# 6

Short description: Natural number
 ← 5 6 7 →
Cardinalsix
Ordinal6th
(sixth)
Numeral systemsenary
Factorization2 × 3
Divisors1, 2, 3, 6
Greek numeralϚ´
Roman numeralVI, vi, ↅ
Greek prefixhexa-/hex-
Latin prefixsexa-/sex-
Binary1102
Ternary203
Quaternary124
Quinary115
Senary106
Octal68
Duodecimal612
Vigesimal620
Base 36636
Greekστ (or ΣΤ or ς)
Arabic, Kurdish, Sindhi, Urdu٦
Persian۶
Amharic
Bengali
Chinese numeral六，陸
Devanāgarī
Gujarati
Hebrewו
Khmer
Thai
Telugu
Tamil
Saraiki٦
Malayalam

6 (six) is the natural number following 5 and preceding 7. It is a composite number and the smallest perfect number.[1]

## In mathematics

Six is the smallest positive integer which is neither a square number nor a prime number; it is the second smallest composite number, behind 4; its proper divisors are 1, 2 and 3.[1]

Since 6 equals the sum of its proper divisors, it is a perfect number; 6 is the smallest of the perfect numbers.[1] It is also the smallest Granville number, or $\displaystyle{ \mathcal{S} }$-perfect number.[2][3]

As a perfect number:

Six is the only number that is both the sum and the product of three consecutive positive numbers.[5]

Unrelated to 6's being a perfect number, a Golomb ruler of length 6 is a "perfect ruler".[6] Six is a congruent number.[7]

Six is the first discrete biprime (2 × 3) and the first member of the (2 × q) discrete biprime family.

Six is a unitary perfect number,[8] a primary pseudoperfect number,[9] a harmonic divisor number[10] and a superior highly composite number,[11] the last to also be a primorial.

There are no Graeco-Latin squares with order 6.[12] If n is a natural number that is not 2 or 6, then there is a Graeco-Latin square with order n.

There is not a prime $\displaystyle{ p }$ such that the multiplicative order of 2 modulo $\displaystyle{ p }$ is 6, that is, $\displaystyle{ ord_p(2) = 6 }$ By Zsigmondy's theorem, if $\displaystyle{ n }$ is a natural number that is not 1 or 6, then there is a prime $\displaystyle{ p }$ such that $\displaystyle{ ord_p(2) = n }$. See A112927 for such $\displaystyle{ p }$.

The ring of integer of the sixth cyclotomic field Q6) , which is called Eisenstein integer, has 6 units: ±1, ±ω, ±ω2, where $\displaystyle{ \omega = \frac{1}{2}(-1 + i\sqrt 3) = e^{2\pi i/3} }$.

The smallest non-abelian group is the symmetric group S3 which has 3! = 6 elements.[1]

S6, 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 Hoffman-Singleton 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 n-set A, invariant under all permutations of A, but not naturally in one-to-one 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 non-trivial functor to itself only for n = 6.

Six 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 two-dimensional kissing number problem.[13] 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.

A cube has six faces.

6 is the largest of the four all-Harshad numbers.[14]

A six-sided polygon is a hexagon,[1] 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 21) with nothing but distinct Fermat primes (specifically 3), a regular hexagon is a constructible polygon.

Six is also an octahedral number.[15] It is a triangular number[16] and so is its square (36).

There are six basic trigonometric functions.[17]

There are six convex regular polytopes in four dimensions.

The six exponentials theorem guarantees (given the right conditions on the exponents) the transcendence of at least one of a set of exponentials.[18]

All primes above 3 are of the form 6n ± 1 for n ≥ 1.

6 is a pronic number and the only semiprime to be.[19]

### 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 25 50 100 1000
6 × x 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 150 300 600 6000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
6 ÷ x 6 3 2 1.5 1.2 1 0.857142 0.75 0.6 0.6 0.54 0.5 0.461538 0.428571 0.4
x ÷ 6 0.16 0.3 0.5 0.6 0.83 1 1.16 1.3 1.5 1.6 1.83 2 2.16 2.3 2.5
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
6x 6 36 216 1296 7776 46656 279936 1679616 10077696 60466176 362797056 2176782336 13060694016
x6 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".[1] Thus:

### The prefix sex-

X-ray of a polydactyl human hand with six fingers

Sex- is a Latin prefix meaning "six".[1] Thus:

• Senary is the ordinal adjective meaning "sixth"[23]
• People with sexdactyly have six fingers on each hand
• The measuring instrument called a sextant got its name because its shape forms one-sixth 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.[24][25]

The SI prefix for 10006 is exa- (E), and for its reciprocal atto- (a).

