24 (number)
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24
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
For other uses, see 24 (disambiguation).

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.

Contents

In mathematics

24 is the factorial of 4 and a composite number, being the first number of the form 23q, 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). Twenty-four is only the aliquot sum of one number the square 529.

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 μ(d)d2 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 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 =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.

24 is the highest number n with the property that every element of the group of units (Z/nZ)* of the commutative ring Z/nZ, apart from the identity element, has order 2; thus the multiplicative group (Z/24Z)* = {1,5,7,11,13,17,19,23} 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 good 3-dimensional analogue.

In science

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

In other fields

Astronomical clock in Prague

24 is also:

  • The number of bits a computer needs to represent truecolour images (for a maximum of 16,777,216 colours). (But greater numbers of bits provide more accurate colors. "Truecolor" is one of many possible color representations.)
  • The number of a broadcast television channel
  • The number of cans or bottles in a beverage case.
  • 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.
Image:Seven-segment 2.svg Image:Seven-segment 4.svg
  • 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 4 standard operations on 4 numbers on a card to get 24 (see Math 24)
  • "24", a song by Welsh musician Jem
  • The maximum number of Knight Companions in the Order of the Garter
  • The Twenty-fourth Amendment to the United States Constitution is one of the Civil Rights-era amendments, abolishing poll tax requirements in federal elections.
  • 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 13, 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'
  • #24 is a recurring character on The Venture Bros.
  • A Straight Edge symbol, for X is the 24th letter of the alphabet
  • The number of the French department Dordogne

Historical years

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

References

  1. ^ 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.
  2. ^ (1864) Chambers's Encyclopædia, p.826. 
  3. ^ Knowshon Moreno

External links
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