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12 (number)

From Wikipedia, the free encyclopedia
Natural number
← 1112 13 →
Cardinaltwelve
Ordinal12th
(twelfth)
Numeral systemduodecimal
Factorization22 × 3
Divisors1, 2, 3, 4, 6, 12
Greek numeralΙΒ´
Roman numeralXII,xii
Greekprefixdodeca-
Latinprefixduodeca-
Binary11002
Ternary1103
Senary206
Octal148
Duodecimal1012
HexadecimalC16
Malayalam൰൨
Bengali১২
Hebrew numeralי"ב
Babylonian numeral𒌋𒐖

12 (twelve) is thenatural number following11 and preceding13.

Twelve is the 3rdsuperior highly composite number,[1] the 3rdcolossally abundant number,[2] the 5thhighly composite number, and is divisible by the numbers from1 to4, and6, a large number of divisors comparatively.

It is central to many systems of timekeeping, including theWestern calendar andunits of time of day, and frequently appears in the world's major religions.

Name

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Twelve is the largest number with asingle-syllable name inEnglish. EarlyGermanic numbers have been theorized to have been non-decimal: evidence includes the unusual phrasing ofeleven and twelve, theformer use of "hundred" to refer to groups of120, 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.[3][4][5] Such uses gradually disappeared with the introduction ofArabic numerals during the12th-century Renaissance.

Derived fromOld English,twelf andtuelf are first attested in the 10th-centuryLindisfarne Gospels'Book of John.[note 1][7] It has cognates in everyGermanic language (e.g. Germanzwölf), whoseProto-Germanic ancestor has beenreconstructed as*twaliƀi..., from*twa ("two") and suffix*-lif- or*-liƀ- of uncertain meaning.[7] It is sometimes compared with theLithuaniandvýlika, although-lika is used as the suffix for all numbers from 11 to 19 (analogous to "-teen").[7] Every otherIndo-European language instead uses a form of "two"+"ten", such as theLatinduōdecim.[7] The usualordinal form is "twelfth" but "dozenth" or "duodecimal" (from the Latin word) is also used in some contexts, particularlybase-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,[8] 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.[7] The Lithuanian suffix is also considered to share a similar development.[7] The suffix*-lif- has also been connected with reconstructions of the Proto-Germanic for ten.[8][9]

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 Frenchdozaine. It is a compound number in many other languages, e.g. Italiandodici (but in Spanish and Portuguese, 16, and in French, 17 is the first compound number),[dubiousdiscuss] Japanese 十二jūni.[clarification needed]

Written representation

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In prose writing,twelve, being the last single-syllable numeral, is sometimes taken as the last number to be written as a word, and13 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 eithernine,ten ortwelve, or evenninety-nine orone hundred. Another system spells out all numbers written in one or two words (sixteen,twenty-seven,fifteen thousand, but372 or15,001).[10]InGerman orthography, there used to be the widely followed (but unofficial) rule of spelling out numbers up to twelve (zwölf). TheDuden[year needed] (the German standard dictionary) mentions this rule as outdated.

In mathematics

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Properties

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12 is acomposite number, the smallestabundant number, asemiperfect number,[11] ahighly composite number,[12] arefactorable number,[13] and aPell number.[14] It is the smallest of two knownsublime numbers, numbers that have aperfect number of divisors whose sum is also perfect.[15]

There are twelveJacobian elliptic functions and twelve cubicdistance-transitive graphs.

Shapes

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A twelve-sidedpolygon is adodecagon. In its regular form, it is the largest polygon that canuniformly tile the plane alongside other regular polygons, as with thetruncated hexagonal tiling or thetruncated trihexagonal tiling.[16]

A regulardodecahedron has twelvepentagonal faces. Regularcubes andoctahedrons both have 12 edges, while regularicosahedrons have 12 vertices.

The cubic close packing and hexagonal close packing, which are the two densest possiblesphere packings in three-dimensional space (theKepler conjecture, proved byThomas Hales), both have each sphere touching twelve other spheres. Twelve is also thekissing number in three dimensions.

There are twelvecomplex apeirotopes in dimensions five and higher, which includevan Oss polytopes in the form of complexn{\displaystyle n}-orthoplexes.[17] There are also twelveparacompact hyperbolicCoxeter groups ofuniform polytopes in five-dimensional space.

Bring's curve is aRiemann surface ofgenus four, with a domain that is a regular hyperbolic 20-sidedicosagon.[18] By theGauss-Bonnet theorem, the area of thisfundamental polygon is equal to12π{\displaystyle 12\pi }.