## Evolution of the Arabic digit

The first appearance of 6 is in the Edicts of Ashoka circa 250 BCE. These are Brahmi numerals, ancestors of Hindu-Arabic numerals.
The first known digit "6" in the number "256" in Ashoka's Minor Rock Edict No.1 in Sasaram, circa 250 BCE

The evolution of our modern digit 6 appears rather simple when compared with the other digits. The modern 6 can be traced back to the Brahmi numerals of India , which are first known from the Edicts of Ashoka circa 250 BCE.[26][27][28][29] It was written 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 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.[30]

On the seven-segment 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 character for the digit 6 usually has an ascender, as, for example, in .[31]

This digit 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

A standard guitar has six strings.

### In artists

• Les Six ("The Six" in English) was a group consisting of the French composers Georges Auric, Louis Durey, Arthur Honegger, Darius Milhaud, Francis Poulenc and Germaine Tailleferre in the 1920s[32]
• Bands with the number six in their name include Six Organs of Admittance,[33] 6 O'Clock Saints, Electric Six,[34] Eve 6, Los Xey (sei is Basque for "six"), Out On Blue Six, Six In Six, Sixpence None the Richer,[35] Slant 6,[36] Vanity 6, and You Me At Six[37]
• #6 is the pseudonym of United States musician Shawn Crahan, when performing with the band Slipknot

### In instruments

• A standard guitar has six strings[38]
• Most woodwind instruments have six 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 six whole tones in an octave.[39]
• There are six semitones in a tritone.[40]

### In works

• "Six geese a-laying" were given as a present on the sixth day in the popular Christmas carol, "The Twelve Days of Christmas."[41]
• Divided in six arias, Hexachordum Apollinis is generally regarded as one of the pinnacles of Johann Pachelbel's oeuvre.[42]
• 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.[43]
• Aristotle gave six elements of tragedy, the first of which is Mythos.[44]

## In science

### Biology

The cells of a beehive are six-sided.

### Medicine

• There are six tastes in traditional Indian medicine (Ayurveda): sweet, sour, salty, bitter, pungent, and astringent. These tastes are used to suggest a diet based on the symptoms of the body.[63]
• Phase 6 is one of six pandemic influenza phases.[64]

### Physics

In the Standard Model of particle physics, there are six types of quarks and six types of leptons.

## In sports

• The Original Six teams in the National Hockey League are Toronto, Chicago, Montreal, New York, Boston, and Detroit.[68] 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:
• In association football (soccer), the number of substitutes combined by both teams, that are allowed in the game.
• In box lacrosse, the number of players per team, including the goaltender, that are on the floor at any one time, excluding penalty situations.[69]
• In ice hockey, the number of players per team, including the goaltender, that are on the ice at any one time during regulation play, excluding penalty situations. (Some leagues reduce the number of players on the ice during overtime.)[70]
• In volleyball:
• Six players from each team on each side play against each other.[71]
• Standard rules only allow six total substitutions per team per set. (Substitutions involving the libero, a defensive specialist who can only play in the back row, are not counted against this limit.)
• Six-man football is a variant of American or Canadian football, played by smaller schools with insufficient enrollment to field the traditional 11-man (American) or 12-man (Canadian) squad.[72]
• Scoring:
• In both American and Canadian football, 6 points are awarded for a touchdown.[73]
• In Australian rules football, 6 points are awarded for a goal, scored when a kicked ball passes between the defending team's two inner goalposts without having been touched by another player.
• In cricket, six runs are scored for the batting team when the ball is hit to the boundary or the ground beyond it without having touched the ground in the field.
• In basketball, the ball used for women's full-court competitions is designated "size 6".[74]
• In most rugby league competitions (but not the Super League, which uses static squad numbering), the jersey number 6 is worn by the starting Template:Rlp (Southern Hemisphere term) or Template:Rlp (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.)[75]

## In technology

• On most phones, the 6 key is associated with the letters M, N, and O, but on the BlackBerry Pearl 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 "6-meter band" in amateur radio includes the frequencies from 50 to 54 MHz
• 6 is the resin identification code used in recycling to identify polystyrene[76]

