# 12 (number)

12 (number)
12
Cardinal twelve
Ordinal 12th
(twelfth)
Numeral system duodecimal
Factorization $2^2 \cdot 3$
Divisors 1, 2, 3, 4, 6, 12
Roman numeral XII
Latin Prefix duodeca-
Binary 11002
Octal 148
Duodecimal 1012
 Mathematical properties φ(12) = 4 τ(12) = 6 σ(12) = 28 π(12) = 5 μ(12) = 0 M(12) = -2

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

The word "twelve" is the largest number with a single-morpheme name in English. Etymology suggests that "twelve" (similar to "eleven") arises from the Germanic compound twalif "two-leftover", so a literal translation would yield "two remaining [after having ten taken]".[1] This compound meaning may have been transparent to speakers of Old English, but the modern form "twelve" is quite opaque. Only the remaining tw- hints that twelve and two are related.

A group of twelve things is called a duodecad. The ordinal adjective is duodenary, twelfth. The adjective referring to a group consisting of twelve things is duodecuple.

The number twelve is often used as a sales unit in trade, and is often referred to as a dozen. Twelve dozen are known as a gross. (Note that there are thirteen items in a baker's dozen.)

As shown below, the number twelve is frequently cited in the Abrahamic religions and is also central to Western calendar and units of time.

## In mathematics

Twelve is a composite number, the smallest number with exactly six divisors, its proper divisors being 1, 2, 3, 4, 6 and 12. Twelve is also a highly composite number, the next one being 24. It is the first composite number of the form p2q; a square-prime, and also the first member of the (p2) family in this form. 12 has an aliquot sum of 16 (133% in abundance). Accordingly, 12 is the first abundant number (in fact a superabundant number) and demonstrates an 8 member aliquot sequence; {12,16,15,9,4,3,1,0} 12 is the 3rd composite number in the 3-aliquot tree. The only number which has 12 as its aliquot sum is the square 121. Only 2 other square primes are abundant (18 and 20).

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. 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.

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

Twelve is a superfactorial, being the product of the first three factorials. Twelve being the product of three and four, the first four positive integers show up in the equation 12 = 3 × 4, which can be continued with the equation 56 = 7 × 8.

Twelve is the ninth Perrin number, preceded in the sequence by 5, 7, 10, and also appears in the Padovan sequence, preceded by the terms 5, 7, 9 (it is the sum of the first two of these). It is the fourth Pell number, preceded in the sequence by 2 and 5 (it is the sum of the former plus twice the latter).

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 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.

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
$12 \times x$ 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
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
$12 ^ x\,$ 12 144 1728 20736 248832 2985984 35831808 429981696 5159780352 61917364224 743008370688 8916100448256 106993205379072
$x ^ {12}\,$ 1 4096 531441 16777216 244140625 2176782336 13841287201 68719476736 282429536481 1000000000000 3138428376721 8916100448256 23298085122481

## In numeral systems

١٢ Arabic ԺԲ Armenian
ιβʹ Ionian Greek ΔΙΙ Attic Greek
יב Hebrew
Egyptian
१२ Indian (Devanāgarī) 十二 Chinese and Japanese
௧௨ Tamil Roman and Etruscan
๑๒ Thai IIX Chuvash
൧൨ Malayalam

## In religion

There are twelve ' Jyotirlingas' (epitome of God Shiva) in Hindu Shaivism. The shaivites (orthodox devotees of God Shiva) treat them with great respect and these 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 Antiquity there are numerous magical/religious uses of twelves.[2] Ancient Greek religion, the Twelve Olympians were the principal gods of the pantheon. 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 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 Shi'a Islam, there are twelve Imams, legitimate successors of the 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 Hinduism, the sun god Surya has 12 names. Also, there are 12 Petals in Anahata(Heart Chakra.)

## In time

• Most calendar systems have twelve months in a year.
• The Western zodiac has twelve signs, as does the Chinese zodiac.
• The Chinese use a 12 years 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. Furthermore, the basic units of time (60 seconds, 60 minutes, 24 hours) can all perfectly divide by twelve.

## In sports

In rugby union one of the centres, most often but not always the inside centre, wears the 12 shirt, while in rugby league, one of the second-row forwards wears the number 12 jersey.

