12 (number)

← 11 12 13 →
← 10 11 12 13 14 15 16 17 18 19 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	twelve
Ordinal	12th (twelfth)
Numeral system	duodecimal
Factorization	22 × 3
Divisors	1, 2, 3, 4, 6, 12
Greek numeral	ΙΒ´
Roman numeral	XII, 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

12 (twelve) is the natural number following 11 and preceding 13.

Twelve is the 3rd superior highly composite number, ^[1] the 3rd colossally abundant number, ^[2] the 5th highly composite number, and is divisible by the numbers from 1 to 4, and 6, a large number of divisors comparatively.

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.

Name

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. ^{[3] [4] [5]} 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] [7]} 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. ^[7] 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"). ^[7] Every other Indo-European language instead uses a form of "two"+"ten", such as the Latin duōdecim. ^[7] 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, ^[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 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 ]}

Written representation

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). ^[10] 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.

In mathematics

12 is a composite number, the smallest abundant number, a semiperfect number, ^[11] a highly composite number, ^[12] a refactorable number, ^[13] and a Pell number. ^[14] It is the smallest of two known sublime numbers, numbers that have a perfect number of divisors whose sum is also perfect. ^[15]

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. ^[16]

A regular dodecahedron has twelve pentagonal faces. Regular cubes and octahedrons both have 12 edges, while regular icosahedrons have 12 vertices.

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 ${\displaystyle n}$-orthoplexes. ^[17] 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. ^[18] By the Gauss-Bonnet theorem, the area of this fundamental polygon is equal to ${\displaystyle 12\pi }$.

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

${\displaystyle \Delta (\tau )=(2\pi )^{12}\eta ^{24}(\tau )}$

This fact is related to a constellation of interesting appearances of the number twelve in mathematics ranging from the fact that the abelianization of special linear group ${\displaystyle \operatorname {SL} (2,\mathrm {Z} )}$ has twelve elements, to the value of the Riemann zeta function at ${\displaystyle -1}$ being ${\displaystyle -{\tfrac {1}{12}}}$, which stems from the Ramanujan summation

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

Although the series is divergent, methods such as Ramanujan summation can assign finite values to divergent series.

List of basic calculations

Multiplication	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25		50	100	1000
12 × x	12	24	36	48	60	72	84	96	108	120	132	144	156	168	180	192	204	216	228	240	252	264	276	288	300		600	1200	12000

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

In other bases

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.

Religion

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

Judaism and Christianity

• The significance is especially pronounced in the Hebrew Bible.
• 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. ^[20] 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).

• According to the New Testament, Jesus had twelve Apostles.
• The "Twelve Days of Christmas" count the interval between Christmas and Epiphany.
• Eastern Orthodoxy observes twelve Great Feasts.

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 ]}

Islam

• Twelve Imams are the spiritual and political successors to the Islamic prophet Muhammad in the Twelver branch of Shia Islam. ^[21]
• The hadith of the twelve successors is a widely-reported prophecy, attributed to the Islamic prophet Muhammad, predicting that there would be twelve successors after him. ^{[22] [23]}

Timekeeping

• The lunar year is 12 lunar months. Adding 11 or 12 days completes the solar year. ^[24]
• Most calendar systems – solar or lunar – have twelve months in a year.
• The Chinese use a 12-year cycle for time-reckoning called Earthly Branches.
• There are twelve hours in a half day, numbered one to twelve for both the ante meridiem (a.m.) and the post meridiem (p.m.). 12:00 p.m. is midday or noon, and 12:00 a.m. is midnight.
• The basic units of time (60 seconds, 60 minutes, 24 hours) are evenly divisible by twelve into smaller units.

In numeral systems

۱۲	Arabic	១២	Khmer	ԺԲ	Armenian
১২	Bangla	ΔΙΙ	Attic Greek	𝋬	Maya
יב	Hebrew		Egyptian
१२	Indian and Nepali (Devanāgarī)	十二	Chinese and Japanese
௧௨	Tamil	XII	Roman and Etruscan
๑๒	Thai	IIX	Chuvash
౧౨	Telugu and Kannada	١٢	Urdu
ιβʹ	Ionian Greek	൧൨	Malayalam

In science

• Force 12 on the Beaufort wind force scale corresponds to the maximum wind speed of a hurricane.

