6 (number)



Cardinal  six  
Ordinal 
6th (sixth) 

Numeral system  senary  
Factorization  2 × 3  
Divisors  1, 2, 3, 6  
Roman numeral  VI  
Roman numeral (unicode)  Ⅵ, ⅵ, ↅ  
Greek prefix  hexa/hex  
Latin prefix  sexa/sex  
Binary  110_{2}  
Ternary  20_{3}  
Quaternary  12_{4}  
Quinary  11_{5}  
Senary  10_{6}  
Octal  6_{8}  
Duodecimal  6_{12}  
Hexadecimal  6_{16}  
Vigesimal  6_{20}  
Base 36  6_{36}  
Greek  στ (or ΣΤ or ς)  
Arabic  ٦  
Persian  ۶  
Urdu  
Amharic  ፮  
Bengali  ৬  
Chinese numeral  六，陆  
Devanāgarī  ६  
Hebrew  ו (Vav)  
Khmer  ៦  
Thai  ๖  
Telugu  ౬  
Tamil  ௬  
Saraiki  ٦ 
6 (six) is the natural number following 5 and preceding 7.
The SI prefix for 1000^{6} is exa (E), and for its reciprocal atto (a).
Contents

In mathematics 1
 In numeral systems 1.1
 List of basic calculations 1.2

Greek and Latin word parts 2
 Hexa 2.1
 The prefix sex 2.2
 Evolution of the glyph 3

In music 4
 In artists 4.1
 In instruments 4.2
 In music theory 4.3
 In works 4.4
 In religion 5

In science 6
 Astronomy 6.1
 Biology 6.2
 Chemistry 6.3
 Medicine 6.4
 Physics 6.5
 In sports 7
 In technology 8
 In calendars 9
 In the arts and entertainment 10
 In other fields 11
 References 12
 External links 13
In mathematics
6 is the smallest positive integer which is neither a square number nor a prime number. Six is the second smallest composite number; its proper divisors are 1, 2 and 3.
Since six equals the sum of its proper divisors, six is the smallest perfect number, Granville number, and \mathcal{S}perfect number.^{[1]}^{[2]}
As a perfect number:
 6 is related to the Mersenne prime 3, since 2^{1}(2^{2}  1) = 6. (The next perfect number is 28.)
 6 is the only even perfect number that is not the sum of successive odd cubes.^{[3]}
 As a perfect number, 6 is the root of the 6aliquot tree, and is itself the aliquot sum of only one number; the square number, 25.
Six is the only number that is both the sum and the product of three consecutive positive numbers.^{[4]}
Unrelated to 6 being a perfect number, a Golomb ruler of length 6 is a "perfect ruler."^{[5]} Six is a congruent number.
Six is the first discrete biprime (2.3) and the first member of the (2.q) discrete biprime family.
Six is the smallest natural number that can be written as the sum of two positive rational cubes which are not integers: 6 = \left(\frac{17}{21}\right)^3 + \left(\frac{37}{21}\right)^3. Others up to 100 are: 7, 9, 12, 13, 15, 17, 19, 20, 22, 26, 28, 30, 31, 33, 34, 35, 37, 42, 43, 48, 49, 50, 51, 53, 56, 58, 61, 62, 63, 65, 67, 68, 69, 70, 71, 72, 75, 78, 79, 84, 85, 86, 87, 89, 90, 91, 92, 94, 96, 97, 98. (sequence A228499 in OEIS)
Six is a unitary perfect number, a harmonic divisor number and a superior highly composite number, the last to also be a primorial. The next superior highly composite number is 12. The next primorial is 30.
5 and 6 form a RuthAaron pair under either definition.
There are no GraecoLatin squares with order 6. If n is a natural number that is not 2 or 6, than there is a GraecoLatin square with order n.
The smallest nonabelian group is the symmetric group S_{3} which has 3! = 6 elements.
S_{6}, with 720 elements, is the only finite symmetric group which has an outer automorphism. This automorphism allows us to construct a number of exceptional mathematical objects such as the S(5,6,12) Steiner system, the projective plane of order 4 and the HoffmanSingleton graph. A closely related result is the following theorem: 6 is the only natural number n for which there is a construction of n isomorphic objects on an nset A, invariant under all permutations of A, but not naturally in 11 correspondence with the elements of A. This can also be expressed category theoretically: consider the category whose objects are the n element sets and whose arrows are the bijections between the sets. This category has a nontrivial functor to itself only for n=6.
6 similar coins can be arranged around a central coin of the same radius so that each coin makes contact with the central one (and touches both its neighbors without a gap), but seven cannot be so arranged. This makes 6 the answer to the twodimensional kissing number problem. The densest sphere packing of the plane is obtained by extending this pattern to the hexagonal lattice in which each circle touches just six others.
6 is the largest of the four allHarshad numbers.
A sixsided polygon is a hexagon, one of the three regular polygons capable of tiling the plane. Figurate numbers representing hexagons (including six) are called hexagonal numbers. Because 6 is the product of a power of 2 (namely 2^{1}) with nothing but distinct Fermat primes (specifically 3), a regular hexagon is a constructible polygon.
Six is also an octahedral number. It is a triangular number and so is its square (36).
There are six basic trigonometric functions.
There are six convex regular polytopes in four dimensions.
Six is the fourbit binary complement of number nine (6 = 0110_{2} and 9 = 1001_{2}).
The six exponentials theorem guarantees (given the right conditions on the exponents) the transcendence of at least one of a set of exponentials.
All primes above 3 are of the form 6n±1 for n≥1.
In numeral systems
Base  Numeral system  Representation 

