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| In ancient [[Indian mathematics]], a '''[[Vedic]] square''' is a variation on a typical 9 × 9 [[multiplication table]]. The entry in each cell is the [[digital root]] of the product of the column and row headings i.e. the [[remainder]] when the product of the row and column headings is divided by 9 (with remainder 0 represented by 9).
| | Im Kina and was born on 22 October 1986. My hobbies are Seashell Collecting and Genealogy.<br><br>Here is my web site ... [http://nsaksi.com/main/index.php?qa=1040&qa_1=easy-plans-in-worpress-examined wordpress backup plugin] |
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| <div align="center"> | |
| {|class="wikitable" style="text-align:center; width:270px; height:270px" border="1"
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| ! !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9
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| |-
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| ! 1
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| |1||2||3||4||5||6||7||8||9
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| |-
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| !2
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| |2||4||6||8||1||3||5||7||9
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| |-
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| !3
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| |3||6||9||3||6||9||3||6||9
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| |-
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| !4
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| |4||8||3||7||2||6||1||5||9
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| |-
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| !5
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| |5||1||6||2||7||3||8||4||9
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| |-
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| !6
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| |6||3||9||6||3||9||6||3||9
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| |-
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| !7
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| |7||5||3||1||8||6||4||2||9
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| |-
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| !8
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| |8||7||6||5||4||3||2||1||9
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| |-
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| !9
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| |9||9||9||9||9||9||9||9||9
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| |}
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| </div> | |
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| [[File:Shapes in the Vedic Square.png|thumb|Highlighting specific numbers within the Vedic square reveals distinct shapes each with some form of [[reflection symmetry]]. ]]
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| Numerous [[geometric]] [[patterns]] and [[symmetries]] can be observed in a Vedic square some of which can be found in traditional [[Islamic art]].{{Harv|Pritchard|2003|pp=119–122}}
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| <div align="center">
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| {|class="wikitable" style="text-align:center; width:270px; height:270px" border="1"
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| ! !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9
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| |-
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| ! 1
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| |style="background-color:red"|1||style="background-color:orange"|2||style="background-color:yellow"|3||style="background-color:lime"|4||style="background-color:green"|5||style="background-color:blue"|6||style="background-color:aqua"|7||style="background-color:purple"|8||style="background-color:fuchsia"|9
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| |-
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| !2
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| |style="background-color:orange"|2||style="background-color:lime"|4||style="background-color:blue"|6||style="background-color:purple"|8||style="background-color:red"|1||style="background-color:yellow"|3||style="background-color:green"|5||style="background-color:aqua"|7||style="background-color:fuchsia"|9
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| |-
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| !3
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| |style="background-color:yellow"|3||style="background-color:blue"|6||style="background-color:fuchsia"|9||style="background-color:yellow"|3||style="background-color:blue"|6||style="background-color:fuchsia"|9||style="background-color:yellow"|3||style="background-color:blue"|6||style="background-color:fuchsia"|9
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| |-
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| !4
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| |style="background-color:lime"|4||style="background-color:purple"|8||style="background-color:yellow"|3||style="background-color:aqua"|7||style="background-color:orange"|2||style="background-color:blue"|6||style="background-color:red"|1||style="background-color:green"|5||style="background-color:fuchsia"|9
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| |-
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| !5
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| |style="background-color:green"|5||style="background-color:red"|1||style="background-color:blue"|6||style="background-color:orange"|2||style="background-color:aqua"|7||style="background-color:yellow"|3||style="background-color:purple"|8||style="background-color:lime"|4||style="background-color:fuchsia"|9
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| |-
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| !6
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| |style="background-color:blue"|6||style="background-color:yellow"|3||style="background-color:fuchsia"|9||style="background-color:blue"|6||style="background-color:yellow"|3||style="background-color:fuchsia"|9||style="background-color:blue"|6||style="background-color:yellow"|3||style="background-color:fuchsia"|9
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| |-
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| !7
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| |style="background-color:aqua"|7||style="background-color:green"|5||style="background-color:yellow"|3||style="background-color:red"|1||style="background-color:purple"|8||style="background-color:blue"|6||style="background-color:lime"|4||style="background-color:orange"|2||style="background-color:fuchsia"|9
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| |-
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| !8
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| |style="background-color:purple"|8||style="background-color:aqua"|7||style="background-color:blue"|6||style="background-color:green"|5||style="background-color:lime"|4||style="background-color:yellow"|3||style="background-color:orange"|2||style="background-color:red"|1||style="background-color:fuchsia"|9
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| |-
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| !9
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| |style="background-color:fuchsia"|9||style="background-color:fuchsia"|9||style="background-color:fuchsia"|9||style="background-color:fuchsia"|9||style="background-color:fuchsia"|9||style="background-color:fuchsia"|9||style="background-color:fuchsia"|9||style="background-color:fuchsia"|9||style="background-color:fuchsia"|9
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| |}
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| </div>
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| ==Algebraic Properties==
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| This table can be viewed as the multiplication table of the [[monoid]] <math>((\mathbb{Z}/9\mathbb{Z})^{\times}, \{1, \circ\})</math> where <math>\mathbb{Z}/9\mathbb{Z}</math> is the set of positive integers partitioned by the [[residue class]]es [[Modular arithmetic|modulo]] nine. Also, the operator ''<math>\circ</math>'' means the abstract "multiplication" between the elements of this monoid. If <math>a,b</math> are elements of <math>((\mathbb{Z}/9\mathbb{Z})^{\times}, \{1, \circ\})</math> then <math>a \circ b</math> can be defined as <math>(a \times b) \mod{9}</math> by using the [[Modulo operation|modulus operator mod]], where we take the element 9 as the representative of the residue class of 0 rather than the traditional choice of 0.
