Main Page | See live article | Alphabetical index

# 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 ū 2ū2 = 4 2 3ū3 = 9 3ū2 = 6 3 4ū4 = 16 4ū3 = 12 4ū2 = 8 4 5ū5 = 25 5ū4 = 20 5ū3 = 15 5ū2 = 10 5 6ū6 = 36 6ū5 = 30 6ū4 = 24 6ū3 = 18 6ū2 = 12 6 7ū7 = 49 7ū6 = 42 7ū5 = 35 7ū4 = 28 7ū3 = 21 7ū2 = 14 7 8ū8 = 64 8ū7 = 56 8ū6 = 48 8ū5 = 40 8ū4 = 32 8ū3 = 24 8ū2 = 16 8 9ū9 = 81 9ū8 = 72 9ū7 = 63 9ū6 = 54 9ū5 = 45 9ū4 = 36 9ū3 = 27 9ū2 = 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, 5ū1=5.

Adding a number to itself is the same as multiplying it by two. For example, 7+7=14, which is the same as 7ū2.

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