# 10

 ← 9 10 11 →
Cardinalten
Ordinal10th
(tenth)
Numeral systemdecimal
Factorization2 × 5
Divisors1, 2, 5, 10
Greek numeralΙ´
Roman numeralX
Roman numeral (unicode)X, x
Greek prefixdeca-/deka-
Latin prefixdeci-
Binary10102
Ternary1013
Senary146
Octal128
DuodecimalA12
Chinese numeral十，拾
Hebrewי (Yod)
Khmer១០
Tamil
Thai๑๐
Devanāgarī१०
Bengali১০
Arabic & Kurdish١٠
Malayalam

10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, by far the most common system of denoting numbers in both spoken and written language. It is the first double-digit number. The reason for the choice of ten is assumed to be that humans have ten fingers (digits).

## Anthropology

### Usage and terms

• A collection of ten items (most often ten years) is called a decade.
• Increasing a quantity by one order of magnitude is most widely understood to mean multiplying the quantity by ten.
• To reduce something by one tenth is to decimate. (In ancient Rome, the killing of one in ten soldiers in a cohort was the punishment for cowardice or mutiny; or, one-tenth of the able-bodied men in a village as a form of retribution, thus causing a labor shortage and threat of starvation in agrarian societies.)

## In mathematics

Ten is the fifth composite number. Ten is the smallest noncototient, a number that cannot be expressed as the difference between any integer and the total number of coprimes below it.[1] It is the second discrete semiprime (${\displaystyle 2\times 5}$) and the second member of the (${\displaystyle 2\times q}$) discrete semiprime family. Ten has an aliquot sum σ(10) of 8 and is accordingly the first discrete semiprime to be in deficit, with all subsequent discrete semiprimes in deficit.[2] Ten is the smallest number that can be written as the sum of two prime numbers in two different ways (${\displaystyle 3+7=5+5=10}$), and the only number whose sum and difference of its prime divisors yield prime numbers (${\displaystyle 2+5=7}$ and ${\displaystyle 5-2=3}$). It is also the smallest semiprime that is the sum of all the distinct prime numbers from its lower factor through its higher factor (${\displaystyle 10=2+3+5=2\times 5}$).[3] In general, powers of 10 contain ${\displaystyle n^{2}}$ divisors, where ${\displaystyle n}$ is the number of digits: 10 has 22 = 4 divisors, 100 has 32 = 9 divisors, 1,000 has 42 = 16 divisors, 10,000 has 52 = 25 divisors, and so forth. Ten is the eighth Perrin number, preceded in the sequence by (5, 5, 7).[4]

According to conjecture, ten is the average sum of the proper divisors of the natural numbers ${\displaystyle \mathbb {N} }$ if the size of the numbers approaches infinity.[5]

As important sums,

• ${\displaystyle 10=2+3+5}$, the sum of the first three prime numbers.
• ${\displaystyle 10=1+2+3+4}$, the sum of the first four positive integers.
• ${\displaystyle 10=0!+1!+2!+3!}$, the sum of the first four factorials.
• ${\displaystyle 10=1^{2}+3^{2}}$, the sum of the squares of the first two odd numbers.

The factorial of ten is equal to the product of the factorials of the first three odd primes, ${\displaystyle 10!=3!\cdot 5!\cdot 7!}$.[6] Ten is also the first number whose fourth power can be written as a sum of two squares in two different ways (${\displaystyle 80^{2}+60^{2}}$ and ${\displaystyle 96^{2}+28^{2}}$).

In the sequence of triangular numbers, indexed powers of 10 in this sequence generate the following sequence of triangular numbers: 55 (10th), 5,050 (100th), 500,500 (1,000th), ...[7][a]

While 55 is the tenth triangular number, it is also the tenth Fibonacci number, and the largest such number to also be a triangular number.[8]

Ten is the first non-trivial decagonal number, the third centered triangular number and tetrahedral number, and the fifth semi-meandric number.[9][10][11][12]

A ${\displaystyle 10\times 10}$ magic square has a magic constant of 505.[13]

The aliquot sequence for 10 comprises five members (10, 8, 7, 1, 0) with this number being the second composite member of the 7-aliquot tree.[14]

It is also the number of ${\displaystyle n}$-queens problem solutions for ${\displaystyle n=5}$.[15]

Ten is the smallest number whose status as a possible friendly number is unknown.[16]

There are ten small Pisot numbers that do not exceed the golden ratio.[17]

### In geometry

A polygon with ten sides is called a decagon. As a constructible polygon with a compass and straight-edge, it has an internal angle of ${\displaystyle 12^{2}=144}$ degrees and a central angle of ${\displaystyle 6^{2}=36}$ degrees. All regular ${\displaystyle n}$-sided polygons with up to ten sides are able to tile a plane-vertex alongside other regular polygons alone; the first regular polygon unable to do so is the eleven-sided hendecagon.[18] A decagon can fill a plane-vertex alongside two regular pentagons, and alongside a fifteen-sided pentadecagon and triangle. Ten of the eleven regular and semiregular (or Archimedean) tilings of the plane are Wythoffian, the elongated triangular tiling is the only exception.[19]

