# 9

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Cardinal nine
Ordinal 9th
(ninth)
Numeral system nonary
Factorization 32
Divisors 1, 3, 9
Greek numeral Θ´
Roman numeral IX
Unicode symbol(s) Ⅸ, ⅸ
Greek prefix ennea-
Latin prefix nona-
Binary 10012
Ternary 1003
Quaternary 214
Quinary 145
Senary 136
Octal 118
Duodecimal 912
Vigesimal 920
Base 36 936
Amharic
Arabicl & Kurdish ٩
Urdu
Armenian numeral Թ
Bengali
Chinese/Japanese
/Korean numeral

Devanāgarī
Greek numeral θ´
Hebrew numeral ט
Tamil numerals
Khmer
Telugu numeral
Thai numeral

9 (nine /nn/) is the natural number following 8 and preceding 10. Nine is the highest one-digit number.

## 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 single-digit 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 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.

Since 9 = 321, 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 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 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]

### 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 the probability of an event, defined as the negative of the base-10 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 −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 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

Multiplication 1 2 3 4 5 6 7 8 9 10 20 25 50 100 1000
9 × x 9 18 27 36 45 54 63 72 81 90 180 225 450 900 9000
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
9x 9 81 729 6561 59049 531441 4782969 43046721 387420489 3486784401
x9 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
x9 1 5 119 169 229 279 339 449 559 669 779 889 1109 1219
1329 1439 1549 1659 1769 2429 3079 6159 13319 146419 1621519 17836619

## 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 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 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.

## Culture and mythology

### Indian culture

Nine is a number that appears often in Indian culture and mythology. Some instances are enumerated below.

### Chinese culture

• Nine ( pinyin jiǔ) is considered a good number in Chinese culture because it sounds the same as the word "long-lasting" ( 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 nine-dotted line (Chinese: 南海九段线; pinyin: nánhǎi jiǔduàn xiàn; literally: "Nine-segment line of the South China Sea") delimits certain island claims by China in the South China Sea.
• The nine-rank 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.

### 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 step-pyramid 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 cat-o'-nine-tails 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 "K-9" 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.

### Technique

Playing cards showing the 9 of all four suits

## 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.
• 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) 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 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.
• Tian's Trigram Number, of Feng Shui, in Taoism.

## Science

### Physiology

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

## Sports

Billiards: A Nine-ball rack with the no. 9 ball at the center

## 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 industry, including former ties to the extremely popular MMORPG World of Warcraft.

## References

1. ^ "Sloane's A001006 : Motzkin numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
2. ^ "Sloane's A049384 : a(0)=1, a(n+1) = (n+1)^a(n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
3. ^ Robert Dixon, Mathographics. New York: Courier Dover Publications: 24
4. ^ Martin Gardner, A Gardner's Workout: Training the Mind and Entertaining the Spirit. New York: A. K. Peters (2001): 155
5. ^ Cajori, Florian (1991, 5e) A History of Mathematics, AMS. ISBN 0-8218-2102-4. p.91
6. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 93
7. ^ Georges Ifrah (1985). From One to Zero: A Universal History of Numbers. Viking. ISBN 0-670-37395-8.
8. ^ Donald Alexander Mackenzie (2005). Myths of China And Japan. Kessinger. ISBN 1-4179-6429-4.
9. ^ Jane Dowson (1996). Women's Poetry of the 1930s: A Critical Anthology. Routledge. ISBN 0-415-13095-6.
10. ^ Anthea Fraser (1988). The Nine Bright Shiners. Doubleday. ISBN 0-385-24323-5.
11. ^ Charles Herbert Malden (1905). Recollections of an Eton Colleger, 1898-1902. Spottiswoode.
12. ^ "Web site for NINE: A Journal of Baseball History & Culture". Retrieved 20 February 2013.