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9 (nine) is the natural number following and preceding .

Evolution of the Arabic digit

In the beginning, 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 -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 a seven-segment display, the number 9 can be constructed in two ways, 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.

Mathematics

Nine is the fourth composite number, 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 (3²), and the second non-unitary square prime of the form ''p''² and first that is odd, with all subsequent squares of this form odd as well.

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.

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

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

Since , 9 is an exponential factorial.

Six recurring nines appear in the decimal places 762 through 767 of . (See six nines in pi).

The first non-trivial magic square is a $3$ x $3$ magic square made of nine cells, with a magic constant of 15; there are no $2$ x $2$ magic squares with four cells. Meanwhile, a $9$ x $9$ magic square has a magic constant of 369.

A polygon 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, or 18-sided polygon.

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 nine 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.

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''. There are also nine uniform demitesseractic ($\mathrm D_$) 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 $\mathrm A_$ simplex groups, the $\mathrm B_$ hypercube groups, and the $\mathrm D_$ 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 $_9$ paracompact groups is the group $_9$ 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 $\mathbb E_$ 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.

There are nine Heegner numbers, or square-free positive integers $n$ that yield an imaginary quadratic field \Q\left sqrt\right/math> whose ring of integers has a unique factorization, or class number of 1.

In decimal

A positive number is divisible by nine if and only if its digital root is nine. That is, if any natural number is multiplied by 9, and the digits of the answer are repeatedly added until it is just one digit, the sum will be nine:
*2 × 9 = 18 (1 + 8 = 9)
*3 × 9 = 27 (2 + 7 = 9)
*9 × 9 = 81 (8 + 1 = 9)
*121 × 9 = 1089 (1 + 0 + 8 + 9 = 18; 1 + 8 = 9)
*234 × 9 = 2106 (2 + 1 + 0 + 6 = 9)
*578329 × 9 = 5204961 (5 + 2 + 0 + 4 + 9 + 6 + 1 = 27; 2 + 7 = 9)
*482729235601 × 9 = 4344563120409 (4 + 3 + 4 + 4 + 5 + 6 + 3 + 1 + 2 + 0 + 4 + 0 + 9 = 45; 4 + 5 = 9)

There are other interesting patterns involving multiples of nine:
*12345679 × 9 = 111111111
*12345679 × 18 = 222222222
*12345679 × 81 = 999999999

This works for all the multiples of 9. is the only other such that a number is divisible by ''n'' if and only if its digital root is divisible by ''n''. In base-''N'', the divisors of have this property. Another consequence of 9 being , is that it is also a Kaprekar number.

The difference between a base-10 positive integer and the sum of its digits is a whole multiple of nine. Examples:
*The sum of the digits of 41 is 5, and 41 − 5 = 36. The digital root of 36 is 3 + 6 = 9, which, as explained above, demonstrates that it is divisible by nine.
*The sum of the digits of 35967930 is 3 + 5 + 9 + 6 + 7 + 9 + 3 + 0 = 42, and 35967930 − 42 = 35967888. The digital root of 35967888 is 3 + 5 + 9 + 6 + 7 + 8 + 8 + 8 = 54, 5 + 4 = 9.

If dividing a number by the amount of 9s corresponding to its number of digits, the number is turned into a repeating decimal. (e.g. )

Casting out nines is a quick way of testing the calculations of sums, differences, products, and quotients of integers known as long ago as the 12th century.

List of basic calculations

Alphabets and codes

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

Culture and mythology

Indian culture

Nine is a number that appears often in Indian culture and mythology. Some instances are enumerated below.
*Nine influencers are attested in Indian astrology.
*In the Vaisheshika branch of Hindu philosophy, there are nine universal substances or elements: Earth, Water, Air, Fire, Ether, Time, Space, Soul<