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==Evolution of the Hindu–Arabic digit==
==Evolution of the Hindu–Arabic digit==
{{See also|Hindu–Arabic numeral system}}
{{See also|Hindu–Arabic numeral system}}{{Verification section|date=September 2024}}[[File:Evo9glyph.svg|200px|left]]
{{Refimprove|date=March 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.
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.


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== Mathematics ==
== 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>
'''Nine''' is the fourth [[composite number]], and the first composite number that is odd. Nine is the third [[square number]] (3<sup>2</sup>), and the second non-unitary square [[prime number|prime]] of the form ''p''<sup>2</sup>, and, the first that is odd, with all subsequent squares of this form odd as well. Nine has the even [[aliquot sum]] of [[4 (number)|4]], and with a composite number sequence of two (9, 4, [[3 (number)|3]], [[1 (number)|1]], [[0]]) within the '''3'''-aliquot tree. It is the first member of the first cluster of two [[semiprime]]s (9, [[10]]), preceding ([[14 (number)|14]], [[15 (number)|15]]).<ref>{{Cite OEIS |A001358 |Semiprimes (or biprimes): products of two primes. |access-date=2024-02-27 }}</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&nbsp;century.<ref>[[Cajori, Florian]] (1991, 5e) ''A History of Mathematics'', AMS. {{ISBN|0-8218-2102-4}}. p.91</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>
By [[Mihăilescu's theorem]], 9 is the only positive [[perfect power]] that is one more than another positive perfect power, since the [[Square (algebra)|square]] of 3 is one more than the [[Cube (algebra)|cube]] of 2.<ref>{{Cite journal |first=Preda |last=Mihăilescu |author-link=Preda Mihăilescu |title=Primary Cyclotomic Units and a Proof of Catalan's Conjecture |url=https://www.degruyter.com/document/doi/10.1515/crll.2004.048/html |journal=[[J. Reine Angew. Math.]] |volume=572 |year=2004 |pages=167–195 |publisher=[[De Gruyter]] |location=Berlin |doi=10.1515/crll.2004.048 |mr=2076124 |s2cid=121389998 }}</ref><ref>{{Cite journal |first=Tauno |last=Metsänkylä |title=Catalan's conjecture: another old Diophantine problem solved |url=https://www.ams.org/journals/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf |journal=[[Bulletin of the American Mathematical Society]] |volume=41 |year=2004 |issue=1 |pages=43–57 |publisher=[[American Mathematical Society]] |location=Providence, R.I. |doi=10.1090/S0273-0979-03-00993-5 |doi-access=free |mr=2015449 |s2cid=17998831 |zbl=1081.11021 }}</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]]
[[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>
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>
It is also the sum of the first three nonzero [[factorial]]s <math>1! + 2! + 3!</math>, and equal to the third [[exponential factorial]], since <math>9 = 3^{2^{1}}.</math><ref>{{Cite web |url=https://oeis.org/A049384 |title=Sloane's A049384 : a(0)=1, a(n+1) = (n+1)^a(n) |website=The On-Line Encyclopedia of Integer Sequences |publisher=OEIS Foundation |access-date=1 June 2016}}</ref>

Nine is the number of [[derangement]]s of 4, or the number of [[permutation]]s of four elements with no [[Fixed point (mathematics)|fixed points]].<ref>{{Cite OEIS |A000166 |Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points. |access-date=10 December 2022}}</ref>

9 is the fourth [[refactorable number]], as it has exactly three positive divisors, and 3 is one of them.<ref>{{Cite OEIS |A033950 |Refactorable numbers: number of divisors of k divides k. Also known as tau numbers. |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>

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

9 is a [[Motzkin number]], for the number of ways of drawing non-intersecting [[Chord (geometry)|chords]] between four points on a [[circle]].<ref>{{Cite web |url=https://oeis.org/A001006 |title=Sloane's A001006 : Motzkin numbers |website=The On-Line Encyclopedia of Integer Sequences |publisher=OEIS Foundation |access-date=1 June 2016}}</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]].]]