In sports

• In both soccer and American football, the number 12 can be a symbolic reference to the fans because of the support they give to the 11 players on the field. Texas A&M University reserves the number 12 jersey for a walk-on player who represents the original "12th Man", a fan who was asked to play when the team's reserves were low in a college American football game in 1922. Similarly, Bayern Munich, Hammarby, Feyenoord, Atlético Mineiro, Flamengo, Seattle Seahawks, Portsmouth and Cork City do not allow field players to wear the number 12 on their jersey because it is reserved for their supporters.
• In rugby league, one of the starting second-row forwards wears the number 12 jersey in most competitions. An exception is in the Super League, which uses static squad numbering.
• In rugby union, one of the starting centres, most often but not always the inside centre, wears the 12 shirt.

In technology

• ASCII and Unicode code point for form feed.

Music

Music theory

• Twelve is the number of pitch classes in an octave, not counting the duplicated (octave) pitch. Also, the total number of major keys, (not counting enharmonic equivalents) and the total number of minor keys (also not counting equivalents). This applies only to twelve tone equal temperament, the most common tuning used today in western influenced music.
• The twelfth is the interval of an octave and a fifth. Instruments such as the clarinet which behave as a stopped cylindrical pipe overblow at the twelfth.
• The twelve-tone technique (also dodecaphony) is a method of musical composition devised by Arnold Schoenberg. Music using the technique is called twelve-tone music.
• The twelve-bar blues is one of the most prominent chord progressions in popular music.

Art theory

• There are twelve basic hues in the color 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

12 stars are featured on the Flag of Europe.

• There are 12 troy ounces in a troy pound (used for precious metals).
• Twelve of something is called a dozen.
• In the former British currency system, there were twelve pence in a shilling. ^[25]
• In English, twelve is the number of greatest magnitude that has just one syllable.

  

  The numerical range on the analog clock ends at 12.

• 12 is the last number featured on the analogue clock, and also the starting point of the transition from A.M. to P.M. hours or vice versa.
• There are twelve months within a year, with the last one being December.
• 12 inches in a foot.
• 12 is slang for Police officers because of the 10-12 Police radio code.

Notes

1. Specially, a passage referring to Judas Iscariot as "one of the twelve" (an of ðæm tuelfum). ^[6]

References

1. "A002201 - OEIS". oeis.org. Retrieved 2024-11-28.
2. "A004490 - OEIS". oeis.org. Retrieved 2024-11-28.
3. Gordon, E. V. (1957). Introduction to Old Norse. Oxford, England: Clarendon Press. pp. 292–293. Archived from the original on 2016-04-15. Retrieved 2017-09-08.
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. Oxford English Dictionary, 1st ed. "twelve, adj. and n." Oxford University Press (Oxford), 1916.
8. Oxford English Dictionary, 1st ed. "eleven, adj. and n." Oxford University Press (Oxford), 1891.
9. Dantzig, Tobias (1930), Number: The Language of Science.
10. "Numbers: Writing Numbers // Purdue Writing Lab". Purdue Writing Lab. Retrieved 25 February 2020.
11. Sloane, N. J. A. (ed.). "SequenceA005835(Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-06-01.
12. Sloane, N. J. A. (ed.). "SequenceA002182(Highly composite numbers, definition (1): numbers n where d(n), the number of divisors of n (A000005), increases to a record.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-19.
13. Sloane, N. J. A. (ed.). "SequenceA033950(Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-15.
14. Sloane, N. J. A. (ed.). "SequenceA000129(Pell numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-01-10.
15. "Sloane's A081357 : Sublime numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
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.
24. "Lunar versus solar calendar".
25. "Shilling | currency". Encyclopedia Britannica. Retrieved 20 May 2021.

Sources

• Collins, Billie Jean (2002), "Necromancy, Fertility and the Dark Earth: The Use of Ritual Pits in Hittite Cult", in Mirecki, Paul; Meyer, Marvin (eds.), Magic and Ritual in the Ancient World, Leiden, The Netherlands: Brill, pp. 224–233, ISBN   90-04-10406-2
• Drews, Robert (January 1972). "Light from Anatolia on the Roman fasces". The American Journal of Philology . 93 (1): 40–51. doi:10.2307/292899. JSTOR   292899..

Further reading

Books

• Schwartzman, Steven (1994). The words of mathematics: An etymological dictionary of mathematical terms used in English . The Mathematical Association of America. ISBN   0-88385-511-9.

Journal articles

• Poonen, Bjorn; Rodriguez-Villegas, Fernando (March 2000). "Lattice Polygons and the Number 12" (PDF). American Mathematical Monthly. 107 (3): 238–250. doi:10.1080/00029890.2000.12005186. S2CID   11433945. Archived from the original (PDF) on 2012-04-14.

External links

• Media related to 12 (number) at Wikimedia Commons