# 6

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 ← 5 6 7 →
Cardinal six
Ordinal 6th
(sixth)
Numeral system senary
Factorization 2 × 3
Divisors 1, 2, 3, 6
Greek numeral Ϛ´
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
Vigesimal 620
Base 36 636
Greek στ (or ΣΤ or ς)
Arabic & Kurdish ٦
Persian ۶
Urdu
Amharic
Bengali
Chinese numeral 六，陆
Devanāgarī
Hebrew ו
Khmer
Thai
Telugu
Tamil
Saraiki ٦

6 (six /sɪks/) 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 ${\displaystyle {\mathcal {S}}}$-perfect number.[1][2]

As a perfect number:

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.[6]

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,[7] a harmonic divisor number[8] 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.

There are no Graeco-Latin squares with order 6. 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 p such that the multiplicative order of 2 modulo p is 6, that is, ordp(2) = 6. By Zsigmondy's theorem, if n is a natural number that is not 1 or 6, then there is a prime p such that ordp(2) = n. See A112927 for such 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.

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. 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.[9] 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.

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.

### 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". 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
• 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.[10]

## 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.[11]

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 instruments

• A standard guitar has six strings
• 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.
• There are six 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 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.
• Aristotle gave six elements of tragedy, the first of which is Mythos.

## In science

### Biology

The cells of a beehive are 6-sided

### Medicine

• There are six 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 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. 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 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, six players from each team on each side play against each other
• 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
• Also in volleyball, 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.)
• Scoring:
• In both American and Canadian football, 6 points are awarded for a touchdown.
• 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 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.)

## In the arts and entertainment

### Games

• The number of sides on a cube, hence the highest number on a standard die
• 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, an 1966 animated cartoon series featuring six different super-powered heroes.
• The Bionic Six are the heroes of the eponymous animated series
• Sinister Six is a group of super villains who appear in American comic books published by Marvel Comics

## In other fields

• Six Flags is an American company running amusement parks and theme parks in the U.S., Canada, and Mexico.
• 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".[12] (For a similar example see also: Rainbow Six.)

## References

1. ^ Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 11. ISBN 978-1-84800-000-1.
2. ^ "Granville number". OeisWiki. The Online Encyclopedia of Integer Sequences. Archived from the original on 29 March 2011. Retrieved 27 March 2011.
3. ^ David Wells, The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin Books (1987): 67
4. ^ Peter Higgins, Number Story. London: Copernicus Books (2008): 12
5. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 72
6. ^ "Sloane's A003273 : Congruent numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
7. ^ "Sloane's A002827 : Unitary perfect numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
8. ^ "Sloane's A001599 : Harmonic or Ore numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
9. ^ "Sloane's A005900 : Octahedral numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
10. ^ Chris K. Caldwell; G. L. Honaker Jr. (2009). Prime Curios!: The Dictionary of Prime Number Trivia. CreateSpace Independent Publishing Platform. p. 11. ISBN 978-1448651702.
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
12. ^ Mason, Robert (1983). Chickenhawk. London: Corgi Books. p. 141. ISBN 978-0-552-12419-5.
• 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