12 (number)

← 11 12 13 →
← 10 11 12 13 14 15 16 17 18 19 → List of numbers — Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	twelve
Ordinal	12th (twelfth)
Numeral system	duodecimal
Factorization	2^2× 3
Divisors	1, 2, 3, 4, 6, 12
Roman numeral	XII
Greek prefix	dodeca-
Latin prefix	duodeca-
Binary	11002
Ternary	1103
Quaternary	304
Quinary	225
Senary	206
Octal	148
Duodecimal	1012
Hexadecimal	C16
Vigesimal	C20
Base 36	C36

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

12 (twelve ⁱ/ˈtwɛlv/) is the natural number following 11 and preceding 13. The product of the first three factorials, twelve is a superior highly composite number, divisible by 2, 3, 4, and 6. It is central to many systems of counting, including the Western calendar and units of time, and frequently appears in the Abrahamic religions.

Name

The word "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.

It derives from the Old English twelf and tuelf, first attested in the 10th-century Lindisfarne Gospels' Book of John.^{[n 1][5]} It has cognates in every Germanic language, 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 "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]

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 twenty-four.^[8] Twelve is also a superior highly composite number, the next one being sixty.^[9] 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^[10]) 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.^[11] 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.^[12]

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.^[13] 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,^[14] and also appears in the Padovan sequence, preceded by the terms 5, 7, 9 (it is the sum of the first two of these).^[15] 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).^[16]

Twelve is probably the last even number that is the sum of only one pair of prime numbers (5+7).^[17] See Goldbach's conjecture.

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 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.

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 × 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	0.083	0.16	0.25	0.3	0.416	0.5	0.583	0.6	0.75	0.83		0.916	1	1.083	1.16	1.25	1.3

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^12	1	4096	531441	16777216	244140625	2176782336	13841287201	68719476736	282429536481	1000000000000		3138428376721	8916100448256	23298085122481

In numeral systems

١٢	Arabic	ԺԲ	Armenian
১২	Bangla	ΔΙΙ	Attic Greek
יב	Hebrew		Egyptian
१२	Indian & Nepali (Devanāgarī)	十二	Chinese and Japanese
௧௨	Tamil	Ⅻ	Roman and Etruscan
๑๒	Thai	IIX	Chuvash
౧౨	Telugu	١٢	Urdu
ιβʹ	Ionian Greek	൧൨	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 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 BC August 20 and ended on 1129 BC 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 BC May 28 and ended on 1062 BC 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.
• The Western zodiac has twelve signs, as does the Chinese zodiac.

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 the Hindu epic Mahabharata, the Pandavas and their wife Draupadi go to the forest for 12 years exile and 1 year disguise-appearance exile.

In antiquity there are numerous ceremonial, magical, and religious uses of twelve.^{[n 2]} The twelve lictors carried fasces of twelve rods. Ancient Greek religion, the Twelve Olympians were the principal gods of the pantheon and Hercules enacted out twelve labours. 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 Saint 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. 12:1 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).

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 Islamic 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 11 brothers, making 12.

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

In time

• Most calendar systems have twelve months in a year.
• 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 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 Major League Baseball:
  • The Tampa Bay Rays, for Hall of Famer Wade Boggs.
  • The Toronto Blue Jays, for Hall of Famer Roberto Alomar.
• In the NFL:
  • The New England Patriots, for Tom Brady.
  • The Buffalo Bills, for Hall of Famer Jim Kelly.
  • The Miami Dolphins, for Hall of Famer Bob Griese.
  • The New York Jets, for Hall of Famer Joe Namath.
  • The San Francisco 49ers, for John Brodie.
  • The Seattle Seahawks, for their fans (the "12th Man").
  • The Dallas Cowboys have a policy of not retiring numbers. However, the team has not issued #12 since the retirement of Hall of Famer Roger Staubach.
  • The Pittsburgh Steelers currently have a policy of not retiring numbers, having retired only two numbers (70 & 75) in their earlier history. However, the Steelers have not issued #12 since the retirement of Hall of Famer Terry Bradshaw.
• In the NBA:
  • The Cincinnati Royals, for Hall of Famer Maurice Stokes, who suffered a career-ending head injury in 1958, the team's first season in Cincinnati. The franchise continues to honor the number in its current incarnation as the Sacramento Kings.
  • The New York Knicks, for Dick Barnett.
  • The Utah Jazz, for Hall of Famer John Stockton.
• In the NHL:
  • The Detroit Red Wings, for Hall of Famer Sid Abel.
  • The Montreal Canadiens, for Hall of Famers Yvan Cournoyer and Dickie Moore.
  • The Vancouver Canucks, for Stan Smyl.
• The jersey number 12 has also been retired by the men's basketball program of the University of North Carolina for Phil Ford.

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 Formula One: Ayrton Senna's racing number at Lotus and McLaren. It is currently used by Felipe Nasr for Sauber.

In Indy racing: Eddie Sachs drove a number 12 car in the 1961 Indianapolis 500.