2  binary  110 
3  ternary  20 
4  quaternary  12 
5  quinary  11 
6  senary  10 
over 6 (decimal, duodecimal, hexadecimal, etc.)  6 
In bases 10, 15 and 30, 6 is a 1automorphic 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  

6 \times x  6  12  18  24  30  36  42  48  54  60  66  72  78  84  90  96  102  108  114  120  126  132  138  144  150  300  600  6000 
Division  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  

6 \div x  6  3  2  1.5  1.2  1  0.\overline{85714}\overline{2}  0.75  0.\overline{6}  0.6  0.\overline{5}\overline{4}  0.5  0.\overline{46153}\overline{8}  0.\overline{42857}\overline{1}  0.4  
x \div 6  0.1\overline{6}  0.\overline{3}  0.5  0.\overline{6}  0.8\overline{3}  1  1.1\overline{6}  1.\overline{3}  1.5  1.\overline{6}  1.8\overline{3}  2  2.1\overline{6}  2.\overline{3}  2.5 
Exponentiation  1  2  3  4  5  6  7  8  9  10  11  12  13  

6 ^ x\,  6  36  216  1296  7776  46656  279936  1679616  10077696  60466176  362797056  2176782336  13060694016  
x ^ 6\,  1  64  729  4096  15625  46656  117649  262144  531441  1000000  1771561  2985984  4826809 
Greek and Latin word parts
Hexa
Hexa is classical Greek for "six". Thus:
 "Hexadecimal" combines hexa with the Latinate decimal to name a number base of 16

