Mathematical properties |
φ = 4 |
τ = 6 |
σ = 28 |
π = 5 |
μ = 0 |
M = -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 isleft 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 translucent. 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
[after having ten taken]".
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, and
6. Twelve is
also a
highly composite
number, the next one being
24. It is
the first
composite number of the
form
p^{2}q; a square prime, and also the
first member of the (
p^{2}) 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 12
k + 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 has twelve elements, and even the properties of
lattice polygons.
There are twelve
Jacobian
elliptic functions.
The
duodecimal system (12
_{10}
[twelve] = 10
_{12}), 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
In numeral systems
In science
Astronomy
In religion
The number 12 is very important in many religions, mainly Judaism
and Christianity, but some uses are to be found in pagan
times.
In Antiquity there are numerous magical/religious uses of twelves.
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. 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).
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
Judaism, 12 signifies the age a girl
matures
(bat mitzvah).
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.
In
Hinduism, the sun god
Surya has 12 names.
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.
In both
soccer and
American Football, the number 12 can be a
symbolic reference to the
fan
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
and the Seattle Seahawks do not
allow field players to wear the number 12 on their jersey because it is reserved for their
supporters. However, several notable
National Football League
quarterbacks have worn number 12 jerseys, including
Tom Brady,
Aaron
Rogers,
Ken Stabler Joe Namath and
Roger
Staubach.
In the Canadian Football League (CFL) 12 is the maximum number of
players that can be on the field of play for each team at any time.
The Saskatchewan Roughriders lost the 2009 Grey Cup Championship
game when the Montreal Alouettes, trailing by 2 points, missed
their last play Field Goal in an attempt to win the game. However,
because Saskatchewan had 13 men on the field they were penalized
for it. This gave Montreal a second chance 10 yards closer in which
this time the FG was made to give Montreal the Grey Cup victory
with no time left on the clock.
In technology
In the arts
Film
Movies with the number twelve or its variations in their titles
include
Television
Theatre
Literature
Music
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
- The Roman form of 12 (XII) is used as the symbol of the
organization MJ12 in the Deus Ex
computer game.
In other fields
- There are 12 in a dozen.
- There are 12 ounces in a troy pound
(used for precious metals), and 12 constellations in the ecliptic
(or signs of the zodiac).
- In the former British currency system,
there were twelve pence in a shilling.
- In astrology, there are 12 signs in
the Zodiac.
- In Greek mythology, the number
of labours of Heracles was increased from ten to make
twelve.
- 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.
- 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 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.
- 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
- 12 Inches in a Foot
- 12 face cards in a card deck.
Historical years
12 A.D.,
12
B.C.,
1912,
2012,
etc.
See also
Notes
- 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
- Lattice Polygons and the Number 12, Bjorn Poonen, Fernando
Rodriguez-Villegas, American Mathematical Monthly, Vol.
107, No. 3 (Mar., 2000) , pp. 238–250 [29897]