24 (number)

24 (number)
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

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). 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 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.
  • For any prime p greater than 3, p2 − 1 is divisible by 24 with no remainder. This results that squares of p expressed in radix 24 end with an 1, e.g. 52 = 1124, 72 = 2124, B2 = 5124.
  • 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, which is divisible with no remainder by all natural numbers smaller or equal to its square root.

In science

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

In other fields

Astronomical clock in Prague

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.
Seven-segment 2.svg 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 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".[citation needed] 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

  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. ^ Chambers's Encyclopædia. Edinburgh: Chambers W. and R., ltd. 1864. pp. 826. 

External links


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