9 ( nine ) is the natural number following and preceding . Nine is the highest onedigit number.
Mathematics
9 is a
composite number, its proper
being and . It is 3 times 3 and hence the third
square number. Nine is a
Motzkin number.
It is the first composite
lucky number, along with the first composite odd number and only singledigit composite odd number.
9 is the only positive perfect power that is one more than another positive perfect power, by Mihăilescu's Theorem.
9 is the highest singledigit number in the decimal. It is the second nonunitary square prime number of the form ( p^{2}) and the first that is odd. All subsequent squares of this form are odd.
Since , 9 is an exponential factorial.
A polygon with nine sides is called a nonagon or enneagon.[Robert Dixon, Mathographics. New York: Courier Dover Publications: 24] A group of nine of anything is called an ennead.
In decimal a positive number is divisible by 9 if and only if its digital root is 9.[Martin Gardner, A Gardner's Workout: Training the Mind and Entertaining the Spirit. New York: A. K. Peters (2001): 155] 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 n. In base N, the of have this property. Another consequence of 9 being , is that it is also a Kaprekar number.
The difference between a base10 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.
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.[Cajori, Florian (1991, 5e) A History of Mathematics, AMS. . p.91]
Six recurring nines appear in the decimal places 762 through 767 of , see Six nines in pi.
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. )
There are nine .[Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 93]
Numeral systems
Probability
In
probability, the
nine is a logarithmic measure of the probability of an event, defined as the negative of the base
logarithm of the probability of the event's complement. For example, an event that is 99% likely to occur has an unlikelihood of 1% or 0.01, which amounts to of probability.
Zero probability gives zero nines . A 100% probability is considered to be impossible in most circumstances: that results in infinite improbability. The effectivity of processes and the
availability of
systems can be expressed (as a rule of thumb, not explicitly) as a series of "nines". For example, "five nines" (99.999%) availability implies a total
downtime of no more than five minutes per year – typically a very high degree of
reliability; but never 100%.
List of basic calculations


9 × x  9  18  27  36  45  54  63  72  81  90
!  180  225  450  900  9000 


9 ÷ x  9  4.5  3  2.25  1.8  1.5  1.  1.125  1  0.9
!  0.  0.75  0.  0.6  0.6 
'x'' ÷ 9  0.  0.  0.  0.  0.  0.  0.  0.  1  1.
!  1.  1.  1.  1.  1. 


9  9  81  729  6561  59049  531441  4782969  43046721  387420489  3486784401 
x  1  512  19683  262144  1953125  10077696  40353607  134217728  387420489  1000000000 



'x''  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} 
Evolution of the glyph
According to Georges Ifrah, the origin of the 9 integers can be attributed to ancient Indian civilization, and was adopted by subsequent civilizations in conjunction with the .
In the beginning, various Indians wrote 9 similar 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 lookalike. The Nagari continued the bottom stroke to make a circle and enclose the 3lookalike, in much the same way that the @ character encircles a lowercase a. As time went on, the enclosing circle became bigger and its line continued beyond the circle downwards, as the 3lookalike became smaller. Soon, all that was left of the 3lookalike 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 9 character has an ascender in most modern , in typefaces with text figures the character usually has a descender, as, for example, in
.
This numeral resembles an inverted 6. To disambiguate the two on objects and documents that can be inverted, the 9 is often underlined, as is done for the 6. Another distinction from the 6 is that it is sometimes handwritten with a straight stem, resembling a raised lowercase letter q.
Alphabets and codes

In the NATO phonetic alphabet, the digit 9 is called "Niner".

Fivedigit produce PLU codes that begin with 9 are Organic food.
Commerce

Common terminal digit in psychological pricing.
Culture and mythology
Indian culture
Nine is a number that appears often in
Indian culture and mythology. Some instances are enumerated below.

Nine Navagraha are attested in Indian astrology.

In the Vaisheshika branch of Hindu philosophy, there are nine universal substances or elements: Prithvi, Water, Vayu, Agni, Akasha, Time, Space, Soul, and Mind.

Navaratri is a nineday festival dedicated to the Navadurga forms of Durga.

Navaratna, meaning "nine jewels" may also refer to Navaratnas – accomplished courtiers, Navratan – a kind of dish, or a form of architecture.

According to Yoga, the human body has nine doors – two eyes, two ears, the mouth, two nostrils, and the openings for defecation and procreation.

In Indian aesthetics, there are nine kinds of Rasa.
Chinese culture

Nine (九 pinyin jiǔ) is considered a good number in Chinese culture because it sounds the same as the word "longlasting" (久 pinyin jiǔ).

Nine is strongly associated with the Chinese dragon, a symbol of magic and power. There are nine forms of the dragon, it is described in terms of nine attributes, and it has nine children. It has 117 scales – 81 yang (masculine, heavenly) and 36 yin (feminine, earthly). All three numbers are multiples of 9 (, , )
as well as having the same digital root of 9.