## 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[77]
• Sextidi was the sixth day of the décade in the French Revolutionary calendar[78]

## In the arts and entertainment

### Games

• The number of sides on a cube, hence the highest number on a standard die[79]
• The six-sided tiles on a hex grid are used in many tabletop and board games.
• The highest number on one end of a standard domino

### Comics and cartoons

• The Super 6, a 1966 animated cartoon series featuring six different super-powered heroes.[80]

### Literature

• The Power of Six is a book written by Pittacus Lore, and the second in the Lorien Legacies series.[81]
• Number 6 is a character in the book series Lorien Legacies

### TV

• Number Six (Tricia Helfer), is a family of fictional characters from the reimagined science fiction television series, Battlestar Galactica
• Number 6, the main protagonist in The Prisoner played by Patrick McGoohan, and portrayed by Jim Caviezel in the remake.
• Six is a character in the television series Blossom played by Jenna von Oÿ.[82]
• Six is the nickname of Kal Varrik, a central character in the television series Dark Matter, played by Roger Cross.[83]
• Six is a History channel series that chronicles the operations and daily lives of SEAL Team Six.[84]
• Six Feet Under, an HBO series that ran from 2005 to 2011.[85]

### Movies

• Number 6 (Teresa Palmer) is a character in the movie I Am Number Four (2011).[86]
• The 6th Day (2000), starring Arnold Schwarzenegger.[87]
• The Sixth Sense (1999), written and directed by M. Night Shyamalan and starring Haley Joel Osment and Bruce Willis.[88]
• Girl 6 (1996), directed by Spike Lee.[89]

### Musicals

• Six is a modern retelling of the lives of the six wives of Henry VIII presented as a pop concert.[90]

## Anthropology

• The name of the smallest group of Cub Scouts and Guiding's equivalent Brownies, traditionally consisting of six people and is led by a "sixer".
• A coffin is traditionally buried six feet under the ground; thus, the phrase "six feet under" means that a person (or thing, or concept) is dead[91]
• There are said to be no more than six degrees of separation between any two people on Earth.[92]
• In Western astrology, Virgo is the 6th astrological sign of the Zodiac[93]
• The Six Dynasties form part of Chinese history[94]
• Six is a lucky number in Chinese culture.[95]
• The Birmingham Six were a British miscarriage of justice, held in prison for 16 years.[96]
• "Six" is used as an informal slang term for the British Secret Intelligence Service, MI6.[97]

## In other fields

• 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.[98]
• In Pythagorean numerology (a pseudoscience), the number 6 is the digit of balance, harmony and organization of the home and family
• The fundamental flight instruments lumped together on a cockpit display are often called the Basic Six or six-pack.
• The number of dots in a braille cell.[99]
• Extrasensory perception is sometimes called the "sixth sense".[100]
• Six Flags is an American company running amusement parks and theme parks in the U.S., Canada, and Mexico.[101]
• In the U.S. Army "Six" as part of a radio call sign is used by the commanding officer of a unit, while subordinate platoon leaders usually go by "One".[102] (For a similar example see also: Rainbow Six.)