The first non-trivial [[magic square]] is a <math>3</math> x <math>3</math> magic square made of nine cells, with a [[magic constant]] of 15.<ref>{{Cite web |url=https://www.math.wichita.edu/~richardson/mathematics/magic%20squares/order3magicsquare.html |title=Magic Squares of Order 3 |author=William H. Richardson |website=Wichita State University Dept. of Mathematics |access-date=6 November 2022}}</ref> Meanwhile, a <math>9</math> x <math>9</math> magic square has a magic constant of [[369 (number)|369]].<ref>{{Cite OEIS |A006003 |Also the sequence M(n) of magic constants for n X n magic squares |access-date=8 December 2022}}</ref>

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 ===
=== 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>
==== Polygons and tilings ====

The regular [[hexagon]] contains a total of nine [[diagonal]]s, and is one of only four [[polytope]]s with [[radial equilateral symmetry]] such that its ''long radius'' (center to vertex [[Length#Use in mathematics|length]]) is the same as the edge-length: (the hexagon), the [[cuboctahedron]], the [[tesseract]], and the [[24-cell]].

A [[polygon]] with nine sides is called a [[nonagon]].<ref>Robert Dixon, ''Mathographics''. New York: Courier Dover Publications: 24</ref> Since 9 can be written in the form <math>2^m3^np</math>, for any nonnegative natural integers <math>m</math> and <math>n</math> with <math>p</math> a product of [[Pierpont prime]]s, a regular nonagon is [[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> Also an ''enneagon'', a regular nonagon is able to fill a [[Euclidean tilings by convex regular polygons#Plane-vertex tilings|plane-vertex]] alongside an [[equilateral triangle]] and a regular 18-sided [[octadecagon]] ([[:File:3.9.18 vertex.png|3.9.18]]), and as such, it is one of only nine polygons that are able to fill a plane-vertex without [[Euclidean tilings by convex regular polygons|uniformly]] tiling the [[Plane (geometry)|plane]].<ref>{{Cite journal |first1=Branko |last1=Grünbaum |author-link=Branko Grünbaum |first2=Geoffrey |last2=Shepard |author-link2=G.C. Shephard |title=Tilings by Regular Polygons |date=November 1977 |url=http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf |journal=[[Mathematics Magazine]] |volume=50 |issue=5 |publisher=Taylor & Francis, Ltd.|pages=228–234 |doi=10.2307/2689529 |jstor=2689529 |s2cid=123776612 |zbl=0385.51006 }}</ref> In total, there are a maximum of nine semiregular [[Euclidean tilings by convex regular polygons|Archimedean tilings]] by convex regular polygons, when including [[Chirality (mathematics)|chiral]] forms of the [[snub hexagonal tiling]]. More specifically, there are nine distinct [[uniform coloring]]s to both the [[Triangular tiling#Uniform colorings|triangular tiling]] and the [[Square tiling#Uniform colorings|square tiling]] (the simplest regular tilings) while the [[Hexagonal tiling#Uniform colorings|hexagonal tiling]], on the other hand, has three distinct uniform colorings.

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

==== Polyhedra ====

There are nine uniform [[edge-transitive]] convex polyhedra in [[Three-dimensional space|three dimensions]]:
* the five [[Platonic solid|regular Platonic solids]]: the [[tetrahedron]], [[Regular octahedron|octahedron]], [[cube]], [[Regular dodecahedron|dodecahedron]] and [[Regular icosahedron|icosahedron]];
* the two [[Quasiregular polyhedron|quasiregular]] [[Archimedean solid]]s: the [[cuboctahedron]] and the [[icosidodecahedron]]; and
* two [[Catalan solid]]s: the [[rhombic dodecahedron]] and the [[rhombic triacontahedron]], which are [[Dual polyhedron|duals]] to the only two quasiregular polyhedra.

Nine distinct [[stellation]]'s by [[Miller's rules#Miller's rules|Miller's rules]] are produced by the [[truncated tetrahedron]].<ref>{{Cite web |url=https://www.software3d.com/Enumerate.php |last=Webb |first=Robert |title=Enumeration of Stellations |website=www.software3d.com |access-date=15 December 2022 |archive-url=https://archive.today/20221126015207/https://www.software3d.com/Enumerate.php |archive-date=26 November 2022}}</ref> It is the simplest Archimedean solid, with a total of four equilateral triangular and four [[hexagon]]al faces.

