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8 (number)

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 Title: 8 (number) Author: World Heritage Encyclopedia Language: English Subject: Collection: Integers Publisher: World Heritage Encyclopedia Publication Date:

8 (number)

 ← 7 8 9 →
Cardinal eight
Ordinal 8th
(eighth)
Numeral system octal
Factorization 23
Divisors 1, 2, 4, 8
Roman numeral VIII
Roman numeral (unicode) Ⅷ, ⅷ
Greek prefix octa-/oct-
Latin prefix octo-/oct-
Binary 10002
Ternary 223
Quaternary 204
Quinary 135
Senary 126
Octal 108
Duodecimal 812
Vigesimal 820
Base 36 836
Greek η (or Η)
Arabic ٨,8
Urdu
Amharic
Bengali
Chinese numeral 八，捌
Devanāgarī
Telugu
Tamil
Hebrew ח (Het)
Hebrew שמונה (shmoneh)
Khmer
Korean
Thai

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

Contents

• In mathematics 1
• In numeral systems 1.1
• List of basic calculations 1.2
• Etymology 2
• Glyph 3
• In science 4
• Physics 4.1
• Astronomy 4.2
• Chemistry 4.3
• Geology 4.4
• Biology 4.5
• In technology 5
• In measurement 5.1
• In culture 6
• Architecture 6.1
• In religion, folk belief and divination 6.2
• As a lucky number 6.2.1
• In astrology 6.2.2
• In music and dance 6.3
• In film and television 6.4
• In sports and other games 6.5
• In foods 6.6
• In literature 6.7
• In slang 6.8
• In other fields 6.9
• References 8

In mathematics

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 2^3 (two cubed), and is the first number of the form p^3, p being an integer greater than 1. It is also the first number which is neither prime nor semiprime. 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.

All powers of 2 ;(2^x), have an aliquot sum of one less than themselves.

A number is divisible by 8 if its last 3 digits are also divisible by 8.

Eight is the first number to be the aliquot sum of two numbers other than itself; the discrete biprime 10, and the square number 49.

8 is the base of the octal number system, which is mostly used with computers. In octal, one digit represents 3 bits. In modern computers, a byte is a grouping of eight bits, also called an octet.

The number 8 is 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.[1]

8 is the only nonzero perfect power that is one less than another perfect power, by Mihăilescu's Theorem.

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.

There are a total of eight convex deltahedra.

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.

A cube has eight vertices.

Sphenic numbers always have exactly eight divisors.

8 is the dimension of the octonions and is the highest possible dimension of a normed division algebra.

The number 8 is involved with a number of interesting mathematical phenomena related to the notion of Bott periodicity. For example, 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 then \pi_{k+8}(O(\infty))\cong\pi_{k}(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 E8 lattice. Even positive definite unimodular lattice 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.

In numeral systems

Base Numeral system Representation
2 binary 1000
3 ternary 22
4 quaternary 20
5 quinary 13
6 senary 12
7 septenary 11
8 octal 10
over 8 (decimal, duodecimal, hexadecimal, etc.) 8

List of basic calculations

Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
8 \times 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 \div 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 \div 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

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 sibling delivered in one birth.

The Semitic numeral is based on a root *θmn-, whence Akkadian smn-, Arabic ṯmn-, Hebrew šmn- etc. The Chinese numeral (pinyin ) is from an Old Chinese *priāt-, ultimately from Sino-Tibetan b-g-ryat or b-r-gyat which also yielded Tibetan brgyat.

It has been argued that, as the cardinal number seven is the highest number of item 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");[2] 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 nine, which might be built on the stem new-, meaning "new" (indicating the beginning of a "new set of numerals" after having counted to eight).[3]

Glyph

The modern 8 glyph, like all modern Hindu-Arabic numerals (other than zero) originates with the Brahmi numerals. The Brahmi numeral 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 eight glyph used in India in the early centuries of the Common Era developed considerable 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 numeral ८; the alternative curved glyph also existed as a variant in Perso-Arabic tradition, where it came to look similar to our glyph 5.

The numerals 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 numerals, 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.[4]

Just as in most modern typefaces, in typefaces with text figures the 8 character usually has an ascender, as, for example, in .

The infinity symbol ∞, described as "sideways figure eight" is unrelated to the 8 glyph 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.

The numeral eight in Greek numerals, developed in Classical Greece by the 5th century BC, was written as Η, the eight letter of the Greek alphabet.

The Chinese numeral eight is written in two strokes, 八; the glyph is also the 12th Kangxi radical.

