Multiplication table

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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	?/td>
							2? = 4	2
						3? = 9	3? = 6	3
					4? = 16	4? = 12	4? = 8	4
				5? = 25	5? = 20	5? = 15	5? = 10	5
			6? = 36	6? = 30	6? = 24	6? = 18	6? = 12	6
		7? = 49	7? = 42	7? = 35	7? = 28	7? = 21	7? = 14	7
	8? = 64	8? = 56	8? = 48	8? = 40	8? = 32	8? = 24	8? = 16	8
9? = 81	9? = 72	9? = 63	9? = 54	9? = 45	9? = 36	9? = 27	9? = 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?=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?.

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