Collectively, there are nine [[regular polyhedra]] in the third dimension, when extending the convex Platonic solids to include the concave [[Star polyhedron|regular star polyhedra]] known as the [[Kepler-Poinsot polyhedron|Kepler-Poinsot polyhedra]].<ref>{{Cite web |last=Weisstein |first= Eric W. |author-link=Eric W. Weisstein |title=Regular Polyhedron |url=https://mathworld.wolfram.com/RegularPolyhedron.html |website=[[Mathworld]] -- A [[WolframAlpha]] Resource |access-date=2024-02-27 }}</ref><ref>{{Cite book |last=Coxeter |first= H. S. M. |author-link=Harold Scott MacDonald Coxeter |title=Regular Polytopes |url=https://archive.org/details/regularpolytopes0000hsmc/page/92/mode/2up |url-access=registration |edition=1st |publisher=[[Methuen Publishing|Methuen & Co., Ltd]] |location=London |year=1948 |page=93 |isbn=0-486-61480-8 |oclc=798003 |mr=0027148 }}</ref>

==== Higher dimensions ====


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>
In [[four-dimensional space]], there are nine [[Uniform 5-polytope#Regular and uniform hyperbolic honeycombs|paracompact hyperbolic]] honeycomb [[Coxeter group]]s, as well as nine [[Uniform 5-polytope#Compact regular tessellations of hyperbolic 4-space|regular compact hyperbolic]] honeycombs from regular [[Regular 4-polytope#Regular convex 4-polytope|convex]] and [[Regular 4-polytope#Regular star (Schläfli–Hess) 4-polytopes|star]] ''[[Uniform 4-polytope|polychora]]''.<ref>{{citation |last=Coxeter |first=H. S. M. |author-link=Harold Scott MacDonald Coxeter |contribution=Regular honeycombs in hyperbolic space |mr=0087114 |pages=167–169 |publisher=North-Holland Publishing Co. |location=Amsterdam |title=Proceedings of the International Congress of Mathematicians |volume=III |year=1956}}</ref> There are also nine uniform [[Demihypercube|demitesseractic]] (<math>\mathrm D_{4}</math>) [[Uniform 5-polytope#Regular and uniform honeycombs|Euclidean honeycombs]] in the fourth dimension.


9 is the largest single-digit number in the [[decimal|decimal system]].
There are only three types of Coxeter groups of [[Uniform polytope|uniform figures]] in dimensions nine and thereafter, aside from the many families of [[Prism (geometry)#Prismatic polytope|prisms]] and [[proprism]]s: the <math>\mathrm A_{n}</math> [[simplex]] groups, the <math>\mathrm B_{n}</math> [[hypercube]] groups, and the <math>\mathrm D_{n}</math> [[demihypercube]] groups. The [[Uniform 9-polytope|ninth dimension]] is also the final dimension that contains [[Coxeter-Dynkin diagram]]s as [[Uniform 10-polytope#Regular and uniform hyperbolic honeycombs|uniform solutions]] in [[hyperbolic space]]. Inclusive of compact hyperbolic solutions, there are a total of '''238''' [[Dynkin diagram#238 Hyperbolic groups (compact and noncompact)|compact and paracompact]] Coxeter-Dynkin diagrams between dimensions two and '''nine''', or equivalently between ranks three and ten. The most important of the last [[E9 honeycomb|<math>{\tilde{E}}_9</math>]] paracompact groups is the group <math>{\tilde{T}}_9</math> with [[1023 (number)|1023]] total honeycombs, the simplest of which is [[E9 honeycomb#621 honeycomb|'''6<sub>21</sub>''']] whose [[vertex figure]] is the [[5 21 honeycomb|'''5<sub>21</sub>''' honeycomb]]: the vertex arrangement of the densest-possible packing of spheres in [[Uniform 8-polytope|8 dimensions]] which forms the [[E8 lattice|<math>\mathbb E_{8}</math> lattice]]. The 6<sub>21</sub> honeycomb is made of [[9-simplex]]es and [[9-orthoplex]]es, with 1023 total [[Simplex#Elements|polytope elements]] making up each 9-simplex. It is the final honeycomb figure with infinite facets and vertex figures in the '''k<sub>21</sub>''' family of [[semiregular polytope]]s, first defined by [[Thorold Gosset]] in 1900.


