10

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.)

Other

* The number of kingdoms in Five Dynasties and Ten Kingdoms period.
* The house number of 10 Downing Street.
* The number of Provinces in Canada.
* Number of dots in a tetractys.
* The number of the French department Aube. 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. It is the second discrete semiprime ($2 \times 5$), as well as the second member of the $2 \times q$ discrete semiprime family. Ten is the only number whose sum and difference of its prime divisors yield prime numbers ($2 + 5 = 7$ and $5 - 2 = 3$). In general, powers of 10 contain $n^2$ divisors, where $n$ 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.

As important sums,

*$10 = 1 + 2 + 3 + 4$ the sum of the first four positive integers.
*$10 = 2 + 3 + 5 = 2 \times 5$, 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.
*$10 = 3 + 7 = 5 + 5$, the smallest number that can be written as the sum of two prime numbers in two different ways.
*$10 = 1^2 + 3^2$, 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, $10!=3!\cdot5!\cdot7!$. Ten is also the first number whose fourth power can be written as a sum of two squares in two different ways ($80^2 + 60^2$ and $96^2 + 28^2$).

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. 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. 10 is also the eighth Perrin number, preceded in the sequence by (5, 5, 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), ...

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. Ten is also the first non-trivial decagonal number, the third centered triangular number and tetrahedral number, and the fifth semi-meandric number.

10 is the fourth telephone number, and the number of Young tableaux with four cells. It is the number of $n$- queens problem solutions for $n = 5$.

A $10 \times 10$ magic square has a magic constant of 505.

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

According to conjecture, ten is the average sum of the proper divisors of the natural numbers $\mathbb N$ if the size of the numbers approaches infinity.

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 $12^2 = 144$ degrees and a central angle of $6^2 = 36$ degrees. All regular $n$-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. 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.

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 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. All of these polychora have orthographic projections in the $\mathrm _$ Coxeter plane that contain various decagrammic symmetries, which include the regular form as well as its three alternate compound forms.

$\mathrm_$ 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 $\mathrm_$, a Mathieu group which has an irreducible faithful complex representation in 10 dimensions.

$\mathrm_$ is an infinite-dimensional Kac–Moody algebra which has the even Lorentzian unimodular lattice II_9,1 of dimension 10 as its root lattice. It is the first $\mathrm_$ 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 $n$ 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 $\tilde I_1$ ''∞ which represents the apeirogonal tiling, as well as the five affine Coxeter groups $\tilde G_2$, $\tilde F_4$, $\tilde E_6$, $\tilde E_7$, and $\tilde E_8$ that are associated with the five exceptional Lie algebras. They also include the four general affine Coxeter groups $\tilde A_n$, $\tilde B_n$, $\tilde C_n$, and $\tilde D_n$ 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 atomic number of neon.
* The number of hydrogen atoms in butane, a hydrocarbon.
* The number of spacetime dimensions in some superstring theories.

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).

Astronomy

* The New General Catalogue object NGC 10, a magnitude 12.5 spiral galaxy

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