# 9 (number)

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9 (number)
 9 Cardinal 9 nine Ordinal 9th ninth Numeral system nonary Factorization 32 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 (Nao) Greek numeral θ´ Hebrew numeral ט (Tet) Tamil numeral ௯ Khmer ៩ Telugu numeral ౯ Thai numeral ๙

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

## Companies

• Nine Lives cat food; its name is derived from the legend that a cat has nine lives
• Nine Network a.k.a. Channel 9, an Australian free-to-air television station
• Nine West, a clothing brand [1]

## Culture and mythology

### Chinese culture

• 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 (9x13=117, 9x9=81, 9x4=36)[2] as well adding up individually to 9 (1+1+7=9, 8+1=9, 3+6=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 nine-rank system was a civil service nomination system used during certain Chinese dynasties.

### Ancient Egypt

• The nine bows is a term used in Ancient Egypt to represent the traditional enemies of Egypt

### European culture

• The Nine Worthies are nine historical, or semi-legendary figures who, in the Middle Ages, were believed to personify the ideals of chivalry

### 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).

### Japanese culture

• The Japanese consider 9 to be unlucky because it sounds similar to the Japanese word for "pain" or "distress" (苦 kunrei ku)[citation needed].
• 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?).[3]

## Evolution of the glyph

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.[4]

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 evolved from the letter "8". 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 often handwritten with a straight stem.

## Games

• 9: The Last Resort was a 1995 computer game
• 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
• Magic: The Gathering has a set of nine rare cards, widely regarded as overpowered, known as the Power Nine
• Nine Mens Morris (Nine Men’s Morris) is a European board game known since Roman times.

## Idioms and popular phrases

• "A cat-o'-nine-tails suggests perfect punishment and atonement." --Robert Ripley.
• The word "K-9" pronounces the same as canine and is used in many U.S. police departments to denote the police dog unit.
• Someone dressed "to the nines" is dressed up as much as they can be.
• In urban culture, "nine" is a slang word for a 9mm pistol or homicide, the latter from the Illinois Criminal Code for homicide.

## Internet

• 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.

## 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[5] and a 1988 fiction book by Anthea Fraser;[6] the name derives from "a very curious old semi-pagan, semi-Christian" song.[7]
• 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 Middle-earth, there are nine rings of power given to men, and consequently, nine ringwraiths

## Mathematics

Nine 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. It is the first composite lucky number, along with the first composite odd 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. Nine is; and can be, the only square prime with an aliquot sum of the same form. The aliquot sequence of nine has 5 members (9,4,3,1,0) this number being the second composite member of the 3-aliquot tree. It is the aliquot sum of only one number the discrete semiprime 15.

There are nine Heegner numbers.[8]

Since 9 = 321, 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.

In bases 12, 18 and 24, nine is a 1-automorphic number and in base 6 a 2-automorphic number (displayed as '13').

A polygon with nine sides is called a nonagon or enneagon.[9] 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.[10] That is, if you multiply nine by any natural number, 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)

There are other interesting patterns involving multiples of nine:

• 12345679 x 9 = 111111111
• 12345679 x 18 = 222222222
• 12345679 x 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 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 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. Examples:

• 41 - 14 = 27 (2 + 7 = 9)
• 36957930 - 35967930 = 990000, 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; 3+6 = 9.
• 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.[11]

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.[12]

Nine is the binary complement of number six:

9 = 1001
6 = 0110


### 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} \$

### 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

### 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). 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 explictly) 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%.

## Organizations

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

## Religion and philosophy

• 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.
• 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 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.

## Science

### Physiology

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

## Sports

A Nine-ball rack with the 9 ball at the center

### Auto racing

• A car in the Sprint Cup Series currently owned by Richard Petty Motorsports. The number was most notably borne by the car that Bill Elliott drove to the Cup Series title in 1988 with Melling Racing. Evernham Motorsports, the predecessor team to Richard Petty Motorsports, acquired the number in 2001 when Elliott joined that team after a brief stint as a driver-owner. Elliott used this number again through the 2003 season. Kasey Kahne has driven the 9 car since 2004.

### Billiards

• Nine-ball is the standard professional pocket billiards variant played in the United States.

### Rugby

• In rugby league, the number generally worn by the hooker.
• In rugby union, the number generally worn by the scrum-half.

### Soccer

• In association football (soccer) the centre-forward/striker traditionally (since at least the fifties) wears the number 9 shirt.

### All sports

The jersey number 9 has been retired by several North American sports teams in honor of past playing greats (or in one case, an owner):

## 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.
• The9 Limited (owner of the9.com) is a company in the video-game game industry, including former ties to the extremely popular MMORPG World of Warcraft

## Other fields

International maritime signal flag for 9
Playing cards showing the 9 of all four suits
• Nine justices sit on the United States Supreme Court.
• Stanines, a method of scaling test scores, range from 1 to 9.

## References

2. ^ Donald Alexander Mackenzie (2005). Myths of China And Japan. Kessinger. ISBN 1417964294.
3. ^ Cirno - TouhouWiki http://en.touhouwiki.net/wiki/Cirno#Other_Information
4. ^ Georges Ifrah (1985). From One to Zero: A Universal History of Numbers. Viking. ISBN 0-670-37395-8.
5. ^ Jane Dowson (1996). Women's Poetry of the 1930s: A Critical Anthology. Routledge. ISBN 0415130956.
6. ^ Anthea Fraser (1988). The Nine Bright Shiners. Doubleday. ISBN 0385243235.
7. ^ Charles Herbert Malden (1905). Recollections of an Eton Colleger, 1898-1902. Spottiswoode.
8. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 93
9. ^ Robert Dixon, Mathographics. New York: Courier Dover Publications: 24
10. ^ Martin Gardner, A Gardner's Workout: Training the Mind and Entertaining the Spirit. New York: A. K. Peters (2001): 155
11. ^ Cajori, Florian (1991, 5e) A History of Mathematics, AMS. ISBN 0-8218-2102-4. p.91
12. ^ Eyob Delele Yirdaw, "Proving Touchard's Theorem from Euler's Form" ArXiv preprint.

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