=== List of basic calculations ===
=== List of basic calculations ===
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|[[1000000000 (number)|1000000000]]
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{|class="wikitable" style="text-align: center; background: white"
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! rowspan="2" style="width:105px;"|[[Radix]]
!1
!5
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!30
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!35
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!45
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!110
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!200
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|-
|rowspan="2"|'''''x''{{sub|9}}'''
|1
|5
|11<sub>9</sub>
|16<sub>9</sub>
|22<sub>9</sub>
|27<sub>9</sub>
|33<sub>9</sub>
|44<sub>9</sub>
|55<sub>9</sub>
|66<sub>9</sub>
|77<sub>9</sub>
|88<sub>9</sub>
|110<sub>9</sub>
|121<sub>9</sub>
|-
|132<sub>9</sub>
|143<sub>9</sub>
|154<sub>9</sub>
|165<sub>9</sub>
|176<sub>9</sub>
|242<sub>9</sub>
|307<sub>9</sub>
|615<sub>9</sub>
|1331<sub>9</sub>
|14641<sub>9</sub>
|162151<sub>9</sub>
|1783661<sub>9</sub>
|}

=== In base 10 ===

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

==== Occurrence ====

9 is related to just 3, 6 and itself. When we do the doubling of all [[natural numbers]] we get, 1, 2, 4, 8, 16, 32, 64... Now if we try to get the [[digital roots]] of two digit numbers, then, 16= 1+6 =7. 32= 3+2 =5. Such way, only 1, 2, 4, 5, 7, 8 occur. Same thing happens when we start [[halving]] the numbers.

==== Nikola tesla ====

[[Nikola Tesla]] was a great scientist. Despite his work on free energy etc. he stated, " If the understand the numbers 3, 6, 9, you understand the universe."

==== Divisibility ====

A positive number is divisible by nine [[if and only if]] its [[digital root]] is nine:
* 9 × 2 = 18 (1 + 8 = 9)
* 9 × 3 = 27 (2 + 7 = 9)
* 9 × 9 = 81 (8 + 1 = 9)
* 9 × 121 = 1089 (1 + 0 + 8 + 9 = 18; 1 + 8 = 9)
* 9 × 234 = 2106 (2 + 1 + 0 + 6 = 9)
* 9 × 578329 = 5204961 (5 + 2 + 0 + 4 + 9 + 6 + 1 = 27; 2 + 7 = 9)
* 9 × 482729235601 = 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.<ref>[[Martin Gardner]], ''A Gardner's Workout: Training the Mind and Entertaining the Spirit''. New York: A. K. Peters (2001): 155</ref>

In [[Positional notation|base]]-<math>N</math>, the [[divisor]]s of <math>N - 1</math> have this property.

==== Multiples of 9 ====

There are other interesting patterns involving multiples of nine:
* 9 × 12345679 = 111111111
* 18 × 12345679 = 222222222
* 81 × 12345679 = 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. {{nowrap|1={{sfrac|274|999}} = 0.274274274274...}})

Another consequence of 9 being {{nowrap|10 − 1}} is that it is a [[Kaprekar number]], preceding the ninth and tenth [[triangle number]]s, [[45 (number)|45]] and [[55 (number)|55]] (where all 9, 99, 999, 9999, ... are Keprekar numbers).<ref>{{Cite OEIS |A006886 |Kaprekar numbers. |access-date=2024-02-27 }}</ref>

Six recurring nines appear in the decimal places 762 through 767 of [[pi|{{pi}}]]. (See [[six nines in pi]]).


==Alphabets and codes==
==Alphabets and codes==
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*[[Stanine]]s, a method of scaling test scores, range from 1 to 9.
*[[Stanine]]s, a method of scaling test scores, range from 1 to 9.
*There are 9 [[Square foot|square feet]] in a [[square yard]].
*There are 9 [[Square foot|square feet]] in a [[square yard]].

