# 6 (number)

6
Cardinal 6
six
Ordinal 6th
sixth
Numeral system senary
Factorization $2 \cdot 3$
Divisors 1, 2, 3, 6
Roman numeral VI
Roman numeral (Unicode) Ⅵ, ⅵ, ↅ
Greek στ (or ΣΤ or ς)
Arabic ٦,6
Arabic (Persian, Urdu) ۶
Amharic
Bengali
Chinese numeral 六，陆
Devanāgarī
Hebrew ו (Vav)
Khmer
Thai
Telugu
Tamil
prefixes hexa-/hex- (from Greek)

sexa-/sex- (from Latin)

Binary 110
Octal 6
Duodecimal 6

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

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 the smallest perfect number, Granville number, and $\mathcal{S}$-perfect number.[1][2] As a perfect number, 6 is related to the Mersenne prime 3, since 21(22 - 1) = 6. (The next perfect number is 28.) It is the only even perfect number that is not the sum of successive odd cubes.[3] Being perfect, six is the root of the 6-aliquot tree, and is itself the aliquot sum of only one number; the square number, 25. Unrelated to 6 being a perfect number, a Golomb ruler of length 6 is a "perfect ruler."[4] 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 only number that is both the sum and the product of three consecutive positive numbers.[5]

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.

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 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).

Six is the smallest number of distinct isosceles right triangles that will tile an isosceles right triangle.

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

Base Numeral system
2 binary 110
3 ternary 20
4 quaternary 12
5 quinary 11
6 senary 10

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
$6 \times x$ 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
$6 \div x$ 6 3 2 1.5 1.2 1 $0.\overline{85714}\overline{2}$ 0.75 $0.\overline{6}$ 0.6 $0.\overline{5}\overline{4}$ 0.5 $0.\overline{46153}\overline{8}$ $0.\overline{42857}\overline{1}$ 0.4
$x \div 6$ $0.1\overline{6}$ $0.\overline{3}$ 0.5 $0.\overline{6}$ $0.8\overline{3}$ 1 $1.1\overline{6}$ $1.\overline{3}$ 1.5 $1.\overline{6}$ $1.8\overline{3}$ 2 $2.1\overline{6}$ $2.\overline{3}$ 2.5
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
$6 ^ x\,$ 6 36 216 1296 7776 46656 279936 1679616 10077696 60466176 362797056 2176782336 13060694016
$x ^ 6\,$ 1 64 729 4096 15625 46656 117649 262144 531441 1000000 1771561 2985984 4826809

## 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.[6]

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 science

### 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 quark and 6 types of lepton
• In statistical mechanics, the six-vertex model has 6 possible configurations of arrows at each vertex

### Chemistry

The cells of a beehive honeycomb are 6-sided

### Medicine

• 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
• Phase 6 is one of six pandemic influenza phases

## In music

A standard guitar has 6 strings
• 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 concerti grossi Opus 3, organ concertos Opus 4 and Opus 7 (each) by Georg Frideric Handel
• 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
• The French composers Georges Auric, Louis Durey, Arthur Honegger, Darius Milhaud, Francis Poulenc and Germaine Tailleferre were part of a group known as Les Six ("The Six" in English) in the 1920s.
• Bands with the number six in their name include 6 O'clock Saints, Eve 6, Slant 6, Sixpence None the Richer, Six In Six, Los Xey (sei is Basque for "six"), Vanity 6, You Me At Six, Out On Blue Six and Electric Six
• The number of semitones in a tritone
• The number that the sixth album by Dream Theater "Six Degrees Of Inner Turbulence" was based around. The album has six songs. The sixth song — that is, the complete second disc — explores the stories of six individuals suffering from various mental illnesses.
• #6, the pseudonym of American musician Shawn Crahan, when performing with Slipknot

## In technology

• The resin identification code used in recycling to identify polystyrene.
• 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 and Curve 8900 with full keyboard, it is the key for F.
• The "six meter band" in amateur radio includes the frequencies from 50 to 54 MHz

## 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
• The name of the smallest group of Cub Scouts, traditionally consisting of six people and is led by a 'sixer'.
• Sixth Officer Hantin in The Pendragon Series
• 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

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
• 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" 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
• 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

## 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. Retrieved 27 March 2011.
3. ^ David Wells, The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin Books (1987): 67
4. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 72
5. ^ Peter Higgins, Number Story. London: Copernicus Books (2008): 12
6. ^ 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

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