9

This article is about the number. For other uses, see 9 (disambiguation).

9
Template:Numbers (digits)
Cardinal	9 nine
Ordinal	9th ninth
Numeral system	nonary
Factorization	$3^{2}$
Divisors	1, 3, 9
Amharic	፱
Roman numeral	IX
Roman numeral (Unicode)	Ⅸ, ⅸ
prefixes	ennea- (from Greek) nona- (from Latin)
Binary	1001
Octal	11
Duodecimal	9
Hexadecimal	9
Arabic-Indic numeral	٩
Armenian numeral	Թ
Bengali	৯
Chinese/Japanese numeral	九 玖 (formal writing)
Devanāgarī	९
Greek numeral	θ´
Hebrew numeral	ט (Tet)
Tamil numeral	௯
Khmer	៩
Thai numeral	๙

9 (nine) is the natural number following 8 and preceding 10. The ordinal adjective is ninth.

Mathematics

Nine is a composite number, its proper divisors being 1 and 3. It is 3 times 3 and hence the third square number. 9 is a Motzkin number. It is the first composite lucky number.

Nine is the highest single-digit number in the decimal system. It is the second non-unitary square prime of the form (p²) and the first that is odd. All subsequent squares of this form are odd. It has a unique aliquot sum 4 which is itself a square prime. 9 is; and can be, the only square prime with an aliquot sum of the same form. The aliquot sequence of 9 has 5 members (9,4,3,1,0) this number being the second composite member of the 3-aliquot tree.

There are nine Heegner numbers.

Since $9=3^{2^{1}}$, 9 is an exponential factorial.

8 and 9 form a Ruth-Aaron pair under the second definition that counts repeated prime factors as often as they occur.

A polygon with nine sides is called an enneagon (technically) or nonagon (in common usage). A group of nine of anything is called an ennead.

In base 10 a number is evenly divisible by nine if and only if its digital root is 9. That is, if you multiply nine by any whole number (except zero), and repeatedly add the digits of the answer until it is just one digit, you will end up with 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))
• (Exception) 0 x 9 = 0 (0 is not equal to 9)(though it can be considered -1 in the tens place and 10 in the ones place(10-1=9))

The only other number with this property is three. 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 evenly 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.

Subtracting two base-10 positive integers that are transpositions of each other yields a number that is a whole multiple of nine. Some examples:

• 41-14 = 27. The digital root of 27 is 2+7 = 9.
• 36957930-35967930 = 990000, which is obviously a multiple of nine.

This works regardless of the number of digits that are transposed. For example, the largest transposition of 35967930 is 99765330 (all digits in descending order) and its smallest transposition is 03356799 (all digits in ascending order); subtracting pairs of these numbers produces:

• 99765330-35967930 = 63797400; 6+3+7+9+7+4+0+0 = 36, 3+6 = 9.
• 99765330-03356799 = 96408531; 9+6+4+0+8+5+3+1 = 36.
• 35967930-03356799 = 32611131; 3+2+6+1+1+1+3+1 = 18, 1+8 = 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.

The Nine Chapters on the Mathematical Art is a Chinese mathematics book, composed by scholars between the 10th century BC, and the 1st century AD; it is the one of the earliest surviving mathematical text from China.

Every prime in a Cunningham chain of the first kind with a length of 4 or greater is congruent to 9 mod 10 (the only exception being the chain 2, 5, 11, 23, 47).

Six recurring nines appear in the decimal places 762 through 767 of pi. This is known as the Feynman point.

If an odd perfect number is of the form 36k + 9, it has at least nine distinct prime factors.

Nine is the binary complement of number six:

9 = 1001
6 = 0110

Probability

In probability, the nine is a logarithmic measure of probability of an event, defined as the negative of the base-10 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 −log₁₀ 0.01 = 2 nines of probability. Zero probability gives zero nines (−log₁₀ 1 = 0). The effectivity of processes and the availability of systems can be expressed in nines. For example, "five nines" (99.999%) availability implies a total downtime of no more than five minutes per year.

