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9

From Wikipedia, the free encyclopedia
Integer number 9
This article is about the number. For the years, see9 BC andAD 9. For other uses, see9 (disambiguation) andNumber nine (disambiguation).

Natural number
← 89 10 →
Cardinalnine
Ordinal9th
(ninth)
Numeral systemnonary
Factorization32
Divisors1,3,9
Greek numeralΘ´
Roman numeralIX, ix
Greekprefixennea-
Latinprefixnona-
Binary10012
Ternary1003
Senary136
Octal118
Duodecimal912
Hexadecimal916
Amharic
Arabic,Kurdish,Persian,Sindhi,Urdu٩
Armenian numeralԹ
Bengali
Chinese numeral九, 玖
Devanāgarī
Santali
Greek numeralθ´
Hebrew numeralט
Tamil numerals
Khmer
Telugu script
Thai numeral
Malayalam
Babylonian numeral𒐝
Egyptian hieroglyph𓐂
Morse code____.
NATO phonetic alphabetNINER

9 (nine) is thenatural number following8 and preceding10.

Evolution of the Hindu–Arabic digit

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See also:Hindu–Arabic numeral system
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Circa 300 BC, as part of theBrahmi numerals, various Indians wrote a digit 9 similar in shape to the modern closingquestion mark without the bottom dot. The Kshatrapa, Andhra and Gupta started curving the bottom vertical line coming up with a3-look-alike.[1] How the numbers got to their Gupta form is open to considerable debate. 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 lowercasea. 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 anascender in most moderntypefaces, in typefaces withtext figures the character usually has adescender, as, for example, in.

The form of the number nine (9) could possibly derived from the Arabic letterwaw, in which its isolated form (و) resembles the number 9.

The modern digit resembles an inverted6. To disambiguate the two on objects and labels that can be inverted, they are often underlined. It is sometimes handwritten with two strokes and a straight stem, resembling a raised lower-case letterq, which distinguishes it from the 6. Similarly, inseven-segment display, the number 9 can be constructed either with a hook at the end of its stem or without one. MostLCD calculators use the former, but someVFD models use the latter.

Mathematics

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9 is the fourthcomposite number, and the first odd composite number. 9 is also arefactorable number.[2]

Casting out nines is a quick way of testing the calculations of sums, differences, products, andquotients ofintegers indecimal, a method known as long ago as the 12th century.[3]

9 is the only square number that is the sum of two consecutive, positive cubes:32=9=13+23{\displaystyle 3^{2}=9=1^{3}+2^{3}}[4]

If anoddperfect number exists, it will have at least nine distinctprime factors.[5]

Non-intersectingchords between four points on a circle

9 is the sum of thecubes of the first two non-zero positive integers13+23{\displaystyle 1^{3}+2^{3}} which makes it the first cube-sum number greater thanone.[6] A number that is 4 or 5modulo 9 cannot be represented as thesum of three cubes.[7]

Four concentricmagic circles with9 in the center (byYang Hui), where numbers on each circle anddiameter around the center generate amagic sum of138.

There are nineHeegner numbers, orsquare-free positive integersn{\displaystyle n} that yield an imaginaryquadratic fieldQ[n]{\displaystyle \mathbb {Q} \left[{\sqrt {-n}}\right]} whosering of integers has aunique factorization, orclass number of 1.[8]

Geometry

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Apolygon with nine sides is called anonagon.[9] A regular nonagon can beconstructed with a regularcompass,straightedge, andangle trisector.[10]

The lowest number ofsquares needed for aperfect tiling of arectangle is 9.[11]

9 is the largest single-digit number in thedecimal system.

List of basic calculations

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Multiplication123456789101112131415162025501001000
9 ×x9182736455463728190991081171261351441802254509009000
Division123456789101112131415
9 ÷x94.532.251.81.51.2857141.12510.90.810.750.6923070.64285710.6
x ÷ 90.10.20.30.40.50.60.70.811.11.21.31.41.51.6
Exponentiation12345678910
9x9817296561590495314414782969430467213874204893486784401
x9151219683262144195312510077696403536071342177283874204891000000000

Culture and mythology

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Indian culture

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Nine is a number that appears often inIndian culture and mythology.[12] For example, there are nineinfluencers attested to inIndian astrology. In theVaisheshika branch ofHindu philosophy, there are nine universal substances or elements:Earth,Water,Air,Fire,Ether,Time,Space,Soul, andMind.[13] AndNavaratri is a nine-day festival dedicated to thenine forms ofDurga.[14][15]