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. 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.
• Timothy Leary identified a hierarchy of eight levels of consciousness.
• 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.

In culture

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

Buddhism
The 8-spoked Dharmacakra represents the Noble Eightfold Path
• 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). 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' (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
Islam
Other

As a lucky number

• The number eight is considered to be a lucky number in Chinese and other Asian cultures.[6] 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.[7] 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.[8]
• Eight ( hachi, ya) is also considered a lucky number in Japanese culture, but the reason is different from that in Chinese culture. Eight gives an idea of growing prosperous, because the letter (八) broadens gradually.
• The Japanese thought 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.
• 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 and The Meaning of 8 by Minnesota indie rock band Cloud Cult
• 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 stagename 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.

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 cooking recipes, there are approximately 8 pinches to a teaspoon

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. Also, an eighth of an ounce of cocaine is commonly referred to as an "8-ball."[12]
• Referring to the shape of the numeral, eight was formerly represented in bingo slang as "One Fat Lady". Eighty-eight was "Two Fat Ladies".
• The numeral "8" is sometimes used in informal writing and Internet slang to represent the syllable "ate", as in writing "H8" for "hate", or "congratul8ions" for "congratulations". 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 one's self.
• 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'.
• Eight is symbolic for lesbian sexual relations.

In other fields

Four playing cards showing the "8" of all four suits
• Eight, West Virginia, a community in the United States
• A double quatrefoil has eight foils used as the cadency mark of a ninth son.
• A duke's coronet, worn by a British or Irish duke, has eight conventional strawberry leaves upon the rim of the circlet.
• Eight extra facets are sometimes added around the culet at the lower pyramid of a brilliant diamond.
• I-8 is the designation of the US interstate highway that runs from San Diego, California to Casa Grande, Arizona
• Eight is the number of categories the VALS system uses to classify consumer groups, and the number of categories used by Fallon-McElligott's system for teen marketing
• War of the Eight Princes, a war in Chinese history
• "88" is the abbreviated terminology used by the Aryan Brotherhood for the Nazi salute, "Heil Hitler" – "H" being the eighth letter of the alphabet, twice
• The silver piece of eight was coined in the Spanish Empire and moved trade around the world. Examples include the Carolus dollar, cob and cut. A doubloon was worth 8 gold escudos or 16 pieces of eight.
• Eight elementary grades are included in the eight-four plan of school organization (along with four secondary grades).
• There are Eight Principles of Yong in Chinese calligraphy
• 8vo is shorthand for "octavo", a book and paper size.
• The Eight - Eight American painters who exhibited together only once in 1908 in New York City. They joined this exhibition to oppose traditions upheld by the National Academy and help advance modernism in the United States. Five of the eight painters were associated with the Ashcan School: Robert Henri (1865–1929), George Luks (1867–1933), William Glackens (1870–1938), John Sloan (1871–1951), and Everett Shinn (1876–1953), along with Maurice Prendergast (1859–1924), Ernest Lawson (1873–1939), and Arthur Bowen Davies (1862–1928).

References

1. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 88
2. ^ Etymological Dictionary of Turkic Languages: Common Turkic and Interturkic stems starting with letters «L», «M», «N», «P», «S», Vostochnaja Literatura RAS, 2003, 241f. (altaica.ru)
3. ^ the hypothesis is discussed critically (and rejected as "without sufficient support") by Werner Winter, 'Some thought about Indo-European numerals' in: Jadranka Gvozdanović (ed.), Indo-European Numerals, Walter de Gruyter, 1992, 14f.
4. ^ Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 395, Fig. 24.68.
5. ^ "Life Application New Testament Commentary", Bruce B. Barton. Tyndale House Publishers, Inc., 2001. ISBN 0-8423-7066-8, ISBN 978-0-8423-7066-0. p. 1257
6. ^ Ang, Swee Hoon (1997). "Chinese consumers’ perception of alpha-numeric brand names". Journal of Consumer Marketing 14 (3): 220–233.
7. ^ Steven C. Bourassa, Vincent S. Peng (1999). "Hedonic Prices and House Numbers: The Influence of Feng Shui" (PDF). International Real Estate Review 2 (1): 79–93.
8. ^ "Patriot games: China makes its point with greatest show" by Richard Williams, The Guardian, published 9 August 2008
9. ^ Barney's burp song
10. ^ Anaheim Ducks retire Teemu Selanne's No. 8
11. ^ A to Z Encyclopaedia of Ice Hockey
12. ^ http://www.thegooddrugsguide.com/cocaine/faq.htm