A hexagon is a regular polygon with six sides
 L’Hexagone is a French nickname for the continental part of Metropolitan France
 A hexahedron is a polyhedron with six faces, with a cube being a special case
 Hexameter is a poetic form consisting of six feet per line
 A "hex nut" is a nut with six sides, and a hex bolt has a sixsided head
 The prefix "hexa" also occurs in the systematic name of many chemical compounds, such as "hexamethyl"
The prefix sex
Sex is a Latin prefix meaning "six". Thus:
 Senary is the ordinal adjective meaning "sixth"
 People with sexdactyly have six fingers on each hand
 The measuring instrument called a sextant got its name because its shape forms onesixth of a whole circle
 A group of six musicians is called a sextet
 Six babies delivered in one birth are sextuplets
 Sexy prime pairs  Prime pairs differing by six are sexy, because sex is the Latin word for six.^{[6]}
Evolution of the glyph
The evolution of our modern glyph for 6 appears rather simple when compared with that for the other numerals. Our modern 6 can be traced back to the Brahmins of India, who wrote it in one stroke like a cursive lowercase e rotated 90 degrees clockwise. Gradually, the upper part of the stroke (above the central squiggle) became more curved, while the lower part of the stroke (below the central squiggle) became straighter. The Ghubar Arabs dropped the part of the stroke below the squiggle. From there, the European evolution to our modern 6 was very straightforward, aside from a flirtation with a glyph that looked more like an uppercase G.^{[7]}
On the sevensegment displays of calculators and watches, 6 is usually written with six segments. Some historical calculator models use just five segments for the 6, by omitting the top horizontal bar. This glyph variant has not caught on; for calculators that can display results in hexadecimal, a 6 that looks like a 'b' is not practical.
Just as in most modern typefaces, in typefaces with text figures the 6 character usually has an ascender, as, for example, in .
This numeral resembles an inverted 9. To disambiguate the two on objects and documents that can be inverted, the 6 has often been underlined, both in handwriting and on printed labels.
In music
In artists
 Louis Durey, Arthur Honegger, Darius Milhaud, Francis Poulenc and Germaine Tailleferre in the 1920s
 Bands with the number six in their name include Electric Six, Eve 6, Los Xey (sei is Basque for "six"), Out On Blue Six, Six In Six, Sixpence None the Richer, Slant 6, Vanity 6, and You Me At Six
 #6 is the pseudonym of American musician Shawn Crahan, when performing with the band Slipknot
Also the number of Tuka's album "Life Death Time Eternal" debuting on the Aria Charts.
In instruments
 A standard guitar has 6 strings
 Most woodwind instruments have 6 basic holes or keys (e.g., bassoon, clarinet, pennywhistle, saxophone); these holes or keys are usually not given numbers or letters in the fingering charts
In music theory
 There are 6 whole tones in an octave.
 There are 6 semitones in a tritone.
In works
 "Six geese alaying" were given as a present on the sixth day in the popular Christmas carol, "The Twelve Days of Christmas."
 Divided in six arias, Hexachordum Apollinis is generally regarded as one of the pinnacles of Johann Pachelbel's oeuvre.
 The concerti grossi Opus 3, organ concertos Opus 4 and Opus 7 (each) by Georg Frideric Handel.
 The theme of the sixth album by Dream Theater, Six Degrees Of Inner Turbulence, was the number six: the album has six songs, and the sixth song — that is, the complete second disc — explores the stories of six individuals suffering from various mental illnesses.
In religion

There are 6:
 Points on a Star of David
 Orders of the Mishnah
 Symbolic foods placed on the Passover Seder Plate
 Yahweh took 6 days to create the world in the Old Testament Book of Genesis; humankind was created on day 6. In the City of God, Augustine of Hippo suggested (book 11, chapter 30) that God's creation of the world took six days because 6 is a perfect number.
 The Jewish holiday of Shavuot starts on the sixth day of the Hebrew month of Sivan
 Seraphs have six wings.

In Islam:
 There are Six articles of belief
 Fasting six days of Shawwal, together with the month of Ramadan, is equivalent to fasting the whole year
 In Hindu theology, a trasarenu is the combination of 6 celestial paramānus (atoms)
See also 666.
In science
Astronomy
 Messier object M6, a magnitude 4.5 open cluster in the constellation Scorpius, also known as the Butterfly Cluster
 The New General Catalogue object NGC 6, a spiral galaxy in the constellation Andromeda
 The Saros number of the solar eclipse series which began on March 16, 2691 BC and ended on May 3, 1393 BC. The duration of Saros series 6 was 1298.1 years, and it contained 73 solar eclipses
 The Saros number of the lunar eclipse series which began on July 25, 2642 BC and ended on February 10, 1091 BC. The duration of Saros series 6 was 1550.6 years, and it contained 87 lunar eclipses

