6 (number)

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Cardinal	6 six
Ordinal	6th sixth
Numeral system	senary
Factorization
Divisors	1, 2, 3, 6
Roman numeral	VI
Roman numeral (Unicode)	Ⅵ, ⅵ
Japanese numeral	六
prefixes	hexa-/hex- (from Greek) sexa-/sex- (from Latin)
Binary	110
Octal	6
Duodecimal	6
Hexadecimal	6
Hebrew	ו (Vav)

6 (six) is the natural number following 5 and preceding 7.

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

Contents 1 In mathematics 1.1 In numeral systems 1.2 List of basic calculations 2 Evolution of the glyph 3 In science 3.1 Astronomy 4 In religion 5 In music 6 In sports 7 In technology 8 In television 9 In other fields 10 Historical years 11 References

[edit] In mathematics

Six is the second smallest composite number, its proper divisors being 1, 2 and 3. Since six equals the sum of these proper divisors, six is a 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. Six is also a unitary perfect number, a harmonic divisor number and a highly composite number. The next highly composite number is 12.

5 and 6 form a Ruth-Aaron pair under either definition.

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 1-1 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. 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 one of the four all-Harshad numbers.

A six-sided polygon is a hexagon, one of the three polygons capable of tiling the plane. Figurate numbers representing hexagons (including six) are called hexagonal numbers. 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 binary complement of number nine:

6 = 0110
9 = 1001

[edit] In numeral systems

Base	Numeral system
2	binary	110
3	ternary	20
4	quaternary	12
5	quinary	11
6	senary	10
over 6 (decimal, hexadecimal)

In base 10, 6 is a 1-automorphic number.

[edit] 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	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	3	2	1.5	1.2	1		0.75		0.6			0.5			0.4
			0.5			1			1.5				2			2.5

Exponentiation	1	2	3	4	5	6	7	8	9	10		11	12	13
	6	36	216	1296	7776	46656	279936	1679616	10077696	60466176		362797056	2176782336	13060694016
	1	64	729	4096	15625	46656	117649	262144	531441	1000000		1771561	2985984	4826809

[edit] 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 Brahmin Indians, who wrote it in one stroke like a cursive lowercase e rotated 45 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 6 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.

In fonts with text figures, 6 usually has an ascender, for example, .

[edit] In science

• The atomic number of carbon
• The number of carbon atoms and carbon-carbon bonds in benzene
• The number of known quarks and leptons
• The number of tastes in traditional Indian Medicine called Ayurveda. They are: sweet, sour, salty, bitter, pungent, and astringent. These tastes are used to suggest a diet based on the symptoms of the body

[edit] 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 -2691 March 16 and ended on -1393 May 3. 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 -2642 July 25 and ended on -1091 February 10. 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.
• The Roman numeral VI (usually) stands for the sixth-discovered satellite of a planet or minor planet (e.g. Jupiter VI)

[edit] In religion

• The Day that God created Man
• The number of points on a Star of David
• The number of orders of the Mishnah
• The number of symbolic foods placed on the Passover Seder Plate
• The Jewish holiday of Shavuot starts on the sixth day of the Hebrew month of Sivan

See also 666.

[edit] In music

• The number of strings on a standard guitar
• The number of basic holes or keys on most woodwind instruments (e.g., pennywhistle, clarinet, saxophone, bassoon). These holes or keys are usually not given numbers or letters in the fingering charts
• "Six geese a-laying" were given as a present on the sixth day in the popular Christmas carol, "The Twelve Days of Christmas"
• The number of completed symphonies by Peter Tchaikovsky and Bohuslav Martinů
• The number of Brandenburg Concertos of Johann Sebastian Bach
• The concerti grossi Opus 3, organ concertos Opus 4 and Opus 7 (each) by Georg Frideric Handel
• The number of string quartets of Béla Bartók
• The name of the second album by Mansun released in 1998. It takes its name in part from the main character in the television series The Prisoner, and from A. A. Milne's poetry book, Now We Are Six