Also in rugby union, the number 12 appears in the current sponsored name of the Celtic League, RaboDirect Pro12.

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 and Cork City do not allow field players to wear the number 12 on their jersey because it is reserved for their supporters.

The jersey number 12 has been retired by several North American sports teams in honor of past playing greats (or, in one case, a team's fans):

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 ten-pin bowling, 12 is the number of strikes needed for a perfect game.

In curling, the House or the circular scoring area, is 12 feet in diameter.

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 association football, 12 was also the number of teams in the finals of the FIFA Women's World Cup in its first two editions in 1991 and 1995.

## In technology

• ASCII and Unicode code point for form feed.
• The number of function keys on most PC keyboards (F1 through F12)
• The number of keys in any standard digital telephone (1 through 9, 0, * and #)
• Microsoft's Rich Text Format specification assigns numbers congruent to 12 mod 256 to variants of the French language.

## In the arts

### Film

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

### Television

• The number twelve plays a significant role in the television franchise Battlestar Galactica. The characters come from the Twelve Colonies of Kobol and worship the twelve lords of Kobol. In the re-imagined series, there are also twelve models of the humanoid version of Cylons.
• Twelve Angry Men, the original 1954 live performance on the anthology television series Studio One
• "Number 12 Looks Just Like You" is an episode of the television show "The Twilight Zone".
• Schoolhouse Rock! portrayed an alien child using base-twelve arithmetic in the short "Little Twelvetoes".
• 12 Oz Mouse was an animated television show on Adult Swim.

### 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

• Twelve is a character in the Street Fighter video game
• The Roman form of 12 (XII) is used as the symbol of the organization MJ12 in the Deus Ex computer game.

## In other fields

12 stars are featured on the Flag of Europe
• There are 12 ounces in a troy pound (used for precious metals)
• There are 12 constellations in the ecliptic (or signs of the zodiac).
• In the former British currency system, there were twelve pence in a shilling.
• In Greek mythology, the number of labours of Heracles was increased from ten to make twelve.
• In English, twelve is the number of greatest magnitude that has just one syllable.
• There are normally twelve pairs of ribs in the human body.
• The Twelve Tables or Lex Duodecim Tabularum, more informally simply Duodecim Tabulae was the ancient legislation underlying Roman law.
• In the United States, twelve people are appointed to sit on a jury for felony trials in all but four states, and in federal and Washington, D.C. courts. The number of jurors gave the title to the play (and subsequent films) Twelve Angry Men.
• Twelve people have walked on Earth's moon.
• The United States of America is divided into twelve Federal Reserve Districts (Boston, New York, Philadelphia, Cleveland, Richmond, Atlanta, Chicago, St. Louis, Minneapolis, Kansas City, Dallas, and San Francisco); American paper currency has serial numbers beginning with one of twelve different letters, A through L, representing the Federal Reserve Bank from which the currency originated.
• According to UFO conspiracy theory, Majestic 12 is a secret committee, allegedly set up by U.S. President Harry S. Truman to investigate the Roswell UFO incident and cover up future extraterrestrial contact.
• 12 is the number of the French department Aveyron.
• King Arthur's round table had 12 knights plus King Arthur himself.
• 12 inches in a foot.
• 12 face cards in a card deck.

## Historical years

12 A.D, 12 B.C., 1812, 1912, 2012, etc.

## Notes

1. ^ http://www.etymonline.com/index.php?term=twelve
2. ^ The classic assembly of instances is Th. Weinreich, "Zwölfgötten' in W.H. Roscher, Ausfurliches Lexikon der Griechischen und Römishen Mythologie vol. vi. col. 764-848, noted in Robert Drews, "Light from Anatolia on the Roman Fasces"The American Journal of Philology 93.1 (January 1972) pp. 40-51) p 43 note 10. The twelve lictors carried fasces of twelve rods.

## References

• 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.
• Lattice Polygons and the Number 12, Bjorn Poonen, Fernando Rodriguez-Villegas, American Mathematical Monthly, Vol. 107, No. 3 (Mar., 2000) , pp. 238–250 [1]

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