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Mathematical properties

φ(12) = 4 τ(12) = 6 σ(12) = 28 π(12) = 5 μ(12) = 0 M(12) = -2

12 (twelve ) is the natural number following 11 and preceding 13. The product of the first three factorials, twelve is a superior highly composite number, divisible by 2, 3, 4, and 6. It is central to many systems of counting, including the Western calendar and units of time, and frequently appears in the Abrahamic religions.

## Name

The word "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.

It derives from the Old English twelf and tuelf, first attested in the 10th-century Lindisfarne Gospels' Book of John.[n 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 "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]

While, as mentioned above, 12 has its own name in Germanic languages such as English, German, and Swedish. 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 ] 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).[citation needed] 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.

## Mathematical properties

Twelve is a composite number, the smallest number with exactly six divisors, its divisors being 1, 2, 3, 4, 6 and 12. Twelve is also a highly composite number, the next one being twenty-four.[8]

Twelve is a sublime number, a number that has a perfect number of divisors, and the sum of its divisors is also a perfect number.[9] Since there is a subset of 12's proper divisors that add up to 12 (all of them but with 4 excluded), 12 is a semiperfect number.[10]

If an odd perfect number is of the form 12k + 1, it has at least twelve distinct prime factors.

A twelve-sided polygon is a dodecagon. A twelve-faced polyhedron is a dodecahedron. Regular cubes and octahedrons both have 12 edges, while regular icosahedrons have 12 vertices. Twelve is a pentagonal number. The densest three-dimensional lattice sphere packing has each sphere touching 12 others, and this is almost certainly true for any arrangement of spheres (the Kepler conjecture). Twelve is also the kissing number in three dimensions.

Twelve is the smallest weight for which a cusp form exists. This cusp form is the discriminant Δ(q) whose Fourier coefficients are given by the Ramanujan τ-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 value of the Riemann zeta function at −1 i.e. ζ(−1) = −1/12, the fact that the abelianization of SL(2,Z) has twelve elements, and even the properties of lattice polygons.

There are twelve Jacobian elliptic functions and twelve cubic distance-transitive graphs.

There are 12 Latin squares of size 3 × 3.

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. In base 10, the number 12 is a Harshad number.

### 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 13
12x 12 144 1728 20736 248832 2985984 35831808 429981696 5159780352 61917364224 743008370688 8916100448256 106993205379072
x12 1 4096 531441 16777216 244140625 2176782336 13841287201 68719476736 282429536481 1000000000000 3138428376721 8916100448256 23298085122481

## Symbolism

The number twelve has important symbolism attached to it in the world's cultures, generally representing perfection or entirety. Notably, twelve is the number of months in a year as traditionally represented both by the Western and the Chinese zodiac and the number of months in a solar calendar such as the Julian one. Ultimately, this is due to the number of full lunations in a solar year. In any case, the number twelve is recurring as the number of a "complete set" or "cosmic order" in religious, mythological and magical traditions since antiquity.[n 2] Several sets of twelve cities are identified in history as a dodecapolis, the most familiar being the Etruscan League.

The significance is especially pronounced in the Hebrew Bible. Ishmael - the first-born son of Abraham - had 12 sons/princes (Genesis 25:16), and Jacob also had 12 sons, who were the progenitors of the Twelve Tribes of Israel.[13] This is reflected in Christian tradition, notably in the twelve Apostles. When Judas Iscariot was disgraced, a meeting was held (Acts) to add Saint Matthias to complete the number twelve once more. The Book of Revelation contains much numerical symbolism, and a lot of the numbers mentioned have 12 as a divisor. 12:1 mentions a woman—interpreted as the people of Israel, the Church or 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, making a total of 144,000 (which is the square of 12 multiplied by a thousand). The "Twelve Days of Christmas" count the interval between Christmas and Epiphany. Eastern Orthodoxy observes twelve Great Feasts.

The number of twelve jurors in jury trials is depicted by Aeschylus in the Eumenides.In the play, the innovation is brought about by the goddess Athena, who summons twelve citizens to sit as jury. In English Common Law, the tradition of twelve jurors harks back to the 10th-century law code introduced by Aethelred the Unready.

