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Cardinal | eight | |||
Ordinal | 8th (eighth) | |||
Numeral system | octal | |||
Factorization | 2^{3} | |||
Divisors | 1, 2, 4, 8 | |||
Greek numeral | Η´ | |||
Roman numeral | VIII, viii | |||
Greek prefix | octa-/oct- | |||
Latin prefix | octo-/oct- | |||
Binary | 1000_{2} | |||
Ternary | 22_{3} | |||
Senary | 12_{6} | |||
Octal | 10_{8} | |||
Duodecimal | 8_{12} | |||
Hexadecimal | 8_{16} | |||
Greek | η (or Η) | |||
Arabic, Kurdish, Persian, Sindhi, Urdu | ٨ | |||
Amharic | ፰ | |||
Bengali | ৮ | |||
Chinese numeral | 八,捌 | |||
Devanāgarī | ८ | |||
Kannada | ೮ | |||
Malayalam | ൮ | |||
Telugu | ౮ | |||
Tamil | ௮ | |||
Hebrew | ח | |||
Khmer | ៨ | |||
Thai | ๘ | |||
Armenian | Ը ը |
8 (eight) is the natural number following 7 and preceding 9.
English eight, from Old English eahta, æhta, Proto-Germanic *ahto is a direct continuation of Proto-Indo-European *oḱtṓ(w)-, and as such cognate with Greek ὀκτώ and Latin octo-, both of which stems are reflected by the English prefix oct(o)-, as in the ordinal adjective octaval or octavary, the distributive adjective is octonary. The adjective octuple (Latin octu-plus) may also be used as a noun, meaning "a set of eight items"; the diminutive octuplet is mostly used to refer to eight siblings delivered in one birth.
The Semitic numeral is based on a root *θmn-, whence Akkadian smn-, Arabic ṯmn-, Hebrew šmn- etc. The Chinese numeral, written 八 (Mandarin: bā; Cantonese: baat), is from Old Chinese *priāt-, ultimately from Sino-Tibetan b-r-gyat or b-g-ryat which also yielded Tibetan brgyat.
It has been argued that, as the cardinal number 7 is the highest number of items that can universally be cognitively processed as a single set, the etymology of the numeral eight might be the first to be considered composite, either as "twice four" or as "two short of ten", or similar. The Turkic words for "eight" are from a Proto-Turkic stem *sekiz, which has been suggested as originating as a negation of eki "two", as in "without two fingers" (i.e., "two short of ten; two fingers are not being held up");^{[1]} this same principle is found in Finnic *kakte-ksa, which conveys a meaning of "two before (ten)". The Proto-Indo-European reconstruction *oḱtṓ(w)- itself has been argued as representing an old dual, which would correspond to an original meaning of "twice four". Proponents of this "quaternary hypothesis" adduce the numeral 9, which might be built on the stem new-, meaning "new" (indicating the beginning of a "new set of numerals" after having counted to eight).^{[2]}
The modern digit 8, like all modern Arabic numerals other than zero, originates with the Brahmi numerals. The Brahmi digit for eight by the 1st century was written in one stroke as a curve └┐ looking like an uppercase H with the bottom half of the left line and the upper half of the right line removed. However, the digit for eight used in India in the early centuries of the Common Era developed considerable graphic variation, and in some cases took the shape of a single wedge, which was adopted into the Perso-Arabic tradition as ٨ (and also gave rise to the later Devanagari form ८); the alternative curved glyph also existed as a variant in Perso-Arabic tradition, where it came to look similar to our digit 5.^{[year needed]}
The digits as used in Al-Andalus by the 10th century were a distinctive western variant of the glyphs used in the Arabic-speaking world, known as ghubār numerals (ghubār translating to "sand table"). In these digits, the line of the 5-like glyph used in Indian manuscripts for eight came to be formed in ghubār as a closed loop, which was the 8-shape that became adopted into European use in the 10th century.^{[3]}
Just as in most modern typefaces, in typefaces with text figures the character for the digit 8 usually has an ascender, as, for example, in .
The infinity symbol ∞, described as a "sideways figure eight", is unrelated to the digit 8 in origin; it is first used (in the mathematical meaning "infinity") in the 17th century, and it may be derived from the Roman numeral for "one thousand" CIƆ, or alternatively from the final Greek letter, ω.
Eight is the third composite number, lying between the fourth prime number (7) and the fourth composite number (9). 8 is the first non-unitary cube prime of the form p^{3}. With proper divisors 1, 2, and 4, it is the third power of two (2^{3}). 8 is the first number which is neither prime nor semiprime and the only nonzero perfect power that is one less than another perfect power, by Mihăilescu's Theorem.
