# 9

 ← 8 9 10 →
Cardinalnine
Ordinal9th
(ninth)
Numeral systemnonary
Factorization32
Divisors1, 3, 9
Greek numeralΘ´
Roman numeralIX, ix
Greek prefixennea-
Latin prefixnona-
Binary10012
Ternary1003
Senary136
Octal118
Duodecimal912
Hexadecimal916
Amharic
Arabic, Kurdish, Persian, Sindhi, Urdu٩
Armenian numeralԹ
Bengali
Chinese numeral九, 玖
Devanāgarī
Greek numeralθ´
Hebrew numeralט
Tamil numerals
Khmer
Telugu numeral
Thai numeral
Malayalam

9 (nine) is the natural number following 8 and preceding 10.

## Evolution of the Arabic digit

In the beginning, various Indians wrote a digit 9 similar in shape 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 sign @ 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 glyph for the digit 9 has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in .

The modern digit resembles an inverted 6. To disambiguate the two on objects and documents that can be inverted, they are often underlined. Another distinction from the 6 is that it is sometimes handwritten with two strokes and a straight stem, resembling a raised lower-case letter q. In seven-segment display, the number 9 can be constructed 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.

## Mathematics

Nine is the fourth composite number, and the first composite number that is odd. 9 is the highest single-digit number in the decimal system. It is the third square number (32), and the second non-unitary square prime of the form p2 and first that is odd, with all subsequent squares of this form odd as well.

By Mihăilescu's theorem, 9 is the only positive perfect power that is one more than another positive perfect power, since the square of 3 is one more than the cube of 2.

Nine is the number of derangements of 4, or the number of permutations of four elements with no fixed points.[1]

A number that is 4 or 5 modulo 9 cannot be represented as the sum of three cubes.[2]

9 is the number of non-intersecting segments between four points on a circle.

Nine is the sum of the first three nonzero factorials ${\displaystyle 1!+2!+3!}$ It is also the third exponential factorial, since ${\displaystyle 9=3^{2^{1}}}$.[3]

Six recurring nines appear in the decimal places 762 through 767 of π. (See six nines in pi).

The first non-trivial magic square is a ${\displaystyle 3}$ x ${\displaystyle 3}$ magic square made of nine cells, with a magic constant of 15.[4] Meanwhile, a ${\displaystyle 9}$ x ${\displaystyle 9}$ magic square has a magic constant of 369.[5]

9 is a Motzkin number, for the number of ways of drawing non-intersecting chords between four points on a circle.[6]

A polygon with nine sides is called a nonagon.[7] Since 9 can be written in the form ${\displaystyle 2^{m}3^{n}p}$, for any nonnegative natural integers ${\displaystyle m}$ and ${\displaystyle n}$ with ${\displaystyle p}$ a product of Pierpont primes, a regular nonagon can be constructed with a regular compass, straightedge, and angle trisector.[8]

Also an enneagon, a regular nonagon is able to fill a plane-vertex alongside an equilateral triangle and a regular 18-sided octadecagon (3.9.18), and as such, it is one of only nine polygons that are able to fill a plane-vertex without uniformly tiling the plane.[9]

There are nine distinct uniform colorings of the triangular tiling and the square tiling, which are the two simplest regular tilings; the hexagonal tiling, on the other hand, has three distinct uniform colorings.

There are a maximum of nine semiregular Archimedean tilings by convex regular polygons, when including chiral forms of the snub hexagonal tiling.

There are nine uniform edge-transitive convex polyhedra in three dimensions:

Nine distinct stellation's by Miller's rules are produced by the truncated tetrahedron.[10] It is the simplest Archimedean solid, with a total of four equilateral triangular and four hexagonal faces.

In four-dimensional space, there are nine paracompact hyperbolic honeycomb Coxeter groups, as well as nine regular compact hyperbolic honeycombs from regular convex and star polychora.[11] There are also nine uniform demitesseractic (${\displaystyle \mathrm {D} _{4}}$) Euclidean honeycombs in the fourth dimension.

