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Natural number From Wikipedia, the free encyclopedia
12 (twelve) is the natural number following 11 and preceding 13. Twelve is a superior highly composite number, divisible by the numbers from 1 to 4, and 6.
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Cardinal | twelve | |||
Ordinal | 12th (twelfth) | |||
Numeral system | duodecimal | |||
Factorization | 22 × 3 | |||
Divisors | 1, 2, 3, 4, 6, 12 | |||
Greek numeral | ΙΒ´ | |||
Roman numeral | XII | |||
Greek prefix | dodeca- | |||
Latin prefix | duodeca- | |||
Binary | 11002 | |||
Ternary | 1103 | |||
Senary | 206 | |||
Octal | 148 | |||
Duodecimal | 1012 | |||
Hexadecimal | C16 | |||
Malayalam | ൰൨ | |||
Bengali | ১২ | |||
Hebrew numeral | י"ב | |||
Babylonian numeral | 𒌋𒐖 |
It is the number of years required for an orbital period of Jupiter. It is central to many systems of timekeeping, including the Western calendar and units of time of day, and frequently appears in the world's major religions.
Twelve is the largest number with a single-syllable name in English. Early Germanic numbers have been theorized to have been non-decimal: evidence includes the unusual phrasing of eleven and twelve, the former use of "hundred" to refer to groups of 120, and the presence of glosses such as "tentywise" or "ten-count" in medieval texts showing that writers could not presume their readers would normally understand them that way.[1][2][3] Such uses gradually disappeared with the introduction of Arabic numerals during the 12th-century Renaissance.
Derived from Old English, twelf and tuelf are first attested in the 10th-century Lindisfarne Gospels' Book of John.[note 1][5] It has cognates in every Germanic language (e.g. German zwölf), whose Proto-Germanic ancestor has been reconstructed as *twaliƀi..., from *twa ("two") and suffix *-lif- or *-liƀ- of uncertain meaning.[5] It is sometimes compared with the Lithuanian dvýlika, although -lika is used as the suffix for all numbers from 11 to 19 (analogous to "-teen").[5] Every other Indo-European language instead uses a form of "two"+"ten", such as the Latin duōdecim.[5] The usual ordinal form is "twelfth" but "dozenth" or "duodecimal" (from the Latin word) is also used in some contexts, particularly base-12 numeration. Similarly, a group of twelve things is usually a "dozen" but may also be referred to as a "dodecad" or "duodecad". The adjective referring to a group of twelve is "duodecuple".
As with eleven,[6] the earliest forms of twelve are often considered to be connected with Proto-Germanic *liƀan or *liƀan ("to leave"), with the implicit meaning that "two is left" after having already counted to ten.[5] The Lithuanian suffix is also considered to share a similar development.[5] The suffix *-lif- has also been connected with reconstructions of the Proto-Germanic for ten.[6][7]
As mentioned above, 12 has its own name in Germanic languages such as English (dozen), Dutch (dozijn), German (Dutzend), and Swedish (dussin), all derived from Old French dozaine. It is a compound number in many other languages, e.g. Italian dodici (but in Spanish and Portuguese, 16, and in French, 17 is the first compound number),[dubious – discuss] Japanese 十二 jūni.[clarification needed]
In prose writing, twelve, being the last single-syllable numeral, is sometimes taken as the last number to be written as a word, and 13 the first to be written using digits. This is not a binding rule, and in English language tradition, it is sometimes recommended to spell out numbers up to and including either nine, ten or twelve, or even ninety-nine or one hundred. Another system spells out all numbers written in one or two words (sixteen, twenty-seven, fifteen thousand, but 372 or 15,001).[8] In German orthography, there used to be the widely followed (but unofficial) rule of spelling out numbers up to twelve (zwölf). The Duden[year needed] (the German standard dictionary) mentions this rule as outdated.
