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{{Short description|Integer number 9}}
{{Hatnote|This article is about the number. For the years, see [[9 BC]] and [[AD 9]]. For other uses, see [[9 (disambiguation)]] and [[Number nine (disambiguation)]].}}
{{Use dmy dates|date=December 2022}}
{{Infobox number
| number        = 9
| numeral       = [[nonary]]
| divisor       = 1,3,9
| roman         = IX, ix
| greek prefix  = [[Wiktionary:ennea-|ennea-]]
| latin prefix  = [[Wiktionary:nona-|nona-]]
| lang1         = [[Amharic language|Amharic]]
| lang1 symbol  = <span style="font-size:100%;">፱</span>
| lang2         = [[Eastern Arabic numerals|Arabic]], [[Central Kurdish|Kurdish]], [[Persian language|Persian]], [[Sindhi language|Sindhi]], [[Urdu numerals|Urdu]]
| lang2 symbol  = {{resize|150%|٩}}
| lang4         = [[Armenian numerals|Armenian numeral]]
| lang4 symbol  = {{resize|150%|Թ}}
| lang5         = [[Bengali language|Bengali]]
| lang5 symbol  = {{resize|150%|৯}}
| lang6         = [[Chinese numeral]]
| lang6 symbol  = 九, 玖
| lang7         = [[Devanāgarī]]
| lang7 symbol  = {{resize|150%|९}}
| lang8         = [[Greek numerals|Greek numeral]]
| lang8 symbol  = θ´
| lang9         = [[Hebrew numerals|Hebrew numeral]]
| lang9 symbol  = {{resize|150%|ט}}
| lang10        = [[Tamil numerals]]
| lang10 symbol = {{resize|150%|௯}}
| lang11        = [[Khmer numerals|Khmer]]
| lang11 symbol = ៩
| lang12        = [[Telugu script]]
| lang12 symbol = {{resize|150%|౯}}
| lang13        = [[Thai numerals|Thai numeral]]
| lang13 symbol = {{resize|150%|๙}}
| lang14        = [[Malayalam numerals|Malayalam]]
| lang14 symbol = ൯
|lang15=[[Babylonian cuneiform numerals|Babylonian numeral]]|lang15 symbol=𒐝|lang16=[[Egyptian numerals|Egyptian hieroglyph]]|lang16 symbol={{resize|200%|𓐂}}|lang18=[[Morse code]]|lang18 symbol={{resize|120%|____.}}
|lang19=[[NATO phonetic alphabet]]
|lang19 symbol=NINER
}}
'''9''' ('''nine''') is the [[natural number]] following {{num|8}} and preceding {{num|10}}.

==Evolution of the Hindu–Arabic digit==
{{See also|Hindu–Arabic numeral system}}{{Verification section|date=September 2024}}[[File:Evo9glyph.svg|200px|left]]
Circa 300 BC, as part of the [[Brahmi numerals]], 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 {{num|3}}-look-alike.<ref>{{Cite web |last=Lippman |first=David |date=2021-07-12 |title=6.0.2: The Hindu-Arabic Number System |url=https://math.libretexts.org/Courses/Cosumnes_River_College/Math_300%3A_Mathematical_Ideas_Textbook_(Muranaka)/06%3A_Miscellaneous_Extra_Topics/6.00%3A_Historical_Counting_Systems/6.0.02%3A_The_Hindu-Arabic_Number_System |access-date=2024-03-31 |website=Mathematics LibreTexts |language=en}}</ref> 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 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 (typography)|ascender]] in most modern [[typeface]]s, in typefaces with [[text figures]] the character usually has a [[descender]], as, for example, in [[File:TextFigs196.png|45px]].

The form of the number nine (9) could possibly derived from the Arabic letter ''[[و|waw]]'', in which its isolated form (و) resembles the number 9.

