# 12 number

 Template:Numbers (10s) Cardinal 12 twelve Ordinal 12th twelfth Numeral system duodecimal Factorization $\displaystyle 2^2 \cdot 3$ 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.

A group of twelve things is called a Duodecad.

## 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.

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.

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.) 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. See also: base 12. 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.)

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

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 [after having ten taken]".

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

### The geometry of twelve

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12-sided dodecahedron
File:Rhombicdodecahedron.jpg
12-sided rhombic dodecahedron

It is possible to construct a perfect circle divided into twelve sectors of 30 degrees each, using only a compass and straightedge. Follow these steps to construct this figure:

1. Using a compass, construct a simple circle.
2. Taking care not to change the radius setting, draw a second circle with center anywhere on the perimeter of the first.
3. Now go to the two points where the second circle crosses the first, and draw two more circles from these centers.
4. Continue this process to draw the final three circles, working your way around the perimeter of the first.
5. 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 science

In 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 time

There are twelve months in a year.

The Western zodiac has twelve signs, as does the Chinese zodiac.

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.

The minute hand of a clock face turns twelve times faster than the hour hand.

## In religion

Jewish tradition and the old testament of the Bible talks of twelve tribes of Israel. The New Testament describes twelve apostles of Jesus. The Book of Revelation 12:1 mentions a woman - usually interpreted as Virgin Mary - wearing a crown of twelve stars.

In modern Jewish practice, a girl gets her bat mitzvah on the Shabbat after her twelfth birthday.

In Islam, especially Shia, there are twelve Imams. These twelve early leaders of Islam are—Ali, Hasan, Husayn, and nine of Husayn's descendants.

The 12 apostles of Christ.

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 as Christmastide.

Eastern Orthodoxy observes 12 Great Feasts.

In Ancient Greek religion, the Twelve Olympians were the principal gods of the pantheon.

## In numeral systems

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

## In other fields

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.

In astrology, Pisces is the 12th and final astrological sign of the Zodiac.

There have always been 12 eggs in a dozen, 12 hours on a clock,12 inches in a foot, 12 months in a year, 12 steps in a staircase, 12 ounces in a Troy pound (used for precious metals), 12 constellations in the ecliptic (or signs of the zodiac), 12 gears on a racing bicycle,, and 12 volts in a car battery.

Other movies with the number twelve in their titles include The Dirty Dozen, 12 Monkeys, Ocean's Twelve and Twelve Chairs.

Twelfth Night, another name for the 12th Day of Christmas or January 5 (the eve of Epiphany), gives its name to a Shakespeare comedy, as well as that of progressive rock band.

At MIT, Course 12 refers to the Department of Earth, Atmospheric, and Planetary Science.