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| This does not form a [[Group (mathematics)|group]] not every non-zero element has a corresponding [[inverse element]], for example <math>6\circ 3 = 9</math> but there is no <math>a \in \{ 1,\cdots,9 \}</math> such that <math>9\circ a = 6.</math>.
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| If we consider the subset <math>\{1,2,4,5,7,8\}</math>, however, this does form a group. It forms a [[cyclic group]] with 2 as one choice of [[Generating set of a group|generator]]. In fact, this is just the group of multiplicative [[Unit (ring theory)|units]] in the [[Ring (mathematics)|ring]] <math>\mathbb{Z}/9\mathbb{Z}</math>.
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| <div align="center">
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| {|class="wikitable" style="text-align:center; width:270px; height:270px" border="1"
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| ! <math>\circ</math> !! 1 !! 2 !! 4 !! 5 !! 7 !! 8
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| |-
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| ! 1
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| |1||2||4||5||7||8
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| |-
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| !2
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| |2||4||8||1||5||7
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| |-
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| !4
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| |4||8||7||2||1||5
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| |-
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| !5
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| |5||1||2||7||8||4
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| |-
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| !7
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| |7||5||1||8||4||2
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| |-
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| !8
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| |8||7||5||4||2||1
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| |}
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| </div>
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| We can see the every columns and rows has all six cells. It shows that <math>\{1, 2, 4, 5, 7, 8\}</math> forms a [[Latin square]].
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| <div align="center">
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| {|class="wikitable" style="text-align:center; width:270px; height:270px" border="1"
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| ! <math>\circ</math> !! 1 !! 2 !! 4 !! 5 !! 7 !! 8
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| |-
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| ! 1
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| |style="background-color:red"|1||style="background-color:orange"|2||style="background-color:lime"|4||style="background-color:green"|5||style="background-color:aqua"|7||style="background-color:purple"|8
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| |-
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| !2
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| |style="background-color:orange"|2||style="background-color:lime"|4||style="background-color:purple"|8||style="background-color:red"|1||style="background-color:green"|5||style="background-color:aqua"|7
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| |-
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| !4
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| |style="background-color:lime"|4||style="background-color:purple"|8||style="background-color:aqua"|7||style="background-color:orange"|2||style="background-color:red"|1||style="background-color:green"|5
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| |-
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| !5
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| |style="background-color:green"|5||style="background-color:red"|1||style="background-color:orange"|2||style="background-color:aqua"|7||style="background-color:purple"|8||style="background-color:lime"|4
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| |-
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| !7
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| |style="background-color:aqua"|7||style="background-color:green"|5||style="background-color:red"|1||style="background-color:purple"|8||style="background-color:lime"|4||style="background-color:orange"|2
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| |-
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| !8
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| |style="background-color:purple"|8||style="background-color:aqua"|7||style="background-color:green"|5||style="background-color:lime"|4||style="background-color:orange"|2||style="background-color:red"|1
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| |}
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| </div>
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| ==See also==
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| *[[Latin square]]
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| *[[Modular arithmetic]]
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| *[[Monoid]]
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| ==References==
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| *{{cite book|last=Deskins|first=W.E.|title=Abstract Algebra|publisher=Dover|location=New York|pages=162–167|isbn=0-486-68888-7|year=1996|ref=harv}}
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| *{{cite book|last=Pritchard|first=Chris|title=The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching|publisher=Cambridge University Press|location=Great Britain|pages=119–122|isbn=0-521-53162-4|year=2003|ref=harv}}
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| *Talal Ghannam: ''The Mystery of Numbers: Revealed Through Their Digital Root''. CreateSpace Publications 2012, ISBN 978-1477678411, pp. 68–73
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| == External links ==
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| *[http://mathormagic.com/vedic_square Vedic Square] - MATH or MAGIC!
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| *[http://people.revoledu.com/kardi/tutorial/DigitSum/Vedic-square.html Digital Root: Vedic Square]
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| *{{YouTube|id=Zx-vnMXS3-E|title="Magic Maths - In a minute - Vedic Square"}}
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| [[Category:Asian mathematics]]
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| [[Category:Modular arithmetic]]
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| {{algebra-stub}}
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Im Kina and was born on 22 October 1986. My hobbies are Seashell Collecting and Genealogy.
Here is my web site ... wordpress backup plugin