# 10

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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.

 ← 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
ArmenianԺ
Tamil
Thai๑๐
Devanāgarī१०
Bengali১০
Arabic & Kurdish & Iranian١٠
Malayalam
Egyptian hieroglyph𓎆
Babylonian numeral𒌋

## Anthropology

### Usage and terms

• A collection of ten items (most often ten years) is called a decade.
• The ordinal adjective is decimal; the distributive adjective is denary.
• 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.)

## Mathematics

Ten is the fifth composite number, and the smallest noncototient, which is a number that cannot be expressed as the difference between any integer and the total number of coprimes below it.[1] Ten is the eighth Perrin number, preceded by 5, 5, and 7.[2]

As important sums,

• ${\displaystyle 10=1^{2}+3^{2}}$ , the sum of the squares of the first two odd numbers[3]
• ${\displaystyle 10=1+2+3+4}$ , the sum of the first four positive integers, equivalently the fourth triangle number[4]
• ${\displaystyle 10=3+7=5+5}$ , the smallest number that can be written as the sum of two prime numbers in two different ways[5][6]
• ${\displaystyle 10=2+3+5}$ , 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[7]

The factorial of ten is equal to the product of the factorials of the first four odd numbers as well: ${\displaystyle 10!=1!\cdot 3!\cdot 5!\cdot 7!}$ ,[8] and 10 is the only number whose sum and difference of its prime divisors yield prime numbers ${\displaystyle (2+5=7}$  and ${\displaystyle 5-2=3)}$ .

10 is also the first number whose fourth power (10,000) 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 of 8, and is the first discrete semiprime ${\displaystyle (2\times 5)}$  to be in deficit, as with all subsequent discrete semiprimes.[9] It is the second composite in the aliquot sequence for ten (10, 8, 7, 1, 0) that is rooted in the prime 7-aliquot tree.[10]

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,[11] and it is the smallest number whose status as a possible friendly number is unknown.[12]

The smallest integer with exactly ten divisors is 48, while the least integer with exactly eleven divisors is 1024, which sets a new record.[13][a]

Figurate numbers that represent regular ten-sided polygons are called decagonal and centered decagonal numbers.[14] On the other hand, 10 is the first non-trivial centered triangular number[15] and tetrahedral number.[16][b]

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.[19][c]

A ${\displaystyle 10\times 10}$  magic square has a magic constant of 505,[23][d] where this is the ninth number to have a reduced totient of 100;[26] the previous such number is 500, which represents the number of planar partitions of ten.[27][e]

10 is the fourth telephone number, and the number of Young tableaux with four cells.[33] it is also the number of ${\displaystyle n}$ -queens problem solutions for ${\displaystyle n=5}$ .[34]

There are precisely ten small Pisot numbers that do not exceed the golden ratio.[35]

### Geometry

#### Decagon

As a constructible polygon with a compass and straight-edge, the regular decagon 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.[36][f] While the regular decagon cannot tile alongside other regular figures, ten of the eleven regular and semiregular tilings of the plane are Wythoffian (the elongated triangular tiling is the only exception);[37] however, the plane can be covered using overlapping decagons, and is equivalent to the Penrose P2 tiling when it is decomposed into kites and rhombi that are proportioned in golden ratio.[38] The regular decagon is also 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 icosidodecahedron.[g]

There are ten regular star polychora in the fourth dimension, all of which have orthographic projections in the ${\displaystyle \mathrm {H} _{3}}$  Coxeter plane that contain various decagrammic symmetries, which include compound forms of the regular decagram.[39]

#### Higher-dimensional spaces

${\displaystyle \mathrm {M} _{10}}$  is a multiply transitive permutation group on ten points. It is an almost simple group, of order,

${\displaystyle 720=2^{4}\cdot 3^{2}\cdot 5=2\cdot 3\cdot 4\cdot 5\cdot 6=8\cdot 9\cdot 10}$

It functions as a point stabilizer of degree 11 inside the smallest sporadic simple group ${\displaystyle \mathrm {M} _{11}}$ , a group with an irreducible faithful complex representation in ten dimensions, and an order equal to  ${\displaystyle 7920=11\cdot 10\cdot 9\cdot 8}$   that is one less than the one-thousandth prime number, 7919.

${\displaystyle \mathrm {E} _{10}}$  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.

## 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).

## Music

• The interval of a major tenth is an octave plus a major third.
• The interval of a minor tenth is an octave plus a minor third.

## Philosophy and religion

In Pythagoreanism, the number 10 played an important role and was symbolized by the tetractys.

There are Ten Sephirot in the Kabbalistic Tree of Life.

## Other fields

In Chinese astrology, the 10 Heavenly Stems, refer to a cyclic number system that is used also for time reckoning.