The regular decagon is the Petrie polygon of the regular dodecahedron and icosahedron, and it is the largest face that an Archimedean solid can contain, as with the truncated dodecahedron and the truncated icosidodecahedron. The decagon is the hemi-face of the icosidodecahedron, such that a plane dissection yields two mirrored pentagonal rotundae. A regular ten-pointed {10/3} decagram is the hemi-face of the great icosidodecahedron, as well as the Petrie polygon of two regular Kepler-Poinsot polyhedra. Ten non-prismatic uniform polyhedra contain regular decagons as faces (U26, U28, U33, U37, U39, ...), and ten contain regular decagrams as faces (U42, U45, U58, U59, U63...). The decagonal prism is also the largest prism that is a facet inside four-dimensional uniform polychora.

There are ten regular star polychora in the fourth dimension.[20] All of these polychora have orthographic projections in the ${\displaystyle \mathrm {H} _{3}}$ Coxeter plane that contain various decagrammic symmetries, which include the regular {10/3} form as well as its three alternate compound forms.

${\displaystyle \mathrm {M} _{10}}$ is a multiply transitive permutation group on 10 points. It is an almost simple group, of order 720 = 24·32·5 = 2·3·4·5·6 = 8·9·10. It functions as a point stabilizer of degree 11 inside the smallest sporadic group ${\displaystyle \mathrm {M} _{11}}$, a Mathieu group which has an irreducible faithful complex representation in 10 dimensions.

${\displaystyle \mathrm {E} _{10}}$ is an infinite-dimensional Kac–Moody algebra which has the even Lorentzian unimodular lattice II9,1 of dimension 10 as its root lattice. It is the first ${\displaystyle \mathrm {E} _{n}}$ Lie algebra with a negative Cartan matrix determinant, of −1.

There are precisely ten affine Coxeter groups that admit a formal description of reflections across ${\displaystyle n}$ dimensions in Euclidean space. These contain infinite facets whose quotient group of their normal abelian subgroups is finite. They include the one-dimensional Coxeter group ${\displaystyle {\tilde {I}}_{1}}$ [], which represents the apeirogonal tiling, as well as the five affine Coxeter groups ${\displaystyle {\tilde {G}}_{2}}$, ${\displaystyle {\tilde {F}}_{4}}$, ${\displaystyle {\tilde {E}}_{6}}$, ${\displaystyle {\tilde {E}}_{7}}$, and ${\displaystyle {\tilde {E}}_{8}}$ that are associated with the five exceptional Lie algebras. They also include the four general affine Coxeter groups ${\displaystyle {\tilde {A}}_{n}}$, ${\displaystyle {\tilde {B}}_{n}}$, ${\displaystyle {\tilde {C}}_{n}}$, and ${\displaystyle {\tilde {D}}_{n}}$ that are associated with simplex, cubic and demihypercubic honeycombs, or tessellations. Regarding Coxeter groups in hyperbolic space, there are infinitely many such groups; however, ten is the highest rank for paracompact hyperbolic solutions, with a representation in nine dimensions. There also exist hyperbolic Lorentzian cocompact groups where removing any permutation of two nodes in its Coxeter-Dynkin diagram leaves a finite or Euclidean graph. The tenth dimension is the highest dimensional representation for such solutions, which share a root symmetry in eleven dimensions. These are of particular interest in M-theory of string theory.

## 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 26 50 100 1000
10 × x 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 500 1000 10000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
10 ÷ x 10 5 3.3 2.5 2 1.6 1.428571 1.25 1.1 1 0.90 0.83 0.769230 0.714285 0.6
x ÷ 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.1 1.2 1.3 1.4 1.5
Exponentiation 1 2 3 4 5 6 7 8 9 10
10x 10 100 1000 10000 100000 1000000 10000000 100000000 1000000000 10000000000
x10 1 1024 59049 1048576 9765625 60466176 282475249 1073741824 3486784401

## In science

The SI prefix for 10 is "deca-".

The meaning "10" is part of the following terms:

• decapoda, an order of crustaceans with ten feet.
• decane, a hydrocarbon with 10 carbon atoms.

Also, the number 10 plays a role in the following:

The metric system is based on the number 10, so converting units is done by adding or removing zeros (e.g. 1 centimeter = 10 millimeters, 1 decimeter = 10 centimeters, 1 meter = 100 centimeters, 1 dekameter = 10 meters, 1 kilometer = 1,000 meters).

## In money

Most countries issue coins and bills with a denomination of 10 (See e.g. 10 dollar note). Of these, the U.S. dime, with the value of ten cents, or one tenth of a dollar, derives its name from the meaning "one-tenth" − see Dime (United States coin)#Denomination history and etymology.

