
Slang for ecstasy
In Brazil, the number is associated with 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) being a slang for homosexuals.

Historical years

24 A.D., 24 B.C., 1924, 2024, etc.

References

^ Neil Sloane, "A046970 formatted as a simple table" The only times n = a(n) are for n = 1, 24. The table only goes up to 60, though.
^ Chambers's Encyclopædia. Edinburgh: Chambers W. and R., ltd. 1864. pp. 826.

External links

My Favorite Numbers: 24, John C. Baez