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# 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.

Adding a number to itself is the same as multiplying it by two. For example, 7+7=14, which is the same as 7×2.

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