9 (nine) is the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
following and preceding .
Evolution of the Arabic digit
![Evo9glyph](https://upload.wikimedia.org/wikipedia/commons/0/05/Evo9glyph.svg)
In the
beginning, various Indians wrote a digit 9 similar in shape to the modern closing
question mark
The question mark (also known as interrogation point, query, or eroteme in journalism) is a punctuation mark that indicates an interrogative clause or phrase in many languages.
History
In the fifth century, Syriac Bible manuscripts used ques ...
without the bottom dot. The Kshatrapa, Andhra and Gupta started curving the bottom vertical line coming up with a -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
typeface
A typeface (or font family) is the design of lettering that can include variations in size, weight (e.g. bold), slope (e.g. italic), width (e.g. condensed), and so on. Each of these variations of the typeface is a font.
There are list of type ...
s, in typefaces with
text figures
Text figures (also known as non-lining, lowercase, old style, ranging, hanging, medieval, billing, or antique figures or numerals) are numerals designed with varying heights in a fashion that resembles a typical line of running text, hence the ...
the character usually has a
descender
In typography and handwriting, a descender is the portion of a letter that extends below the baseline of a font.
For example, in the letter ''y'', the descender is the "tail", or that portion of the diagonal line which lies below the ''v'' c ...
, as, for example, in
![TextFigs196](https://upload.wikimedia.org/wikipedia/commons/5/58/TextFigs196.png)
.
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 a
seven-segment display
A seven-segment display is a form of electronic display device for displaying decimal numerals that is an alternative to the more complex dot matrix displays.
Seven-segment displays are widely used in digital clocks, electronic meters, basic ...
, the number 9 can be constructed in two ways, either with a hook at the end of its stem or without one. Most
LCD
A liquid-crystal display (LCD) is a flat-panel display or other electronically modulated optical device that uses the light-modulating properties of liquid crystals combined with polarizers. Liquid crystals do not emit light directly but in ...
calculators use the former, but some
VFD
VFD may refer to:
* Factory Workers' Union of Germany, (german: Verband der Fabrikarbeiter Deutschlands, link=no), a former trade union in Germany
* Vacuum fluorescent display, a display device on consumer electronics equipment
* Variable-freque ...
models use the latter.
Mathematics
Nine is the fourth
composite number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
, and the first composite number that is odd. 9 is the highest single-digit number in the
decimal system. It is the third
square number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
(3
2), and the second non-unitary square
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
of the form ''p''
2 and first that is odd, with all subsequent squares of this form odd as well.
By
Mihăilescu's theorem
Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 are ...
, 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
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
of 2.
A number that is 4 or 5
modulo
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is t ...
9 cannot be represented as the
sum of three cubes
In the mathematics of sums of powers, it is an open problem to characterize the numbers that can be expressed as a sum of three cubes of integers, allowing both positive and negative cubes in the sum. A necessary condition for n to equal such a s ...
.
![MotzkinChords4](https://upload.wikimedia.org/wikipedia/commons/4/4a/MotzkinChords4.svg)
9 is a
Motzkin number, for the number of ways of drawing non-intersecting
chords between four points on a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
.
Since , 9 is an
exponential factorial
The exponential factorial is a positive integer ''n'' raised to the power of ''n'' − 1, which in turn is raised to the power of ''n'' − 2, and so on and so forth in a right-grouping manner. That is,
: n^
The expon ...
.
Six recurring nines appear in the decimal places 762 through 767 of
. (See
six nines in pi
A sequence of six consecutive nines occurs in the decimal representation of the number pi (), starting at the 762nd decimal place.. It has become famous because of the mathematical coincidence and because of the idea that one could memorize the ...
).
The first non-trivial
magic square
In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number ...
is a
x
magic square made of nine cells, with a
magic constant
The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. For a normal magic square of order ''n'' – that is ...
of 15; there are no
x
magic squares with four cells. Meanwhile, a
x
magic square has a magic constant of
369
__NOTOC__
Year 369 (Roman numerals, CCCLXIX) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Galates and Victor (or, less frequentl ...
.
A
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
with nine sides is called a
nonagon. Also an ''enneagon'', it is able to fill a
plane-vertex alongside an
equilateral triangle and an regular
octadecagon
In geometry, an octadecagon (or octakaidecagon) or 18-gon is an eighteen-sided polygon.
Regular octadecagon
A '' regular octadecagon'' has a Schläfli symbol and can be constructed as a quasiregular truncated enneagon, t, which alternates two ...
, or 18-sided
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
.
There are nine distinct
uniform colorings of the
triangular tiling
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...
and the
square tiling, which are the two simplest
regular tilings
This article lists the regular polytopes and regular polytope compounds in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces.
The Schläfli symbol describes every regular tessellation of an ' ...
; the
hexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling).
English mathemat ...
, on the other hand, has three distinct uniform colorings.
There are nine
edge-transitive convex polyhedra in
three dimensions
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
:
*the five
regular Platonic solids: the
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
,
octahedron,
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
,
dodecahedron and
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
;
*the two
quasiregular Archimedean solid
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
s: the
cuboctahedron and the
icosidodecahedron; and
*two
Catalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865.
The Catalan sol ...
s: the
rhombic dodecahedron and the
rhombic triacontahedron, which are
duals to the only two quasiregular polyhedra.
In
four-dimensional space
A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...
, there are nine
paracompact hyperbolic honeycomb
Coxeter groups, as well as nine
regular compact hyperbolic honeycombs from regular
convex and
star
A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
''
polychora''. There are also nine uniform
demitesseractic (
)
Euclidean honeycombs in the fourth dimension.
There are only three types of
Coxeter groups
In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, H. S. M. Coxeter, is an group (mathematics), abstract group that admits a group presentation, formal description in terms of Reflection (mathematics), reflections (or Kal ...
of
uniform figures in dimensions
nine and thereafter, aside from the many families of
prisms and
proprisms: the
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
groups, the
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...
groups, and the
demihypercube
In geometry, demihypercubes (also called ''n-demicubes'', ''n-hemicubes'', and ''half measure polytopes'') are a class of ''n''-polytopes constructed from alternation of an ''n''-hypercube, labeled as ''hγn'' for being ''half'' of the hype ...
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
paracompact groups is the group
with
1023
Year 1023 ( MXXIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* The Judge-Governor of Seville in Al-Andalus (modern Spain) takes advantage of the disinte ...
total honeycombs, the simplest of which is
621 whose
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
is the
521 honeycomb: the vertex arrangement of the densest-possible packing of spheres in
8 dimensions which forms the
lattice. The 6
21 honeycomb is made of
9-simplexes and
9-orthoplex
In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells ''4-faces'', 5376 5-simplex ''5-faces'', 4608 6-simplex ''6-faces'', 2304 7-simplex '' ...
es, with 1023 total
polytope elements making up each 9-simplex. It is the final honeycomb figure with infinite facets and vertex figures in the k
21 family of
semiregular polytopes, first defined by
Thorold Gosset
John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, and ...
in 1900.
There are nine
Heegner number In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer ''d'' such that the imaginary quadratic field \Q\left sqrt\right/math> has class number 1. Equivalently, its ring of integers has unique factoriza ...
s, or
square-free positive integers
that yield an imaginary
quadratic field