[edit] In sports

• In American and Canadian Football, the number of points received for a touchdown
• In rugby union, the position of blindside flanker
• In rugby league, the position of 5/8
• In football/soccer, the number of Arsenal's legendary centre back and captain Tony Adams. The shirt was retired in 2002 on his retirement but has recently been passed on to Philippe Senderos.
• In baseball, six represents the shortstop's position
• In Australian Rules football, six points are received for a goal
• In cricket, a "six" or "sixer" is a shot in which the ball clears the boundary without bouncing, scoring six runs. Also, there are six balls to an over in cricket. See cricket terminology
• Retired number of former baseball players Stan Musial, Al Kaline, Tony Oliva and Steve Garvey.
• The number of goals Arsenal scored at Anfield to knock Liverpool out of the 2006-07 Carling Cup (only one more in numbers than the five times that Liverpool FC have won the European Cup, more than any other British club)
• Until recently 6 was the number borne on the car of NASCAR driver Mark Martin.

[edit] 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. On BlackBerry 8700 series with full keyboard, it is the key for F.
• The "six meter band" in amateur radio includes the frequencies from 50 to 54 MHz

[edit] In television

• The audio component of any broadcast channel 6 is located at 87.7 MHz. This means that people can listen to TV stations such as WSYX using an FM radio when a television is not nearby
• Number Six was the main character in the television series The Prisoner
• Number Six is a character in the television mini-series and series Battlestar Galactica (re-imagining)
• Six is the name of a character on Blossom
• On the show Sesame Street, six is Bert's favorite number.
• There are six friends on the television show "Friends"
• Six (along with twenty-three and eight hundred and 1) is a recurring number on the revived series of Doctor Who.

[edit] In other fields

• 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
• The standard term in office for a United States senator is six years
• Mark Martin's NASCAR racing number. 6 was also the number of David Pearson's car when he won his Nextel Cup Series Championship in 1966
• The name of the smallest group of Cub Scouts, traditionally consisting of six people and is led by a 'sixer'. Logically speaking, this isn't always the case, particularly in packs with less than 6 Cub Scouts in it
• The number of cans of soda or beer in a six-pack
• 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
• The number of inhabited continents, and the total number of continents if Eurasia (Europe plus Asia) is considered a single continent
• The number of large geyser fields in the world

• In Astrology, Virgo is the 6th astrological sign of the Zodiac
• The number of sides on a cube, hence the highest number on a standard dice
• The highest number on one end of a standard domino
• The number of dots in a Braille cell.
• Les Six were a group of Lost Generation writers
• Extra-sensory perception is sometimes called the "sixth sense"
• Bands with the number six in their name include Eve 6, Slant 6, Sixpence None the Richer, Los Xey (sei is Basque for "six"), Vanity 6, and Electric Six
• Six Cardinal Directions: north, south, east, west, heaven, and land
• The Six Dynasties form part of Chinese history
• 6 is a lucky number in Chinese culture
• The Bionic Six are the heroes of an animated series
• The Birmingham Six were held in prison for 16 years
• Six Degrees of Kevin Bacon is a game that can be played with the names of actors and actresses
• 666 is considered the Number of the Beast
• There are 6 balls to an Over in Cricket
• In tarot, trump card No. 6 is "The Lovers".
• "Six" is used as an informal slang term for the British Secret Intelligence Service, MI6
• Six Flags amusement parks and theme parks
• the number of Australian states

Hexa is Greek for "six". Thus:

• A hexahedron is a polyhedron with six faces, with a cube being a special case
• An hexagon is a regular polygon with six sides
  • L' Hexagone is a French nickname for the continental part of France

• A hexapod is an animal with six legs; this includes all insects
• Hexameter is a poetic form consisting of six feet per line
• "Hexadecimal" combines hexa- with the Latinate decimal to name a number base of 16
• A "hex nut", of course, 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"

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

• A group of six musicians is called a sextet
• Six babies delivered in one birth are sextuplets. The first set of sextuplets of whom all six survived are the Dilley sextuplets
• 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
• The ordinal adjective senary

• Number-6 is also a French Hip-Hop production label : http://myspace.com/n6records

[edit] Historical years

A.D. 6, 6 B.C., 1906, 2006, etc. The 06/06/06 is the day of the devil in some religions

[edit] 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
• Some Thoughts on the Number Six, John Baez