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In ancient [[Indian mathematics]], a '''[[Vedic]] square''' is a variation on a typical 9&nbsp;&times;&nbsp;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).
 
<div align="center">
{|class="wikitable" style="text-align:center; width:270px; height:270px" border="1"
!  !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9
|-
! 1
|1||2||3||4||5||6||7||8||9
|-
!2
|2||4||6||8||1||3||5||7||9
|-
!3
|3||6||9||3||6||9||3||6||9
|-
!4
|4||8||3||7||2||6||1||5||9
|-
!5
|5||1||6||2||7||3||8||4||9
|-
!6
|6||3||9||6||3||9||6||3||9   
|-
!7
|7||5||3||1||8||6||4||2||9
|-
!8
|8||7||6||5||4||3||2||1||9
|-
!9
|9||9||9||9||9||9||9||9||9
|}
</div>
 
[[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]]. ]]
 
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}}
 
<div align="center">
{|class="wikitable" style="text-align:center; width:270px; height:270px" border="1"
!  !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9
|-
! 1
|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
|-
!2
|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
|-
!3
|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
|-
!4
|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
|-
!5
|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
|-
!6
|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
|-
!7
|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
|-
!8
|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
|-
!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||style="background-color:fuchsia"|9
|}
</div>
 
==Algebraic Properties==
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.
 
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>.
 
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>.
<div align="center">
{|class="wikitable" style="text-align:center; width:270px; height:270px" border="1"
! <math>\circ</math> !! 1 !! 2 !! 4 !! 5 !! 7 !! 8
|-
! 1
|1||2||4||5||7||8
|-
!2
|2||4||8||1||5||7
|-
!4
|4||8||7||2||1||5
|-
!5
|5||1||2||7||8||4
|-
!7
|7||5||1||8||4||2
|-
!8
|8||7||5||4||2||1
|}
</div>
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]].
<div align="center">
{|class="wikitable" style="text-align:center; width:270px; height:270px" border="1"
! <math>\circ</math> !! 1 !! 2 !! 4 !! 5 !! 7 !! 8
|-
! 1
|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
|-
!2
|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
|-
!4
|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
|-
!5
|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
|-
!7
|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
|-
!8
|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
|}
</div>
 
==See also==
*[[Latin square]]
*[[Modular arithmetic]]
*[[Monoid]]
 
==References==
*{{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}}
*{{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}}
*Talal Ghannam: ''The Mystery of Numbers: Revealed Through Their Digital Root''. CreateSpace Publications 2012, ISBN 978-1477678411, pp.&nbsp;68–73
 
== External links ==
*[http://mathormagic.com/vedic_square Vedic Square] - MATH or MAGIC!
*[http://people.revoledu.com/kardi/tutorial/DigitSum/Vedic-square.html Digital Root: Vedic Square]
*{{YouTube|id=Zx-vnMXS3-E|title="Magic Maths - In a minute - Vedic Square"}}
 
[[Category:Asian mathematics]]
[[Category:Modular arithmetic]]
 
 
{{algebra-stub}}

Revision as of 14:59, 28 January 2014

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).

1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 4 6 8 1 3 5 7 9
3 3 6 9 3 6 9 3 6 9
4 4 8 3 7 2 6 1 5 9
5 5 1 6 2 7 3 8 4 9
6 6 3 9 6 3 9 6 3 9
7 7 5 3 1 8 6 4 2 9
8 8 7 6 5 4 3 2 1 9
9 9 9 9 9 9 9 9 9 9
Highlighting specific numbers within the Vedic square reveals distinct shapes each with some form of reflection symmetry.

Numerous geometric patterns and symmetries can be observed in a Vedic square some of which can be found in traditional Islamic art.Template:Harv

1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 4 6 8 1 3 5 7 9
3 3 6 9 3 6 9 3 6 9
4 4 8 3 7 2 6 1 5 9
5 5 1 6 2 7 3 8 4 9
6 6 3 9 6 3 9 6 3 9
7 7 5 3 1 8 6 4 2 9
8 8 7 6 5 4 3 2 1 9
9 9 9 9 9 9 9 9 9 9

Algebraic Properties

This table can be viewed as the multiplication table of the monoid where is the set of positive integers partitioned by the residue classes modulo nine. Also, the operator means the abstract "multiplication" between the elements of this monoid. If are elements of then can be defined as by using the 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.

This does not form a group not every non-zero element has a corresponding inverse element, for example but there is no such that .

If we consider the subset , however, this does form a group. It forms a cyclic group with 2 as one choice of generator. In fact, this is just the group of multiplicative units in the ring .

1 2 4 5 7 8
1 1 2 4 5 7 8
2 2 4 8 1 5 7
4 4 8 7 2 1 5
5 5 1 2 7 8 4
7 7 5 1 8 4 2
8 8 7 5 4 2 1

We can see the every columns and rows has all six cells. It shows that forms a Latin square.

1 2 4 5 7 8
1 1 2 4 5 7 8
2 2 4 8 1 5 7
4 4 8 7 2 1 5
5 5 1 2 7 8 4
7 7 5 1 8 4 2
8 8 7 5 4 2 1

See also

References

  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • Talal Ghannam: The Mystery of Numbers: Revealed Through Their Digital Root. CreateSpace Publications 2012, ISBN 978-1477678411, pp. 68–73

External links


Template:Algebra-stub

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