Natural number

Look up nine in Wiktionary, the free dictionary. 
9 (nine) is the natural number following 8 and preceding 10.
Mathematics
9 is a composite number, its proper divisors being 1 and 3. It is 3 times 3 and hence the third square number. Nine is a Motzkin number.^{[1]} 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 system. It is the second nonunitary square prime of the form (p^{2}) and the first that is odd. All subsequent squares of this form are odd.
Since 9 = 3^{21}, 9 is an exponential factorial.^{[2]}
A polygon with nine sides is called a nonagon or enneagon.^{[3]} A group of nine of anything is called an ennead.
In base 10 a positive number is divisible by 9 if and only if its digital root is 9.^{[4]} 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 baseN, 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 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.^{[5]}
Six recurring nines appear in the decimal places 762 through 767 of π, 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. 274/999 = 0.274274274274...)
There are nine Heegner numbers.^{[6]}
Probability
In probability, the nine is a logarithmic measure of the probability of an event, defined as the negative of the base10 logarithm of the probability of the event's complement.^{[citation needed]} For example, an event that is 99% likely to occur has an unlikelihood of 1% or 0.01, which amounts to −log_{10} 0.01 = 2 nines of probability. Zero probability gives zero nines (−log_{10} 1 = 0). 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
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

0.6428571

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

9^{x}

9

81

729

6561

59049

531441

4782969

43046721

387420489

3486784401

x^{9}

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}

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 0.^{[7]}
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 3lookalike. 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 typefaces, 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
Commerce
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, and Mind.
 Navaratri is a nineday festival dedicated to the nine forms of Durga.
 Navaratna, meaning "nine jewels" may also refer to Navaratnas – accomplished courtiers, Navratan – a kind of dish, or a form of architecture.
 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).^{[citation needed]}
 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 (9 × 13 = 117, 9 × 9 = 81, 9 × 4 = 36)^{[8]} 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 81 = 9 × 9 plates.
 The name of the area called Kowloon in Hong Kong literally means: nine dragons.
 The ninedotted line (Chinese: 南海九段线; pinyin: nánhǎi jiǔduàn xiàn; literally: 'Ninesegment line of the South China Sea') 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 (Heal) / 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 Muses 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.
 Leto labored for nine days and nine nights for Apollo, according to the Homeric Hymn to Delian Apollo.
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 underworld 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 U.S. 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
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^{[9]} and a 1988 fiction book by Anthea Fraser;^{[10]} the name derives from "a very curious old semipagan, semiChristian" song.^{[11]}
 The Nine Tailors is a 1934 mystery novel by British 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 ringwraiths. 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
Religion and philosophy
 Nine, as the highest singledigit number (in base ten), symbolizes completeness in the Bahá'í Faith. In addition, the word Bahá' in the Abjad notation 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 month 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.
 In Christianity there are nine Fruit of the Holy Spirit which followers are expected to have: love, joy, peace, patience, kindness, goodness, faithfulness, gentleness, and selfcontrol.
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^{[12]}
 In rugby league, the jersey number assigned to the hooker in most competitions. (An exception is the 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
See also
References
Further reading