The Roman numeral VI:
 Stands for subdwarfs in the Yerkes spectral classification scheme
 (Usually) stands for the sixthdiscovered satellite of a planet or minor planet (e.g. Jupiter VI)
Biology
Chemistry
 A benzene molecule has a ring of 6 carbon atoms
 6 is the atomic number of carbon
 The sixfold symmetry of snowflakes arises from the hexagonal crystal structure of ordinary ice
 A hexamer is an oligomer made of six subunits
Medicine
 There are 6 tastes in traditional Indian Medicine called Ayurveda: sweet, sour, salty, bitter, pungent, and astringent. These tastes are used to suggest a diet based on the symptoms of the body
 Phase 6 is one of six pandemic influenza phases
Physics
 In the Standard Model of particle physics, there are 6 types of quarks and 6 types of leptons
 In statistical mechanics, the sixvertex model has 6 possible configurations of arrows at each vertex
In sports

6 entities in a grouping:
 The National Basketball Association has six divisions (the National Hockey League had six divisions until realigning into four divisions for the 201314 season)
 The Original Six teams in the National Hockey League are Toronto, Chicago, Montreal, New York, Boston, and Detroit. They are the oldest remaining teams in the league, though not necessarily the first six; they comprised the entire league from 1942 to 1967

Number of players:
 Sixman football is a variant of American or Canadian football, played by smaller schools with insufficient enrollment to field the traditional 11man (American) or 12man (Canadian) squad
 In football (soccer), the number of substitutes combined by both teams, that are allowed in the game.
 In ice hockey, the number of players per team, including the goaltender, that are on the ice at any one time, excluding penalty situations.
 In volleyball, 6 players from each team on each side play against each other

Scoring and related terminology:
 In some sports, 6 goals is known as a doublehattrick, but is very hard to accomplish
 In American and Canadian football, a touchdown earns 6 points
 In Australian Rules football, six points are received for a goal
 In baseball scorekeeping, the number 6 represents the shortstop's position.
 In cricket (See cricket terminology):

Jersey number 6 and car number 6:

In football/soccer:
 Is the number of AC Milan's centre back and captain Franco Baresi; the shirt was retired in 1997

In NASCAR:
 The number 6 is currently owned by Roush Fenway Racing. Since the 2007 season, the first year in which Roush Racing was merged with the Fenway Sports Group that owns the Boston Red Sox, the Cup Series version of the car has been driven by David Ragan. From 1988 to 2006, Mark Martin drove the #6 in the Cup Series for what was then Roush Racing.

In rugby:
 In most rugby league competitions (but not the European Super League, which uses static squad numbering), the jersey number 6 is worn by the starting fiveeighth (Southern Hemisphere term) or standoff half (Northern Hemisphere term).
 In rugby union, the starting blindside flanker wears jersey number 6. (Some teams use "left" and "right" flankers instead of "openside" and "blindside", with 6 being worn by the starting left flanker.)

The jersey number 6 has been retired by several North American sports teams in honor of past playing greats or other key figures (or, in two cases, a team's fans):

In Major League Baseball:
 The Atlanta Braves, for manager Bobby Cox
 The Boston Red Sox, for Johnny Pesky
 The Detroit Tigers, for Hall of Famer Al Kaline
 The Minnesota Twins, for Tony Oliva
 The New York Yankees, for manager Joe Torre
 The St. Louis Cardinals, for Hall of Famer Stan Musial
 The San Diego Padres, for Steve Garvey

In the NBA:
 The Boston Celtics, for Hall of Famer Bill Russell
 The Orlando Magic, for their fans (the "sixth man")
 The Philadelphia 76ers, for Hall of Famer Julius Erving
 The Phoenix Suns, for Walter Davis
 The Sacramento Kings, also for their fans

In the NFL:
 The Kansas City Chiefs, for Warren McVea

In the NHL:
 The Detroit Red Wings, for Larry Aurie
 The Pittsburgh Penguins, for Ian Ackerman
 The Toronto Maple Leafs, for Hall of Famer Ace Bailey. The Leafs have a unique policy of not retiring numbers unless the player honoured either died or suffered a careerending incident while a member of the team. Bailey suffered a fractured skull during a game in 1933; while he recovered and lived for nearly 60 years after the incident, he never played again. The Leafs would issue the number to Ron Ellis in 1968 at Bailey's personal request, and Ellis wore it until his own retirement in 1981.