The dragon often symbolizes the Emperor, and the number nine can be found in many ornaments in the Forbidden City.

The circular altar platform ( Earthly Mount) of the Temple of Heaven has one circular marble plate in the center, surrounded by a ring of nine plates, then by a ring of 18 plates, and so on, for a total of nine rings, with the outermost having

The name of the area called Kowloon in Hong Kong literally means: nine Chinese dragon.

The ninedotted line () delimits certain island claims by China in the South China Sea.

The ninerank system was a civil service nomination system used during certain Chinese dynasties.

9 Points of the Heart (Healing) / Heart Master (Immortality) Channels in Traditional Chinese Medicine.
Ancient Egypt

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
Greek mythology

The nine in Greek mythology are Calliope (epic poetry), Clio (history), Erato (erotic poetry), Euterpe (lyric poetry), Melpomene (tragedy), Polyhymnia (song), Terpsichore (dance), Thalia (comedy), and Urania (astronomy).

It takes nine days (for an anvil) to fall from heaven to earth, and nine more to fall from earth to Tartarus – a place of torment in the Greek underworld.

Leto labored for nine days and nine nights for Apollo, according to the Homeric Hymn.
Mesoamerican mythology

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

Mictlan the underwold in Aztec mythology, consists of nine levels.
Mayan mythology

The Mayan underworld Xibalba consists of nine levels.

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

"to go the whole nine yards"

"A cato'ninetails 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 "K9" pronounces the same as canine and is used in many United States 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.

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

The 9 on Yahoo!, hosted by Maria Sansone, was a daily video compilation show, or vlog, on Yahoo! featuring the nine top "web finds" of the day.

Nine justices sit on the United States Supreme Court.
Technique
Literature

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
and a 1988 fiction book by Anthea Fraser; the name derives from "a very curious old semipagan, semiChristian" song.

The Nine Tailors is a 1934 mystery novel by United Kingdom 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 Middleearth, there are nine rings of power given to men, and consequently, nine Nazgul. 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 citystates in western Essos known collectively as the Free Cities (Braavos, Lorath, Lys, Myr, Norvos, Pentos, Qohor, Tyrosh and Volantis).
Organizations

Divine Nine – The National PanHellenic Council (NPHC) is a collaborative organization of nine historically African American, international Greeklettered fraternities and sororities.
Places and thoroughfares

List of highways numbered 9

Ninth Avenue is a major avenue in Manhattan.
Religion and philosophy

Nine, as the highest singledigit number (in decimal), symbolizes completeness in the Bahá'í Faith. In addition, the word Bahá' in the Abjad numerals has a value of 9, and a 9pointed star is used to symbolize the religion.

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.

In Buddhism, Gautama Buddha, was believed to have nine virtues, which he was (1) Accomplished, (2) Perfectly Enlightened, (3) Endowed with knowledge and Conduct or Practice, (4) Wellgone or Wellspoken, (5) the Knower of worlds, (6) the Guide Unsurpassed of men to be tamed, (7) the Teacher of gods and men, (8) Enlightened, and (9) Blessed.

Important Buddhist rituals usually involve nine monks.

The first nine days of the Hebrew calendar of Av are collectively known as "The Nine Days" ( Tisha HaYamim), and are a period of semimourning leading up to Tisha B'Av, the ninth day of Av on which both Temples in Jerusalem were destroyed.

Nine is a significant number in Norse Mythology. Odin hung himself on an ash tree for nine days to learn the runes.

The Fourth Way Enneagram is one system of knowledge which shows the correspondence between the 9 integers and the circle.

In the Christian angelic hierarchy there are 9 choirs of angels.

Ramadan, the month of fasting and prayer, is the ninth month of the Islamic calendar.

Tian's Trigram Number, of Feng Shui, in Taoism.
Science
Astronomy
Chemistry
Physiology
A human
pregnancy normally lasts nine months, the basis of Naegele's rule.
Sports

Nineball is the standard professional pocket billiards variant played in the United States.

In association football (soccer), the centreforward/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

In rugby league, the jersey number assigned to the hooker in most competitions. (An exception is the European Super League, which uses static squad numbering.)

In rugby union, the number worn by the starting scrumhalf.
Technology

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.

The sevensegment display allows the number 9 to be constructed 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.

The9 Limited (owner of the9.com) is a company in the videogame industry, including former ties to the extremely popular MMORPG World of Warcraft.
Music

"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".

There are 9 semitones in a Major 6th interval in music.
See also

9 (disambiguation)

0.999...

Further reading

Cecil Balmond, "Number 9, the search for the sigma code" 1998, Prestel 2008, ,