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Why is 12 special?

Twelve is the smallest abundant number, since it is the smallest integer for which the sum of its proper divisors (1 + 2 + 3 + 4 + 6 = 16) is greater than itself. Twelve is the smallest of two known sublime numbers, numbers that has a perfect number of divisors, and the sum of its divisors is also a perfect number.

en.wikipedia.org - 12 (number) - Wikipedia
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Natural number

12 (twelve) is the natural number following 11 and preceding 13. Twelve is a superior highly composite number, divisible by 2, 3, 4, and 6. It is the number of years required for an orbital period of Jupiter. It is central to many systems of timekeeping, including the Western calendar and units of time of day and frequently appears in the world's major religions.

Name [ edit ]

Twelve is the largest number with a single-syllable name in English. Early Germanic numbers have been theorized to have been non-decimal: evidence includes the unusual phrasing of eleven and twelve, the former use of "hundred" to refer to groups of 120, and the presence of glosses such as "tentywise" or "ten-count" in medieval texts showing that writers could not presume their readers would normally understand them that way.[1][2][3] Such uses gradually disappeared with the introduction of Arabic numerals during the 12th-century Renaissance. Derived from Old English, twelf and tuelf are first attested in the 10th-century Lindisfarne Gospels' Book of John.[note 1][5] It has cognates in every Germanic language (e.g. German zwölf), whose Proto-Germanic ancestor has been reconstructed as *twaliƀi..., from *twa ("two") and suffix *-lif- or *-liƀ- of uncertain meaning.[5] It is sometimes compared with the Lithuanian dvýlika, although -lika is used as the suffix for all numbers from 11 to 19 (analogous to "-teen").[5] Every other Indo-European language instead uses a form of "two"+"ten", such as the Latin duōdecim.[5] The usual ordinal form is "twelfth" but "dozenth" or "duodecimal" (from the Latin word) is also used in some contexts, particularly base-12 numeration. Similarly, a group of twelve things is usually a "dozen" but may also be referred to as a "dodecad" or "duodecad". The adjective referring to a group of twelve is "duodecuple". As with eleven,[6] the earliest forms of twelve are often considered to be connected with Proto-Germanic *liƀan or *liƀan ("to leave"), with the implicit meaning that "two is left" after having already counted to ten.[5] The Lithuanian suffix is also considered to share a similar development.[5] The suffix *-lif- has also been connected with reconstructions of the Proto-Germanic for ten.[6][7] As mentioned above, 12 has its own name in Germanic languages such as English (dozen), Dutch (dozijn), German (Dutzend), and Swedish (dussin), all derived from Old French dozaine. It is a compound number in many other languages, e.g. Italian dodici (but in Spanish and Portuguese, 16, and in French, 17 is the first compound number),[dubious – discuss] Japanese 十二 jūni.[clarification needed]

Written representation [ edit ]

In prose writing, twelve, being the last single-syllable numeral, is sometimes taken as the last number to be written as a word, and 13 the first to be written using digits. This is not a binding rule, and in English language tradition, it is sometimes recommended to spell out numbers up to and including either nine, ten or twelve, or even ninety-nine or one hundred. Another system spells out all numbers written in one or two words (sixteen, twenty-seven, fifteen thousand, but 372 or 15,001).[8] In German orthography, there used to be the widely followed (but unofficial) rule of spelling out numbers up to twelve (zwölf). The Duden[year needed] (the German standard dictionary) mentions this rule as outdated.

Mathematical properties [ edit ]

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 twenty-four.[9] Twelve is the smallest abundant number, since it is the smallest integer for which the sum of its proper divisors (1 + 2 + 3 + 4 + 6 = 16) is greater than itself. Twelve is the smallest of two known sublime numbers, numbers that has a perfect number of divisors, and the sum of its divisors is also a perfect number.[10] 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.[11] If an odd perfect number is of the form 12k + 1, it has at least twelve distinct prime factors. 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.

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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 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.

There are 12 Latin squares of size 3 × 3.

There are 12 two-dimensional regular tilings when enantiomorphic forms are counted separately.

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.

12 is a Harshad number in all bases except octal.

12 is a pronic number[12] and a Pell number.

12 is the negative reciprocal of the Ramanujan summation of the infinite series 1 + 2 + 3 + 4 + ⋯. Although the series is divergent, methods such as Ramanujan summation can assign finite values to divergent series.