## Notes

1. ^ The initial largest span of numbers for a new maximum record of divisors to appear lies between numbers with 1 and 5 divisors, respectively.
This is also the next greatest such span, set by the numbers with 7 and 11 divisors, and followed by numbers with 13 and 17 divisors; these are maximal records set by successive prime counts.
Powers of 10 contain ${\displaystyle n^{2}}$  divisors, where ${\displaystyle n}$  is the number of digits: 10 has 22 = 4 divisors, 102 has 32 = 9 divisors, 103 has 42 = 16 divisors, and so forth.
2. ^ 10 is also the first member in the coordination sequence for body-centered tetragonal lattices,[17][18] also found by
"... reading the segment (1, 10) together with the line from 10, in the direction 10, 34, ..., in the square spiral whose vertices are the generalized hexagonal numbers (A000217)."[17]
Aside from the zeroth term, this sequence matches the sums of squares of consecutive odd numbers.[3]
3. ^ 55 is also the fourth doubly triangular number.[20] In the sequence of triangular numbers, indexed powers of 10 in this sequence generate the following sequence of triangular numbers, in decimal representation: 55 (10th), 5,050 (100th), 500,500 (1,000th), ...[21]
19 is another number that is the first member of a sequence displaying a similar uniform property, where the 19th triangular number is 190, the 199th triangular number is 19900, etc.[22]
4. ^ Where 55 is the sum of the first four terms in Sylvester's sequence (2, 3, 7, and 43), the product of these is 1806, whose sum with the fifth term 1807 yields the 505th indexed prime number and 42nd square number, 3613.[24][25]
Unit fractions from terms in this sequence form an infinite series that converges to 1, where successive terms from Sylvester's sequence will always multiply to one less the value of the following term (i.e., 42 and 43 for the first three and fourth terms).
5. ^ Meanwhile, 504 represents ninth semi-miandric number, where 10 is the third such non-trivial semi-meander.[28] The former is also the arithmetic mean of the divisors of 5005,[29][30] which is the magic constant of a ${\displaystyle 10\times 10}$  magic cube.[31]
5005 is also the tenth non-unitary convolution of triangular numbers and square numbers, equivalently five-dimensional pyramidal numbers.[32]
6. ^ Specifically, a decagon can fill a plane-vertex alongside two regular pentagons, and alongside a fifteen-sided pentadecagon and triangle.
7. ^ 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.
In total, 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, ...). Also, the decagonal prism is the largest prism that is a facet inside four-dimensional uniform polychora.

## References

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 A001608 (Perrin sequence (or Ondrej Such sequence))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
3. ^ a b Sloane, N. J. A. (ed.). "Sequence A108100 ((2*n-1)^2+(2*n+1)^2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-07.
4. ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers: a(n) is the binomial(n+1,2) equal to n*(n+1)/2 or 0 + 1 + 2 + ... + n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-02.
5. ^ Sloane, N. J. A. (ed.). "Sequence A001172 (Smallest even number that is an unordered sum of two odd primes in exactly n ways.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-07.
6. ^ Sloane, N. J. A. (ed.). "Sequence A067188 (Numbers that can be expressed as the (unordered) sum of two primes in exactly two ways.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-07.
7. ^ 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.
8. ^ "10". PrimeCurios!. PrimePages. Retrieved 2023-01-14.
9. ^ 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.
10. ^ Sloane, N. J. A. (1975). "Aliquot sequences". Mathematics of Computation. 29 (129). OEIS Foundation: 101–107. Retrieved 2022-12-08.
11. ^ 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.
12. ^ Sloane, N. J. A. (ed.). "Sequence A074902 (Known friendly numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
13. ^ Sloane, N. J. A. (ed.). "Sequence A005179 (Smallest number with exactly n divisors.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-07.
14. ^ "Sloane's A001107 : 10-gonal (or decagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
15. ^ "Sloane's A005448 : Centered triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
16. ^ "Sloane's A000292 : Tetrahedral numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
17. ^ a b Sloane, N. J. A. (ed.). "Sequence A008527 (Coordination sequence for body-centered tetragonal lattice.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-07.
18. ^ O'Keeffe, Michael (1995). "Coordination sequences for lattices" (PDF). Zeitschrift für Kristallographie. 210 (12). Berlin: De Grutyer: 905–908. Bibcode:1995ZK....210..905O. doi:10.1524/zkri.1995.210.12.905. S2CID 96758246.
19. ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
20. ^ Sloane, N. J. A. (ed.). "Sequence A002817". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-18.
21. ^ 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
...
22. ^
23. ^ Andrews, W.S. (1917). Magic Squares and Cubes (2nd ed.). Open Court Publishing. p. 30.
24. ^ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-18.
25. ^ Sloane, N. J. A. (ed.). "Sequence A001844 (Centered square numbers...Sums of two consecutive squares)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-18.
26. ^
27. ^ Sloane, N. J. A. (ed.). "Sequence A000219 (Number of planar partitions (or plane partitions) of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-08.
28. ^ 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.
29. ^
30. ^
31. ^ Sloane, N. J. A. (ed.). "Sequence A027441 (a(n) equal to (n^4 + n)/2 (Row sums of an n X n X n magic cube, when it exists).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-13.
32. ^ Sloane, N. J. A. (ed.). "Sequence A005585 (5-dimensional pyramidal numbers: a(n) is equal to n*(n+1)*(n+2)*(n+3)*(2n+3)/5!.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-13.
33. ^ 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.
34. ^ 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.
35. ^ 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.
36. ^ Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine. 50 (5). Taylor & Francis, Ltd.: 230, 231. doi:10.2307/2689529. JSTOR 2689529. S2CID 123776612. Zbl 0385.51006.
37. ^ 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.
38. ^ Gummelt, Petra (1996). "Penrose tilings as coverings of congruent decagons". Geometriae Dedicata. 62 (1). Berlin: Springer: 1–17. doi:10.1007/BF00239998. MR 1400977. S2CID 120127686. Zbl 0893.52011.
39. ^ Coxeter, H. S. M (1948). "Chapter 14: Star-polytopes". Regular Polytopes. London: Methuen & Co. LTD. p. 263.