008.07.02

8 (eight) is the natural number following 7 and preceding 9.

In mathematics

8 is:
* a composite number, its proper divisors being , , and . It is twice 4 or four times 2.
* a power of two, being 2 (two cubed), and is the first number of the form , being an integer greater than 1.
* the first number which is neither prime nor semiprime.
* the base of the octal number system, which is mostly used with computers. In octal, one digit represents three bits. In modern computers, a byte is a grouping of eight bits, also called an octet.
* a Fibonacci number, being plus . The next Fibonacci number is . 8 is the only positive Fibonacci number, aside from 1, that is a perfect cube.
* the only nonzero perfect power that is one less than another perfect power, by Mihăilescu's Theorem.
* the order of the smallest non-abelian group all of whose subgroups are normal.
* the dimension of the octonions and is the highest possible dimension of a normed division algebra.
* the first number to be the aliquot sum of two numbers other than itself; the discrete biprime , and the square number .

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.

A polygon with eight sides is an octagon. The sides and span of a regular octagon, or truncated square, are in silver ratio, and its circumscribing square has a side and diagonal length ratio of ; 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. With a central angle of 45 degrees and an internal angle of 135 degrees, regular octagons are able to tessellate two-dimensional space alongside squares in the truncated square tiling, as well as fill a plane-vertex with a regular triangle and a regular icositetragon. The Ammann–Beenker tiling is a nonperiodic tesselation of prototiles that feature prominent octagonal ''silver'' eightfold symmetry, and is the two-dimensional orthogonal projection of the 8-8 duoprism. In number theory, figurate numbers representing octagons are called octagonal numbers.

A cube is a regular polyhedron with eight vertices that also forms the cubic honeycomb, the only regular honeycomb in three-dimensional space. Through various truncation operations, the cubic honeycomb generates eight other convex uniform honeycombs under the group $_3$. The octahedron, with eight equilateral triangles as faces, is the dual polyhedron to the cube and one of eight convex deltahedra. The stella octangula, or ''eight-pointed star'', is the only stellation with octahedral symmetry. It has eight triangular faces alongside eight vertices that form a cubic faceting, composed of two self-dual tetrahedra that makes it the simplest of five regular compound polyhedra. 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 alongside six squares, whose first stellation is the cube-octahedron compound. The hexagonal prism, which classifies as an irregular octahedron that is a parallelohedron, like the cube, is able to tessellate space as a three-dimensional analogue of the hexagon. The gyrobifastigium, with four square faces and four triangular faces, is the only Johnson solid that is able to tessellate space, while the truncated octahedron, also a parallelohedron, is the permutohedron of order four, with eight hexagonal faces alongside six squares that is likewise the only Archimedean solid that can generate a honeycomb on its own.

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 k₂₁ polytopes 2₂₁, 3₂₁, and 4₂₁ are the only semiregular (Gosset) polytopes. Collectively, the k₂₁ 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 5₂₁ honeycomb that represents the arrangement of vertices of the eight-dimensional $\mathrm E_$ lattice, and made of 4₂₁ facets. The culminating figure is the ninth-dimensional 6₂₁ honeycomb, which is the only affine semiregular paracompact hyperbolic honeycomb with infinite facets and vertex figures in the k₂₁ family. There are no other finite semiregular polytopes or honeycombs in dimensions ''n'' > 8.

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. If $O(\infty)$ is the direct limit of the inclusions of real orthogonal groups $O(1)\hookrightarrow O(2)\hookrightarrow\ldots\hookrightarrow O(k)\hookrightarrow\ldots$, the following holds:

:$\pi_(O(\infty))\cong\pi_(O(\infty))$.

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.

The spin group Spin(8) is the unique such group that exhibits the phenomenon of triality.

The lowest-dimensional even unimodular lattice is the 8-dimensional $\mathrm E_$ lattice. Even positive definite unimodular lattices exist only in dimensions divisible by 8.

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.