## In sports and games

• Decathlon is a combined event in athletics consisting of ten track and field events.
• In association football, the number 10 is traditionally worn by the team's advanced playmaker. This use has led to "Number 10" becoming a synonym for the player in that particular role, even if they do not wear that number.[24]
• In gridiron football, a team has a limited number of downs to advance the ball ten yards or more from where it was on its last first down; doing this is referred to as gaining another first down.
• In auto racing, driving a car at ten-tenths is driving as fast as possible, on the limit.
• In a regular basketball game, two teams playing against each other have 5 members each, for a total of 10 players on court. Under FIBA, WNBA, and NCAA women's rules, each quarter runs for 10 minutes.
• In blackjack, the Ten, Jack, Queen and King are all worth 10 points.
• In boxing, if the referee counts to 10 whether the boxer is unconscious or not, it will declare a winner by knockout.
• In men's field lacrosse, each team has 10 players on the field at any given time, except in penalty situations.
• Ten-ball is a pool game played with a cue ball and ten numbered balls.
• In most rugby league competitions, the number 10 is worn by one of the two starting props. One exception to this rule is the Super League, which uses static squad numbering.
• In rugby union, the starting fly-half wears the 10 shirt.
• In ten-pin bowling, 10 pins are arranged in a triangular pattern and there are 10 frames per game.

## In technology

• Ten-codes are commonly used on emergency service radio systems.
• Ten refers to the "meter band" on the radio spectrum between 28 and 29.7 MHz, used by amateur radio.
• ASCII and Unicode code point for line feed.
• In MIDI, Channel 10 is reserved for unpitched percussion instruments.
• In the Rich Text Format specification, all language codes for regional variants of the Spanish language are congruent to 10 mod 256.
• In macOS, the F10 function key tiles all the windows of the current application and grays the windows of other applications.
• The IP addresses in the range 10.0.0.0/8 (meaning the interval between 10.0.0.0 and 10.255.255.255) are reserved for use by private networks by RFC 1918.

## Age 10

• This is generally the age when a child enters the preteen stage and also a denarian (someone within the age range of 10–19).
• The ESRB recommends video games with an E10+ rating to children aged 10 and up.

## In other fields

10 playing cards of all four suits

## References

1. ^ 19 is another number that is the first member of a sequence displaying a similar uniform property. The 19th triangular number is 190, the 199th triangular number is 19900, ... (sequence A186076 in the OEIS)
1. ^ "Sloane's A005278 : Noncototients". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
2. ^ Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
3. ^ Sloane, N. J. A. (ed.). "Sequence A055233 (Composite numbers equal to the sum of the primes from their smallest prime factor to their largest prime factor.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
4. ^ Sloane, N. J. A. (ed.). "Sequence A001608 (Perrin sequence (or Ondrej Such sequence))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
5. ^ Sloane, N. J. A. (ed.). "Sequence A297575 (Numbers whose sum of divisors is divisible by 10.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
6. ^ "10". PrimeCurios!. PrimePages. Retrieved 2023-01-14.
7. ^ Sloane, N. J. A. (ed.). "Sequence A037156". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
For n = 0; a(0) = 1 = 1 * 1 = 1
For n = 1; a(1) = 1 + 2 + ...... + 10 = 11 * 5 = 55
For n = 2; a(2) = 1 + 2 + .... + 100 = 101 * 50 = 5050
For n = 3; a(3) = 1 + 2 + .. + 1000 = 1001 * 500 = 500500
...
8. ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
9. ^ "Sloane's A001107 : 10-gonal (or decagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
10. ^ "Sloane's A005448 : Centered triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
11. ^ "Sloane's A000292 : Tetrahedral numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
12. ^ Sloane, N. J. A. (ed.). "Sequence A000682 (Semi-meanders: number of ways a semi-infinite directed curve can cross a straight line n times.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
13. ^ Andrews, W.S. (1917). Magic Squares and Cubes (2nd ed.). Open Court Publishing. p. 30.
14. ^ Sloane, N. J. A. "Aliquot sequences". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
15. ^ Sloane, N. J. A. (ed.). "Sequence A000170 (Number of ways of placing n nonattacking queens on an n X n board.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
16. ^ Sloane, N. J. A. (ed.). "Sequence A074902 (Known friendly numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
17. ^ M.J. Bertin; A. Decomps-Guilloux; M. Grandet-Hugot; M. Pathiaux-Delefosse; J.P. Schreiber (1992). Pisot and Salem Numbers. Birkhäuser. ISBN 3-7643-2648-4.
18. ^ Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine. Taylor & Francis, Ltd. 50 (5): 230, 231. doi:10.2307/2689529. JSTOR 2689529. S2CID 123776612. Zbl 0385.51006.
19. ^ Grünbaum, Branko; Shephard, G. C. (1987). "Section 2.1: Regular and uniform tilings". Tilings and Patterns. New York: W. H. Freeman and Company. p. 64. ISBN 0-7167-1193-1. OCLC 13092426. S2CID 119730123.
20. ^ Coxeter, H. S. M (1948). "Chapter 14: Star-polytopes". Regular Polytopes. London: Methuen & Co. LTD. p. 263.
21. ^
22. ^
23. ^ [1] Archived February 23, 2006, at the Wayback Machine
24. ^ Khalil Garriot (21 June 2014). "Mystery solved: Why do the best soccer players wear No. 10?". Yahoo. Retrieved 19 May 2015.