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