List of basic calculations

Etymology

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 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 (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 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");
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 ', which might be built on the stem ''new-'', meaning "new" (indicating the beginning of a "new set of numerals" after having counted to eight).

Evolution of the Arabic digit

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.

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.

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.

In science

Physics

* In nuclear physics, the second magic number.
* In particle physics, the eightfold way is used to classify sub-atomic particles.
* In statistical mechanics, the eight-vertex model has 8 possible configurations of arrows at each vertex.

Astronomy

* Messier object M8, a magnitude 5.0 nebula in the constellation of Sagittarius.
* The New General Cataloguebr>object
NGC 8, a double star in the constellation Pegasus.
* Since the demotion of Pluto to a dwarf planet on 24 August 2006, in our Solar System, eight of the bodies orbiting the Sun are considered to be planets.

Chemistry

* The atomic number of oxygen.
* The most stable allotrope of a sulfur molecule is made of eight sulfur atoms arranged in a rhombic form.
* The maximum number of electrons that can occupy a valence shell.
* The red pigment lycopene consists of eight isoprene units.

Geology

* A disphenoid crystal is bounded by eight scalene triangles arranged in pairs. A ditetragonal prism in the tetragonal crystal system has eight similar faces whose alternate interfacial angles only are equal.

Biology

* All spiders, and more generally all arachnids, have eight legs. Orb-weaver spiders of the cosmopolitan family Areneidae have eight similar eyes.
* The octopus and its cephalopod relatives in genus ''Argonauta'' have eight arms (tentacles).
* Compound coelenterates of the subclass or order Alcyonaria have polyps with eight-branched tentacles and eight septa.
* Sea anemones of genus '' Edwardsia'' have eight mesenteries.
* Animals of phylum Ctenophora swim by means of eight meridional bands of transverse ciliated plates, each plate representing a row of large modified cilia.
* The eight-spotted forester (genus ''Alypia'', family Zygaenidae) is a diurnal moth having black wings with brilliant white spots.
* The ascus in fungi of the class Ascomycetes, following nuclear fusion, bears within it typically eight ascospores.
* Herbs of genus '' Coreopsis'' (tickseed) have showy flower heads with involucral bracts in two distinct series of eight each.
* In human adult dentition there are eight teeth in each quadrant. The eighth tooth is the so-called wisdom tooth.
* There are eight cervical nerves on each side in man and most mammals.

Psychology

* There are eight Jungian cognitive functions, according to the MBTI models by John Beebe and Linda Berens.
* Timothy Leary identified a hierarchy of eight levels of consciousness.

In technology

* A byte is eight bits.
* Many (mostly historic) computer architectures are eight-bit, among them the Nintendo Entertainment System.
* Standard-8 and Super-8 are 8 mm film formats.
* Video8, Hi8 and Digital8 are related 8 mm video formats.
* On most phones, the 8 key is associated with the letters T, U, and V, but on the BlackBerry Pearl it is the key for B and N.
* An eight may refer to an eight-cylinder engine or automobile. A V8 engine is an internal combustion engine with eight cylinders configured in two banks (rows) of four forming a "V" when seen from the end.
* A figure-eight knot (so named for its configuration) is a kind of stopper knot.
* The number eight written in parentheses is the code for the musical note in Windows Live Messenger.
* In a seven-segment display, when an 8 is illuminated, all the display bulbs are on.

In measurement

* The SI prefix for 1000⁸ is yotta (Y), and for its reciprocal, yocto (y).
* In liquid measurement (United States customary units), there are eight fluid ounces in a cup, eight pints in a gallon and eight tablespoonfuls in a gill.
* There are eight furlongs in a mile.
* The clove, an old English unit of weight, was equal to eight pounds when measuring cheese.
* An eight may be an article of clothing of the eighth size.
* Force eight is the first wind strength attributed to a gale on the Beaufort scale when announced on a Shipping Forecast.

In culture

Currency

* Sailors and civilians alike from the 1500s onward referred to evenly divided parts of the Spanish dollar as "pieces of eight", or "bits".