==Literature==
* There are [[Divine Comedy#The Circles of Hell|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<ref>{{cite book |title=Women's Poetry of the 1930s: A Critical Anthology |author=Jane Dowson |year=1996 |publisher=Routledge |isbn=0-415-13095-6 |url=https://books.google.com/books?id=fVTQPI3ZIHcC&q=nine-bright-shiners+ridler&pg=RA1-PA103}}</ref> and a 1988 fiction book by Anthea Fraser;<ref>{{cite book |title=The Nine Bright Shiners |author=Anthea Fraser |publisher=Doubleday |year=1988 |isbn=0-385-24323-5 |url=https://archive.org/details/ninebrightshiner00fras}}</ref> the name derives from "a very curious old semi-pagan, semi-Christian" song.<ref>{{cite book |title=Recollections of an Eton Colleger, 1898–1902 |author=Charles Herbert Malden |publisher=Spottiswoode |year=1905 |url=https://archive.org/details/recollectionsan00maldgoog |page=[https://archive.org/details/recollectionsan00maldgoog/page/n193 182] |quote=nine-bright-shiners.}}</ref>
* ''[[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 [[Nazgûl|ringwraiths]]. Additionally, [[The Fellowship of the Ring#Members of the Fellowship of the Ring|The Fellowship of the Ring]]{{Broken anchor|date=2024-05-27|bot=User:Cewbot/log/20201008/configuration|target_link=The Fellowship of the Ring#Members of the Fellowship of the Ring|reason= The anchor (Members of the Fellowship of the Ring) [[Special:Diff/950491470|has been deleted]].}} 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.

==Places and thoroughfares==
* [[List of highways numbered 9]]
* [[Ninth Avenue (Manhattan)|Ninth Avenue]] is a major avenue in Manhattan.
* [[Provinces of South Africa|South Africa]] has 9 provinces
* [[Negeri Sembilan]], a [[Malaysia]]n [[States in Malaysia|state]] located in [[Peninsular Malaysia]], is named as such as it was historically a confederation of nine ({{lang-ms|sembilan}}) settlements (''[[nagari (settlement)|nagari]]'') of the [[Minangkabau people|Minangkabau]] migrated from [[West Sumatra]].


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


* Nine, as the highest 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]].
* 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]].
* 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 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.
* 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.
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===Psychology===
===Psychology===
Common terminal digit in [[psychological pricing]].
Common terminal digit in [[psychological pricing]].

==Sports==
[[File:9ball rack 2.jpg|thumb|right|200px|Billiards: A [[Nine-ball]] [[Rack (billiards)|rack]] with the no. 9 ball at the center]]

* [[Nine-ball]] is the standard professional pocket [[billiards]] variant played in the United States.
* In [[association football]] (soccer), the centre-forward/striker traditionally (since at least the fifties) wears the number 9 shirt.
* In [[baseball]]:
** There are nine players on the field including the pitcher.
** There are nine innings in a standard game.
** 9 represents the [[right fielder]]'s position.
** ''NINE: A Journal of Baseball History and Culture'', published by the [[University of Nebraska Press]]<ref>{{cite web |url=http://nine.iweb.bsu.edu/ |title=Web site for NINE: A Journal of Baseball History & Culture |access-date=20 February 2013 |archive-url=https://web.archive.org/web/20091104001109/http://nine.iweb.bsu.edu/ |archive-date=4 November 2009 |url-status=dead}}</ref>
* In [[rugby league]], the jersey number assigned to the [[Hooker (rugby league)|hooker]] in most competitions. (An exception is the [[Super League]], which uses static squad numbering.)
* In [[rugby union]], the number worn by the starting [[Scrum-half (rugby union)|scrum-half]].

==Technology==
{{Unreferenced section|date=October 2021}}
* [[ISO 9]] is the [[International Organization for Standardization|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.
* [[The9]] Limited (owner of [https://web.archive.org/web/20191120011629/https://www.the9.com// the9.com]) is a company in the video-game industry, including former ties to the extremely popular [[MMORPG]] ''[[World of Warcraft]]''.