[[File:Seven-segment 9.svg|20px|left]]
The modern digit resembles an inverted ''6''. 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 letter '''q''', which distinguishes it from the 6. Similarly, in [[seven-segment display]], the number 9 can be constructed either with a hook at the end of its stem or without one. Most [[liquid crystal display|LCD]] calculators use the former, but some [[vacuum fluorescent display|VFD]] models use the latter.

== Mathematics ==

9 is the fourth [[composite number]], and the first odd composite number. 9 is also a [[refactorable number]].<ref>{{Cite OEIS|A033950|Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.|access-date=2023-06-19}}</ref> 

[[Casting out nines]] is a quick way of testing the calculations of sums, differences, products, and [[quotient]]s of [[integer]]s in [[decimal]], a method known as long ago as the 12th century.<ref>[[Cajori, Florian]] (1991, 5e) ''A History of Mathematics'', AMS. {{ISBN|0-8218-2102-4}}. p.91</ref>

9 is the only square number that is the sum of two consecutive, positive cubes: <math>3^2 = 9 = 1^3 + 2^3</math><ref>{{cite book |last1=Wells |first1=David |title=The Penguin Dictionary of Curious and Interesting Numbers |date=1997 |publisher=Penguin |isbn=0-14-026149-4 |edition=2nd |language=English}}</ref>

If an [[Parity (mathematics)|odd]] [[perfect number]] exists, it will have at least nine distinct [[prime factor]]s.<ref>{{Cite journal |last=Pace P. |first=Nielsen |year=2007 |title=Odd perfect numbers have at least nine distinct prime factors |url=https://www.ams.org/journals/mcom/2007-76-260/S0025-5718-07-01990-4/ |journal=[[Mathematics of Computation]] |location=Providence, R.I. |publisher=[[American Mathematical Society]] |volume=76 |issue=260 |pages=2109–2126 |arxiv=math/0602485 |bibcode=2007MaCom..76.2109N |doi=10.1090/S0025-5718-07-01990-4 |mr=2336286 |s2cid=2767519 |zbl=1142.11086 |doi-access=free}}</ref>

[[File:MotzkinChords4.svg|left|165px|thumb|Non-intersecting [[Chord (geometry)|chords]] between four points on a circle]]

9 is the sum of the [[Cube (algebra)|cubes]] of the first two non-zero positive integers <math>1^{3} + 2^{3}</math> which makes it the first cube-sum number greater than [[1|one]].<ref>{{Cite OEIS |A000537 |Sum of first n cubes; or n-th triangular number squared. |access-date=2023-06-19 }}</ref> A number that is 4 or 5 [[Modular arithmetic|modulo]] 9 cannot be represented as the [[Sums of three cubes|sum of three cubes]].<ref>{{citation |last=Davenport|first=H. |author-link=Harold Davenport |title=On Waring's problem for cubes |journal=[[Acta Mathematica]] |publisher=International Press of Boston |location=Somerville, MA |volume=71 |year=1939 |pages=123–143 |doi=10.1007/BF02547752 |doi-access=free |mr=0000026 |zbl=0021.10601 |s2cid=120792546 }}</ref>

[[File:Yang Hui magic circle.svg|right|200px|thumb|Four concentric [[Magic circle (mathematics)|magic circle]]s with '''9''' in the center (by [[Yang Hui]]), where numbers on each circle and [[diameter]] around the center generate a [[magic constant|magic sum]] of [[138 (number)|138]].]]There are nine [[Heegner number]]s, or [[Square-free integer|square-free positive integers]] <math>n</math> that yield an imaginary [[quadratic field]] <math>\Q\left[\sqrt{-n}\right]</math> whose [[ring of integers]] has a [[unique factorization]], or [[ideal class group|class number]] of 1.<ref>Bryan Bunch, ''The Kingdom of Infinite Number''. New York: W. H. Freeman & Company (2000): 93</ref> 

