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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:
9×9=81 9×8=72 9×7=63 9×6=54 9×5=45 9×4=36 9×3=27 9×2=18
8×8=64 8×7=56 8×6=48 8×5=40 8×4=32 8×3=24 8×2=16
7×7=49 7×6=42 7×5=35 7×4=28 7×3=21 7×2=14
6×6=36 6×5=30 6×4=24 6×3=18 6×2=12
5×5=25 5×4=20 5×3=15 5×2=10
4×4=16 4×3=12 4×2=8
3×3=9 3×2=6
2×2=4

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



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