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,

• ${\displaystyle 10=1^{2}+3^{2))$, 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 II_9,1 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 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).

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

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

• 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.

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