=== Geometry ===

A [[polygon]] with nine sides is called a [[nonagon]].<ref>Robert Dixon, ''Mathographics''. New York: Courier Dover Publications: 24</ref> A regular nonagon can be [[Constructible polygon|constructed]] with a regular [[compass]], [[straightedge]], and [[angle trisector]].<ref>{{Cite journal|last=Gleason |first=Andrew M. |author-link=Andrew M. Gleason |url=https://www.tandfonline.com/doi/abs/10.1080/00029890.1988.11971989?journalCode=uamm20 |title=Angle trisection, the heptagon, and the triskaidecagon |journal=[[American Mathematical Monthly]] |volume= 95 |issue=3 |year= 1988 |publisher=[[Taylor & Francis, Ltd]] |pages=191–194 |doi=10.2307/2323624 |mr=935432 |jstor=2323624 |s2cid=119831032 }}</ref>

The lowest number of [[square]]s needed for a [[Rectangle#Squared, perfect, and other tiled rectangles|perfect tiling]] of a [[rectangle]] is 9.<ref>{{cite OEIS|A219766|Number of nonsquare simple perfect squared rectangles of order n up to symmetry}}</ref>

9 is the largest single-digit number in the [[decimal|decimal system]].

=== List of basic calculations ===
{|class="wikitable" style="text-align: center; background: white"
|-
! style="width:105px;"|[[Multiplication]]
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
!13
!14
!15
!16
!20
!25
!50
!100
!1000
|-
|'''9 × ''x'''''
|'''9'''
|[[18 (number)|18]]
|[[27 (number)|27]]
|[[36 (number)|36]]
|[[45 (number)|45]]
|[[54 (number)|54]]
|[[63 (number)|63]]
|[[72 (number)|72]]
|[[81 (number)|81]]
|[[90 (number)|90]]
|[[99 (number)|99]]
|[[108 (number)|108]]
|[[117 (number)|117]]
|[[126 (number)|126]]
|[[135 (number)|135]]
|[[144 (number)|144]]
|[[180 (number)|180]]
|[[225 (number)|225]]
|450
|[[900 (number)|900]]
|[[9000 (number)|9000]]
|}

{|class="wikitable" style="text-align: center; background: white"
|-
! style="width:105px;"|[[Division (mathematics)|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.{{overline|285714}}
|1.125
|1
|0.9
|0.{{overline|81}}
|0.75
|0.{{overline|692307}}
|0.6{{overline|428571}}
|0.6
|-
|'''''x'' ÷ 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}}
|}

{|class="wikitable" style="text-align: center; background: white"
|-
! style="width:105px;"|[[Exponentiation]]
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
|-
|'''9{{sup|''x''}}'''
|'''9'''
|81
|729
|6561
|59049
|531441
|4782969
|43046721
|387420489
|3486784401
|-
|'''''x''{{sup|9}}'''
|1
|[[512 (number)|512]]
|19683
|262144
|1953125
|10077696
|40353607
|134217728
|387420489
|[[1000000000 (number)|1000000000]]
|}

==Culture and mythology==
{{More citations needed|date=November 2023}}

===Indian culture===
Nine is a number that appears often in [[Indian culture]] and mythology.<ref>{{Cite web |last=DHAMIJA |first=ANSHUL |date=May 16, 2018 |title=The Auspiciousness Of Number 9 |url=https://www.forbesindia.com/article/9th-anniversary-special/the-auspiciousness-of-number-9/50205/1 |access-date=2024-04-01 |website=Forbes India |language=en}}</ref> For example, there are nine [[Navagraha|influencers]] attested to in [[Indian astrology]]. In the [[Vaisheshika]] branch of [[Hindu philosophy]], there are nine universal substances or elements: [[Prithvi|Earth]], [[Ap (water)|Water]], [[Vayu|Air]], [[Agni|Fire]], [[Akasha|Ether]], [[Kāla (time)|Time]], [[Vaisheshika#The Categories or Padārtha|Space]], [[Ātman (Hinduism)|Soul]], and [[Manas (early Buddhism)|Mind]].<ref>{{Cite web |title=Vaisheshika {{!}} Atomism, Realism, Dualism {{!}} Britannica |url=https://www.britannica.com/topic/Vaisheshika |access-date=2024-04-13 |website=www.britannica.com |language=en}}</ref> And [[Navaratri]] is a nine-day festival dedicated to the [[Navadurga|nine]] forms of [[Durga]].<ref>{{Cite web |date=2024-04-11 |title=Navratri {{!}} Description, Importance, Goddess, & Facts {{!}} Britannica |url=https://www.britannica.com/topic/Navratri |access-date=2024-04-13 |website=www.britannica.com |language=en}}</ref><ref>{{Cite book |last=Lochtefeld |first=James G. |title=The illustrated encyclopedia of Hinduism |date=2002 |publisher=the Rosen publ. group |isbn=978-0-8239-2287-1 |location=New York}}</ref>

