|This article may contain excessive, poor, or irrelevant examples. (March 2010) |
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).
6 is the smallest number 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 -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 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.
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.
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. 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 = 0110 9 = 1001
The six exponentials theorem guarantees (given the right conditions on the exponents) the transcendence of at least one of a set of exponentials.
In numeral systems
In bases 10, 15 and 30, 6 is a 1-automorphic number.
List of basic calculations
Greek and Latin word parts
Hexa is classical Greek for "six". Thus:
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 (see above photo)
- 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
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.
- 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
- 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 semitones in a tritone
- "Six geese a-laying" were given as a present on the sixth day in the popular Christmas carol, "The Twelve Days of Christmas"
- The concerti grossi Opus 3, organ concertos Opus 4 and Opus 7 (each) by Georg Frideric Handel
- The sixth album by Dream Theater, Six Degrees Of Inner Turbulence, was based around 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
See also 666.
- 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
- 6 entities in a grouping:
- 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:
- Jersey number 6 and car number 6:
- In football/soccer:
- Is the number of AC Milan's legendary centre back and captain Franco Baresi; the shirt was retired in 1997
- Is 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 NASCAR:
- In the NBA, number 6 is worn by several players, most notably Julius Erving.
- In rugby:
- 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.)
- 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 stand-off half (Southern Hemisphere term) or five-eighth (Northern Hemisphere term).
- 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 the arts and entertainment
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'.
- 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
- 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
- ^ Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 11. ISBN 978-1-84800-000-1.
- ^ "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
- ^ 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 0-521-42385-6
- Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 67 - 69