6 (number)

This article is about the mathematical number. For other uses, see 6 (disambiguation).

← 5 6 7 →
−1 0 1 2 3 4 5 6 7 8 9 → List of numbers — Integers ← 0 10 20 30 40 50 60 70 80 90 →
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	1102
Ternary	203
Quaternary	124
Quinary	115
Senary	106
Octal	68
Duodecimal	612
Hexadecimal	616
Vigesimal	620
Base 36	636
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 /ˈsɪks/) is the natural number following 5 and preceding 7.

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

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.

As a perfect number:

• 6 is related to the Mersenne prime 3, since 2¹(2² − 1) = 6. (The next perfect number is 28.)
• 6 is the only even perfect number that is not the sum of successive odd cubes.
• As a perfect number, 6 is the root of the 6-aliquot 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.

Unrelated to 6 being a perfect number, a Golomb ruler of length 6 is a "perfect ruler." 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 Ruth-Aaron pair under either definition.

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

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

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.

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 two-dimensional 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.

A cube has 6 faces

6 is the largest of the four all-Harshad numbers.

A six-sided 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 2¹) 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 four-bit binary complement of number nine (6 = 0110₂ and 9 = 1001₂).

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 1-automorphic 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 × 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 ÷ 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

X-ray of a polydactyl human hand with six fingers

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 six-sided 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 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.

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.

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 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

A standard guitar has 6 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
• Bands with the number six in their name include Six Organs of Admittance, 6 O'Clock Saints, 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 a-laying" 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 sixth-discovered satellite of a planet or minor planet (e.g. Jupiter VI)

Biology

• The cells of a beehive are 6-sided
• Insects have 6 legs

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

The cells of a beehive are 6-sided

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 quark and 6 types of lepton

• In the Standard Model of particle physics, there are 6 types of quarks and 6 types of leptons
• In statistical mechanics, the six-vertex 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 2013-14 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:
  • 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
  • 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 double-hat-trick, 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):
    • a "six" or "sixer" is a shot in which the ball clears the boundary without bouncing, scoring six runs
    • there are six balls to an over
• 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 five-eighth (Southern Hemisphere term) or stand-off 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 career-ending 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 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 "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

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 super-powered 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

International maritime signal flag for 6

• 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 six-pack.
• 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 (disambiguation).
• Extra-sensory 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

• 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 0-521-42385-6
• 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