===Chinese culture===
* Nine ({{lang|zh|九}}; {{Lang-zh|p=jiǔ}}) is considered a good [[Numbers in Chinese culture|number in Chinese culture]] because it sounds the same as the word "long-lasting" ({{lang|zh|久}}; {{Lang-zh|p=jiǔ}}).<ref>{{Cite web |title=Lucky Number Nine, Meaning of Number 9 in Chinese Culture |url=https://www.travelchinaguide.com/intro/lucky-number9.htm |access-date=15 January 2021 |website=www.travelchinaguide.com}}</ref>
* 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&nbsp;– 81 [[yin and yang|yang]] (masculine, heavenly) and 36 [[yin and yang|yin]] (feminine, earthly). All three numbers are multiples of 9 ({{nowrap|1=9 × 13 = 117}}, {{nowrap|1=9 × 9 = 81}}, {{nowrap|1=9 × 4 = 36}}).<ref>{{cite book |title=Myths of China And Japan |author=Donald Alexander Mackenzie |url=https://books.google.com/books?id=vzbeLy4TBa4C&q=chinese+dragon+scales+yin+36&pg=PA46 |publisher=Kessinger |year=2005 |isbn=1-4179-6429-4}}</ref>

==Anthropology==
===Idioms===
* "To go the whole nine yards"
* "A cat has nine lives"
* "To be on cloud nine"
* 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.
[[File:ICS Niner.svg|right|thumb|100px|[[International maritime signal flag]] for 9]]

==Religion and philosophy==
[[File:Bahai star.svg|100px|right|A nine-pointed star]]

* Nine, as the largest single-digit number (in [[decimal|base ten]]), symbolizes completeness in the [[Baháʼí Faith]]. In addition, the word Baháʼ in the [[Abjad numerals|Abjad notation]] has a value of 9, and a 9-pointed star is used to [[Baháʼí symbols|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 [[30th century BC|3000 BC]].
* In [[Norse mythology]], the number nine is associated with [[Odin]], as that is how many days he hung from the [[world tree]] [[Yggdrasil]] before attaining knowledge of the [[runes]].
* Nine is the number associated with Satan in [[LaVeyan Satanism]]. [[Anton LaVey]] wrote in The Satanic Rituals that this is because nine is the number of the [[Ego (Freudian)|ego]] since it "always returns to itself" even after being multiplied by any number.

==Science==
===Chemistry===
The purity of chemicals (see [[Nine (purity)]]).

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

===Psychology===
Common terminal digit in [[psychological pricing]].

== See also ==
{{Wiktionary|nine}}
{{Portal|Mathematics}}
* [[9 (disambiguation)]]
* [[0.999...]]
* [[wikt:cloud nine|Cloud Nine]]

==References==
{{Reflist|35em}}

==Further reading==
* Cecil Balmond, "Number 9, the search for the sigma code" 1998, Prestel 2008, {{ISBN|3-7913-1933-7}}, {{ISBN|978-3-7913-1933-9}}

{{Integers|zero}}
{{Authority control}}

{{DEFAULTSORT:9 (Number)}}
[[Category:Integers]]
[[Category:9 (number)]]
[[Category:Superstitions about numbers]]