• List of highways numbered 6

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30. 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
31. Negru, John (1988) (in en). Computer Typesetting. Van Nostrand Reinhold. pp. 59. ISBN 978-0-442-26696-7. "slight ascenders that rise above the cap height ( in 4 and 6 )"
32. Auric, Georges; Durey, Louis; Honegger, Arthur; Milhaud, Darius; Poulenc, Francis; Tailleferre, Germaine (2014-08-20) (in fr). Caramel Mou and Other Great Piano Works of "Les Six": Pieces by Auric, Durey, Honegger, Milhaud, Poulenc and Tailleferre. Courier Corporation. ISBN 978-0-486-49340-4.
33.
34. D'Amante, Elvo (1994-01-01) (in en). Music Fundamentals: Pitch Structures and Rhythmic Design. Scarecrow Press. pp. 194. ISBN 978-1-4616-6985-2. "The division of an octave into six equal parts is referred to as the whole-tone scale"
35. Horsley, Charles Edward (1876) (in en). A Text Book of Harmony: For the Use of Schools and Students. Sampson Low, Marston, Searle, & Rivington. pp. 4. "Like the Tritone , it contains six semitones"
36. Tribble, Mimi (2004) (in en). 300 Ways to Make the Best Christmas Ever!: Decorations, Carols, Crafts & Recipes for Every Kind of Christmas Tradition. Sterling Publishing Company, Inc.. pp. 145. ISBN 978-1-4027-1685-0. "Six geese a-laying"
37. Staines, Joe (2010-05-17) (in en). The Rough Guide to Classical Music. Penguin. pp. 393. ISBN 978-1-4053-8321-9. "...the six arias with variations collected under the title Hexachordum Apollinis (1699)..."
38. Hegarty, Paul; Halliwell, Martin (2011-06-23) (in en). Beyond and Before: Progressive Rock since the 1960s. Bloomsbury Publishing USA. pp. 169. ISBN 978-1-4411-1480-8. "Six Degrees of Inner Turbulence"
39. Curran, Angela (2015-10-05) (in en). Routledge Philosophy Guidebook to Aristotle and the Poetics. Routledge. pp. 133. ISBN 978-1-317-67706-2. "THE SIX QUALITATIVE ELEMENTS OF TRAGEDY"
40. Plaut, W. Gunther (1991) (in en). The Magen David: How the Six-pointed Star Became an Emblem for the Jewish People. B'nai B'rith Books. ISBN 978-0-910250-16-0. "How the Six-pointed Star Became an Emblem for the Jewish People"
41. Lauterbach, Jacob Zallel (1916) (in en). Midrash and Mishnah: A Study in the Early History of the Halakah. Bloch. pp. 9. "Six orders of Mishnah"
42. Rosen, Ceil; Rosen, Moishe (2006-05-01) (in en). Christ in the Passover. Moody Publishers. pp. 79. ISBN 978-1-57567-480-3. "Six symbolic foods"
43. Repcheck, Jack (2008-12-15) (in en). The Man Who Found Time: James Hutton And The Discovery Of Earth's Antiquity. Basic Books. ISBN 978-0-7867-4399-5. "it actually took only six days to create the earth"
44. "CHURCH FATHERS: City of God, Book XI (St. Augustine)". "These works are recorded to have been completed in six days (the same day being six times repeated), because six is a perfect number"
45. Grossman, Grace Cohen; Ahlborn, Richard E.; Institution, Smithsonian (1997) (in en). Judaica at the Smithsonian: Cultural Politics as Cultural Model. Smithsonian Institution Press. pp. 228. "Shavuot falls on the sixth day of the Hebrew month of Sivan"
46. Robertson, William Archibald Scott (1880) (in en). The crypt of Canterbury cathedral; its architecture, its history, and its frescoes. Mitchell & Hughes. pp. 91. "...but seraphs , with six wings"
47. Shapera, Paul M. (2009-08-15) (in en). Iran's Religious Leaders. The Rosen Publishing Group, Inc. pp. 10. ISBN 978-1-4358-5283-9. "Islam has six articles of faith"
48. Algül, Hüseyin (2005) (in en). The Blessed Days and Nights of the Islamic Year. Tughra Books. pp. 65. ISBN 978-1-932099-93-5. "...it was blessed to fast for six days in the month of Shawwal..."
49. Bary, William Theodore De; DeBary, William T.; Chan, Wing-tsit; Lufrano, Richard; Ching, Julia; Johnson, David; Liu, Kwang-Ching; Mungello, David (1999) (in en). Sources of Chinese Tradition. Columbia University Press. ISBN 978-0-231-11270-3. "...and the Six Ministries were made..."
50. Rhoads, Samuel E. (1996) (in en). The Sky Tonight: A Guided Tour of the Stars Over Hawai'i. Bishop Museum Press. ISBN 978-0-930897-93-2. "Three Messier objects are visible in this part of the sky : M6 , M7 and M8 ."
51. Sedgwick, Marcus (2011-07-05) (in en). White Crow. Roaring Brook Press. pp. 145. ISBN 978-1-4299-7634-3. "The cells of honeycombs are six-sided because a hexagon is the most material-efficient tessellation"