Architecture

* Various types of buildings are usually eight-sided (octagonal), such as single-roomed gazebos and multi-roomed pagodas (descended from stupas; see religion section below).
* Eight caulicoles rise out of the leafage in a Corinthian capital, ending in leaves that support the volutes.

In religion, folk belief and divination

Hinduism

* Also known as Ashtha, Aṣṭa, or Ashta in Sanskrit, it is the number of wealth and abundance.
* The goddess of wealth and prosperity, Lakshmi, has eight forms known as Ashta Lakshmi and worshipped as:
"''Maha-lakshmi, Dhana-lakshmi, Dhanya-lakshmi, Gaja-lakshmi,
Santana-lakshmi, Veera-lakshmi, Vijaya-lakshmi and Vidhya-lakshmi''"
*There are eight ''nidhi'', or seats of wealth, according to Hinduism.
*There are eight guardians of the directions known as ''Astha-dikpalas''.
*There are eight Hindu monasteries established by the saint Madhvacharya in Udupi, India popularly known as the '' Ashta Mathas of Udupi''.

Buddhism

* The Dharmacakra, a Buddhist symbol, has eight spokes. The Buddha's principal teaching—the Four Noble Truths—ramifies as the Noble Eightfold Path and the Buddha emphasizes the importance of the eight attainments or jhanas.
* In Mahayana Buddhism, the branches of the Eightfold Path are embodied by the Eight Great Bodhisattvas: (Manjusri, Vajrapani, Avalokiteśvara, Maitreya, Ksitigarbha, Nivaranavishkambhi, Akasagarbha, and Samantabhadra (Bodhisattva), Samantabhadra). These are later (controversially) associated with the Eight Consciousnesses according to the Yogacara school of thought: consciousness in the five senses, thought-consciousness, self-consciousness, and unconsciousness-"consciousness" or "store-house consciousness" (alaya-vijñana). The "irreversible" state of enlightenment, at which point a Bodhisattva goes on "autopilot", is the Eight Ground or ''bhūmi''. In general, "eight" seems to be an auspicious number for Buddhists, e.g., the "eight auspicious symbols" (the jewel-encrusted parasol; the goldfish (always shown as a pair, e.g., the glyph of Pisces); the self-replenishing amphora; the white ''kamala'' lotus-flower; the white conch; the eternal (Celtic-style, infinitely looping) knot; the banner of imperial victory; the eight-spoked wheel that guides the ship of state, or that symbolizes the Buddha's teaching). Similarly, Buddha's birthday falls on the 8th day of the 4th month of the Chinese calendar.

Judaism

* The religious rite of brit milah (commonly known as circumcision) is held on a baby boy's eighth day of life.
* Hanukkah is an eight-day Jewish holiday that starts on the 25th day of Kislev.
* Shemini Atzeret ( Hebrew: "Eighth Day of Assembly") is a one-day Jewish holiday immediately following the seven-day holiday of Sukkot.

Christianity

* The spiritual Eighth Day, because the number 7 refers to the days of the week (which repeat themselves).
* The number of Beatitudes.
* 1 Peter 3:20 states that there were eight people on Noah's Ark.
* The Antichrist is the eighth king in the Book of Revelation.

Islam

* In Islam, eight is the number of angels carrying the throne of Allah in heaven.
* The number of gates of heaven

Taoism

* Ba Gua
* Ba Xian
* Ba Duan Jin

Other

* In Wicca, there are eight Sabbats, festivals, seasons, or spokes in the Wheel of the Year.
* In Ancient Egyptian mythology, the Ogdoad represents the eight primordial deities of creation.
* In Scientology there are eight dynamics of existence.
* There is also the Ogdoad in Gnosticism.

