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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:
99=81 98=72 97=63 96=54 95=45 94=36 93=27 92=18
88=64 87=56 86=48 85=40 84=32 83=24 82=16
77=49 76=42 75=35 74=28 73=21 72=14
66=36 65=30 64=24 63=18 62=12
55=25 54=20 53=15 52=10
44=16 43=12 42=8
33=9 32=6

This table does not give the ones and zeros. That is because:

  • Anything times zero is zero.
  • Anything times one is itself. For example, 51=5.

Adding a number to itself is the same as multiplying it by two. For example, 7+7=14, which is the same as 72.

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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