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 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

Film

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

• 12
• 12 Angry Men (1957 and 1997)
• Cheaper by the Dozen
• Ocean's Twelve
• 12 Monkeys
• The Dirty Dozen
• 12 Rounds
• Twelve
• Twelve Years a Slave

Television

• The number twelve plays a significant role in the television franchise Battlestar Galactica. The characters come from the Twelve Colonies of Kobol and worship the twelve lords of Kobol. In the re-imagined series, there are also twelve models of the humanoid version of Cylons.
• Twelve Angry Men, the original 1954 live performance on the anthology television series Studio One
• "Number 12 Looks Just Like You" is an episode of the television show The Twilight Zone.
• Schoolhouse Rock! portrayed an alien child using base-twelve arithmetic in the short "Little Twelvetoes".
• 12 Oz Mouse was an animated television show on Adult Swim.

Theatre

• Twelfth Night is a comedy by William Shakespeare
• Twelve Angry Men, by Reginald Rose, adapted from his own teleplay (see above)

Literature

• The Twelve is a poem by Aleksandr Blok
• Twelve is a novel by Nick McDonell
• The Twelve Chairs is a satirical novel by the Soviet authors Ilf and Petrov
• Cheaper by the Dozen is a 1946 novel by Frank Bunker Gilbreth, Jr. and Ernestine Gilbreth Carey
• The Twelve Dancing Princesses is a folk tale
• The Aeneid, an epic poem by Virgil is divided into two halves composed of twelve books.
• Paradise Lost, an epic poem by John Milton is divided into twelve books perhaps in imitation of the Aeneid

Music

• Twelve Girls Band
• 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.
• The 12-inch single is a vinyl record format.
• Twelfth Night is a progressive rock band
• 12 Play is an R. Kelly album.
• The Number 12 Looks Like You is a mathcore band.
• "12", a Song from Brave Murder Day by Katatonia.
• 12 is a studio album by German singer Herbert Grönemeyer.
• Twelve is an album by Patti Smith.
• Twelve Deadly Cyns...and Then Some is an album by Cyndi Lauper.
• D12 a rap group also known as the Dirty Dozen.
• Musical group named 12 Stones
• 12 Hundred is a song by band Mushroomhead of their Savior Sorrow album.
• 12 is the 12th studio album by Keller Williams.
• "12" ("Dodeka" in Greek) is one of the most well-known hits by Anna Vissi
• 12 is the number of studio albums the band The Beatles released.
• The 1812 overture
• "Twelve drummers drumming" is the gift on the twelfth day of Christmas in the carol "The Twelve Days of Christmas"
• The Twelve-bar blues is one of the most prominent chord progressions in popular 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

• 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.
• In Mario Kart Wii and Mario Kart 8, the starting grid can carry twelve characters.

In other fields

12 stars are featured on the Flag of Europe

• There are 12 troy ounces in a troy pound (used for precious metals)
• In the former British currency system, there were twelve pence in a shilling.
• In Greek mythology, the number of Labours of Heracles was increased from ten to make twelve.
• 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 people have walked on 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.
• King Arthur's round table had 12 knights plus King Arthur himself.
• 12 inches in a foot.
• 12 face cards in a card deck.
• Alcoholics Anonymous has 12 steps, 12 traditions and 12 concepts for world service.
• Wilhelm Heinrich Schüßler developed a list of 12 biochemical cell salts, also known as tissue salts.

See also

• Thirteen
• The Twelfth

Notes

1. Specially, a passage referring to Judas Iscariot as "one of the twelve" (an of ðæm tuelfum).^[4]
2. Drews^[18] considered the classic assembly of instances to be Th. Weinreich's Zwölfgötten.^[19]

References

Citations

1. Gordon, E V (1957). Introduction to Old Norse. Oxford: Claredon Press. pp. 292–293.
2. Stevenson, W. H. (December 1899). "The Long Hundred and its Use in England". Archaeological Review. 4 (5): 313–317.
3. Goodare, Julian (1993). "The Long Hundred in medieval and early modern Scotland" (PDF). Proceedings of the Society of Antiquaries of Scotland. 123: 395–418.
4. John 6:71.
5. Oxford English Dictionary, 1st ed. "twelve, adj. and n." Oxford University Press (Oxford), 1916.
6. Oxford English Dictionary, 1st ed. "eleven, adj. and n." Oxford University Press (Oxford), 1891.
7. Dantzig, Tobias (1930), Number: The Language of Science.
8. "Sloane's A002182 : Highly composite numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
9. "Sloane's A02201 : Superior highly composite numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
10. "Sloane's A004394 : Superabundant numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
11. "Sloane's A081357 : Sublime numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
12. "Sloane's A005835 : Pseudoperfect (or semiperfect) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
13. "Sloane's A000178 : Superfactorials". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
14. "Sloane's A001608 : Perrin sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
15. "Sloane's A000931 : Padovan sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
16. "Sloane's A000129 : Pell numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
17. See graph.
18. Drews (1972), p. 43, n. 10.
19. Weinreich, Th., "Zwölfgötten", Ausfurliches Lexikon der Griechischen und Römishen Mythologie, Vol. VI, col. 764-848.

Bibliography

• Drews, Robert (January 1972), "Light from Anatolia on the Roman Fasces", The American Journal of Philology, Vol. 93, No. 1, pp. 40–51.
• Lattice Polygons and the Number 12, Bjorn Poonen, Fernando Rodriguez-Villegas, American Mathematical Monthly, Vol. 107, No. 3 (March, 2000), pp. 238–250
• Schwartzman, Steven (1994). The words of mathematics: An etymological dictionary of mathematical terms used in English. The Mathematical Association of America. ISBN 0-88385-511-9.