As a lucky number

* The number eight is considered to be a lucky number in Chinese and other Asian cultures. Eight (; accounting ; pinyin ''bā'') is considered a lucky number in Chinese culture because it sounds like the word meaning to generate wealth (; Pinyin: ''fā''). Property with the number 8 may be valued greatly by Chinese. For example, a Hong Kong number plate with the number 8 was sold for $640,000. The opening ceremony of the Summer Olympics in Beijing started at 8 seconds and 8 minutes past 8 pm (local time) on 8 August 2008.
* In Pythagorean numerology (a pseudoscience) the number 8 represents victory, prosperity and overcoming.
* is also considered a lucky number in Japan, but the reason is different from that in Chinese culture. Eight gives an idea of growing prosperous, because the letter () broadens gradually.
* The Japanese thought of as a holy number in the ancient times. The reason is less well-understood, but it is thought that it is related to the fact they used eight to express large numbers vaguely such as (literally, eightfold and twentyfold), (literally, eight clouds), (literally, eight millions of Gods), etc. It is also guessed that the ancient Japanese gave importance to pairs, so some researchers guess twice as , which is also guessed to be a holy number in those times because it indicates the world (north, south, east, and west) might be considered a very holy number.
* In numerology, 8 is the number of building, and in some theories, also the number of destruction.

In astrology

* In astrology, Scorpio is the 8th astrological sign of the Zodiac.
* In the Middle Ages, 8 was the number of "unmoving" stars in the sky, and symbolized the perfection of incoming planetary energy.

In music and dance

* A note played for one-eighth the duration of a whole note is called an eighth note, or quaver.
* An octave, the interval between two musical notes with the same letter name (where one has double the frequency of the other), is so called because there are eight notes between the two on a standard major or minor diatonic scale, including the notes themselves and without chromatic deviation. The ecclesiastical modes are ascending diatonic musical scales of eight notes or tones comprising an octave.
* There are eight notes in the octatonic scale.
* There are eight musicians in a double quartet or an octet. Both terms may also refer to a musical composition for eight voices or instruments.
* Caledonians is a square dance for eight, resembling the quadrille.
* Albums with the number eight in their title include ''8'' by the Swedish band Arvingarna, '' 8'' by the American rock band Incubus, '' The Meaning of 8'' by Minnesota indie rock band Cloud Cult and ''8ight'' by Anglo-American singer-songwriter Beatie Wolfe.
* Dream Theater's eighth album '' Octavarium'' contains many different references to the number 8, including the number of songs and various aspects of the music and cover artwork.
* "Eight maids a-milking" is the gift on the eighth day of Christmas in the carol " The Twelve Days of Christmas".
* The 8-track cartridge is a musical recording format.
* "#8" is the stage name of Slipknot vocalist Corey Taylor.
* "Too Many Eights" is a song by Athens, Georgia's Supercluster.
* Eight Seconds, a Canadian musical group popular in the 1980s with their most notable song "Kiss You (When It's Dangerous)".
* " Eight Days a Week" is a #1 single for the music group The Beatles.
* ''Figure 8'' is the fifth studio album by singer-songwriter Elliott Smith, released in the year 2000, an album released by Julia Darling in 1999, and an album released by Outasight in 2011.
* Ming Hao from the k-pop group Seventeen goes by the name "The8".
* "8 (circle)" is the eighth song on the album ''22, A Million'' by the American band Bon Iver.
* "8" is the eighth song on the album '' When We All Fall Asleep, Where Do We Go?'' by Billie Eilish.

In film and television

* '' 8 Guys'' is a 2003 short film written and directed by Dane Cook.
* '' 8 Man'' (or ''Eightman''): 1963 Japanese manga and anime superhero.
* ''8 Mile'' is a 2002 film directed by Curtis Hanson.
* ''8 mm'' is a 1999 film directed by Joel Schumacher.
* ''8 Women'' (Original French title: ) is a 2001 film directed by François Ozon.
* '' Eight Below'' is a 2006 film directed by Frank Marshall.
* '' Eight Legged Freaks'' is a 2002 film directed by Ellory Elkayem.
* '' Eight Men Out'' is a 1988 film directed by John Sayles.
* '' Jennifer Eight'', also known as ''Jennifer 8'', is a 1992 film written and directed by Bruce Robinson.
* '' Eight Is Enough'' is an American television comedy-drama series.
* In '' Stargate SG-1'' and '' Stargate Atlantis'', dialing an 8-chevron address will open a wormhole to another galaxy.
* '' The Hateful Eight'' is a 2015 American western mystery film written and directed by Quentin Tarantino.
* '' Kate Plus 8'' is an American reality television show.