Numeral systems

Base	Numeral system
2	binary	1001
3	ternary	100
4	quaternary	21
5	quinary	14
6	senary	13
7	septenary	12
8	octal	11
9	novenary	10
over 9 (decimal, hexadecimal)		9

List of basic calculations

Multiplication	1	2	3	4	5	6	7	8	9	10		11	12	13	14	15	16	17	18	19	20		21	22	23	24	25		50	100	1000
$9\times x$	9	18	27	36	45	54	63	72	81	90		99	108	117	126	135	144	153	162	171	180		189	198	207	216	225		450	900	9000

Division	1	2	3	4	5	6	7	8	9	10		11	12	13	14	15
$9\div x$	9	4.5	3	2.25	1.6	1.5	$1.{\overline {285714}}$	1.125	1	0.9		$0.{\overline {81}}$	0.75	$0.{\overline {692307}}$	$0.6{\overline {428571}}$	0.6
$x\div 9$	$0.{\overline {1}}$	$0.{\overline {2}}$	$0.{\overline {3}}$	$0.{\overline {4}}$	$0.{\overline {5}}$	$0.{\overline {6}}$	$0.{\overline {7}}$	$0.{\overline {8}}$	1	$1.{\overline {1}}$		$1.{\overline {2}}$	$1.{\overline {3}}$	$1.{\overline {4}}$	$1.{\overline {5}}$	$1.{\overline {6}}$

Exponentiation	1	2	3	4	5	6	7	8	9	10		11	12	13
$9^{x}\,$	9	81	729	6561	59049	531441	4782969	43046721	387420489	3486784401		31381059609	282429536481	2541865828329
$x^{9}\,$	1	512	19683	262144	1953125	10077696	40353607	134217728	387420489	1000000000		2357947691	5159780352	10604499373

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

See also: Hindu-Arabic numeral system

According to Georges Ifrah, the origin of the 9 integers can be attributed to the ancient Indian civilization, and was adopted by subsequent civilizations in conjunction with the 0.

In the beginning, various Indians wrote 9 similar to the modern closing question mark without the bottom dot. The Kshtrapa, 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 @ 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 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 has often been underlined as is done for the 6. Another distinction from the 6 is that it is often handwritten with a straight stem.

Science

Chemistry

• The purity of chemicals (see Nine (purity))
• Nine is the atomic number of fluorine.

Astronomy

• Before 2006 (when Pluto was officially designated as a non-planet), there were nine planets in the solar system.
• Messier object M9 is a magnitude 9.0 globular cluster in the constellation Ophiuchus.
• The New General Catalogue object NGC 9, a spiral galaxy in the constellation Pegasus

• The Saros number of the solar eclipse series which began on -2568 February 6 and ended on -1252 April 4. The duration of Saros series 9 was 1316.2 years, and it contained 74 solar eclipses.
• The Saros number of the lunar eclipse series which began on -2501 June 26 and ended on -1149 September 16. The duration of Saros series 9 was 1352.2 years, and it contained 76 lunar eclipses.

Physiology

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

Religion and philosophy

• 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.
• 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 semi-mourning leading up to Tisha B'Av, the ninth day of Av on which both Temples in Jerusalem were destroyed.
• Nine, as the highest single-digit 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 9-pointed star is used to symbolize the religion.
• 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.
• Anton LaVey applied the number to Satan.
• Ramadan, the month of fasting and prayer, is the ninth month of the Islamic calendar.

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 9×13 scales, 9×9 being yang (masculine, or bad influence) and 9×4 being yin (feminine, or good influence).

• 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 nine-rank system was a civil service nomination system used during certain Chinese dynasties.

Sports

• Nine ball is the standard professional pocket billiards 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.
• The car number of the 1988 NASCAR Nextel Cup championship-winning team, when Bill Elliott won the title. He used this number again from 2001 to 2003. Kasey Kahne has driven the 9 car since 2004.
• In baseball, nine represents the right fielder's position.
• The retired uniform numbers of former baseball players Ted Williams and Roger Maris. Reggie Jackson also wore 9 with the Oakland A's for nine years and is now retired.
• The San Francisco 49ers are sometimes referred to as the "niners".
• The number worn by current NBA player Tony Parker of the San Antonio Spurs, MVP of the 2007 NBA Finals.
• The number worn by Roy Hobbs in the movie The Natural.
• In rugby union, the number generally worn by the scrum half.
• In rugby league, the number generally worn by the hooker.
• The number worn by both Michael Jordan and Dwyane Wade in the U.S. Olympics Team.
• Nine Men’s Morris is a European board game known since Roman times.
• In professional ice hockey, number 9 is perhaps best known as the jersey number of Gordie Howe and Maurice Richard, and the number has been retired in their honour by the Detroit Red Wings and Montreal Canadiens, respectively.
• 99 is the highest valid jersey number in the NFL

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 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.

