Natural number
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 .
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,
10
=
1
2
+
3
2
{\displaystyle 10=1^{2}+3^{2}}
, the sum of the
squares of the first two
odd numbers
[3]
10
=
1
+
2
+
3
+
4
{\displaystyle 10=1+2+3+4}
, the sum of the first four positive
integers , equivalently the fourth
triangle number
[4]
10
=
3
+
7
=
5
+
5
{\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]
10
=
2
+
3
+
5
{\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:
10
!
=
1
!
⋅
3
!
⋅
5
!
⋅
7
!
{\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
(
2
+
5
=
7
{\displaystyle (2+5=7}
and
5
−
2
=
3
)
{\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,
80
2
+
60
2
{\displaystyle 80^{2}+60^{2}}
and
96
2
+
28
2
.
{\displaystyle 96^{2}+28^{2}.}
Ten has an
aliquot sum of
8 , and is the first discrete
semiprime
(
2
×
5
)
{\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
N
{\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
10
×
10
{\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
n
{\displaystyle n}
-
queens problem solutions for
n
=
5
{\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
12
2
=
144
{\displaystyle 12^{2}=144}
degrees and a
central angle of
6
2
=
36
{\displaystyle 6^{2}=36}
degrees. All regular
n
{\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
H
3
{\displaystyle \mathrm {H} _{3}}
Coxeter plane that contain various
decagrammic symmetries, which include
compound forms of the regular decagram.
[39]
Higher-dimensional spaces
M
10
{\displaystyle \mathrm {M} _{10}}
is a multiply transitive
permutation group on ten points. It is an
almost simple group, of
order ,
720
=
2
4
⋅
3
2
⋅
5
=
2
⋅
3
⋅
4
⋅
5
⋅
6
=
8
⋅
9
⋅
10
{\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
M
11
{\displaystyle \mathrm {M} _{11}}
, a group with an irreducible
faithful complex representation in ten dimensions, and an order equal to
7920
=
11
⋅
10
⋅
9
⋅
8
{\displaystyle 7920=11\cdot 10\cdot 9\cdot 8}
that is one less than the
one-thousandth prime number, 7919.
E
10
{\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
E
n
{\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
n
{\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
I
~
1
{\displaystyle {\tilde {I}}_{1}}
[∞ ], which represents the
apeirogonal tiling , as well as the five affine Coxeter groups
G
~
2
{\displaystyle {\tilde {G}}_{2}}
,
F
~
4
{\displaystyle {\tilde {F}}_{4}}
,
E
~
6
{\displaystyle {\tilde {E}}_{6}}
,
E
~
7
{\displaystyle {\tilde {E}}_{7}}
, and
E
~
8
{\displaystyle {\tilde {E}}_{8}}
that are associated with the five
exceptional Lie algebras . They also include the four general affine Coxeter groups
A
~
n
{\displaystyle {\tilde {A}}_{n}}
,
B
~
n
{\displaystyle {\tilde {B}}_{n}}
,
C
~
n
{\displaystyle {\tilde {C}}_{n}}
, and
D
~
n
{\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.
Religion
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
^ 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
n
2
{\displaystyle n^{2}}
divisors, where
n
{\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.
^ 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]
^ 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]
^ 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 505 th 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).
^ 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
10
×
10
{\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]
^ Specifically, a decagon can fill a plane-vertex alongside two regular
pentagons , and alongside a fifteen-sided
pentadecagon and
triangle .
^ 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
^
"Sloane's A005278 : Noncototients" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-06-01 .
^
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 .
^
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 .
^
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 .
^
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 .
^
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 .
^
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 .
^
"10" . PrimeCurios! .
PrimePages . Retrieved 2023-01-14 .
^
Sloane, N. J. A. (ed.).
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On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-12-08 .
^
Sloane, N. J. A. (1975).
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^
Sloane, N. J. A. (ed.).
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^
Sloane, N. J. A. (ed.).
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^
Sloane, N. J. A. (ed.).
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On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-11-07 .
^
"Sloane's A001107 : 10-gonal (or decagonal) numbers" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-06-01 .
^
"Sloane's A005448 : Centered triangular numbers" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-06-01 .
^
"Sloane's A000292 : Tetrahedral numbers" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-06-01 .
^
a
b
Sloane, N. J. A. (ed.).
"Sequence A008527 (Coordination sequence for body-centered tetragonal lattice.)" . The
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^
O'Keeffe, Michael (1995).
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De Grutyer : 905–908.
Bibcode :
1995ZK....210..905O .
doi :
10.1524/zkri.1995.210.12.905 .
S2CID
96758246 .
^
Sloane, N. J. A. (ed.).
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^
Sloane, N. J. A. (ed.).
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^
Sloane, N. J. A. (ed.).
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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
...
^
Sloane, N. J. A. (ed.).
"Sequence A186076 (Numbers m such that m equal to Sum_{i equal to x..y} i being (10^k)*y + x, where 0 is less than or equal to x less than y, 0 less than or equal to x less than 10^k for some positive integers k.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-11-07 .
^ Andrews, W.S. (1917).
Magic Squares and Cubes (2nd ed.). Open Court Publishing. p. 30.
^
Sloane, N. J. A. (ed.).
"Sequence A000040 (The prime numbers.)" . The
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^
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 .
^
Sloane, N. J. A. (ed.).
"Sequence A002322 (Reduced totient function psi(n): least k such that x^k is congruent 1 (mod n) for all x prime to n; also known as the Carmichael lambda function (exponent of unit group mod n); also called the universal exponent of n.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-11-07 .
^
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 .
^
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
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^
Sloane, N. J. A. (ed.).
"Sequence A003601 (Numbers j such that the average of the divisors of j is an integer: sigma_0(j) divides sigma_1(j). Alternatively, numbers j such that tau(j) (A000005(j)) divides sigma(j) (A000203(j)).)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-12-13 .
^
Sloane, N. J. A. (ed.).
"Sequence A102187 (Arithmetic means of divisors of arithmetic numbers (arithmetic numbers, A003601, are those for which the average of the divisors is an integer).)" . The
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^
Sloane, N. J. A. (ed.).
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On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-12-13 .
^
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 .
^
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 .
^
Sloane, N. J. A. (ed.).
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^ M.J. Bertin; A. Decomps-Guilloux; M. Grandet-Hugot; M. Pathiaux-Delefosse; J.P. Schreiber (1992). Pisot and Salem Numbers . Birkhäuser.
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3-7643-2648-4 .
^
Grünbaum, Branko ;
Shepard, Geoffrey (November 1977).
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doi :
10.2307/2689529 .
JSTOR
2689529 .
S2CID
123776612 .
Zbl
0385.51006 .
^
Grünbaum, Branko ;
Shephard, G. C. (1987). "Section 2.1: Regular and uniform tilings".
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doi :
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ISBN
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JSTOR
2323457 .
OCLC
13092426 .
S2CID
119730123 .
^ Gummelt, Petra (1996).
"Penrose tilings as coverings of congruent decagons" .
Geometriae Dedicata . 62 (1). Berlin:
Springer : 1–17.
doi :
10.1007/BF00239998 .
MR
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S2CID
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Zbl
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^
Coxeter, H. S. M (1948). "Chapter 14: Star-polytopes". Regular Polytopes . London: Methuen & Co. LTD. p. 263.
External links
Look up
ten in Wiktionary, the free dictionary.
100,000
1,000,000
10,000,000
100,000,000
1,000,000,000