12 (number)


Cardinal	12 twelve
Ordinal	12th twelfth
Numeral system	duodecimal
Factorization	$2^2 \cdot 3$
Divisors	1, 2, 3, 4, 6, 12
Roman numeral	XII
Unicode representation of Roman numeral	Ⅻ, ⅻ
Binary	1100
Octal	14
Duodecimal	10
Hexadecimal	C
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 word "twelve" is a native English word that presumably arises from the Germanic compound twa-lif "two-leave", meaning that two is left after one takes away the base, ten. 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. Etymology (Weekley, Skeat) suggests that "twelve" (similar to "eleven") consists of two parts, the first meaning "two" and the second "leftover", so a literal translation would yield "two remaining having ten taken".

A group of twelve things is called a Duodecad. The ordinal adjective is duodenary.

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 loaves in a baker's dozen.)

As shown below, the number twelve is frequently referenced 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 and 6. Twelve is also a highly composite number, the next one being 24.

The duodecimal system (12₁₀ ^* = 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. As Schoolhouse Rock explains in its song "Little Twelvetoes", if humankind had been born with twelve fingers, they would have counted and multiplied using the duodecimal system. (There is no need to have twelve fingers though, since one will easily arrive at the duodecimal system by simply counting the phalanx bones of fingers with the same hand's thumb, a practice in use with some people until the present day.)

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.

A twelve-sided polygon is a dodecagon. A twelve-sided polyhedron is a dodecahedron. 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.

In base thirteen and higher bases (such as hexadecimal), twelve is represented as C.

Twelve is superabundant, sparsely totient, a Harshad number and a Pell number.

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.

Geometry

It is possible to construct a perfect circle divided into twelve sectors of 30 degrees each, using only a compass and straightedge.

Using a compass, construct a simple circle.
Taking care not to change the radius setting, draw a second circle with center anywhere on the perimeter of the first.
Now go to the two points where the second circle crosses the first, and draw two more circles from these centers.
Continue this process to draw the final three circles, working your way around the perimeter of the first.
Finally connect the center of the original circle with all other points of intersection, using a straightedge.

You now have twelve radial lines emanating from the center, separated by twelve 30-degree sectors, all constructed from seven equal-sized circles.

In numeral systems

Arabic		Armenian
Greek (Ionian)		Greek (Attic)
Hebrew	V20-Z1-Z1	Egyptian
Indian (Devanāgarī)		Chinese character
Tamil		Roman and Etruscan
Thai		Chuvash

In science

The atomic number of magnesium
There are twelve cranial nerves
The duodenum (from Latin duodecim, twelve) is the first part of the intestine that is about twelve inches long.
Force 12 on the Beaufort wind force scale corresponds to the minimum wind speed of a hurricane.

Astronomy

Messier object M12, a magnitude 8.0 globular cluster in the constellation Ophiuchus
The New General Catalogue object NGC 12, a magnitude 13.1 spiral galaxy in the constellation Pisces
The Saros number of the solar eclipse series which began on -2680 August 8 and ended on -1129 February 25. The duration of Saros series 12 was 1550.5 years, and it contained 87 solar eclipses.
The Saros number of the lunar eclipse series which began on -2360 May 28 and ended on -1062 July 17. 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

The biblical Jacob had 12 sons, who were the progenitors of the Twelve Tribes of Israel
In traditional Jewish practice, a girl becomes a bat mitzvah on her twelfth birthday
The New Testament describes twelve apostles of Jesus
In the Church of Jesus Christ of Latter-day Saints, there is a Quorum of the Twelve Apostles
The Book of Revelation 12:1 mentions a woman — usually interpreted as Virgin Mary — wearing a crown of twelve stars
In Shi'a Islam, there are twelve Imams. These twelve early leaders of Islam are—Ali, Hasan, Husayn, and nine of Husayn's descendants
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
Twelfth Night is another name for the 12th day of Christmas or January 5 (the eve of Epiphany)
There are 12,000 people sealed from each of the twelve tribes of Judah in the Book of Revelation prophecies
Eastern Orthodoxy observes 12 Great Feasts
In Ancient Greek religion, the Twelve Olympians were the principal gods of the pantheon

In time

There are 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 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.
The minute hand of a clock face turns twelve times as fast as the hour hand.

In sports

In rugby union one of the centres, most often but not always the inside centre, wears the 12 shirt.
The retired jersey number of American football quarterback Roger Staubach and hockey player Sid Abel.
In football the number 12 a reference to the fans because of the support they give to the 11 players on the field. Bayern Munich, for example, does not allow field players to wear the number 12 on their jersey because it is reserved for their supporters.

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 dialling keys in any standard digital telephone (1 through 9, 0, * and #)

In the arts

Film

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

Theater

'Twelfth Night is a comedy by William Shakespeare

Music

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.
Twelfth Night is a progressive rock band

Games

Twelve is a character in the Street Fighter video game

The roman form of 12 (XII) is used as the symbol of the organisation MJ12 in the Deus Ex computer game.

In other fields

There are 12 ounces in a troy pound (used for precious metals), and 12 constellations in the ecliptic (or signs of the zodiac),
In astrology, Pisces is the 12th sign of the Zodiac.
In Greek mythology, the number of labours of Heracles (Hercules).
The number of dan (master) grades in judo.
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.
Twelve stars are featured on the European flag.
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 D.C. courts. The number of jurors in the movie gave the title to the film 12 Angry Men.
Twelve men have walked on the 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.
In tarot, card no. 12 is "the Hanged Man"

Historical years

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

References

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

External links

This Week's Finds in Mathematical Physics Week 125 (discusses "12-ness" of SL(2,Z))
http://www.number12secret.com
Why twelve Article on the number 12

Integers

Gourt Home::Gourt :: 12 (number)