In sports and other games

* Eight-ball pool is played with a cue ball and 15 numbered balls, the black ball numbered 8 being the middle and most important one, as the winner is the player or side that legally pockets it after first pocketing its numerical group of 7 object balls (for other meanings see '' Eight ball (disambiguation)'').
* In chess, each side has eight pawns and the board is made of 64 squares arranged in an eight by eight lattice. The eight queens puzzle is a challenge to arrange eight queens on the board so that none can capture any of the others.
* In the game of eights or Crazy Eights, each successive player must play a card either of the same suit or of the same rank as that played by the preceding player, or may play an eight and call for any suit. The object is to get rid of all one's cards first.
* In association football, the number 8 has historically been the number of the Central Midfielder.
* In Australian rules football, the top eight teams at the end of the Australian Football League regular season qualify for the finals series (i.e. playoffs).
* In baseball:
** The center fielder is designated as number 8 for scorekeeping purposes.
** The College World Series, the final phase of the NCAA Division I tournament, features eight teams.
* In rugby union, the only position without a proper name is the Number 8, a forward position.
* In rugby league:
** Most competitions (though not the Super League, which uses static squad numbering) use a position-based player numbering system in which one of the two starting props wears the number 8.
** The Australia-based National Rugby League has its own 8-team finals series, similar but not identical in structure to that of the Australian Football League.
* In rowing, an "eight" refers to a sweep-oar racing boat with a crew of eight rowers plus a coxswain.
* In the 2008 Games of the XXIX Olympiad held in Beijing, the official opening was on 08/08/08 at 8:08:08 p.m. CST.
* In Rock Climbing, climbers frequently use the Figure Eight knot to tie into their harnesses.

In foods

* Nestlé sells a brand of chocolates filled with peppermint-flavoured cream called After Eight, referring to the time 8 p.m.
* There are eight vegetables in V8 juice.

In literature

* Eights may refer to octosyllabic, usually iambic, lines of verse.
* The drott-kvaett, an Old Icelandic verse, consisted of a stanza of eight regular lines.
* In Terry Pratchett's '' Discworld'' series, eight is a magical number and is considered taboo. Eight is not safe to be said by wizards on the Discworld and is the number of Bel-Shamharoth. Also, there are eight days in a Disc week and eight colours in a Disc spectrum, the eighth one being octarine.
* Lewis Carroll's poem '' The Hunting of the Snark'' has 8 "fits" (cantos), which is noted in the full name "The Hunting of the Snark – ''An Agony, in Eight Fits''."
* Eight apparitions appear to Macbeth in Act 4 scene 1 of Shakespeare's ''Macbeth'' as representations of the eight descendants of Banquo.

In slang

* An "eighth" is a common measurement of marijuana, meaning an eighth of an ounce. It is also a common unit of sale for psilocybin mushrooms.
* Avril Lavigne's song " Sk8er Boi" uses this convention in the title.
* "Section 8" is common U.S. slang for "crazy", based on the U.S. military's Section 8 discharge for mentally unfit personnel.
* The Housing Choice Voucher Program, operated by the United States Department of Housing and Urban Development, is commonly referred to as the Section 8 program, as this was the original section of the Act which instituted the program.
* In Colombia and Venezuela, "volverse un ocho" (meaning to tie oneself in a figure 8) refers to getting in trouble or contradicting oneself.
* In China, "8" is used in chat speak as a term for parting. This is due to the closeness in pronunciation of "8" (bā) and the English word "bye".

See also

* The Magical Number Seven, Plus or Minus Two
* List of highways numbered 8

References

External links

The Octonions
John C. Baez

{{DEFAULTSORT:8 (Number)
Integers
8 (number)