12 (number)
 

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  1100_{2}  
Ternary  110_{3}  
Quaternary  30_{4}  
Quinary  22_{5}  
Senary  20_{6}  
Octal  14_{8}  
Duodecimal  10_{12}  
Hexadecimal  C_{16}  
Vigesimal  C_{20}  
Base 36  C_{36} 
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 singlesyllable name in English. Early Germanic numbers have been theorized to have been nondecimal: 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 "tencount" 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 12thcentury Renaissance.
It derives from the Old English twelf and tuelf, first attested in the 10thcentury Lindisfarne Gospels' Book of John.^{[n 1]}^{[5]} It has cognates in every Germanic language, whose ProtoGermanic 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 IndoEuropean 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 base12 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 ProtoGermanic *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 ProtoGermanic 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 twentyfour.^{[8]} Twelve is also a superior highly composite number, the next one being sixty.^{[9]} It is the first composite number of the form p^{2}q; a squareprime, 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^{[10]}) and demonstrates an 8member aliquot sequence; {12,16,15,9,4,3,1,0} 12 is the 3rd composite number in the 3aliquot 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 twelvesided polygon is a dodecagon. A twelvefaced 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 threedimensional 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 τfunction 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 distancetransitive graphs.
There are 12 Latin squares of size 3 × 3.
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
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 "twelvefinger 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 disguiseappearance 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 Latterday 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 alBayt, 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 12year cycle for timereckoning called Earthly Branches.
 There are twentyfour 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 12team 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 secondrow 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 walkon 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 careerending 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 tenpin 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 reimagined 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 basetwelve 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 twelvetone technique (also dodecaphony) is a method of musical composition devised by Arnold Schoenberg. Music using the technique is called twelvetone music.
 The 12inch 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 wellknown 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 Twelvebar 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 comeout 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
 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
Wikimedia Commons has media related to 12 (number). 
Notes
 ↑ Specially, a passage referring to Judas Iscariot as "one of the twelve" (an of ðæm tuelfum).^{[4]}
 ↑ Drews^{[18]} considered the classic assembly of instances to be Th. Weinreich's Zwölfgötten.^{[19]}
References
Citations
 ↑ Gordon, E V (1957). Introduction to Old Norse. Oxford: Claredon Press. pp. 292–293.
 ↑ Stevenson, W. H. (December 1899). "The Long Hundred and its Use in England". Archaeological Review. 4 (5): 313–317.
 ↑ Goodare, Julian (1993). "The Long Hundred in medieval and early modern Scotland" (PDF). Proceedings of the Society of Antiquaries of Scotland. 123: 395–418.
 ↑ John 6:71.
 1 2 3 4 5 6 Oxford English Dictionary, 1st ed. "twelve, adj. and n." Oxford University Press (Oxford), 1916.
 1 2 Oxford English Dictionary, 1st ed. "eleven, adj. and n." Oxford University Press (Oxford), 1891.
 ↑ Dantzig, Tobias (1930), Number: The Language of Science.
 ↑ "Sloane's A002182 : Highly composite numbers". The OnLine Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 20160601.
 ↑ "Sloane's A02201 : Superior highly composite numbers". The OnLine Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 20160601.
 ↑ "Sloane's A004394 : Superabundant numbers". The OnLine Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 20160601.
 ↑ "Sloane's A081357 : Sublime numbers". The OnLine Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 20160601.
 ↑ "Sloane's A005835 : Pseudoperfect (or semiperfect) numbers". The OnLine Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 20160601.
 ↑ "Sloane's A000178 : Superfactorials". The OnLine Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 20160601.
 ↑ "Sloane's A001608 : Perrin sequence". The OnLine Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 20160601.
 ↑ "Sloane's A000931 : Padovan sequence". The OnLine Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 20160601.
 ↑ "Sloane's A000129 : Pell numbers". The OnLine Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 20160601.
 ↑ See graph.
 ↑ Drews (1972), p. 43, n. 10.
 ↑ Weinreich, Th., "Zwölfgötten", Ausfurliches Lexikon der Griechischen und Römishen Mythologie, Vol. VI, col. 764848.
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 RodriguezVillegas, 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 0883855119.