Chinese culture

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  • Nine (;pinyin:jiǔ) is considered a goodnumber in Chinese culture because it sounds the same as the word "long-lasting" (;pinyin:jiǔ).[16]
  • Nine is strongly associated with theChinese 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 – 81yang (masculine, heavenly) and 36yin (feminine, earthly). All three numbers are multiples of 9 (9 × 13 = 117,9 × 9 = 81,9 × 4 = 36).[17]

Anthropology

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Idioms

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  • "To go the whole nine yards"
  • "A cat has nine lives"
  • "To be on cloud nine"
  • The word "K-9" pronounces the same ascanine and is used in many US police departments to denote thepolice dog unit. Despite not sounding like the translation of the wordcanine 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 Americanurban culture, "nine" is a slang word for a9mmpistol orhomicide, the latter from the Illinois Criminal Code for homicide.
International maritime signal flag for 9

Religion and philosophy

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A nine-pointed star
A nine-pointed star
  • Nine, as the largest single-digit number (inbase ten), symbolizes completeness in theBaháʼí Faith. In addition, the word Baháʼ in theAbjad notation has a value of 9, and a 9-pointed star is used tosymbolize 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 thedecimal system, which originated from the Indian subcontinent as early as3000 BC.
  • InNorse mythology, the number nine is associated withOdin, as that is how many days he hung from theworld treeYggdrasil before attaining knowledge of therunes.
  • Nine is the number associated with Satan inLaVeyan Satanism.Anton LaVey wrote in The Satanic Rituals that this is because nine is the number of theego since it "always returns to itself" even after being multiplied by any number.

Science

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Chemistry

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The purity of chemicals (seeNine (purity)).

Physiology

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A humanpregnancy normally lasts nine months, the basis ofNaegele's rule.

Psychology

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Common terminal digit inpsychological pricing.

See also

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Look upnine in Wiktionary, the free dictionary.

References

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  1. ^Lippman, David (12 July 2021)."6.0.2: The Hindu-Arabic Number System".Mathematics LibreTexts. Retrieved31 March 2024.
  2. ^Sloane, N. J. A. (ed.)."Sequence A033950 (Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved19 June 2023.
  3. ^Cajori, Florian (1991, 5e)A History of Mathematics, AMS.ISBN 0-8218-2102-4. p.91
  4. ^Wells, David (1997).The Penguin Dictionary of Curious and Interesting Numbers (2nd ed.). Penguin.ISBN 0-14-026149-4.
  5. ^Pace P., Nielsen (2007)."Odd perfect numbers have at least nine distinct prime factors".Mathematics of Computation.76 (260). Providence, R.I.:American Mathematical Society:2109–2126.arXiv:math/0602485.Bibcode:2007MaCom..76.2109N.doi:10.1090/S0025-5718-07-01990-4.MR 2336286.S2CID 2767519.Zbl 1142.11086.
  6. ^Sloane, N. J. A. (ed.)."Sequence A000537 (Sum of first n cubes; or n-th triangular number squared.)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved19 June 2023.
  7. ^Davenport, H. (1939), "On Waring's problem for cubes",Acta Mathematica,71, Somerville, MA: International Press of Boston:123–143,doi:10.1007/BF02547752,MR 0000026,S2CID 120792546,Zbl 0021.10601
  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. ^Gleason, Andrew M. (1988)."Angle trisection, the heptagon, and the triskaidecagon".American Mathematical Monthly.95 (3).Taylor & Francis, Ltd:191–194.doi:10.2307/2323624.JSTOR 2323624.MR 0935432.S2CID 119831032.
  11. ^Sloane, N. J. A. (ed.)."Sequence A219766 (Number of nonsquare simple perfect squared rectangles of order n up to symmetry)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  12. ^DHAMIJA, ANSHUL (16 May 2018)."The Auspiciousness Of Number 9".Forbes India. Retrieved1 April 2024.
  13. ^"Vaisheshika | Atomism, Realism, Dualism | Britannica".www.britannica.com. Retrieved13 April 2024.
  14. ^"Navratri | Description, Importance, Goddess, & Facts | Britannica".www.britannica.com. 11 April 2024. Retrieved13 April 2024.
  15. ^Lochtefeld, James G. (2002).The illustrated encyclopedia of Hinduism. New York: the Rosen publ. group.ISBN 978-0-8239-2287-1.
  16. ^"Lucky Number Nine, Meaning of Number 9 in Chinese Culture".www.travelchinaguide.com. Retrieved15 January 2021.
  17. ^Donald Alexander Mackenzie (2005).Myths of China And Japan. Kessinger.ISBN 1-4179-6429-4.

Further reading

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