An example from Hinduism are the twelve Jyotirlinga in Shaiva tradition. The Sun god Surya has 12 names. Also, there are 12 Petals in Anahata or "heart chakra". The chief Norse god, Odin, had twelve sons. In the King Arthur Legend, Arthur is said to have subdued 12 rebel princes and to have won 12 great battles against Saxon invaders[14]

In Twelver Shi'a Islam, there are twelve Imams, legitimate successors of the Islamic prophet, Muhammad. These twelve early leaders of Islam are—Ali, Hasan, Husayn, and nine of Husayn's descendants. Sura 12 in the Quran is sura Yusuf, narrating the story of the sons of Jacob.

## Timekeeping

• The lunar year is 12 lunar months. Add 11 or 12 days and we have the solar year[15].
• Most calendar systems - whether solar or lunar - have twelve months in a year.
• The Chinese use a 12-year cycle for time-reckoning called Earthly Branches.
• There are twenty-four hours in a day in all, with twelve hours for a half a day. The hours are numbered from one to twelve for both the ante meridiem (a.m.) half of the day and the post meridiem (p.m.) half of the day. 12:00 after a.m. and before p.m. (in the middle of the day) is midday or noon, and 12:00 after p.m. and before a.m. (in the middle of the night) is midnight. A new day is considered to start with the stroke of midnight. The basic units of time (60 seconds, 60 minutes, 24 hours) can all perfectly divide by twelve.

## In numeral systems

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

## 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 Canadian football, 12 is the maximum number of players that can be on the field of play for each team at any time.
• In cricket, another sport with eleven players per team, teams may select a "12th man", who may replace an injured player for the purpose of fielding (but not batting, bowling or keeping wicket).
• In women's lacrosse, each team has 12 players on the field at any given time, except in penalty situations.
• In rugby league, one of the starting second-row forwards wears the number 12 jersey in most competitions. An exception is in the European 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 the arts

### Film

Movies with the number twelve or its variations in their titles include

Music Theory

Pop Music

### Art theory

• There are twelve basic hues in the color wheel; 3 primary colors (red, yellow, blue), 3 secondary colors (orange, green & purple) and 6 tertiary colors (names for these vary, but are intermediates between the primaries and secondaries).

### Games

• In the game of craps, a dice roll of two sixes (value 12) on the come-out roll constitutes a “craps” and the shooter (dice thrower) loses immediately.
• Twelve is a character in the Street Fighter video game series.
• In Mario Kart Wii and Mario Kart 8, the starting grid can carry twelve characters.

## In other fields

12 stars are featured on the Flag of Europe

## Notes

1. ^ Specially, a passage referring to Judas Iscariot as "one of the twelve" (an of ðæm tuelfum).[4]
2. ^ Drews[11] The twelve lictors carried fasces of twelve rods. In Ancient Greek religion, the Twelve Olympians were the principal gods of the pantheon, and Hercules enacted out twelve labours.[12]

## References

### Citations

1. ^ Gordon, E V (1957). Introduction to Old Norse. Oxford: Claredon Press. pp. 292–293.
2. ^ Stevenson, W. H. (December 1899). "The Long Hundred and its Use in England". Archaeological Review. 4 (5): 313–317.
3. ^ Goodare, Julian (1993). "The Long Hundred in medieval and early modern Scotland" (PDF). Proceedings of the Society of Antiquaries of Scotland. 123: 395–418.
4. ^
5. Oxford English Dictionary, 1st ed. "twelve, adj. and n." Oxford University Press (Oxford), 1916.
6. ^ a b Oxford English Dictionary, 1st ed. "eleven, adj. and n." Oxford University Press (Oxford), 1891.
7. ^ Dantzig, Tobias (1930), Number: The Language of Science.
8. ^ "Sloane's A002182 : Highly composite numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
9. ^ "Sloane's A081357 : Sublime numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
10. ^ "Sloane's A005835 : Pseudoperfect (or semiperfect) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
11. ^ Drews (1972), p. 43, n. 10.
12. ^ Weinreich, Th., "Zwölfgötter", Ausführliches Lexikon der Griechischen und Römischen Mythologie, Vol. VI, col. 764-848.
13. ^ "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.
14. ^ Benet's Reader's Encyclopedia, 3d ed.
15. ^

### Bibliography

• Drews, Robert (January 1972), "Light from Anatolia on the Roman Fasces", The American Journal of Philology, Vol. 93, No. 1, pp. 40–51.
• Lattice Polygons and the Number 12, Bjorn Poonen, Fernando Rodriguez-Villegas, American Mathematical Monthly, Vol. 107, No. 3 (March, 2000), pp. 238–250 [1]
• 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.