==Music==
* "[[Revolution 9]]", a sound collage which appears on The Beatles' eponymous 1968 album [[The Beatles (album)|''The Beatles'' (aka ''The White Album'')]], prominently features a loop of a man's voice repeating the phrase "Number nine".<ref>{{cite web |url=https://www.beatlesstory.com/blog/2019/10/09/9-dream-john-lennon-and-numerology/ |title=#9 Dream: John Lennon and numerology |last=Glover |first=Diane |date=9 October 2019 |website=www.beatlesstory.com |publisher=Beatles Story |access-date=6 November 2022 |quote=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 Lennon|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.'}}</ref>
* There are 9 semitones in a [[Major 6th]] interval in music.<ref>{{cite book |last=Truax |first=Barry |date=2001 |title=Handbook for Acoustic Ecology'' (Interval)'' |url=https://www.sfu.ca/sonic-studio-webdav/handbook/Interval.html |location=Burnaby |publisher=Simon Fraser University |isbn=1-56750-537-6}}.</ref>
* There was a [[Curse of the ninth|superstition]] among some notable classical music composers that they would die after completing their ninth symphony. Some composers who died after composing their ninth symphony include [[Ludwig van Beethoven]], [[Anton Bruckner]], [[Antonin Dvorak]] and [[Gustav Mahler]].<ref>{{Cite web |title=The Curse of the Ninth Haunted These Composers {{!}} WQXR Editorial |url=https://www.wqxr.org/story/how-curse-ninth-symphony-haunted-these-composers |access-date=16 January 2022 |website=WQXR |date=17 October 2016 |language=en}}</ref>
* Beethoven's [[Symphony No. 9 (Beethoven)|Symphony No. 9]] is regarded as a masterpiece, and one of the most frequently performed symphonies in the world.
* A [[staff (music)|staff]] of a [[Sheet music|score]] contains nine notes who comprise five lines and four spaces.


== See also ==
== See also ==

Revision as of 01:12, 19 September 2024

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).

← 8 9 10 →
−1 0 1 2 3 4 5 6 7 8 9 →
Cardinalnine
Ordinal9th
(ninth)
Numeral systemnonary
Factorization32
Divisors1,3,9
Greek numeralΘ´
Roman numeralIX, ix
Greek prefixennea-
Latin prefixnona-
Binary10012
Ternary1003
Senary136
Octal118
Duodecimal912
Hexadecimal916
Amharic፱
Arabic, Kurdish, Persian, Sindhi, Urdu٩
Armenian numeralԹ
Bengali৯
Chinese numeral九, 玖
Devanāgarī९
Greek numeralθ´
Hebrew numeralט
Tamil numerals௯
Khmer៩
Telugu numeral౯
Thai numeral๙
Malayalam൯
Babylonian numeral𒐝
Egyptian hieroglyph𓐂
Morse code____.

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

Evolution of the Hindu–Arabic digit

See also: Hindu–Arabic numeral system

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 3-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 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 form of the number nine (9) could possibly derived from the Arabic letter waw, in which its isolated form (و) resembles the number 9.

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 LCD calculators use the former, but some VFD models use the latter.

Mathematics

9 is the fourth composite number, and the first odd composite number. 9 is also a refactorable number.[2]

Casting out nines is a quick way of testing the calculations of sums, differences, products, and quotients of integers in decimal, a method known as long ago as the 12th century.[3]

If an odd perfect number exists, it will have at least nine distinct prime factors.[4]

Non-intersecting chords between four points on a circle

9 is the sum of the cubes of the first two non-zero positive integers which makes it the first cube-sum number greater than one.[5] A number that is 4 or 5 modulo 9 cannot be represented as the sum of three cubes.[6]

Four concentric magic circles with 9 in the center (by Yang Hui), where numbers on each circle and diameter around the center generate a magic sum of 138.

There are nine Heegner numbers, or square-free positive integers that yield an imaginary quadratic field whose ring of integers has a unique factorization, or class number of 1.[7]

Geometry

A polygon with nine sides is called a nonagon.[8] A regular nonagon can be constructed with a regular compass, straightedge, and angle trisector.[9]

The lowest number of squares needed for a perfect tiling of a rectangle is 9.[10]

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

List of basic calculations

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 27 36 45 54 63 72 81 90 99 108 117 126 135 144 180 225 450 900 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

Alphabets and codes

Culture and mythology

Indian culture

Nine is a number that appears often in Indian culture and mythology.[11] Some instances are enumerated below.

Chinese culture

  • Nine (九; pinyin: jiǔ) is considered a good number in Chinese culture because it sounds the same as the word "long-lasting" (久; 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.[17]
  • 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.

European culture

Greek mythology

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.

