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# 8

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

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## 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, 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.[1]
• 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.

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.

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. 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. Figure-eight turns of a rope or cable around a cleat, pin, or bitt are used to belay something.

### 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 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, 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 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 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).[3]

## Glyph

Evolution of the numeral 8 from the Indians to the Europeans

The modern 8 glyph, like all modern 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 ); the alternative curved glyph also existed as a variant in Perso-Arabic tradition, where it came to look similar to our glyph 5.[year needed]

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 a "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 eighth letter of the Greek alphabet.

The Chinese numeral eight is written in two strokes, ; 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.

## 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

In Buddhism, the 8-spoked Dharmacakra represents the Noble Eightfold Path
• 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). 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.

#### Islam

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

#### 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 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 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 sports and other games

An 8-ball in billiards

### 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.