Sphenic numbers always have exactly eight divisors.^{[10]}
A polygon with eight sides is an octagon.^{[11]} The sides and span of a regular octagon, or truncated square, are in 1 : 1 + √2 silver ratio, and its circumscribing square has a side and diagonal length ratio of 1 : √2; with both the silver ratio and the square root of two intimately interconnected through Pell numbers, where in particular the quotient of successive Pell numbers generates rational approximations for coordinates of a regular octagon.^{[12]}^{[13]} With a central angle of 45 degrees and an internal angle of 135 degrees, a regular octagon can fill a plane-vertex with a regular triangle and a regular icositetragon, as well as tessellate two-dimensional space alongside squares in the truncated square tiling. This tiling is one of eight Archimedean tilings that are semi-regular, or made of more than one type of regular polygon, and the only tiling that can admit a regular octagon.^{[14]} The Ammann–Beenker tiling is a nonperiodic tesselation of prototiles that feature prominent octagonal silver eightfold symmetry, that is the two-dimensional orthographic projection of the four-dimensional 8-8 duoprism.^{[15]} In number theory, figurate numbers representing octagons are called octagonal numbers.^{[16]}
A cube is a regular polyhedron with eight vertices that also forms the cubic honeycomb, the only regular honeycomb in three-dimensional space.^{[17]} Through various truncation operations, the cubic honeycomb generates eight other convex uniform honeycombs under the cubic group .^{[18]} The octahedron, with eight equilateral triangles as faces, is the dual polyhedron to the cube and one of eight convex deltahedra.^{[19]}^{[20]} The stella octangula, or eight-pointed star, is the only stellation with octahedral symmetry. It has eight triangular faces alongside eight vertices that forms a cubic faceting, composed of two self-dual tetrahedra that makes it the simplest of five regular compounds. The cuboctahedron, on the other hand, is a rectified cube or rectified octahedron, and one of only two convex quasiregular polyhedra. It contains eight equilateral triangular faces, whose first stellation is the cube-octahedron compound.^{[21]}^{[22]} There are also eight uniform polyhedron compounds made purely of octahedra, including the regular compound of five octahedra, and an infinite amount of polyhedron compounds made only of octahedra as triangular antiprisms (UC_{22} and UC_{23}, with p = 3 and q = 1).
The truncated tetrahedron is the simplest Archimedean solid, made of four triangles and four hexagons, the hexagonal prism, which classifies as an irregular octahedron and parallelohedron, is able to tessellate space as a three-dimensional analogue of the hexagon, and the gyrobifastigium, with four square faces and four triangular faces, is the only Johnson solid that is able to tessellate space. The truncated octahedron, also a parallelohedron, is the permutohedron of order four, with eight hexagonal faces alongside six squares is likewise the only Archimedean solid that can generate a honeycomb on its own.
A tesseract or 8-cell is the four-dimensional analogue of the cube. It is one of six regular polychora, with a total of eight cubical cells, hence its name. Its dual figure is the analogue of the octahedron, with twice the amount of cells and simply termed the 16-cell, that is the orthonormal basis of vectors in four dimensions. Whereas a tesseractic honeycomb is self-dual, a 16-cell honeycomb is dual to a 24-cell honeycomb that is made of 24-cells. The 24-cell is also regular, and made purely of octahedra whose vertex arrangement represents the ring of Hurwitz integral quaternions. Both the tesseract and the 16-cell can fit inside a 24-cell, and in a 24-cell honeycomb, eight 24-cells meet at a vertex. Also, the Petrie polygon of the tesseract and the 16-cell is a regular octagon.
Vertex-transitive semiregular polytopes whose facets are finite exist up through the 8th dimension. In the third dimension, they include the Archimedean solids and the infinite family of uniform prisms and antiprisms, while in the fourth dimension, only the rectified 5-cell, the rectified 600-cell, and the snub 24-cell are semiregular polytopes. For dimensions five through eight, the demipenteract and the k_{21} polytopes 2_{21}, 3_{21}, and 4_{21} are the only semiregular (Gosset) polytopes. Collectively, the k_{21} family of polytopes contains eight figures that are rooted in the triangular prism, which is the simplest semiregular polytope that is made of three cubes and two equilateral triangles. It also includes one of only three semiregular Euclidean honeycombs: the affine 5_{21} honeycomb that represents the arrangement of vertices of the eight-dimensional lattice, and made of 4_{21} facets. The culminating figure is the ninth-dimensional 6_{21} honeycomb, which is the only affine semiregular paracompact hyperbolic honeycomb with infinite facets and vertex figures in the k_{21} family. There are no other finite semiregular polytopes or honeycombs in dimensions n > 8.
