Multiplication table
A multiplication table is used to define a 'multiplication' operation for an algebraic system. Multiplication tables as they are used to teach schoolchildren multiplication are a grid where rows and columns are headed by the numbers to multiply, and the entry in each cell is the product of the column and row headings:| 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | × |
| 2×2 = 4 | 2 | |||||||
| 3×3 = 9 | 3×2 = 6 | 3 | ||||||
| 4×4 = 16 | 4×3 = 12 | 4×2 = 8 | 4 | |||||
| 5×5 = 25 | 5×4 = 20 | 5×3 = 15 | 5×2 = 10 | 5 | ||||
| 6×6 = 36 | 6×5 = 30 | 6×4 = 24 | 6×3 = 18 | 6×2 = 12 | 6 | |||
| 7×7 = 49 | 7×6 = 42 | 7×5 = 35 | 7×4 = 28 | 7×3 = 21 | 7×2 = 14 | 7 | ||
| 8×8 = 64 | 8×7 = 56 | 8×6 = 48 | 8×5 = 40 | 8×4 = 32 | 8×3 = 24 | 8×2 = 16 | 8 | |
| 9×9 = 81 | 9×8 = 72 | 9×7 = 63 | 9×6 = 54 | 9×5 = 45 | 9×4 = 36 | 9×3 = 27 | 9×2 = 18 | 9 |
This table does not give the ones and zeros. That is because:
- Anything times zero is zero.
- Anything times one is itself. For example, 5×1=5.
Multiplication tables can define 'multiplication' operations for groups, fields, rings, and other algebraic systems.
The following table is an example of a multiplication table for the unit octonions (see octonion, from which this example is taken).
| · | 1 | e1 | e2 | e3 | e4 | e5 | e6 | e7 |
| 1 | 1 | e1 | e2 | e3 | e4 | e5 | e6 | e7 |
| e1 | e1 | -1 | e4 | e7 | -e2 | e6 | -e5 | -e3 |
| e2 | e2 | -e4 | -1 | e5 | e1 | -e3 | e7 | -e6 |
| e3 | e3 | -e7 | -e5 | -1 | e6 | e2 | -e4 | e1 |
| e4 | e4 | e2 | -e1 | -e6 | -1 | e7 | e3 | -e5 |
| e5 | e5 | -e6 | e3 | -e2 | -e7 | -1 | e1 | e4 |
| e6 | e6 | e5 | -e7 | e4 | -e3 | -e1 | -1 | e2 |
| e7 | e7 | e3 | e6 | -e1 | e5 | -e4 | -e2 | -1 |