Functions

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Twelve is the smallest weight for which acusp form exists. This cusp form is the discriminantΔ(q){\displaystyle \Delta (q)} whose Fourier coefficients are given by theRamanujanτ{\displaystyle \tau }-function and which is (up to a constant multiplier) the 24th power of theDedekind eta function:

Δ(τ)=(2π)12η24(τ){\displaystyle \Delta (\tau )=(2\pi )^{12}\eta ^{24}(\tau )}

This fact is related to a constellation of interesting appearances of the number twelve in mathematics ranging from the fact that theabelianization ofspecial linear groupSL(2,Z){\displaystyle \operatorname {SL} (2,\mathrm {Z} )} has twelve elements, to the value of theRiemann zeta function at1{\displaystyle -1} being112{\displaystyle -{\tfrac {1}{12}}}, which stems from theRamanujan summation

1+2+3+4+=112(R){\displaystyle 1+2+3+4+\cdots =-{\frac {1}{12}}\quad ({\mathfrak {R}})}

Although the series isdivergent, methods such as Ramanujan summation can assign finite values todivergent series.

List of basic calculations

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Multiplication12345678910111213141516171819202122232425501001000
12 ×x1224364860728496108120132144156168180192204216228240252264276288300600120012000
Division12345678910111213141516
12 ÷x126432.421.7142851.51.31.21.0910.9230760.8571420.80.75
x ÷ 120.0830.160.250.30.4160.50.5830.60.750.830.91611.0831.161.251.3
Exponentiation123456789101112
12x121441728207362488322985984358318084299816965159780352619173642247430083706888916100448256
x12140965314411677721624414062521767823361384128720168719476736282429536481100000000000031384283767218916100448256

In other bases

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Theduodecimal system (1210 [twelve] = 1012), which is the use of 12 as a division factor for manyancient andmedieval weights and measures, includinghours, probably originates fromMesopotamia.

Religion

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The number twelve carries religious, mythological and magicalsymbolism; since antiquity, the number has generally represented perfection, entirety, or cosmic order.[19]

Judaism and Christianity

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This section mayrequirecleanup to meet Wikipedia'squality standards. The specific problem is:Explain the meaning of 12 to Christians and Jewish people, perWP:NUM/NOT. This section cannot be just a list of times 12 appears in Biblical stories because this article is about the number 12. What do scholars say Christians and Jewish people understand when they see "12"? This sort of standard applies to all non-math content for number pages. Please helpimprove this section if you can.(January 2025) (Learn how and when to remove this message)

The number 12 is notable within theHebrew Bible, and in Christianity:

Ishmael—the first-born son ofAbraham—has 12 sons/princes (Genesis 25:16), andJacob also has 12 sons, who are the progenitors of theTwelve Tribes of Israel.[20] This is reflected in Christian tradition, notably in thetwelve Apostles. WhenJudas Iscariot is disgraced, a meeting is held (Acts) to addSaint Matthias to complete the number twelve once more.

The Old Testament containsTwelve Minor Prophets.

TheBook of Revelation contains much numerical symbolism, and many of the numbers mentioned have 12 as a divisor.12:1 mentions a woman—interpreted as thepeople of Israel, theChurch and theVirgin 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 (theTribe of Dan is omitted whileManasseh is mentioned), making a total of144,000 (which is the square of 12 multiplied by a thousand).

Islam

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Astrology

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Timekeeping

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  • Thelunar year is 12lunar months. Adding 11 or 12 days completes thesolar year.[24]
  • Mostcalendar systems – solar or lunar – have twelve months in ayear.
  • The Chinese use a 12-year cycle for time-reckoning calledEarthly Branches.
  • There are twelvehours in a half day, numbered one to twelve for both theante meridiem (a.m.) and thepost meridiem (p.m.). 12:00 p.m. is midday ornoon, and 12:00 a.m. ismidnight.
  • The basic units of time (60 seconds, 60 minutes, 24 hours) are evenly divisible by twelve into smaller units.

In numeral systems

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۱۲Arabic១២KhmerԺԲArmenian
১২BanglaΔΙΙAtticGreek𝋬Maya
יבHebrew
V20Z1Z1
Egyptian
१२Indian and Nepali (Devanāgarī)十二Chinese andJapanese
௧௨TamilXIIRoman andEtruscan
๑๒ThaiIIXChuvash
౧౨Telugu andKannada١٢Urdu
ιβʹIonianGreek൧൨Malayalam

In science

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In sports

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In technology

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Music

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Music theory

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Art theory

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  • There are twelve basichues in thecolor wheel: three primary colors (red, yellow, blue), three secondary colors (orange, green, purple) and six tertiary colors (names for these vary, but are intermediates between the primaries and secondaries).

