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
22 = 4 2
33 = 9 32 = 6 3
44 = 16 43 = 12 42 = 8 4
55 = 25 54 = 20 53 = 15 52 = 10 5
66 = 36 65 = 30 64 = 24 63 = 18 62 = 12 6
77 = 49 76 = 42 75 = 35 74 = 28 73 = 21 72 = 14 7
88 = 64 87 = 56 86 = 48 85 = 40 84 = 32 83 = 24 82 = 16 8
99 = 81 98 = 72 97 = 63 96 = 54 95 = 45 94 = 36 93 = 27 92 = 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, 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