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