# 8

Short description: Natural number
 ← 7 8 9 →
Cardinaleight
Ordinal8th
(eighth)
Numeral systemoctal
Factorization23
Divisors1, 2, 4, 8
Greek numeralΗ´
Roman numeralVIII, viii
Greek prefixocta-/oct-
Latin prefixocto-/oct-
Binary10002
Ternary223
Quaternary204
Quinary135
Senary126
Octal108
Duodecimal812
Vigesimal820
Base 36836
Greekη (or Η)
Arabic, Kurdish, Persian, Sindhi, Urdu٨
Amharic
Bengali
Chinese numeral八,捌
Devanāgarī
Malayalam
Telugu
Tamil
Hebrewח
Khmer
Thai
ArmenianԸ ը

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

## In mathematics

8 is:

• a composite number, its proper divisors being 1, 2, and 4. It is twice 4 or four times 2.
• a power of two, being 23 (two cubed), and is the first number of the form p3, p being an integer greater than 1.
• the first number which is neither prime nor semiprime.
• the base of the octal number system,[1] 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 3 plus 5. The next Fibonacci number is 13. 8 is the only positive Fibonacci number, aside from 1, that is a perfect cube.[2]
• 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 10, and the square number 49.

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.[3] A regular octagon (or truncated square) has a central angle of 45 degrees and an internal angle angle of 135 degrees, and is therefore able to tessellate two-dimensional space alongside squares in the truncated square tiling.[4] The sides and span of a regular octagon 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.[5] In particular, the quotient of successive Pell numbers generates rational approximations for coordinates of a regular octagon.[6] The Ammann–Beenker tiling features prominent octagonal silver eightfold symmetry.[7] In number theory, figurate numbers representing octagons are called octagonal numbers.[8]

A cube is a regular polyhedron with 8 vertices that forms a cubic honeycomb in three-dimensional space.[9] Through various truncation operations, the cubic honeycomb generates 7 other convex uniform honeycombs, altogether forming 8 honeycombs under the group $\displaystyle{ {\tilde{C}}_3 }$.[10] The dual polyhedron to the cube is the octahedron, with 8 equilateral triangles as faces.[11] It is one of 8 convex deltahedra.[12][13] The cuboctahedron, one of only two convex quasiregular polyhedra, has 8 equilateral triangles as faces (alongside 6 squares); it is equal to a rectified cube or rectified octahedron, and its first stellation is the cube-octahedron compound.[14][15]

Vertex-transitive semiregular polytopes 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. There are no other finite semiregular polytopes in dimensions n > 8.

Sphenic numbers always have exactly eight divisors.[16]

The number 8 is involved with a number of interesting mathematical phenomena related to the notion of Bott periodicity. For example, if O(∞) is the direct limit of the inclusions of real orthogonal groups

$\displaystyle{ O(1)\hookrightarrow O(2)\hookrightarrow\ldots\hookrightarrow O(k)\hookrightarrow\ldots }$,

then

$\displaystyle{ \pi_{k+8}(O(\infty))\cong\pi_{k}(O(\infty)) }$.

Clifford algebras also display a periodicity of 8.[17] 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 E8 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.[18] Figure-eight turns of a rope or cable around a cleat, pin, or bitt are used to belay something.[19]

### List of basic calculations

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
8x 8 64 512 4096 32768 262144 2097152 16777216 134217728 1073741824 8589934592 68719476736 549755813888
x8 1 256 6561 65536 390625 1679616 5764801 16777216 43046721 100000000 214358881 429981696 815730721

## 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 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 (Mandarin: ; 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");[20] 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).[21]

## Evolution of the Arabic digit

Evolution of the numeral 8 from the Indians to the Europeans

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.[22]

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

### 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.[30] Orb-weaver spiders of the cosmopolitan family Areneidae have eight similar eyes.[31]
• 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.[32]
• Sea anemones of genus Edwardsia have eight mesenteries.[33]
• Animals of phylum Ctenophora swim by means of eight meridional bands of transverse ciliated plates, each plate representing a row of large modified cilia.[34]
• 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.[35]
• 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.[36] The eighth tooth is the so-called wisdom tooth.
• There are eight cervical nerves on each side in man and most mammals.[37]

### Psychology

• There are eight Jungian cognitive functions, according to the MBTI models by John Beebe and Linda Berens.[38]
• Timothy Leary identified a hierarchy of eight levels of consciousness.

