Vedic square
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 |
Numerous geometric patterns and symmetries can be observed in a Vedic square some of which can be found in traditional Islamic art (Pritchard, 2003).
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 seen as, 
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 since 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 .