The octonions are a hypercomplex normed division algebra that are an extension of the complex numbers. They are realized in eight dimensions, where they have an isotopy group over the real numbers that is spin group Spin(8), the unique such group that exhibits a phenomenon of triality. As a double cover of special orthogonal group SO(8), Spin(8) contains the special orthogonal Lie algebra D_{4} as its Dynkin diagram, whose order-three outer automorphism is isomorphic to the symmetric group S_{3}, giving rise to its triality. Over finite fields, the eight-dimensional Steinberg group ^{3}D_{4}(q^{3}) is simple, and one of sixteen such groups in the classification of finite simple groups. As is Lie algebra E_{8}, whose complex form in 248 dimensions is the largest of five exceptional Lie algebras that include E_{7} and E_{6}, which are held inside E_{8}. The smallest such algebra is G_{2}, that is the automorphism group of the octonions. In mathematical physics, special unitary group SO(3) has an eight-dimensional adjoint representation whose colors are ascribed gauge symmetries that represent the vectors of the eight gluons in the Standard Model.
The number 8 is involved with a number of interesting mathematical phenomena related to the notion of Bott periodicity. If is the direct limit of the inclusions of real orthogonal groups , the following holds:
Clifford algebras also display a periodicity of 8.^{[23]} 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, which occupy the highest possible dimension for a normed division algebra.
The lattice Γ_{8} is the smallest positive even unimodular lattice. As a lattice, it holds the optimal structure for the densest packing of 240 spheres in eight dimensions, whose lattice points also represent the root system of Lie group E_{8}. This honeycomb arrangement is shared by a unique complex tessellation of Witting polytopes, also with 240 vertices. Each complex Witting polytope is made of Hessian polyhedral cells that have Möbius–Kantor polygons as faces, each with eight vertices and eight complex equilateral triangles as edges, whose Petrie polygons form regular octagons. In general, positive even unimodular lattices only exist in dimensions proportional to eight. In the 16th dimension, there are two such lattices : Γ_{8} ⊕ Γ_{8} and Γ_{16}, while in the 24th dimension there are precisely twenty-four such lattices that are called the Niemeier lattices, the most important being the Leech lattice, which can be constructed using the octonions as well as with three copies of the ring of icosians that are isomorphic to the lattice.^{[24]}^{[25]} The order of the smallest non-abelian group all of whose subgroups are normal is 8.
In the classification of sporadic groups, the third generation consists of eight groups, four of which are centralizers of (itself the largest group of this generation), with another three transpositions of Fischer group .^{[26]} 8 is the difference between 53 and 61, which are the two smallest prime numbers that do not divide the order of any sporadic group. The largest supersingular prime that divides the order of is 71, which is the eighth self-convolution of Fibonacci numbers (where 744, which is essential to Moonshine theory, is the twelfth).^{[27]}^{[28]} While only two sporadic groups have eight prime factors in their order (Lyons group and Fischer group ), Mathieu group holds a semi-presentation whose order is equal to .^{[29]}
Multiplication | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
8 × x | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 | 88 | 96 | 104 | 112 | 120 |
Division | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
8 ÷ x | 8 | 4 | 2.6 | 2 | 1.6 | 1.3 | 1.142857 | 1 | 0.8 | 0.8 | 0.72 | 0.6 | 0.615384 | 0.571428 | 0.53 | |
x ÷ 8 | 0.125 | 0.25 | 0.375 | 0.5 | 0.625 | 0.75 | 0.875 | 1 | 1.125 | 1.25 | 1.375 | 1.5 | 1.625 | 1.75 | 1.875 |
Exponentiation | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
8^{x} | 8 | 64 | 512 | 4096 | 32768 | 262144 | 2097152 | 16777216 | 134217728 | 1073741824 | 8589934592 | 68719476736 | 549755813888 | |
x^{8} | 1 | 256 | 6561 | 65536 | 390625 | 1679616 | 5764801 | 16777216 | 43046721 | 100000000 | 214358881 | 429981696 | 815730721 |
A number is divisible by 8 if its last three digits, when written in decimal, are also divisible by 8, or its last three digits are 0 when written in binary.
8 is the base of the octal number system, which is mostly used with computers.^{[30]} In octal, one digit represents three bits. In modern computers, a byte is a grouping of eight bits, also called an octet.