9

Circa 300 BCE, as part of the Brahmi numerals, various Indians wrote a digit 9 similar in shape to the modern closing question mark without the bottom dot. The Kshatrapa, Andhra and Gupta started curving the bottom vertical line coming up with a 3-look-alike. The Nagari continued the bottom stroke to make a circle and enclose the 3-look-alike, in much the same way that the sign @ encircles a lowercase a. As time went on, the enclosing circle became bigger and its line continued beyond the circle downwards, as the 3-look-alike became smaller. Soon, all that was left of the 3-look-alike was a squiggle. The Arabs simply connected that squiggle to the downward stroke at the middle and subsequent European change was purely cosmetic.

While the shape of the glyph for the digit 9 has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in .

The modern digit resembles an inverted 6. To disambiguate the two on objects and documents that can be inverted, they are often underlined. Another distinction from the 6 is that it is sometimes handwritten with two strokes and a straight stem, resembling a raised lower-case letter q. In seven-segment display, the number 9 can be constructed either with a hook at the end of its stem or without one. Most LCD calculators use the former, but some VFD models use the latter.

Nine is the fourth composite number, and the first composite number that is odd. Nine is the third square number (3²), and the second non-unitary square prime of the form p², and, the first that is odd, with all subsequent squares of this form odd as well. Nine has the even aliquot sum of 4, and with a composite number sequence of two (9, 4, 3, 1, 0) within the 3-aliquot tree. There are nine Heegner numbers, or square-free positive integers ${\displaystyle n}$ that yield an imaginary quadratic field ${\displaystyle \mathbb {Q} \left[{\sqrt {-n}}\right]}$ whose ring of integers has a unique factorization, or class number of 1.[1] By Mihăilescu's theorem, 9 is the only positive perfect power that is one more than another positive perfect power, since the square of 3 is one more than the cube of 2.[2][3]

9 is the sum of the cubes of the first two non-zero positive integers ${\displaystyle 1^{3}+2^{3}}$ which makes it the first cube-sum number greater than one.[4] It is also the sum of the first three nonzero factorials ${\displaystyle 1!+2!+3!}$ and equal to the third exponential factorial, since ${\displaystyle 9=3^{2^{1}}.}$[5]

Non-intersecting chords between four points on a circle

Nine is the number of derangements of 4, or the number of permutations of four elements with no fixed points.[6]

9 is the fourth refactorable number, as it has exactly three positive divisors, and 3 is one of them.[7]

A number that is 4 or 5 modulo 9 cannot be represented as the sum of three cubes.[8]

If an odd perfect number exists, it will have at least nine distinct prime factors.[9]

9 is a Motzkin number, for the number of ways of drawing non-intersecting chords between four points on a circle.[10]

Four concentric magic circles with 9 in the center (by Yang Hui), where numbers on each circle and diameter around the center generate a magic sum of 138.

The first non-trivial magic square is a ${\displaystyle 3}$ x ${\displaystyle 3}$ magic square made of nine cells, with a magic constant of 15.[11] Meanwhile, a ${\displaystyle 9}$ x ${\displaystyle 9}$ magic square has a magic constant of 369.[12]

A polygon with nine sides is called a nonagon.[13] Since 9 can be written in the form ${\displaystyle 2^{m}3^{n}p}$, for any nonnegative natural integers ${\displaystyle m}$ and ${\displaystyle n}$ with ${\displaystyle p}$ a product of Pierpont primes, a regular nonagon can be constructed with a regular compass, straightedge, and angle trisector.[14]

Also an enneagon, a regular nonagon is able to fill a plane-vertex alongside an equilateral triangle and a regular 18-sided octadecagon (3.9.18), and as such, it is one of only nine polygons that are able to fill a plane-vertex without uniformly tiling the plane.[15]

There are nine distinct uniform colorings of the triangular tiling and the square tiling, which are the two simplest regular tilings; the hexagonal tiling, on the other hand, has three distinct uniform colorings.

There are a maximum of nine semiregular Archimedean tilings by convex regular polygons, when including chiral forms of the snub hexagonal tiling.

There are nine uniform edge-transitive convex polyhedra in three dimensions:

• the five regular Platonic solids: the tetrahedron, octahedron, cube, dodecahedron and icosahedron;
• the two quasiregular Archimedean solids: the cuboctahedron and the icosidodecahedron; and
• two Catalan solids: the rhombic dodecahedron and the rhombic triacontahedron, which are duals to the only two quasiregular polyhedra.

Nine distinct stellation's by Miller's rules are produced by the truncated tetrahedron.[16] It is the simplest Archimedean solid, with a total of four equilateral triangular and four hexagonal faces.

In four-dimensional space, there are nine paracompact hyperbolic honeycomb Coxeter groups, as well as nine regular compact hyperbolic honeycombs from regular convex and star polychora.[17] There are also nine uniform demitesseractic (${\displaystyle \mathrm {D} _{4}}$) Euclidean honeycombs in the fourth dimension.

