# 6 (number)

 Cardinal 6 six Ordinal 6th sixth Numeral system senary Factorization Divisors 1, 2, 3, 6 Roman numeral VI Roman numeral (Unicode) Ⅵ, ? Arabic ٦ Arabic (Urdu) ۶ Bengali ৬ Chinese numeral ? Devanāgarī ६ Hebrew ו (Vav) Khmer ? prefixes hexa-/hex- (from Greek) sexa-/sex- (from Latin) Binary 110 Octal 6 Duodecimal 6 Hexadecimal 6

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).

## 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 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```

### In numeral systems

Base Numeral system
2binary110
3ternary20
4quaternary12
5quinary11
6senary10

In base 10, 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
61218243036424854606672788490961021081141201261321381441503006006000

Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
6321.51.210.750.60.50.4
0.511.522.5

Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
636216129677764665627993616796161007769660466176362797056217678233613060694016
164729409615625466561176492621445314411000000177156129859844826809

## 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.[1]

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 .

## 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

## 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

## In other fields

Hexa is Greek for "six". Thus:
All insects have six legs.
• 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

## Historical years

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

## References

1. ^ 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

Number Six is the name used by two notable fictional characters:
• Number Six (The Prisoner), lead character from the television series The Prisoner.

0 (zero) is both a number and a numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures.

0 1 2 3 4 5 6 7 8 9

2 (two) is a number, numeral, and glyph. It is the natural number following 1 and preceding 3.

## In mathematics

Two has many properties in mathematics.[1] An integer is called even if it is divisible by 2.
3 (three) is a number, numeral, and glyph. It is the natural number following 2 and preceding 4.

## In mathematics

Three is the first odd prime number, and the second smallest positive prime.
This article discusses the number Four. For the year 4 AD, see 4. For other uses of 4, see 4 (disambiguation)

0 1 2 3 4 5 6 7 8 9

This article discusses the number five. For the year 5 AD, see 5. For other uses of 5, see 5 (disambiguation).

0 1 2 3 4 5 6 7 8 9

7 (seven) is the natural number following 6 and preceding 8.

## In mathematics

• Seven is the 4th prime number.
• Seven is a Mersenne prime, since 2³ - 1 = 7.
• Seven is the fourth Mersenne prime exponent.

8 (eight) is the natural number following 7 and preceding 9. The SI prefix for 10008 is yotta (Y), and for its reciprocal yocto (y).

## In mathematics

8 is a composite number, its proper divisors being 1, 2, and 4.
9 (nine) is the natural number following 8 and preceding 10.

## In mathematics

Nine is a composite number, its proper divisors being 1 and 3. It is 3 times 3 and hence the third square number. 9 is a Motzkin number. It is the first composite lucky number.
10 (ten) is an even natural number following 9 and preceding 11.

0

## Rational numbers

### Notable rational numbers

#### Natural numbers

0 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 26 27 28 29
30 31 32 33 34 35 36 37 38 39
The integers (from the Latin integer, which means with untouched integrity, whole, entire) are the set of numbers including the whole numbers (0, 1, 2, 3, …) and their negatives (0, −1, −2, −3, …).
0 (zero) is both a number and a numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures.
10 (ten) is an even natural number following 9 and preceding 11.

0

30 (thirty) is the natural number following 29 and preceding 31.

## In mathematics

It is a primorial as well as the sum of the squares of the integers 1, 2, 3 and 4. It is the smallest Giuga number.
40 (forty) is the natural number following 39 and preceding 41.

Despite being related to the word "four" (4), 40 is spelled as "forty", not "fourty". The letters of the word "forty" are in alphabetical order; this is the only number that has this linguistic property
50 (fifty) is the integer following 49 and preceding 51.
(the number fifty derives from the two numerical figures of both 0 and 5 placed in order from greatest to least)

## In mathematics

Fifty
60 (sixty) is the natural number following 59 and preceding 61. Being thrice twenty, 60 is called "three score" in some older literature.

## In mathematics

Sixty
70 (seventy) is the natural number following 69 and preceding 71. The French do not have a word for 70, instead using "soixante-dix" (60 + 10). Other French-speaking countries such as Belgium and Switzerland do have a word for it, using "septante.
80 (eighty) is the natural number following 79 and preceding 81.

## In mathematics

The sum of Euler's totient function φ(x) over the first sixteen integers is 80.

Adding up some subsets of its divisors (e.g.
90 (ninety) is the natural number preceded by 89 and followed by 91.

## In mathematics

90 is the sum of the squares of the integers 2 to 6. Because 90 is the sum of its unitary divisors (excluding itself), it is a unitary perfect number, and because it is equal
100 (one hundred) (the Roman numeral is C for centum) is the natural number following 99 and preceding 101.

## In mathematics

One hundred is the square of 10 (in scientific notation it is written as ).
cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. For finite sets the cardinality is given by a natural number, being simply the number of elements in the set.
ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them.
numeral system (or system of numeration) is a framework where a set of numbers are represented by numerals in a consistent manner. It can be seen as the context that allows the numeral "11" to be interpreted as the binary numeral for three