24 (number)

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

 ← 23 24 25 →
Cardinaltwenty-four
Ordinal24th
(twenty-fourth)
Numeral systemtetravigesimal
Factorization23 × 3
Divisors1, 2, 3, 4, 6, 8, 12, 24
Greek numeralΚΔ´
Roman numeralXXIV
Binary110002
Ternary2203
Octal308
Duodecimal2012

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.

In mathematics

• 24 is a composite number, being the first number of the form 23q, where q is an odd prime. 24 is an abundant number as its proper divisors total 36 and a semiperfect number, since adding up all the proper divisors of 24 except 4 and 8 gives 24.[1]
• 24 is the factorial of 4, 24 = 4!. Hence, it follows that 24 is the number of ways to order 4 distinct items: (1,2,3,4), (1,2,4,3), (1,3,2,4), (1,3,4,2), (1,4,2,3), (1,4,3,2), (2,1,3,4), (2,1,4,3), (2,3,1,4), (2,3,4,1), (2,4,1,3), (2,4,3,1), (3,1,2,4), (3,1,4,2), (3,2,1,4), (3,2,4,1), (3,4,1,2), (3,4,2,1), (4,1,2,3), (4,1,3,2), (4,2,1,3), (4,2,3,1), (4,3,1,2), (4,3,2,1). It is the largest factorial that does not contain a trailing zero at the end of its digits.
• It is the smallest number with exactly eight positive divisors: 1, 2, 3, 4, 6, 8, 12, and 24. Thus it is a highly composite number, having more divisors than any smaller number.[2]
• 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.
• 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.[3]
• 24 is a nonagonal number.[4]
• 24 is a Harshad number.[5]
• 24 is a semi-meandric number.
• 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 at least one multiple of three.
• The tesseract has 24 two-dimensional faces (which are all squares).
• 24 is the only nontrivial solution to the cannonball problem, that is: 12 + 22 + 32 + … + 242 is a perfect square (702). (The trivial case is just 12 = 12.)[6]
• 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 12 + 22 + 32 + ... + 242 = 702.)
• The modular discriminant Δ(τ) is proportional to the 24th power of the Dedekind eta function η(τ): Δ(τ) = (2π)12η(τ)24.
• The Barnes–Wall lattice contains 24 lattices.
• 24 is the only number whose divisors — 1, 2, 3, 4, 6, 8, 12, 24 — are exactly those numbers n for which every invertible element of the commutative ring Z/nZ is a square root of 1. It follows that the multiplicative group of invertible elements (Z/24Z)× = {±1, ±5, ±7, ±11} is isomorphic to the additive group (Z/2Z)3. This fact plays a role in monstrous moonshine.
It follows that any number n relatively prime to 24 (that is, any number of the form 6K ± 1), and in particular any prime n greater than 3, has the property that n2 – 1 is divisible by 24.
• The 24-cell, with 24 octahedral cells and 24 vertices, is a self-dual convex regular 4-polytope. It possesses 576 (24×24) rotational symmetries and 1152 isometries altogether. It tiles 4-dimensional space in a 24-cell honeycomb, in which each 24-cell is surrounded by 24 24-cells.
The vertices of the 24-cell honeycomb can be chosen so that in 4-dimensional space, identified with the ring of quaternions, they are precisely the elements of the subring generated by the binary tetrahedral group as represented by the set of 24 quaternions ${\displaystyle \{\pm 1,\pm i,\pm j,\pm k,{\tfrac {1}{2}}(\pm 1\pm i\pm j\pm k)\}}$ . This set of 24 quaternions forms the set of vertices of a single 24-cell, all lying on the sphere S3 of radius one centered as the origin. S3 is the Lie group of unit quaternions (isomorphic to the Lie groups SU(2) and Spin(3)), and so the binary tetrahedral group — of order 24 — is a subgroup of S3.
• 24 is the order of the octahedral group — the group of rotations of the regular octahedron and the group of rotations of the cube. The binary octahedral group is a subgroup of the 3-sphere S3 consisting of the 24 elements {±1, ±i, ±j, ±k, (±1±i±j±k)/2} of the binary tetrahedral group and the 24 elements contained in its coset {(±1±i)/√2, (±1±j)/√2, (±1±k)/√2, (±i±j)/√2, (±i±k)/√2, (±j±k)/√2}. These two cosets each form the vertices of a 24-cell, and the two 24-cells are dual to each other.
• 24 is the kissing number in 4-dimensional space: the maximum number of unit spheres that can all touch another unit sphere without overlapping. (The centers of 24 such spheres form the vertices of a 24-cell.)
• 24 is the largest integer that is divisible by all natural numbers no larger than its square root.
• 24 is the Euler characteristic of a K3 surface
• 24 is the smallest 5-hemiperfect number
• 24 is the order of the cyclic group equal to the stable 3-stem in homotopy groups of spheres: πn+3(Sn) = Z/24Z for all n ≥ 5.

In music

• There are a total of 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 other fields

Astronomical clock in Prague

24 is also:

• The number of bits a computer needs to represent true-color 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 karats representing 100% pure gold.[9]
• The number of cycles in the Chinese solar year.
• The number of years from the start of the Cold War until the signing of the Seabed Arms Control Treaty, which banned the placing of nuclear weapons on the ocean floor within certain coastal distances.
• The number of frames per second at which motion picture film is usually projected, as this is sufficient to allow for persistence of vision.
• The number of letters in both the modern and classical Greek alphabet.[10] For the latter reason, also the number of chapters or "books" into which Homer's Odyssey and Iliad came to be divided.
• The number of runes in the Elder Futhark.
• The number of points on a backgammon board.[11]
• 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.
• The number of the French department Dordogne.
• Four and twenty is the number of blackbirds baked in a pie in the traditional English nursery rhyme "Sing a Song of Sixpence".

References

1. ^ "Sloane's A005835 : Pseudoperfect (or semiperfect) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
2. ^ "Sloane's A002182 : Highly composite numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
3. ^ "Sloane's A097942 : Highly totient numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
4. ^ "Sloane's A001106 : 9-gonal (or enneagonal or nonagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
5. ^ "Sloane's A005349 : Niven (or Harshad) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
6. ^ Weisstein, Eric W. "Cannonball Problem". mathworld.wolfram.com. Retrieved 2020-08-19.
7. ^ Meija, Juris; Coplen, Tyler B.; Berglund, Michael; Brand, Willi A.; Bièvre, Paul De; Gröning, Manfred; Holden, Norman E.; Irrgeher, Johanna; Loss, Robert D.; Walczyk, Thomas; Prohaska, Thomas (2016-03-01). "Atomic weights of the elements 2013 (IUPAC Technical Report)". Pure and Applied Chemistry. 88 (3): 265–291. doi:10.1515/pac-2015-0305. ISSN 0033-4545. S2CID 101719914.
8. ^ "Revelation 4:4, New International Version (1984)". Bible.cc. Retrieved 2013-05-03.
9. ^ "Is 24K gold pure?". Scientific American. Retrieved 2020-08-12.
10. ^ "Greek alphabet | History, Definition, & Facts". Encyclopedia Britannica. Retrieved 2020-08-12.
11. ^ "GammonSite - Rules of backgammon". www.gammonsite.com. Retrieved 2020-08-12.