There are only three types of Coxeter groups of uniform figures in dimensions nine and thereafter, aside from the many families of prisms and proprisms: the ${\displaystyle \mathrm {A} _{n}}$ simplex groups, the ${\displaystyle \mathrm {B} _{n}}$ hypercube groups, and the ${\displaystyle \mathrm {D} _{n}}$ demihypercube groups. The ninth dimension is also the final dimension that contains Coxeter-Dynkin diagrams as uniform solutions in hyperbolic space. Inclusive of compact hyperbolic solutions, there are a total of 238 compact and paracompact Coxeter-Dynkin diagrams between dimensions two and nine, or equivalently between ranks three and ten. The most important of the last ${\displaystyle {\tilde {E}}_{9}}$ paracompact groups is the group ${\displaystyle {\tilde {T}}_{9}}$ with 1023 total honeycombs, the simplest of which is 621 whose vertex figure is the 521 honeycomb: the vertex arrangement of the densest-possible packing of spheres in 8 dimensions which forms the ${\displaystyle \mathbb {E} _{8}}$ lattice. The 621 honeycomb is made of 9-simplexes and 9-orthoplexes, with 1023 total polytope elements making up each 9-simplex. It is the final honeycomb figure with infinite facets and vertex figures in the k21 family of semiregular polytopes, first defined by Thorold Gosset in 1900.

There are nine Heegner numbers, or square-free positive integers ${\displaystyle n}$ that yield an imaginary quadratic field ${\displaystyle \mathbb {Q} \left[{\sqrt {-n}}\right]}$ whose ring of integers has a unique factorization, or class number of 1.[12]

### In decimal

A positive number is divisible by nine if and only if its digital root is 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)

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.[13]

In base-${\displaystyle N}$, the divisors of ${\displaystyle N-1}$ have such a property, which makes 3 the only other number aside from 9 in decimal that shares this property. Another consequence of 9 being 10 − 1 is that it is a Kaprekar number.

There are other interesting patterns involving multiples of nine:

• 12345679 × 9 = 111111111
• 12345679 × 18 = 222222222
• 12345679 × 81 = 999999999

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.
• 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.

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...)

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.[14]

### List of basic calculations

Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 20 25 50 100 1000
9 × x 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 180 225 450 900 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

## 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).[15]
• 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)[16] 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; lit. '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 underworld 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 US 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

### Pseudoscience

• In Pythagorean numerology the number 9 symbolizes the end of one cycle and the beginning of another.

## 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[17] and a 1988 fiction book by Anthea Fraser;[18] the name derives from "a very curious old semi-pagan, semi-Christian" song.[19]
• 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. 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 city-states in western Essos known collectively as the Free Cities (Braavos, Lorath, Lys, Myr, Norvos, Pentos, Qohor, Tyrosh and Volantis).
• In The Wheel of Time series, Daughter of the Nine Moons is the title given to the heir to the throne of Seanchan, and the Court of the Nine Moons serves as the throne room of the Seanchan rulers themselves. Additionally, the nation of Illian is partially governed by a body known as the Council of Nine, and the flag of Illian displays nine golden bees on it. Furthermore, in the Age of Legends, the Nine Rods of Dominion were nine regional governors who administered individual areas of the world under the ruling world government.

## 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 Baháʼí Faith. In addition, the word Baháʼ 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.
• 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 self-control.
• The Bible recorded that Christ died at the 9th hour of the day (3 pm).[20]

## Science

### Physiology

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

### Psychology

Common terminal digit in psychological pricing.