52. Parker, Steve (2005) (in en). Ant Lions, Wasps & Other Insects. Capstone. pp. 16. ISBN 978-0-7565-1250-7. "Insects have six legs each..."
53. Pendarvis, Murray P.; Crawley, John L. (2019-02-01) (in en). Exploring Biology in the Laboratory: Core Concepts. Morton Publishing Company. pp. 10. ISBN 978-1-61731-899-3. "...presently at least six kingdoms are recognized;"
54. Mader, Sylvia S. (2004) (in en). Biology. McGraw-Hill. pp. 20. ISBN 978-0-07-291934-9. "The acronym CHNOPS helps us remember these six elements"
55. DESS, Fritz Dufour, MBA (2018-09-19) (in en). The Realities of Reality - Part II: Making Sense of Why Modern Science Advances (Volume 1). Fritz Dufour. pp. 100. "The benzene molecule has its six carbon atoms in a ring"
56. Starr, Cecie; Evers, Christine (2012-05-10) (in en). Biology Today and Tomorrow without Physiology. Cengage Learning. pp. 25. ISBN 978-1-133-36536-5. "For example, the atomic number of carbon is 6,"
57. Webb, Stephen; Webb, Professor of Australian Studies Stephen (2004-05-25) (in en). Out of this World: Colliding Universes, Branes, Strings, and Other Wild Ideas of Modern Physics. Springer Science & Business Media. pp. 16. ISBN 978-0-387-02930-6. "snowflake, with its familiar sixfold rotational symmetry"
58. Woo, Teri Moser; Robinson, Marylou V. (2015-08-03) (in en). Pharmacotherapeutics For Advanced Practice Nurse Prescribers. F.A. Davis. pp. 145. ISBN 978-0-8036-4581-3. "Ayurvedic herbology is based on the tridoshic theory that there exist six basic tastes"
59. (in en) Pandemic Influenza Preparedness and Response Guidance for Healthcare Workers and Healthcare Employers. OSHA, U.S. Department of Labor. 2007. pp. 8. "The WHO Plan describes six phases of increasing public health risk associated with the emergence of a new influenza"
60. Sanghera, Paul (2011-03-08) (in en). Quantum Physics for Scientists and Technologists: Fundamental Principles and Applications for Biologists, Chemists, Computer Scientists, and Nanotechnologists. John Wiley & Sons. pp. 64. ISBN 978-0-470-92269-9. "...there are six types of quarks and six types of leptons."
61. Jimbo, M.; Jimbo, Michio; Miwa, Tetsuji; Tsuchiya, Akihiro (1989) (in en). Integrable Systems in Quantum Field Theory and Statistical Mechanics. Academic Press. pp. 588. ISBN 978-0-12-385342-4. "Allowed configurations in the six-vertex model and their statistical weights"
62. Sloan, Robin James Stuart (2015-05-07) (in en). Virtual Character Design for Games and Interactive Media. CRC Press. pp. 34. ISBN 978-1-4665-9820-1. "placing six primaries around the wheel in the following order: red, yellow, green, cyan, blue, magenta."
63. Bamford, Tab (2016-10-15) (in en). 100 Things Blackhawks Fans Should Know & Do Before They Die. Triumph Books. ISBN 978-1-63319-638-4. "the Original Six"
64. Stillwell, Jim L. (1987) (in en). Making and Using Creative Play Equipment. Human Kinetics Publishers. pp. 36. ISBN 978-0-87322-084-2. "Indoor Lacrosse . This is played with six players per team"
65. Williams, Heather (2019) (in en). Hockey: A Guide for Players and Fans. Capstone. pp. 16. ISBN 978-1-5435-7458-6. "There are six players per team on the ice at one time."
66. Sports, The National Alliance For Youth (2009-05-11) (in en). Coaching Volleyball For Dummies. John Wiley & Sons. pp. 48. ISBN 978-0-470-53398-7. "In a regulation volleyball match with six players on each side of the court,"
67. "Rugby flanker (#6 & #7): A position specific guide" (in en-US). 2019-02-01.
68. Stevens, E. S. (2002) (in en). Green Plastics: An Introduction to the New Science of Biodegradable Plastics. Princeton University Press. pp. 45. ISBN 978-0-691-04967-0.
69. Bunson, Matthew (2014-05-14) (in en). Encyclopedia of the Roman Empire. Infobase Publishing. pp. 90. ISBN 978-1-4381-1027-1. "Augustus was also originally called Sextilis, the sixth month."
70. Nicolas, Sir Nicholas Harris (1833) (in en). The Chronology of History: Containing Tables, Calculations and Statements, Indispensable for Ascertaining the Dates of Historical Events and of Public and Private Documents from the Earliest Period to the Present Time. Longham, Rees, Orme, Brown, Green, & Longman and John Taylor. pp. 172. "SEXTIDI , or “ Jour de la Révolution , ”"
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