Editing 9 (section)

== 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]]
|}