6 (number)

From Wikipedia, the free encyclopedia - View original article

567
−1 0 1 2 3 4 5 6 7 8 9
Cardinalsix
Ordinal6th
(sixth)
Factorization2 · 3
Divisors1, 2, 3, 6
Roman numeralVI
Roman numeral (unicode)Ⅵ, ⅵ, ↅ
Greek prefixhexa-/hex-
Latin prefixsexa-/sex-
Binary1102
Ternary203
Quaternary124
Quinary115
Senary106
Octal68
Duodecimal612
Hexadecimal616
Vigesimal620
Base 36636
Greekστ (or ΣΤ or ς)
Arabic٦
Persian۶
Urdu۶
Amharic
Bengali
Chinese numeral六,陆
Devanāgarī
Hebrewו (Vav)
Khmer
Thai
Telugu
Tamil
 
Jump to: navigation, search
567
−1 0 1 2 3 4 5 6 7 8 9
Cardinalsix
Ordinal6th
(sixth)
Factorization2 · 3
Divisors1, 2, 3, 6
Roman numeralVI
Roman numeral (unicode)Ⅵ, ⅵ, ↅ
Greek prefixhexa-/hex-
Latin prefixsexa-/sex-
Binary1102
Ternary203
Quaternary124
Quinary115
Senary106
Octal68
Duodecimal612
Hexadecimal616
Vigesimal620
Base 36636
Greekστ (or ΣΤ or ς)
Arabic٦
Persian۶
Urdu۶
Amharic
Bengali
Chinese numeral六,陆
Devanāgarī
Hebrewו (Vav)
Khmer
Thai
Telugu
Tamil

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[edit]

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.[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.

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.

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. 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[edit]

BaseNumeral systemRepresentation
2binary110
3ternary20
4quaternary12
5quinary11
6senary10
over 6 (decimal, hexadecimal)6

In bases 10, 15 and 30, 6 is a 1-automorphic number.

List of basic calculations[edit]

Multiplication12345678910111213141516171819202122232425501001000
6 \times x61218243036424854606672788490961021081141201261321381441503006006000
Division123456789101112131415
6 \div x6321.51.210.\overline{85714}\overline{2}0.750.\overline{6}0.60.\overline{5}\overline{4}0.50.\overline{46153}\overline{8}0.\overline{42857}\overline{1}0.4
x \div 60.1\overline{6}0.\overline{3}0.50.\overline{6}0.8\overline{3}11.1\overline{6}1.\overline{3}1.51.\overline{6}1.8\overline{3}22.1\overline{6}2.\overline{3}2.5
Exponentiation12345678910111213
6 ^ x\,636216129677764665627993616796161007769660466176362797056217678233613060694016
x ^ 6\,164729409615625466561176492621445314411000000177156129859844826809

Greek and Latin word parts[edit]

Hexa[edit]

Hexa is classical Greek for "six". Thus:

The prefix sex-[edit]

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

Evolution of the glyph[edit]

Evolution6glyph.png

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 Text figures 036.svg.

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[edit]

A standard guitar has 6 strings

In artists[edit]

In instruments[edit]

In music theory[edit]

In works[edit]

In religion[edit]

Black Star of David.svg

See also 666.

In science[edit]

Astronomy[edit]

Biology[edit]

Chemistry[edit]

A molecule of benzene has a ring of 6 carbon atoms
The cells of a beehive are 6-sided

Medicine[edit]

Physics[edit]

In the Standard Model of particle physics, there are 6 types of quark and 6 types of lepton

In sports[edit]

In technology[edit]

6 as a resin identification code, used in recycling.

In calendars[edit]

In the arts and entertainment[edit]

In other fields[edit]

References[edit]

  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. ^ 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

External links[edit]