8 (eight;/ɛit/) is the natural number following 7 and preceding 9. The SI prefix for 10008 is yotta (Y), and for its reciprocal yocto (y). It is the root of two other numbers: eighteen (eight and ten) and eighty (eight tens). Linguistically, it is derived from Middle English eighte.
8 is a composite number, its proper divisors being 1, 2, and 4. It is twice 4 or four times 2. Eight is a power of two, being 23 (two cubed), and is the first number of the form p3. It has an aliquot sum of 7 in the 4 member aliquot sequence (8,7,1,0) being the first member of 7-aliquot tree. It is symbolized by the Arabic numeral (figure)
All powers of 2 ;(2x), have an aliquot sum of one less than themselves.
8 is the order of the smallest non-abelian group all of whose subgroups are normal.
8 and 9 form a Ruth–Aaron pair under the second definition in which repeated prime factors are counted as often as they occur.
A polygon with eight sides is an octagon. Figurate numbers representing octagons (including eight) are called octagonal numbers. A polyhedron with eight faces is an octahedron. A cuboctahedron has as faces six equal squares and eight equal regular triangles.
Sphenic numbers always have exactly eight divisors.
The number 8 is involved with a number of interesting mathematical phenomena related to the notion of Bott periodicity. For example if is the direct limit of the inclusions of real orthogonal groups then . Clifford algebras also display a periodicity of 8. For example the algebra Cl(p + 8,q) is isomorphic to the algebra of 16 by 16 matrices with entries in Cl(p,q). We also see a period of 8 in the K-theory of spheres and in the representation theory of the rotation groups, the latter giving rise to the 8 by 8 spinorial chessboard. All of these properties are closely related to the properties of the octonions.
A figure 8 is the common name of a geometric shape, often used in the context of sports, such as skating. Figure-eight turns of a rope or cable around a cleat, pin, or bitt are used to belay something.