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 | e_{1} | e_{2} | e_{3} | e_{4} | e_{5} | e_{6} | e_{7} |
1 | 1 | e_{1} | e_{2} | e_{3} | e_{4} | e_{5} | e_{6} | e_{7} |
e_{1} | e_{1} | -1 | e_{4} | e_{7} | -e_{2} | e_{6} | -e_{5} | -e_{3} |
e_{2} | e_{2} | -e_{4} | -1 | e_{5} | e_{1} | -e_{3} | e_{7} | -e_{6} |
e_{3} | e_{3} | -e_{7} | -e_{5} | -1 | e_{6} | e_{2} | -e_{4} | e_{1} |
e_{4} | e_{4} | e_{2} | -e_{1} | -e_{6} | -1 | e_{7} | e_{3} | -e_{5} |
e_{5} | e_{5} | -e_{6} | e_{3} | -e_{2} | -e_{7} | -1 | e_{1} | e_{4} |
e_{6} | e_{6} | e_{5} | -e_{7} | e_{4} | -e_{3} | -e_{1} | -1 | e_{2} |
e_{7} | e_{7} | e_{3} | e_{6} | -e_{1} | e_{5} | -e_{4} | -e_{2} | -1 |