Australian culture

The Pintupi Nine, a group of 9 Aboriginal Australian women who remained unaware of European colonisation of Australia and lived a traditional desert-dwelling life in Australia's Gibson Desert until 1984.

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

International maritime signal flag for 9
Playing cards showing the 9 of all four suits

Religion and philosophy

Christianity

Islam

There are three verses that refer to nine in the Quran.

We surely gave Moses nine clear signs.1 ˹You, O Prophet, can˺ ask the Children of Israel. When Moses came to them, Pharaoh said to him, “I really think that you, O Moses, are bewitched.”

— Surah Al-Isra (The Night Journey/Banī Isrāʾīl):101[21]

Note 1: The nine signs of Moses are: the staff, the hand (both mentioned in Surah Ta-Ha 20:17-22), famine, shortage of crops, floods, locusts, lice, frogs, and blood (all mentioned in Surah Al-A'raf 7:130-133). These signs came as proofs for Pharaoh and the Egyptians. Otherwise, Moses had some other signs such as water gushing out of the rock after he hit it with his staff, and splitting the sea.

Now put your hand through ˹the opening of˺ your collar, it will come out ˹shining˺ white, unblemished.2 ˹These are two˺ of nine signs for Pharaoh and his people. They have truly been a rebellious people.”

— Surah Al-Naml (The Ant):12[22]

Note 2: Moses, who was dark-skinned, was asked to put his hand under his armpit. When he took it out it was shining white, but not out of a skin condition like melanoma.

And there were in the city nine ˹elite˺ men who spread corruption in the land, never doing what is right.

— Surah Al-Naml (The Ant):48[23]

Other

A nine-pointed star
A nine-pointed star
  • Nine, as the largest 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.
  • Tian's Trigram Number, of Feng Shui, in Taoism.

Science

Astronomy

Chemistry

Physiology

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

Psychology

Common terminal digit in psychological pricing.

See also

Look up nine in Wiktionary, the free dictionary.

References

  1. ^ Lippman, David (12 July 2021). "6.0.2: The Hindu-Arabic Number System". Mathematics LibreTexts. Retrieved 31 March 2024.
  2. ^ Sloane, N. J. A. (ed.). "Sequence A033950 (Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 19 June 2023.
  3. ^ Cajori, Florian (1991, 5e) A History of Mathematics, AMS. ISBN 0-8218-2102-4. p.91
  4. ^ 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.
  5. ^ Sloane, N. J. A. (ed.). "Sequence A000537 (Sum of first n cubes; or n-th triangular number squared.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 19 June 2023.
  6. ^ 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
  7. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 93
  8. ^ Robert Dixon, Mathographics. New York: Courier Dover Publications: 24
  9. ^ 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.
  10. ^ Sloane, N. J. A. (ed.). "Sequence A219766 (Number of nonsquare simple perfect squared rectangles of order n up to symmetry)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  11. ^ DHAMIJA, ANSHUL (16 May 2018). "The Auspiciousness Of Number 9". Forbes India. Retrieved 1 April 2024.
  12. ^ "Vaisheshika | Atomism, Realism, Dualism | Britannica". www.britannica.com. Retrieved 13 April 2024.
  13. ^ "Navratri | Description, Importance, Goddess, & Facts | Britannica". www.britannica.com. 11 April 2024. Retrieved 13 April 2024.
  14. ^ Lochtefeld, James G. (2002). The illustrated encyclopedia of hinduism. New York: the Rosen publ. group. ISBN 978-0-8239-2287-1.
  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. ^ "The Global Egyptian Museum | Nine Bows". www.globalegyptianmuseum.org. Retrieved 16 November 2023.
  18. ^ Mark, Joshua J. "Nine Realms of Norse Cosmology". World History Encyclopedia. Retrieved 16 November 2023.
  19. ^ Galatians 5:22–23
  20. ^ "Meaning of Numbers in the Bible The Number 9". Bible Study. Archived from the original on 17 November 2007.
  21. ^ "Surah Al-Isra - 101". Quran.com. Retrieved 17 August 2023.
  22. ^ "Surah An-Naml - 12". Quran.com. Retrieved 17 August 2023.
  23. ^ "Surah An-Naml - 48". Quran.com. Retrieved 17 August 2023.

Further reading

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