Brands

• Channel 9, an Australian free-to-air television station.
• Nine Lives Cat Food got its name from the legend that a cat has nine lives.
• Nine West is a brand for clothing.

Music

• There is a rapper named Tech N9ne whose name stands for "Tech Nine" because 9 is the number of completion

• In music theory a ninth is the ninth note of a musical scale or the interval between the first note and the ninth. A ninth chord is a chord with a ninth.

• In classical music the curse of the ninth refers to the superstition that a composer who writes a ninth symphony will die soon. Beethoven, who left his Tenth Symphony unfinished, is regarded by the superstition as the first victim of the curse.

• Bands with the number nine in their name include Nine, Stroke 9, Nine Days, Nine Inch Nails, Nine Black Alps and Alice Nine.

• The Beatles released a song called "Revolution 9" which appears on The White Album (more properly known as The Beatles); its principal vocal feature is a voice repeating, "Number nine...number nine...number nine..." John Lennon, the primary composer of "Revolution 9", also released a solo recording entitled "#9 Dream".

• "Love Potion No. 9 originally performed by The Clovers in 1955. Love Potion No. 9 was a 1992 movie starring Sandra Bullock and Tate Donovan.

• The first single of Leeds-based band ¡Forward, Russia! was called Nine and was released in April 2005. The title did not carry any symbolism, but followed the then "tradition" of the band for naming their songs in the order in which they were written.

• Composer Maury Yeston adapted Federico Fellini's 8½ for Broadway as the musical Nine, which won several Tony Awards in 1982, and was successfully revived in 2003 starring Antonio Banderas.

• 9 is also the name of various albums, including 9 by Mercyful Fate, released in 1999, and 9 by Public Image Limited. Fairport Convention also released an album with the number for a title. Irish singer/songwriter Damien Rice released an album called 9 in November 2006.

• In urban culture, 'nine' is a slang word for a 9mm pistol or homicide, the latter from the Illinois Criminal Code for homicide.

• In hip-hop, the rapper Nine, who got his big break in late 1993 as a featured guest on First Funkmaster Flex and the Ghetto Celebs, released 2 albums (Nine Livez in 1995 and Cloud Nine in 1996) and is now making a comeback.

• Dublin pop-punk band Nine IX Lives formed in 2005 and use 'IX' in their logo.

• "Nine" is a song title on the Neurosis album Given to the Rising.

• Cloud Number Nine is a popular song by Canadian rock artist Bryan Adams.

Other fields

International maritime signal flag for 9

Playing cards showing the 9 of all four suits

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

• Five-digit produce PLU codes that begin with 9 are organic.

• Nine justices sit on the United States Supreme Court.
• The word "K-9" pronounces the same as canine and is used in many U.S. police departments to denote the police dog unit.

• There are nine circles of Hell in Dante's Divine Comedy.

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

• Someone dressed "to the nines" is dressed up as much as they can be.

• The Japanese consider 9 to be unlucky because it sounds similar to the Japanese word for "pain" or "distress" (苦 kunrei ku).

• Nine Worthies
• 9: The Last Resort was a 1995 computer game.
• 9 is the name of Shane Acker's short film, and its 2009 adaption.
• Star Trek: Deep Space Nine employs the number "9" in its title and is often abbreviated as ST:DS9
• Magic: The Gathering has a set of nine rare cards, widely regarded as overpowered, known as the Power Nine.

• The number of hostages in the TV show The Nine
• Ninth Avenue is a major avenue in Manhattan.
• Nine Unknown Men are, in occult legend, the custodians of the sciences of the world since ancient times.
• The 9 on Yahoo!, hosted by Maria Sansone.
• The Nines is a film written and directed by John August.
• In the game of Craps, 9 is known as the center field because it is in the middle of the seven numbers on the field bet.
• In Middle Earth, there are nine rings of power given to men, and consequently, nine ringwraiths.
• "A cat-o'-nine-tails suggests perfect punishment and atonement." --Robert Ripley.
• 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 semi-pagan, semi-Christian" song.

• The character Cirno from the Touhou series is often called "nine," "⑨," "circle-nine," or "nineball," because in the game manual for "Phantasmagoria of Flower View," she was labeled (9) Idiot (⑨ バカ, ⑨　baka).

• 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).

See also

• 0.999...

References

Specific references:

General references:

• Cecil Balmond, "Number 9, the search for the sigma code" 1998, Prestel 2008, ISBN-10: 3791319337, ISBN-13: 9783791319339