## In technology

NATO signal flag for 8
• A byte is eight bits.[39]
• Many (mostly historic) computer architectures are eight-bit, among them the Nintendo Entertainment System.
• Standard-8 and Super-8 are 8 mm film formats.[40]
• Video8, Hi8 and Digital8 are related 8 mm video formats.[41]
• 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.[42] 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.[43]
• 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.[44]
• There are eight furlongs in a mile.[45]
• The clove, an old English unit of weight, was equal to eight pounds when measuring cheese.[46]
• 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.[47]

## 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
"[48]
• There are eight nidhi, or seats of wealth, according to Hinduism.
• There are eight guardians of the directions known as Astha-dikpalas.[49]
• There are eight Hindu monasteries established by the saint Madhvacharya in Udupi, India popularly known as the Ashta Mathas of Udupi.[50]

#### Buddhism

In Buddhism, the 8-spoked Dharmacakra represents the Noble Eightfold Path
• The Dharmacakra, a Buddhist symbol, has eight spokes.[51] 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).[52] 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.[53]
• Hanukkah is an eight-day Jewish holiday that starts on the 25th day of Kislev.[54]
• Shemini Atzeret (Hebrew: "Eighth Day of Assembly") is a one-day Jewish holiday immediately following the seven-day holiday of Sukkot.[55]

#### Christianity

• The spiritual Eighth Day, because the number 7 refers to the days of the week (which repeat themselves).
• The number of Beatitudes.[56]
• 1 Peter 3:20 states that there were eight people on Noah's Ark.[57]
• The Antichrist is the eighth king in the Book of Revelation.[58]

#### Other

• In Wicca, there are eight Sabbats, festivals, seasons, or spokes in the Wheel of the Year.[62]
• In Ancient Egyptian mythology, the Ogdoad represents the eight primordial deities of creation.[63]
• In Scientology there are eight dynamics of existence.[64]
• There is also the Ogdoad in Gnosticism.[65]

#### As a lucky number

• The number eight is considered to be a lucky number in Chinese and other Asian cultures.[66] Eight (; accounting ; pinyin ) is considered a lucky number in Chinese culture because it sounds like the word meaning to generate wealth (發(T) 发(S); Pinyin: ). 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.[67] 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.[68]
• In Pythagorean numerology (a pseudoscience) the number 8 represents victory, prosperity and overcoming.
• Eight ( hachi, ya) is also considered a lucky number in Japan , but the reason is different from that in Chinese culture.[69] Eight gives an idea of growing prosperous, because the letter () broadens gradually.
• The Japanese thought of eight ( ya) 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 manyfold (やえはたえ Yae Hatae) (literally, eightfold and twentyfold), many clouds (やくも Yakumo) (literally, eight clouds), millions and millions of Gods (やおよろずのかみ Yaoyorozu no Kami) (literally, eight millions of Gods), etc. It is also guessed that the ancient Japanese gave importance to pairs, so some researchers guess twice as four ( yo), 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 music and dance

• A note played for one-eighth the duration of a whole note is called an eighth note, or quaver.[71]
• 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.[72] 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.[73] Both terms may also refer to a musical composition for eight voices or instruments.[74]
• 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,[75] The Meaning of 8 by Minnesota indie rock band Cloud Cult and 8ight by Anglo-American singer-songwriter Beatie Wolfe.[76]
• 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".[77]
• 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.[78]
• Eight Seconds, a Canadian musical group popular in the 1980s with their most notable song "Kiss You (When It's Dangerous)".[79]
• "Eight Days a Week" is a #1 single for the music group The Beatles.[80]
• Figure 8 is the fifth studio album by singer-songwriter Elliott Smith, released in the year 2000,[81] an album released by Julia Darling in 1999,[82] and an album released by Outasight in 2011.[83]
• Ming Hao from the k-pop group Seventeen goes by the name "The8".[84]
• "8 (circle)" is the eighth song on the album 22, A Million by the American band Bon Iver.[85]
• "8" is the eighth song on the album When We All Fall Asleep, Where Do We Go? by Billie Eilish.[86]

### 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.[87]
• 8mm is a 1999 film directed by Joel Schumacher.[88]
• 8 Women (Original French title: 8 femmes) is a 2001 film directed by François Ozon.[89]
• Eight Below is a 2006 film directed by Frank Marshall.[90]
• Eight Legged Freaks is a 2002 film directed by Ellory Elkayem.[91]
• Eight Men Out is a 1988 film directed by John Sayles.[92]
• Jennifer Eight, also known as Jennifer 8, is a 1992 film written and directed by Bruce Robinson.[93]
• 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.[94]
• Kate Plus 8 is an American reality television show.[95]

### In sports and other games

An 8-ball in pool
• 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.[96]
• 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.[97]
• 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.[98]
• In Terry Pratchett's Discworld series, eight is a magical number[99] 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."[100]
• 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.[101]
• 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".

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

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26. Choppin, Gregory R.; Johnsen, Russell H. (1972) (in en). Introductory chemistry. Addison-Wesley Pub. Co.. pp. 366. ISBN 9780201010220. "under normal conditions the most stable allotropic form (Fig. 23-8a). Sulfur molecules within the crystal consist of puckered rings of eight sulfur atoms linked by single..."
27. Puri, Basant; Hall, Anne (1998-12-16) (in en). Phytochemical Dictionary: A Handbook of Bioactive Compounds from Plants, Second Edition. CRC Press. pp. 810. ISBN 978-0-203-48375-6. "The chemical structure of lycopene consists of a long chain of eight isoprene units joined head to tail"
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53. Little, Stephen; Eichman, Shawn; Shipper, Kristofer; Ebrey, Patricia Buckley (2000-01-01) (in en). Taoism and the Arts of China. University of California Press. pp. 139. ISBN 978-0-520-22785-9. "Evidence for the early use of the Eight Trigrams in a religious Taoist..."
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