12 is the sixth composite number and the superfactorial of 3.[9][10] It is the fourth pronic number (equal to 3 × 4),[11] whose digits in decimal are also successive. It is the smallest abundant number, since it is the smallest integer for which the sum of its proper divisors (1 + 2 + 3 + 4 + 6 = 16) is greater than itself,[12] and the second semiperfect number, since there is a subset of the proper divisors of 12 that add up to itself.[13] It is equal to the sum between the second pair of twin primes (5 + 7),[14] while it is also the smallest number with exactly six divisors (1, 2, 3, 4, 6 and 12) which makes it the fifth highly composite number,[15] and since 6 is also one of them, twelve is also the fifth refactorable number.[16] 12, as a number with a perfect number of divisors (six), has a sum of divisors that yields the second perfect number, σ(12) = 28,[17] and as such it is the smallest of two known sublime numbers, which are numbers that have a perfect number of divisors whose sum is also perfect.[18] 12 is the fifth Pell number (preceded by 0, 1, 2, and 5)[19] as well as the third pentagonal number,[20] and a Harshad number in all bases except octal.
Twelve is the number of divisors of 60 and 90, the second and third unitary perfect numbers (6 is the first). It is also the number of distinct prime factors that belong to the fifth unitary perfect number, the largest known,
The second perfect number, 28, is the arithmetic mean of the twelve divisors of the fourth harmonic divisor number, 140 (like 6, and 28), which generates an integer harmonic mean of 5.[23][24][25]
If an odd perfect number is not divisible by 3, it will have at least twelve distinct prime factors.[26]
There are 12 Latin squares of size 3 × 3, where symbols appear exactly once in each row and exactly once in each column.[27]
There are twelve Jacobian elliptic functions and twelve cubic distance-transitive graphs.
A twelve-sided polygon is a dodecagon. In its regular form, it is the largest polygon that can uniformly tile the plane alongside other regular polygons, as with the truncated hexagonal tiling or the truncated trihexagonal tiling. There are 12 regular and semiregular tilings when enantiomorphic forms of the snub hexagonal tiling are counted separately.[28]
A regular dodecahedron has twelve pentagonal faces. Regular cubes and octahedrons both have 12 edges, while regular icosahedrons have 12 vertices. The rhombic dodecahedron has twelve rhombic faces and is able to tessellate three-dimensional space; it is the only Catalan solid to generate a honeycomb with copies of itself. Its dual polyhedron, the cuboctahedron, has 12 vertices with radial equilateral symmetry, and is one of two quasiregular polyhedra.
The densest three-dimensional lattice sphere packing has each sphere touching twelve other spheres, and this is almost certainly true for any arrangement of spheres (the Kepler conjecture). Twelve is also the kissing number in three dimensions.
There are twelve complex apeirotopes in dimensions five and higher, which include van Oss polytopes in the form of complex -orthoplexes.[29] There are also twelve paracompact hyperbolic Coxeter groups of uniform polytopes in five-dimensional space.
Bring's curve is a Riemann surface of genus four, with a domain that is a regular hyperbolic 20-sided icosagon.[30] By the Gauss-Bonnet theorem, the area of this fundamental polygon is equal to .
The Leech lattice, which holds the solution to the kissing number in twenty-four dimensions,[31] has a density equal to:
Its quaternionic representation contains vectors modulo that are congruent to either one of coordinate-frames, or zero;[33][34] with 1,365 the twelfth Jacobsthal number, and 144 equal to 122.
Fischer group is a sporadic group with a total of twelve maximal subgroups, the smallest of which is Mathieu group .[35][36] holds standard generators equal to (2A, 13, 11),[37] with a further condition where .[38] Furthermore, its faithful complex representation is 78-dimensional,[39] where 78 is the twelfth triangular number.[40] Otherwise, the largest alternating group represented inside any sporadic groups is , as a maximal subgroup inside the third-largest third generation sporadic group, Harada-Norton group .[41][42] While or are not maximal subgroups of the largest sporadic group, the friendly giant , one of its maximal subgroup is .[43] More deeply, the double cover is a maximal subgroup of ,[44][45] which is the third-largest maximal subgroup inside ;[46][47] with the double cover as the largest maximal subgroup inside .[43] The smallest second generation sporadic group, Janko group , holds standard generators (2A, 3B, 7) that yield .[38]
Twelve is the smallest weight for which a cusp form exists. This cusp form is the discriminant whose Fourier coefficients are given by the Ramanujan -function and which is (up to a constant multiplier) the 24th power of the Dedekind eta function:
This fact is related to a constellation of interesting appearances of the number twelve in mathematics ranging from the fact that the abelianization of special linear group has twelve elements, to the value of the Riemann zeta function at being , which stems from the Ramanujan summation
Although the series is divergent, methods such as Ramanujan summation can assign finite values to divergent series.