## Sports

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

## References

1. ^ Sloane, N. J. A. (ed.). "Sequence A000166 (Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 10 December 2022.
2. ^ Avagyan, Armen; Dallakyan, Gurgen (2018). "A new method in the problem of three cubes". Universal Journal of Computational Mathematics. 5 (3): 45–56. arXiv:1802.06776. doi:10.13189/ujcmj.2017.050301. S2CID 36818799.
3. ^ "Sloane's A049384 : a(0)=1, a(n+1) = (n+1)^a(n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 1 June 2016.
4. ^ William H. Richardson. "Magic Squares of Order 3". Wichita State University Dept. of Mathematics. Retrieved 6 November 2022.
5. ^ Sloane, N. J. A. (ed.). "Sequence A006003 (Also the sequence M(n) of magic constants for n X n magic squares)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 8 December 2022.
6. ^ "Sloane's A001006 : Motzkin numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 1 June 2016.
7. ^ Robert Dixon, Mathographics. New York: Courier Dover Publications: 24
8. ^
9. ^ Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine. Taylor & Francis, Ltd. 50 (5): 228-234. doi:10.2307/2689529. JSTOR 2689529. S2CID 123776612. Zbl 0385.51006.
10. ^ Webb, Robert. "Enumeration of Stellations". www.software3d.com. Archived from the original on 25 November 2022. Retrieved 15 December 2022.
11. ^ Coxeter, H. S. M. (1956), "Regular honeycombs in hyperbolic space", Proceedings of the International Congress of Mathematicians, vol. III, Amsterdam: North-Holland Publishing Co., pp. 167–169, MR 0087114
12. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 93
13. ^ Martin Gardner, A Gardner's Workout: Training the Mind and Entertaining the Spirit. New York: A. K. Peters (2001): 155
14. ^ Cajori, Florian (1991, 5e) A History of Mathematics, AMS. ISBN 0-8218-2102-4. p.91
15. ^ "Lucky Number Nine, Meaning of Number 9 in Chinese Culture". www.travelchinaguide.com. Retrieved 15 January 2021.
16. ^ Donald Alexander Mackenzie (2005). Myths of China And Japan. Kessinger. ISBN 1-4179-6429-4.
17. ^ Jane Dowson (1996). Women's Poetry of the 1930s: A Critical Anthology. Routledge. ISBN 0-415-13095-6.
18. ^ Anthea Fraser (1988). The Nine Bright Shiners. Doubleday. ISBN 0-385-24323-5.
19. ^ Charles Herbert Malden (1905). Recollections of an Eton Colleger, 1898–1902. Spottiswoode. p. 182. nine-bright-shiners.
20. ^ "Meaning of Numbers in the Bible The Number 9". Bible Study. Archived from the original on 17 November 2007.
21. ^ "Web site for NINE: A Journal of Baseball History & Culture". Archived from the original on 4 November 2009. Retrieved 20 February 2013.
22. ^ Glover, Diane (9 October 2019). "#9 Dream: John Lennon and numerology". www.beatlesstory.com. Beatles Story. Retrieved 6 November 2022. Perhaps the most significant use of the number 9 in John's music was the White Album's 'Revolution 9', an experimental sound collage influenced by the avant-garde style of Yoko Ono and composers such as Edgard Varèse and Karlheinz Stockhausen. It featured a series of tape loops including one with a recurring 'Number Nine' announcement. John said of 'Revolution 9': 'It's an unconscious picture of what I actually think will happen when it happens; just like a drawing of a revolution. One thing was an engineer's testing voice saying, 'This is EMI test series number nine.' I just cut up whatever he said and I'd number nine it. Nine turned out to be my birthday and my lucky number and everything. I didn't realise it: it was just so funny the voice saying, 'number nine'; it was like a joke, bringing number nine into it all the time, that's all it was.'
23. ^ Truax, Barry (2001). Handbook for Acoustic Ecology (Interval). Burnaby: Simon Fraser University. ISBN 1-56750-537-6..
24. ^ "The Curse of the Ninth Haunted These Composers | WQXR Editorial". WQXR. Retrieved 16 January 2022.