In Major League Baseball:

In football/soccer:
In technology
 On most phones, the 6 key is associated with the letters M, N, and O, but on the BlackBerry it is the key for J and K, and on the BlackBerry 8700 series and Curve 8900 with full keyboard, it is the key for F
 The "6meter band" in amateur radio includes the frequencies from 50 to 54 MHz
 6 is the resin identification code used in recycling to identify polystyrene
In calendars
 In the ancient Roman calendar, Sextilis was the sixth month. After the Julian reform, June became the sixth month and Sextilis was renamed August
 Sextidi was the sixth day of the decade in the French Revolutionary calendar
In the arts and entertainment
 The Super 6  1966 animated cartoon series featuring six different superpowered heroes.
 The Bionic Six are the heroes of the eponymous animated series
 The number of sides on a cube, hence the highest number on a standard die
 The highest number on one end of a standard domino
 The Power of Six is a book written by Pittacus Lore, and the second in the Lorien Legacies series.
 Number Six (Tricia Helfer), is a family of fictional characters from the reimagined science fiction television series, Battlestar Galactica
 Number 6 is a character in the book series Lorien Legacies
 Number 6 (Teresa Palmer) is a character in the movie I Am Number Four (2011)
 Number 6, the main protagonist in The Prisoner played by Patrick McGoohan, portrayed by Jim Caviezel in the remake.
 Six is a character in the television series Blossom played by Jenna von Oÿ.
In other fields
 The standard term in office for a United States senator is six years
 The name of the smallest group of Cub Scouts, traditionally consisting of six people and is led by a 'sixer'.
 Six pack is a common form of packaging for six bottles or cans of drink (especially beer), and by extension, other assemblages of six items.
 The number of fundamental flight instruments lumped together on a cockpit display, called the sixpack.
 The number of feet below ground level a coffin is traditionally buried; thus, the phrase "six feet under" means that a person (or thing, or concept) is dead
 In Astrology, Virgo is the 6th astrological sign of the Zodiac
 The number of dots in a Braille cell.
 The number of Royal Stickers that Mario must collect in the Nintendo 3DS game Paper Mario: Sticker Star
 There are said to be no more than six degrees of separation between any two people on Earth. See also Six degrees.
 Extrasensory perception is sometimes called the "sixth sense"
 Six Cardinal Directions: north, south, east, west, up, and down
 The Six Dynasties form part of Chinese history
 6 is a lucky number in Chinese culture
 The Birmingham Six were held in prison for 16 years
 "Six" is used as an informal slang term for the British Secret Intelligence Service, MI6
 Six Flags amusement parks and theme parks
References
 ^ Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 11.
 ^ "Granville number". OeisWiki. The Online Encyclopedia of Integer Sequences. Archived from the original on 29 March 2011. Retrieved 27 March 2011.
 ^ David Wells, The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin Books (1987): 67
 ^ Peter Higgins, Number Story. London: Copernicus Books (2008): 12
 ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 72
 ^ Chris K. Caldwell; G. L. Honaker Jr. (2009). Prime Curios!: The Dictionary of Prime Number Trivia. CreateSpace Independent Publishing Platform. p. 11.
 ^ Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 395, Fig. 24.66
 The Odd Number 6, JA Todd, Math. Proc. Camb. Phil. Soc. 41 (1945) 66—68
 A Property of the Number Six, Chapter 6, P Cameron, JH v. Lint, Designs, Graphs, Codes and their Links ISBN 0521423856
 Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 67  69
External links
 The Number 6
 The Positive Integer 6
 Prime curiosities: 6
