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# 9  ( Integers )

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9 ( nine ) is the following and preceding . Nine is the highest one-digit number in a system of numbers using base 10.

Mathematics
9 is a , its proper being and . It is 3 times 3 and hence the third . Nine is a . It is the first composite , along with the first composite odd number and only single-digit composite odd number.

9 is the only positive that is one more than another positive perfect power, by Mihăilescu's Theorem.

9 is the highest single-digit number in the . It is the second non-unitary square of the form ( p2) and the first that is odd. All subsequent squares of this form are odd.

Since , 9 is an exponential factorial.

A with nine sides is called a or enneagon.Robert Dixon, Mathographics. New York: Courier Dover Publications: 24 A group of nine of anything is called an ennead.

In a positive number is divisible by 9 if and only if its is 9., A Gardner's Workout: Training the Mind and Entertaining the Spirit. New York: A. K. Peters (2001): 155 That is, if any 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 of have this property. Another consequence of 9 being , is that it is also a .

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.

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

Probability
In , the nine is a logarithmic measure of the probability of an event, defined as the negative of the base- 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. 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 of 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 of no more than five minutes per year – typically a very high degree of ; but never 100%.

List of basic calculations
9 × x9182736455463728190 !1802254509009000

9 ÷ x94.532.251.81.51.1.12510.9 !0.0.750.0.60.6
'x'' ÷ 90.0.0.0.0.0.0.0.11. !1.1.1.1.1.

99817296561590495314414782969430467213874204893486784401
x151219683262144195312510077696403536071342177283874204891000000000

'x''1511916922927933944955966977988911091219
1329143915491659176924293079615913319146419162151917836619

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 .
(1985). 9780670373956, Viking.

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 -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 @ character 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 9 character has an ascender in most modern , in typefaces with the character usually has a , 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 lower-case letter q.

Alphabets and codes
• In the NATO phonetic alphabet, the digit 9 is called "Niner".
• Five-digit PLU codes that begin with 9 are .

Commerce
• Common terminal digit in psychological pricing.

Culture and mythology

Indian culture
Nine is a number that appears often in and mythology. Some instances are enumerated below.
• Nine are attested in .
• In the branch of , there are nine universal substances or elements: , Water, , , , Time, Space, Soul, and Mind.
• is a nine-day festival dedicated to the forms of .
• , meaning "nine jewels" may also refer to – accomplished courtiers, Navratan – a kind of dish, or a form of architecture.
• According to , 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 jiǔ) is considered a good number in Chinese culture because it sounds the same as the word "long-lasting" (久 jiǔ).
• Nine is strongly associated with the , 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 (, , )
(2018). 9781417964291, Kessinger. .
as well as having the same of 9.
• The dragon often symbolizes the Emperor, and the number nine can be found in many ornaments in the .
• 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 in literally means: nine .
• The () delimits certain island claims by China in the South China Sea.
• The was a civil service nomination system used during certain Chinese dynasties.
• 9 Points of the () / Heart Master () Channels in Traditional Chinese Medicine.

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

European culture

Greek mythology

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

Mayan mythology
• The Mayan underworld 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 .

Anthropology

Idioms
• "to go the whole nine yards-"
• "A cat-o'-nine-tails suggests perfect punishment and atonement." – .
• "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 police departments to denote the 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 , "nine" is a word for a 9mm or , the latter from the Illinois Criminal Code for homicide.

Society
• The 9 on Yahoo!, hosted by , 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 .
• 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
(1996). 9780415130950, Routledge. .
and a 1988 fiction book by Anthea Fraser;
(1988). 9780385243230, Doubleday.
the name derives from "a very curious old semi-pagan, semi-Christian" song.
• The Nine Tailors is a 1934 by 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 , there are nine rings of power given to men, and consequently, nine . Additionally, The Fellowship of the Ring consists of nine companions.
• In there are nine Garde sent to Earth.
• Number Nine is a character in .
• 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).

Organizations
• 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
• List of highways numbered 9
• Ninth Avenue is a major avenue in Manhattan.

Religion and philosophy
• Nine, as the highest single-digit number (in ), symbolizes completeness in the Bahá'í Faith. In addition, the word Bahá' in the has a value of 9, and a 9-pointed 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 system, which originated from the Indian subcontinent as early as 3000 BC.
• In Buddhism, , was believed to have nine virtues, which he was (1) Accomplished, (2) Perfectly Enlightened, (3) Endowed with knowledge and Conduct or Practice, (4) Well-gone or Well-spoken, (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 of Av are collectively known as "The Nine Days" ( Tisha HaYamim), and are a period of semi-mourning 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 . 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 .
• Tian's Trigram Number, of Feng Shui, in .

Science

Astronomy

Chemistry

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

Sports
• is the standard professional pocket 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 :
• There are nine players on the field including the pitcher.
• There are nine innings in a standard game.
• 9 represents the 's position.
• In , the jersey number assigned to the hooker in most competitions. (An exception is the European , which uses static squad numbering.)
• In , the number worn by the starting scrum-half.

Technology
• ISO 9 is the ISO's standard for the transliteration of characters into characters
• In the Rich Text Format specification, 9 is the language code for the . All codes for regional variants of English are congruent to 9 mod 256.
• The seven-segment 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 video-game industry, including former ties to the extremely popular 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.

• 9 (disambiguation)
• 0.999...

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

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