# 12 (number)

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 ← 11 12 13 →
Cardinal twelve
Ordinal 12th
(twelfth)
Numeral system duodecimal
Factorization 22× 3
Divisors 1, 2, 3, 4, 6, 12
Roman numeral XII
Greek prefix dodeca-
Latin prefix duodeca-
Binary 11002
Ternary 1103
Quaternary 304
Quinary 225
Senary 206
Octal 148
Duodecimal 1012
Vigesimal C20
Base 36 C36
 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. Late 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]

## In languages

### Grammar

While, as mentioned above, 12 has its own name in Germanic languages such as English and German, 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), Japanese 十二 jūni.

### Spelling

In Germany, according to an old tradition, the numbers 0 (null) through 12 (zwölf) were spelt out, and 13 (dreizehn) as the first compound number was the first number written in digits. The Duden (the German standard dictionary) now calls this tradition (which was actually never written down as an official rule) outdated and no longer valid, but many writers still follow it.

For the English language, different systems are used: Sometimes it is recommended to spell out numbers up to and including nine or ten or twelve, like formerly in German, 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).

## In mathematics

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

## In numeral systems

12 Arabic 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 religion and mythology

There are twelve "Jyotirlingas" in Hindu Shaivism. The Shaivites (orthodox devotees of God Shiva) treat them with great respect and they are visited by almost every pious Hindu at least once in a lifetime. The number 12 is very important in many religions, mainly Judaism, Christianity, and Islam, and also found in some older religions and belief systems. In the Hindu epic Mahabharata, the Pandavas and their wife Draupadi go to the forest for 12 years exile and 1 year disguise-appearance exile.

In antiquity there are numerous ceremonial, magical, and religious uses of twelve.[n 2] The twelve lictors carried fasces of twelve rods. Ancient Greek religion, the Twelve Olympians were the principal gods of the pantheon and Hercules enacted out twelve labours. The chief Norse god, Odin, had 12 sons. Several sets of twelve cities are identified in history as a dodecapolis, the most familiar being the Etruscan League. In the King Arthur Legend, Arthur is said to have subdued 12 rebel princes and to have won 12 great battles against Saxon invaders. [source: Benet's Reader's Encyclopedia, 3d ed]

The importance of 12 in Judaism and Christianity can be found in the Bible. The biblical Jacob had 12 sons, who were the progenitors of the Twelve Tribes of Israel, while the New Testament describes twelve apostles of Jesus; when Judas Iscariot was disgraced, a meeting was held (Acts) to add Saint Matthias to complete the number twelve once more. (Today, The Church of Jesus Christ of Latter-day Saints has a Quorum of the Twelve Apostles.)

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).

In Orthodox Judaism, 12 signifies the age a girl matures (bat mitzvah)

There are 12 days of Christmas. The song Twelve Days of Christmas came from the traditional practice of extending Yuletide celebrations over the twelve days from Christmas day to the eve of Epiphany; the period of thirteen days including Epiphany is sometimes known as Christmastide. Thus Twelfth Night is another name for the twelfth day of Christmas or January 5 (the eve of Epiphany). Similarly, Eastern Orthodoxy observes 12 Great Feasts.

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.

Imāmah (Arabic: إمامة) is the Shī‘ah doctrine of religious, spiritual and political leadership of the Ummah. The Shī‘ah believe that the A'immah ("Imams") are the true Caliphs or rightful successors of Muḥammad, and Twelver and Ismā‘īlī Shī‘ah further that Imams are possessed of supernatural knowledge, authority, and infallibility (‘Iṣmah) as well as being part of the Ahl al-Bayt, the family of Muhammad.[1] Both beliefs distinguish the Shī‘ah from Sunnis.

In Quran, the Sura number 12 is Sura Yusuf (Joseph), and it is located in Juz'a (Arabic : الجزء) number 12. This Sura narrates the story of Prophet Yusuf and his 11 brothers, making 12.

In Hinduism, the sun god Surya has 12 names. Also, there are 12 Petals in Anahata (Heart Chakra.)

Taoism

## In time

• Most calendar systems 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 one solar year there are roughly 12 lunar months: ( one solar year / one average lunar month ≈ 365.2422 solar days / 29.5305 solar days ≈ 12.3683 ≈ 12 ) [13]

## In the arts

### Film

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

### 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] considered the classic assembly of instances to be Th. Weinreich's Zwölfgötten.[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., "Zwölfgötten", Ausfurliches Lexikon der Griechischen und Römishen Mythologie, Vol. VI, col. 764-848.
13. ^

### 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.