### 12th

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

12 (twelve 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 duodecimal, 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 the Western calendar and units of time.

## 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 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
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
$12 \div x$ 12 6 4 3 2.4 2 $\mathrm\left\{1.\overline\left\{714285\right\}\right\}$ 1.5 $\mathrm\left\{1.\overline\left\{3\right\}\right\}$ 1.2 $\mathrm\left\{1.\overline\left\{09\right\}\right\}$ 1 $\mathrm\left\{0.\overline\left\{923076\right\}\right\}$ $\mathrm\left\{0.\overline\left\{857142\right\}\right\}$ 0.8
$x \div 12$ $\mathrm\left\{0.08\overline\left\{3\right\}\right\}$ $\mathrm\left\{0.1\overline\left\{6\right\}\right\}$ 0.25 $\mathrm\left\{0.\overline\left\{3\right\}\right\}$ $\mathrm\left\{0.41\overline\left\{6\right\}\right\}$ 0.5 $\mathrm\left\{0.58\overline\left\{3\right\}\right\}$ $\mathrm\left\{0.\overline\left\{6\right\}\right\}$ 0.75 $\mathrm\left\{0.8\overline\left\{3\right\}\right\}$ $\mathrm\left\{0.91\overline\left\{6\right\}\right\}$ 1 $\mathrm\left\{1.08\overline\left\{3\right\}\right\}$ $\mathrm\left\{1.1\overline\left\{6\right\}\right\}$ 1.25
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
$x ^ \left\{12\right\}\,$ 1 4096 531441 16777216 244140625 2176782336 13841287201 68719476736 282429536481 1000000000000 3138428376721 8916100448256 23298085122481

## In numeral systems

١٢ ԺԲ ১২ Arabic Armenian Bangla Attic Greek Hebrew V20-Z1-Z1 Egyptian Indian & Nepali (Devanāgarī) Chinese and Japanese Tamil Roman and Etruscan Thai Chuvash Telugu Urdu Ionian Greek Malayalam

## In science

• The atomic number of magnesium in the periodic table.
• The human body has twelve cranial nerves.
• The duodenum (from Latin duodecim, "twelve") is the first part of the small intestine, that is about twelve inches (30 cm) long. More precisely, this section of the intestine was measured not in inches but in fingerwidths. In fact, in German the name of the duodenum is Zwölffingerdarm and in Dutch the name is twaalfvingerige darm, both meaning "twelve-finger bowel".
• Force 12 on the Beaufort wind force scale corresponds to the maximum wind speed of a hurricane.

### Astronomy

• Messier object M12, a magnitude 8.0 globular cluster in the constellation Ophiuchus
• The Pisces
• The solar eclipse series which began on 2680 BC BC August and ended on 1129 BC BC February. The duration of Saros series 12 was 1550.5 years, and it contained 87 solar eclipses.
• The Saros lunar eclipse series which began on 2360 BC May and ended on 1062 BC July. The duration of Saros series 12 was 1298.1 years, and it contained 73 lunar eclipses.
• The 12th moon of Jupiter is Lysithea.

## 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 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 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 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 12 brothers.

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

Greek mythology has the twelve labours of Hercules.

## 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-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 sports

In the rugby codes:

• In rugby union:
• One of the starting centres, most often but not always the inside centre, wears the 12 shirt.
• The competition that was founded in 2001 as the Celtic League changed its name in 2011 to Pro12, reflecting its status as a 12-team league after it expanded in 2010 to include teams from Italy.
• The Southern Hemisphere competition now known as Super Rugby was known from 1996 through 2005, an era in which it had 12 teams, as Super 12.
• 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) sevens, the IRB Sevens World Series had 12 "core teams" that competed in all events through the 2011–12 edition. The number increased to 15 starting in 2012–13.

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.

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 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).

## References

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

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