# 10

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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 & Kurdish & Iranian١٠
Malayalam

10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, the most common system of denoting numbers in both spoken and written language.

## 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. It is also 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 (${\displaystyle 2\times 5}$), as well as the second member of the ${\displaystyle 2\times q}$ discrete semiprime family. Ten is the only number whose sum and difference of its prime divisors yield prime numbers (${\displaystyle 2+5=7}$ and ${\displaystyle 5-2=3}$). 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 smallest number whose status as a possible friendly number is unknown.[2]

As important sums,

• ${\displaystyle 10=1+2+3+4}$ the sum of the first four positive integers.
• ${\displaystyle 10=2+3+5=2\times 5}$, the sum of the first three prime numbers, and the smallest semiprime that is the sum of all the distinct prime numbers from its lower factor through its higher factor.[3]
• ${\displaystyle 10=3+7=5+5}$, the smallest number that can be written as the sum of two prime numbers in two different ways.
• ${\displaystyle 10=1^{2}+3^{2}}$, 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, ${\displaystyle 10!=3!\cdot 5!\cdot 7!}$.[4] 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 ${\displaystyle 96^{2}+28^{2}}$).

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.[5] 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.[6] 10 is also the eighth Perrin number, preceded in the sequence by (5, 5, 7).[7] In the sequence of triangular numbers, indexed powers of 10 in this sequence generate the following sequence of triangular numbers in decimal: 55 (10th), 5,050 (100th), 500,500 (1,000th), ...[8][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.[9] Ten is also the first non-trivial decagonal number,[10] the third centered triangular number[11] and tetrahedral number,[12] and the fifth semi-meandric number.[13]

10 is the fourth telephone number, and the number of Young tableaux with four cells.[14] It is the number of ${\displaystyle n}$-queens problem solutions for ${\displaystyle n=5}$.[15]

A ${\displaystyle 10\times 10}$ magic square has a magic constant of 505.[16]

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

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.[18]

### 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.[19] 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.[20]

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.[21] 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.

## 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 religion and philosophy

• References in the Bible and the Quran, Judaism, Christianity, and Islam:
• The Ten Commandments of Exodus[22] and Deuteronomy[23] are considered a cornerstone of Judaism and Christianity.
• People traditionally tithed one-tenth of their produce. The practice of tithing is still common in Christian churches today, though it is disputed in some circles as to whether or not it is required of Christians.
• In Deuteronomy 26:12, the Torah commands Jews to give one-tenth of their produce to the poor (Maaser Ani). From this verse and from an earlier verse (Deut. 14:22) there derives a practice for Jews to give one-tenth of all earnings to the poor.[24]
• Ten Plagues were inflicted on Egypt in Exodus 7–12.
• Jews observe the annual Ten Days of Repentance beginning on Rosh Hashanah and ending on Yom Kippur.
• In Jewish liturgy, Ten Martyrs are singled out as a group.
• There are said to be Ten Lost Tribes of Israel (those other than Judah and Benjamin).
• There are Ten Sephirot in the Kabbalistic Tree of Life.
• In Judaism, ten men are the required quorum, called a minyan, for prayer services.
• In Genesis 18:23-32, Abraham pleads on behalf of Sodom and Gomorrah, asking to save the cities if there are enough righteous people there. He starts at 10 per city, and ends with 10 total in all cities.
• Interpretations of Genesis in Talmudic and Midrashic teachings suggest that on the first day, God drew forth ten primal elements from the abyss in order to construct all of Creation: Heaven (or Fire), Earth, Chaos, Void, Light, Darkness, Wind (or Spirit), Water, Day, and Night. See also Bereshit (parsha).
• Jesus tells the Parable of the Ten Virgins in Matthew 25:1–13.
• Complete the pilgrimage and minor pilgrimage for Allah.1 But if prevented ˹from proceeding˺, then ˹offer˺ whatever sacrificial animals you can afford. And do not shave your heads until the sacrificial animal reaches its destination. But if any of you is ill or has a scalp ailment ˹requiring shaving˺, then compensate either by fasting, charity, or a sacrificial offering. In times of peace, you may combine the pilgrimage and minor pilgrimage then make the sacrificial offering you can afford. Whoever cannot afford that ˹offering˺, let them fast three days during pilgrimage and seven after returning ˹home˺—completing ten. These offerings are for those who do not live near the Sacred House. And be mindful of Allah, and know that Allah is severe in punishment.

— Surah al-Baqarah (The Heifer):196[25]
• Note 1: Ḥajj, the pilgrimage to Mecca, is required at least once in a Muslim’s lifetime if they are physically and financially able to. Umrah, which is a short version of ḥajj, is recommended, but not obligatory.
• Allah will not call you to account for your thoughtless oaths, but He will hold you accountable for deliberate oaths. The penalty for a broken oath is to feed ten poor people from what you normally feed your own family, or to clothe them, or to free a bondsperson. But if none of this is affordable, then you must fast three days. This is the penalty for breaking your oaths. So be mindful of your oaths. This is how Allah makes things clear to you, so perhaps you will be grateful.

— Surah al-Maidah (The Table):89[26]
• In Pythagoreanism, the number 10 played an important role and was symbolized by the tetractys.
• In Hinduism, Lord Vishnu appeared on the earth in 10 incarnations, popularly known as Dashaavathar.
• In Sikhism, there are ten human Gurus.

## 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.[27]
• 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

• The ESRB recommends video games with an E10+ rating to children aged 10 and up.

## 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 A074902 (Known friendly numbers)". 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. ^ "10". PrimeCurios!. PrimePages. Retrieved 2023-01-14.
5. ^ 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.
6. ^ Sloane, N. J. A. (1975). "Aliquot sequences". Mathematics of Computation. OEIS Foundation. 29 (129): 101–107. Retrieved 2022-12-08.
7. ^ 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.
8. ^ 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
...
9. ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
10. ^ "Sloane's A001107 : 10-gonal (or decagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
11. ^ "Sloane's A005448 : Centered triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
12. ^ "Sloane's A000292 : Tetrahedral numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
13. ^ 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.
14. ^ Sloane, N. J. A. (ed.). "Sequence A000085 (Number of self-inverse permutations on n letters, also known as involutions; number of standard Young tableaux with four cells.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-17.
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. ^ Andrews, W.S. (1917). Magic Squares and Cubes (2nd ed.). Open Court Publishing. p. 30.
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. ^ 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.
19. ^ 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.
20. ^ 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. doi:10.2307/2323457. ISBN 0-7167-1193-1. JSTOR 2323457. OCLC 13092426. S2CID 119730123.
21. ^ Coxeter, H. S. M (1948). "Chapter 14: Star-polytopes". Regular Polytopes. London: Methuen & Co. LTD. p. 263.
22. ^
23. ^
24. ^ [1] Archived February 23, 2006, at the Wayback Machine
25. ^ "Surah Al-Baqarah - 196". Quran.com. Retrieved 2023-08-25.
26. ^ "Surah Al-Ma'idah - 89". Quran.com. Retrieved 2023-08-25.
27. ^ Khalil Garriot (21 June 2014). "Mystery solved: Why do the best soccer players wear No. 10?". Yahoo. Retrieved 19 May 2015.