24 (number)

24 (twenty-four) is the natural number following 23 and preceding 25. In English, it is the smallest integer with ten letters in its spelled out name.

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.


Cardinal	24 twenty-four
Ordinal	24th twenty-fourth
Factorization	$2^3 \cdot 3$
Divisors	1, 2, 3, 4, 6, 8, 12, 24
Roman numeral	XXIV
Binary	11000
Hexadecimal	18

In mathematics

It is the factorial of 4 and a composite number, the smallest number with exactly eight divisors, its proper divisors being 1, 2, 3, 4, 6, 8, and 12, making it a highly composite number. Subtracting one from each of its divisors (except 1 and 2, but including itself) yields a prime number; 24 is the largest number with this property.

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.

24 is one of the few numbers for which the sum of $\mu(d)d^2$ for each of its divisors is equal to itself. (This it has in common with 1).

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 sphere packing 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 =70²is 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 24'th power of the Dedekind eta function.

24 is the largest number n with the property that every element of the group (Z/nZ)^* has order 1 or 2; this strange 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 good 3-dimensional analogue.

In science

The atomic number of chromium

Astronomy

The Messier object M24, a magnitude 11.5 star cloud in the constellation Sagittarius, known among other names as Delle Caustiche and the Sagittarius Star Cloud
The New General Catalogue object NGC 24, a magnitude 11.6 spiral galaxy in the constellation Sculptor
The Saros number of the solar eclipse series which began on -2134 April 6 and ended on -854 May 14. The duration of Saros series 24 was 1280.1 years, and it contained 72 solar eclipses. Further, the Saros number of the lunar eclipse series which began on -2049 September 6 and ended on -516 March 14. The duration of Saros series 24 was 1532.5 years, and it contained 86 lunar eclipses.
The 24th moon of Jupiter is Pasiphae

In religion

The number of books in the Tanakh

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 retired number of Willie Mays, former baseball star
The car number of 4-time NASCAR Nextel Cup Series champion Jeff Gordon

In other fields

24 is also:

The number of hours in a day.
The number of frames per second at which motion picture film is usually projected .
The number of cycles in the Chinese solar year.
The usual number of teeth in a human child's mouth.
The number of karats representing 100% pure gold.
The number of bits a computer needs to represent truecolour images (for a maximum of 16,777,216 colours).
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 24-episode season of the show covers 24 hours.
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 4 standard operations on 4 numbers on a card to get 24 (see Math 24)
"24", a song by Welsh musician Jem
The nursery rhyme Sing a Song of Sixpence has the line "four and twenty blackbirds, baked in a pie"
The maximum number of Knight Companions in the Order of the Garter
The Twenty-fourth Amendment to the United States Constitution
The designation of US Interstate 24, a freeway that runs from Tennessee to Illinois. Also many other roads called Route 24
24 is considered an unlucky number in Cantonese culture because its pronounication is similar to that of "easy to die". For this reason, many buildings skip this floor number. The same goes for 14, 34, 44, etc.
In Brazil, the number 24 holds a strong popular association with homosexuality, as there is a popular betting game, the Jogo do Bicho (Animal Game) in which 24 is the "deer" number. The Brazilian word for deer, "veado", is also used as a depreciative slang term for homosexual, although the latter is sometimes spelled with an 'i' as opposed to an 'e'

Historical years

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

Integers

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