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Why is 9 the perfect number?

The number 9 is revered in Hinduism and considered a complete, perfected and divine number because it represents the end of a cycle in the decimal system, which originated from the Indian subcontinent as early as 3000 BC.

en.wikipedia.org - 9 - Wikipedia
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Natural number

9 (nine) is the natural number following 8 and preceding 10.

Evolution of the Arabic digit [ edit ]

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 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 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 [ edit ]

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 p2 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. 9 is the number of derangements of 4, or the number of permutations of four elements with no fixed points.[1] A number that is 4 or 5 modulo 9 cannot be represented as the sum of three cubes.[2] 9 is the number of non-intersecting segments between four points on a is the number of non-intersecting segments between four points on a circle

9 is an exponential factorial, since 9 = 321.[3]

Six recurring nines appear in the decimal places 762 through 767 of π. (See six nines in pi). The first non-trivial magic square is a 3 {displaystyle 3} x 3 {displaystyle 3} magic square made of nine cells, with a magic constant of 15.[4] Meanwhile, a 9 {displaystyle 9} x 9 {displaystyle 9} magic square has a magic constant of 369.[5] 9 is a Motzkin number, for the number of ways of drawing non-intersecting chords between four points on a circle.[6] A polygon with nine sides is called a nonagon.[7] 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:

Nine distinct stellation's by Miller's rules are produced by the truncated tetrahedron.[8] It is the simplest Archimedian solid, with a total of four 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.[9] There are also nine uniform demitesseractic ( D 4 {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 A n {displaystyle mathrm {A} _{n}} simplex groups, the B n {displaystyle mathrm {B} _{n}} hypercube groups, and the D n {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 E ~ 9 {displaystyle { ilde {E}}_{9}} paracompact groups is the group T ~ 9 {displaystyle { ilde {T}}_{9}} with 1023 total honeycombs, the simplest of which is 6 21 whose vertex figure is the 5 21 honeycomb: the vertex arrangement of the densest-possible packing of spheres in 8 dimensions which forms the E 8 {displaystyle mathbb {E} _{8}} lattice. The 6 21 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 k 21 family of semiregular polytopes, first defined by Thorold Gosset in 1900. There are nine Heegner numbers, or square-free positive integers n {displaystyle n} that yield an imaginary quadratic field Q [ − n ] {displaystyle mathbb {Q} left[{sqrt {-n}} ight]} whose ring of integers has a unique factorization, or class number of 1.[10]

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In decimal [ edit ]

A positive number is divisible by nine if and only if its digital root is nine.[11] 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. n = 3 is the only other n > 1 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 N − 1 have this property. Another consequence of 9 being 10 − 1, 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. 274/999 = 0.274274274274...) 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.[12]

List of basic calculations [ edit ]

Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 20 25 50 100 1000 9 × x 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 180 225 450 900 9000 Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 9 ÷ x 9 4.5 3 2.25 1.8 1.5 1. 285714 1.125 1 0.9 0. 81 0.75 0. 692307 0.6 428571 0.6 x ÷ 9 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 1 1. 1 1. 2 1. 3 1. 4 1. 5 1. 6 Exponentiation 1 2 3 4 5 6 7 8 9 10 9x 9 81 729 6561 59049 531441 4782969 43046721 387420489 3486784401 x9 1 512 19683 262144 1953125 10077696 40353607 134217728 387420489 1000000000 Radix 1 5 10 15 20 25 30 40 50 60 70 80 90 100 110 120 130 140 150 200 250 500 1000 10000 100000 1000000 x 9 1 5 11 9 16 9 22 9 27 9 33 9 44 9 55 9 66 9 77 9 88 9 110 9 121 9 132 9 143 9 154 9 165 9 176 9 242 9 307 9 615 9 1331 9 14641 9 162151 9 1783661 9

Alphabets and codes [ edit ]

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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 [ edit ]

Indian culture [ edit ]

Nine is a number that appears often in Indian culture and mythology. Some instances are enumerated below.

Chinese culture [ edit ]

Ancient Egypt [ edit ]

The nine bows is a term used in Ancient Egypt to represent the traditional enemies of Egypt. The Ennead is a group of nine Egyptian deities, who, in some versions of the Osiris myth, judged whether Horus or Set should inherit Egypt.

European culture [ edit ]

Greek mythology [ edit ]

Mesoamerican mythology [ edit ]

The Lords of the Night, is a group of nine deities who each ruled over every ninth night forming a calendrical cycle

Aztec mythology [ edit ]

Mictlan the underworld in Aztec mythology, consists of nine levels.

Mayan mythology [ edit ]

The Mayan underworld Xibalba consists of nine levels.

El Castillo the Mayan step-pyramid in Chichén Itzá, consists of nine steps. It is said that this was done to represent the nine levels of Xibalba.

Anthropology [ edit ]

Idioms [ edit ]

"to go the whole nine yards-"

"A cat-o'-nine-tails suggests perfect punishment and atonement." – Robert Ripley.

"A cat has nine lives"

"to be on cloud nine"

"A stitch in time saves nine"

"found true 9 out of 10 times"

"possession is nine tenths of the law"

The word "K-9" pronounces the same as canine and is used in many US police departments to denote the police dog unit. Despite not sounding like the translation of the word canine in other languages, many police and military units around the world use the same designation. and is used in many US police departments to denote the police dog unit. Despite not sounding like the translation of the word in other languages, many police and military units around the world use the same designation. Someone dressed "to the nines" is dressed up as much as they can be. In North American urban culture, "nine" is a slang word for a 9mm pistol or homicide, the latter from the Illinois Criminal Code for homicide.

Society [ edit ]

Technique [ edit ]

Stanines, a method of scaling test scores, range from 1 to 9.

There are 9 square feet in a square yard.

Pseudoscience [ edit ]

In Pythagorean numerology the number 9 symbolizes the end of one cycle and the beginning of another.

Literature [ edit ]

Organizations [ edit ]

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

Places and thoroughfares [ edit ]

Religion and philosophy [ edit ]

Science [ edit ]

Astronomy [ edit ]

Chemistry [ edit ]

The purity of chemicals (see Nine (purity)).

Nine is the atomic number of fluorine.

Physiology [ edit ]

A human pregnancy normally lasts nine months, the basis of Naegele's rule.

Psychology [ edit ]

Common terminal digit in psychological pricing.

Sports [ edit ]

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

Technology [ edit ]

Music [ edit ]

See also [ edit ]

References [ edit ]

Further reading [ edit ]

en.wikipedia.org - 9 - Wikipedia
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