10

Last updated
9 10 11
Cardinal ten
Ordinal 10th
(tenth)
Numeral system decimal
Factorization 2 × 5
Divisors 1, 2, 5, 10
Greek numeral Ι´
Roman numeral X
Roman numeral (unicode)X, x
Greek prefix deca-/deka-
Latin prefix deci-
Binary 10102
Ternary 1013
Senary 146
Octal 128
Duodecimal A12
Hexadecimal A16
Chinese numeral 十,拾
Hebrew י (Yod)
Khmer ១០
Armenian Ժ
Tamil
Thai ๑๐
Devanāgarī १०
Bengali ১০
Arabic & Kurdish & Iranian ١٠
Malayalam
Egyptian hieroglyph 𓎆
Babylonian numeral 𒌋

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.

Contents

Anthropology

Usage and terms

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,

The factorial of ten is equal to the product of the factorials of the first four odd numbers as well: , [8] and 10 is the only number whose sum and difference of its prime divisors yield prime numbers and .

10 is also the first number whose fourth power (10,000) can be written as a sum of two squares in two different ways, and

Ten has an aliquot sum of 8, and is the first discrete semiprime 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 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] [lower-alpha 1]

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] 10 is also the first member in the coordination sequence for body-centered tetragonal lattices. [17] [18] [lower-alpha 2]

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] 55 is also the fourth doubly triangular number. [20]

10 is the fourth telephone number, and the number of Young tableaux with four cells. [21] It is also the number of -queens problem solutions for . [22]

There are precisely ten small Pisot numbers that do not exceed the golden ratio. [23]

Geometry

Decagon

As a constructible polygon with a compass and straight-edge, the regular decagon has an internal angle of degrees and a central angle of degrees. All regular -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. [24] [lower-alpha 3] 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); [25] 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. [26] 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. [lower-alpha 4]

There are ten regular star polychora in the fourth dimension, all of which have orthographic projections in the Coxeter plane that contain various decagrammic symmetries, which include compound forms of the regular decagram. [27]

Higher-dimensional spaces

is a multiply transitive permutation group on ten points. It is an almost simple group, of order,

It functions as a point stabilizer of degree 11 inside the smallest sporadic simple group , a group with an irreducible faithful complex representation in ten dimensions, and an order equal to that is one more than the one-thousandth prime number, 7919.

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 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 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 [], which represents the apeirogonal tiling, as well as the five affine Coxeter groups , , , , and that are associated with the five exceptional Lie algebras. They also include the four general affine Coxeter groups , , , and 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:

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 centimetre = 10 millimetres, 1 decimetre = 10 centimetres, 1 meter = 100 centimetres, 1 dekametre = 10 meters, 1 kilometre = 1,000 meters).

Music

Religion

The tetractys Tetractys.svg
The tetractys

Abrahamic religions

The Ten Commandments in the Hebrew Bible are ethical commandments decreed by God (to Moses) for the people of Israel to follow.

Mysticism

See also

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 divisors, where is the number of digits: 10 has 22 = 4 divisors, 102 has 32 = 9 divisors, 103 has 42 = 16 divisors, and so forth.
  2. 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. Specifically, a decagon can fill a plane-vertex alongside two regular pentagons, and alongside a fifteen-sided pentadecagon and triangle.
  4. 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.

Related Research Articles

15 (fifteen) is the natural number following 14 and preceding 16.

17 (seventeen) is the natural number following 16 and preceding 18. It is a prime number.

19 (nineteen) is the natural number following 18 and preceding 20. It is a prime number.

70 (seventy) is the natural number following 69 and preceding 71.

90 (ninety) is the natural number following 89 and preceding 91.

29 (twenty-nine) is the natural number following 28 and preceding 30. It is a prime number.

72 (seventy-two) is the natural number following 71 and preceding 73. It is half a gross or six dozen.

32 (thirty-two) is the natural number following 31 and preceding 33.

57 (fifty-seven) is the natural number following 56 and preceding 58.

58 (fifty-eight) is the natural number following 57 and preceding 59.

104 is the natural number following 103 and preceding 105.

<span class="mw-page-title-main">120 (number)</span> Natural number

120 is the natural number following 119 and preceding 121. It is five sixths of a gross, or ten dozens.

220 is the natural number following 219 and preceding 221.

1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.

<span class="mw-page-title-main">360 (number)</span> Natural number

360 is the natural number following 359 and preceding 361.

144 is the natural number following 143 and preceding 145.

135 is the natural number following 134 and preceding 136.

168 is the natural number following 167 and preceding 169.

1728 is the natural number following 1727 and preceding 1729. It is a dozen gross, or one great gross. It is also the number of cubic inches in a cubic foot.

288 is the natural number following 287 and preceding 289. Because 288 = 2 · 12 · 12, it may also be called "two gross" or "two dozen dozen".

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.). "SequenceA001608(Perrin sequence (or Ondrej Such sequence))". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-12-08.
  3. 1 2 Sloane, N. J. A. (ed.). "SequenceA108100((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.). "SequenceA000217(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.). "SequenceA001172(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.). "SequenceA067188(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.). "SequenceA055233(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.). "SequenceA001065(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.). "SequenceA297575(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.). "SequenceA074902(Known friendly numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-12-08.
  13. Sloane, N. J. A. (ed.). "SequenceA005179(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. 1 2 Sloane, N. J. A. (ed.). "SequenceA008527(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.). "SequenceA000217(Triangular numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-12-08.
  20. Sloane, N. J. A. (ed.). "SequenceA002817(Doubly triangular numbers: a(n) as n*(n+1)*(n^2+n+2)/8.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-12-18.
  21. Sloane, N. J. A. (ed.). "SequenceA000085(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.
  22. Sloane, N. J. A. (ed.). "SequenceA000170(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.
  23. 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.
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  25. 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.
  26. 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.
  27. Coxeter, H. S. M (1948). "Chapter 14: Star-polytopes". Regular Polytopes. London: Methuen & Co. LTD. p. 263.