12 is the superfactorial of 3. [13]

List of basic calculations [ edit ]

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 × 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 16 12 ÷ x 12 6 4 3 2.4 2 1. 714285 1.5 1. 3 1.2 1. 09 1 0. 923076 0. 857142 0.8 0.75 x ÷ 12 0.08 3 0.1 6 0.25 0. 3 0.41 6 0.5 0.58 3 0. 6 0.75 0.8 3 0.91 6 1 1.08 3 1.1 6 1.25 1. 3 Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 12x 12 144 1728 20736 248832 2985984 35831808 429981696 5159780352 61917364224 743008370688 8916100448256 106993205379072 x12 1 4096 531441 16777216 244140625 2176782336 13841287201 68719476736 282429536481 1000000000000 3138428376721 8916100448256 23298085122481

In nature [ edit ]

Notably, twelve is the number of full lunations in a solar year, hence the number of months in a solar calendar, as well as the number of signs in the Western and the Chinese zodiac. Twelve is also the number of years for an orbital period of Jupiter.

Religion [ edit ]

The number twelve carries religious, mythological and magical symbolism, generally representing perfection, entirety, or cosmic order in traditions since antiquity.

Ancient Greek religion [ edit ]

The Twelve Olympians are the principal gods of the pantheon, they were preceded by twelve Titans, and Hercules carries out twelve labours. [15] Several sets of twelve cities are identified in history as a dodecapolis, the most familiar being the Etruscan League. In ancient Rome, the twelve lictors carried fasces of twelve rods.

Judaism and Christianity [ edit ]

The significance is especially pronounced in the Hebrew Bible.

Ishmael - the first-born son of Abraham - has 12 sons/princes (Genesis 25:16), and Jacob also has 12 sons, who are the progenitors of the Twelve Tribes of Israel.[16] This is reflected in Christian tradition, notably in the twelve Apostles. When Judas Iscariot is disgraced, a meeting is held (Acts) to add Saint Matthias to complete the number twelve once more. The Book of Revelation contains much numerical symbolism, and many of the numbers mentioned have 12 as a divisor. 12:1 mentions a woman—interpreted as the people of Israel, the Church and 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 (the Tribe of Dan is omitted while Manasseh is mentioned), making a total of 144,000 (which is the square of 12 multiplied by a thousand).

Hinduism [ edit ]

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There are twelve Jyotirlinga( Self-formed Lingas ) of Lord Shiva in Hindu temples across India according to the Shaiva tradition. ) of Lord Shiva in Hindu temples across India according to the Shaiva tradition. The Sun god Surya has 12 names.

The Monkey god Hanuman has 12 names.

There are 12 Petals in Anahata or "heart chakra".

There are frequently said to be 12 Âdityas.

Others [ edit ]

Law [ edit ]

The number of twelve jurors in jury trials is depicted by Aeschylus in the Eumenides . In the play, the innovation is brought about by the goddess Athena, who summons twelve citizens to sit as jury. . In the play, the innovation is brought about by the goddess Athena, who summons twelve citizens to sit as jury. In English Common Law, the tradition of twelve jurors harks back to the 10th-century law code introduced by Aethelred the Unready.

Timekeeping [ edit ]

The lunar year is 12 lunar months. Adding 11 or 12 days completes the solar year. [18] Most calendar systems – solar or lunar – have twelve months in a year.

The Chinese use a 12-year cycle for time-reckoning called Earthly Branches.

There are twelve hours in a half day, numbered one to twelve for both the ante meridiem (a.m.) and the post meridiem (p.m.). 12:00 p.m. is midday or noon, and 12:00 a.m. is midnight. (a.m.) and the (p.m.). 12:00 p.m. is midday or noon, and 12:00 a.m. is midnight. The basic units of time (60 seconds, 60 minutes, 24 hours) are evenly divisible by twelve into smaller units.

In numeral systems [ edit ]

In science [ edit ]

In sports [ edit ]

In technology [ edit ]

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 [ edit ]

Film [ edit ]

Films with the number twelve or its variations in their titles include:

Television [ edit ]

Theatre [ edit ]

Literature [ edit ]

Music [ edit ]

Music theory [ edit ]

Pop music [ edit ]

Art theory [ edit ]

There are twelve basic hues in the color wheel: three primary colors (red, yellow, blue), three secondary colors (orange, green & purple) and six tertiary colors (names for these vary, but are intermediates between the primaries and secondaries).

Games [ edit ]

In the game of craps, a dice roll of two sixes (value 12) on the come-out roll constitutes a "craps" and the shooter (dice thrower) loses immediately.

Twelve is a character in the Street Fighter video game series.

video game series. Games such as Backgammon have a long history of 12 points on each side of the gaming board, as evidenced in the XII scripta board in the museum at Ephesus.[19]

In other fields [ edit ]

The numerical range on the analogue clock ends at 12.

See also [ edit ]

Notes [ edit ]

^ an of ðæm tuelfum ).[4] Specially, a passage referring to Judas Iscariot as "one of the twelve " ().

References [ edit ]

Citations [ edit ]

Journal articles

en.wikipedia.org - 12 (number) - Wikipedia
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