2^2 * 3^1 = 12
Division | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
12 ÷ x | 12 | 6 | 4 | 3 | 2.4 | 2 | 1.714285 | 1.5 | 1.3 | 1.2 | 1.09 | 1 | 0.923076 | 0.857142 | 0.8 | 0.75 | |
x ÷ 12 | 0.083 | 0.16 | 0.25 | 0.3 | 0.416 | 0.5 | 0.583 | 0.6 | 0.75 | 0.83 | 0.916 | 1 | 1.083 | 1.16 | 1.25 | 1.3 |
Exponentiation | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
12x | 12 | 144 | 1728 | 20736 | 248832 | 2985984 | 35831808 | 429981696 | 5159780352 | 61917364224 | 743008370688 | 8916100448256 | |
x12 | 1 | 4096 | 531441 | 16777216 | 244140625 | 2176782336 | 13841287201 | 68719476736 | 282429536481 | 1000000000000 | 3138428376721 | 8916100448256 |
The duodecimal system (1210 [twelve] = 1012), which is the use of 12 as a division factor for many ancient and medieval weights and measures, including hours, probably originates from Mesopotamia.
In base thirteen and higher bases (such as hexadecimal), twelve is represented as C.
Notably, twelve is the number of full lunations in a solar year, hence the number of months in a solar calendar, as well as the number of signs in the Western, Islamic and the Chinese zodiac. Twelve is also the number of years for an orbital period of Jupiter.
The number twelve carries religious, mythological and magical symbolism, generally representing perfection, entirety, or cosmic order in traditions since antiquity.[48]
Ishmael – the first-born son of Abraham – has 12 sons/princes (Genesis 25:16), and Jacob also has 12 sons, who are the progenitors of the Twelve Tribes of Israel.[50] This is reflected in Christian tradition, notably in the twelve Apostles. When Judas Iscariot is disgraced, a meeting is held (Acts) to add Saint Matthias to complete the number twelve once more. The Book of Revelation contains much numerical symbolism, and many of the numbers mentioned have 12 as a divisor. 12:1 mentions a woman—interpreted as the people of Israel, the Church and the Virgin Mary—wearing a crown of twelve stars (representing each of the twelve tribes of Israel). Furthermore, there are 12,000 people sealed from each of the twelve tribes of Israel (the Tribe of Dan is omitted while Manasseh is mentioned), making a total of 144,000 (which is the square of 12 multiplied by a thousand).
12 was the only number considered to be religiously divine in the 1600s causing many Catholics to wear 12 buttons to church every Sunday. Some extremely devout Catholics would always wear this number of buttons to any occasion on any type of clothing.[citation needed]
Twelve is referred to in a few different verses of the Quran. Two are in reference to the Twelve Tribes of Israel.
And ˹remember˺ when Moses prayed for water for his people, We said, "Strike the rock with your staff." Then twelve springs gushed out, ˹and˺ each tribe knew its drinking place. ˹We then said,˺ "Eat and drink of Allah’s provisions, and do not go about spreading corruption in the land."
The second reference is:
We divided them into twelve tribes—each as a community. And We revealed to Moses, when his people asked for water, "Strike the rock with your staff." Then twelve springs gushed out. Each tribe knew its drinking place. We shaded them with clouds and sent down to them manna and quails,1 ˹saying˺, "Eat from the good things We have provided for you." They ˹certainly˺ did not wrong Us, but wronged themselves.
Note 1: Manna (heavenly bread) and quails (chicken-like birds) sustained the children of Israel in the wilderness after they left Egypt.
The third reference is to the number of months and the sacred ones amongst them:
Indeed, the number of months with Allāh is twelve [lunar] months in the register of Allāh [from] the day He created the heavens and the earth; of these, four are sacred.2
Note 2: The four sacred months of the Islamic calendar are Dhu al-Qa'dah, Dhu al-Hijjah, Muharram, and Rajab (months 11, 12, 1 and 7).
Films with the number twelve or its variations in their titles include:
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