In other fields

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12 stars are featured on theFlag of Europe.

Notes

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  1. ^Specially, a passage referring toJudas Iscariot as "one of thetwelve" (an of ðæm tuelfum).[6]

References

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  1. ^"A002201 - OEIS".oeis.org. Retrieved2024-11-28.
  2. ^"A004490 - OEIS".oeis.org. Retrieved2024-11-28.
  3. ^Gordon, E. V. (1957).Introduction to Old Norse. Oxford, England: Clarendon Press. pp. 292–293. Archived fromthe original on 2016-04-15. Retrieved2017-09-08.
  4. ^Stevenson, W. H. (December 1899). "The Long Hundred and its Use in England".Archaeological Review.4 (5):313–317.
  5. ^Goodare, Julian (1994)."The long hundred in medieval and early modern Scotland"(PDF).Proceedings of the Society of Antiquaries of Scotland.123:395–418.doi:10.9750/PSAS.123.395.418.S2CID 162146336.
  6. ^John 6:71.
  7. ^abcdefOxford English Dictionary, 1st ed. "twelve,adj. andn." Oxford University Press (Oxford), 1916.
  8. ^abOxford English Dictionary, 1st ed. "eleven,adj. andn." Oxford University Press (Oxford), 1891.
  9. ^Dantzig, Tobias (1930),Number: The Language of Science.
  10. ^"Numbers: Writing Numbers // Purdue Writing Lab".Purdue Writing Lab. Retrieved25 February 2020.
  11. ^Sloane, N. J. A. (ed.)."Sequence A005835 (Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n.)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved2016-06-01.
  12. ^Sloane, N. J. A. (ed.)."Sequence A002182 (Highly composite numbers, definition (1): numbers n where d(n), the number of divisors of n (A000005), increases to a record.)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved2023-06-19.
  13. ^Sloane, N. J. A. (ed.)."Sequence A033950 (Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved2023-06-15.
  14. ^Sloane, N. J. A. (ed.)."Sequence A000129 (Pell numbers)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved2023-01-10.
  15. ^"Sloane's A081357 : Sublime numbers".The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved2016-06-01.
  16. ^Grünbaum, Branko;Shephard, G. C. (1987). "Section 2.1: Regular and uniform tilings".Tilings and Patterns. New York: W. H. Freeman and Company. p. 59.doi:10.2307/2323457.ISBN 0-7167-1193-1.JSTOR 2323457.OCLC 13092426.S2CID 119730123.
  17. ^H. S. M. Coxeter (1991).Regular Complex Polytopes (2 ed.). Cambridge University Press. pp. 144–146.doi:10.2307/3617711.ISBN 978-0-521-39490-1.JSTOR 3617711.S2CID 116900933.Zbl 0732.51002.
  18. ^Weber, Matthias (2005)."Kepler's small stellated dodecahedron as a Riemann surface"(PDF).Pacific Journal of Mathematics.220 (1): 172.doi:10.2140/pjm.2005.220.167.MR 2195068.S2CID 54518859.Zbl 1100.30036.
  19. ^Drews (1972), p. 43, n. 10.
  20. ^"And it is thought that there is a special significance in the number twelve. It was typified, we know, by many things in the Old Testament; by the twelve sons of Jacob, by the twelve princes of the children of Israel, by the twelve fountains in Elim, by the twelve stones in Aaron's breast-plate, by the twelve loaves of the shew-bread, by the twelve spies sent by Moses, by the twelve stones of which the altar was made, by the twelve stones taken out of Jordan, by the twelve oxen which bare"P. Young,Daily readings for a year (1863),p. 150.
  21. ^Olsson, Tord; Ozdalga, Elisabeth; Raudvere, Catharina (2005).Alevi Identity: Cultural, Religious and Social Perspectives. Routledge.ISBN 978-1-135-79725-6.
  22. ^Hussain, J.M. (1982).Occultation of the Twelfth Imam: A Historical Background. Muhammadi Trust.ISBN 9780710301581.
  23. ^Kohlberg, E. (2009)."From Imāmiyya to Ithnā-'ashariyya".Bulletin of the School of Oriental and African Studies, University of London.39 (3):521–534.doi:10.1017/S0041977X00050989.JSTOR 614712.
  24. ^"Lunar versus solar calendar".
  25. ^"Shilling | currency".Encyclopedia Britannica. Retrieved20 May 2021.

Sources

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Further reading

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Books

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Journal articles

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External links

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