There are only three types of Coxeter groups of uniform figures in dimensions nine and thereafter, aside from the many families of prisms and proprisms: the ${\displaystyle \mathrm {A} _{n}}$ simplex groups, the ${\displaystyle \mathrm {B} _{n}}$ hypercube groups, and the ${\displaystyle \mathrm {D} _{n}}$ demihypercube groups. The ninth dimension is also the final dimension that contains Coxeter-Dynkin diagrams as uniform solutions in hyperbolic space. Inclusive of compact hyperbolic solutions, there are a total of 238 compact and paracompact Coxeter-Dynkin diagrams between dimensions two and nine, or equivalently between ranks three and ten. The most important of the last ${\displaystyle {\tilde {E}}_{9}}$ paracompact groups is the group ${\displaystyle {\tilde {T}}_{9}}$ with 1023 total honeycombs, the simplest of which is 6₂₁ whose vertex figure is the 5₂₁ honeycomb: the vertex arrangement of the densest-possible packing of spheres in 8 dimensions which forms the ${\displaystyle \mathbb {E} _{8}}$ lattice. The 6₂₁ honeycomb is made of 9-simplexes and 9-orthoplexes, with 1023 total polytope elements making up each 9-simplex. It is the final honeycomb figure with infinite facets and vertex figures in the k₂₁ family of semiregular polytopes, first defined by Thorold Gosset in 1900.

• In the NATO phonetic alphabet, the digit 9 is called "Niner".
• Five-digit produce PLU codes that begin with 9 indicate organic foods.

• There are nine circles of Hell in Dante's Divine Comedy.
• The Nine Bright Shiners, characters in Garth Nix's Old Kingdom trilogy. The Nine Bright Shiners was a 1930s book of poems by Anne Ridler[22] and a 1988 fiction book by Anthea Fraser;[23] the name derives from "a very curious old semi-pagan, semi-Christian" song.[24]
• The Nine Tailors is a 1934 mystery novel by British writer Dorothy L. Sayers, her ninth featuring sleuth Lord Peter Wimsey.
• Nine Unknown Men are, in occult legend, the custodians of the sciences of the world since ancient times.
• In J. R. R. Tolkien's Middle-earth, there are nine rings of power given to men, and consequently, nine ringwraiths. Additionally, The Fellowship of the Ring consists of nine companions.
• In Lorien Legacies there are nine Garde sent to Earth.
• Number Nine is a character in Lorien Legacies.
• In the series A Song of Ice and Fire, there are nine regions of Westeros (the Crownlands, the North, the Riverlands, the Westerlands, the Reach, the Stormlands, the Vale of Arryn, the Iron Islands and Dorne). Additionally, there is a group of nine city-states in western Essos known collectively as the Free Cities (Braavos, Lorath, Lys, Myr, Norvos, Pentos, Qohor, Tyrosh and Volantis).
• In The Wheel of Time series, Daughter of the Nine Moons is the title given to the heir to the throne of Seanchan, and the Court of the Nine Moons serves as the throne room of the Seanchan rulers themselves. Additionally, the nation of Illian is partially governed by a body known as the Council of Nine, and the flag of Illian displays nine golden bees on it. Furthermore, in the Age of Legends, the Nine Rods of Dominion were nine regional governors who administered individual areas of the world under the ruling world government.

• Divine Nine – The National Pan-Hellenic Council (NPHC) is a collaborative organization of nine historically African American, international Greek-lettered fraternities and sororities.

• List of highways numbered 9
• Ninth Avenue is a major avenue in Manhattan.
• South Africa has 9 provinces
• Negeri Sembilan, a Malaysian state located in Peninsular Malaysia, is named as such as it was historically a confederation of nine (Malay: sembilan) settlements (nagari) of the Minangkabau migrated from West Sumatra.

Billiards: A Nine-ball rack with the no. 9 ball at the center

• Nine-ball is the standard professional pocket billiards variant played in the United States.
• In association football (soccer), the centre-forward/striker traditionally (since at least the fifties) wears the number 9 shirt.
• In baseball:
  • There are nine players on the field including the pitcher.
  • There are nine innings in a standard game.
  • 9 represents the right fielder's position.
  • NINE: A Journal of Baseball History and Culture, published by the University of Nebraska Press[29]
• In rugby league, the jersey number assigned to the hooker in most competitions. (An exception is the Super League, which uses static squad numbering.)
• In rugby union, the number worn by the starting scrum-half.

• ISO 9 is the ISO's standard for the transliteration of Cyrillic characters into Latin characters
• In the Rich Text Format specification, 9 is the language code for the English language. All codes for regional variants of English are congruent to 9 mod 256.
• The9 Limited (owner of the9.com) is a company in the video-game industry, including former ties to the extremely popular MMORPG World of Warcraft.

• "Revolution 9", a sound collage which appears on The Beatles' eponymous 1968 album The Beatles (aka The White Album), prominently features a loop of a man's voice repeating the phrase "Number nine".[30]
• There are 9 semitones in a Major 6th interval in music.[31]
• There was a superstition among some notable classical music composers that they would die after completing their ninth symphony. Some composers who died after composing their ninth symphony include Ludwig van Beethoven, Anton Bruckner, Antonin Dvorak and Gustav Mahler.[32]
• Beethoven's Symphony No. 9 is regarded as a masterpiece, and one of the most frequently performed symphonies in the world.

Look up nine in Wiktionary, the free dictionary.

• 9 (disambiguation)
• 0.999...
• Cloud Nine
• List of highways numbered 9

• Cecil Balmond, "Number 9, the search for the sigma code" 1998, Prestel 2008, ISBN 3-7913-1933-7, ISBN 978-3-7913-1933-9