Multiplication Table

A multiplication table is an array showing the result of applying a binary operator to elements of a given set . For example, the following table is the multiplication table for ordinary multiplication.

	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

The ordinary multiplication table extends to the plane by assigning to each point . If , , ..., , are the vertices of a regular polygon with and center , then their assigned values have arithmetic mean (Delaporte 2023, p. 37)

The Chinese table illustrated above consists of 21 bamboo strips and dates to around 305 BC. It is the oldest known multiplication table using base 10 notation. Much earlier Babylonian multiplication tables use base 60 (Qiu 2014).

The results of any binary mathematical operation can be written as a multiplication table. For example, groups have multiplication tables, where the group operation is understood as multiplication. However, different labelings and orderings of a multiplication table may describe the same abstract group. For example, the multiplication table for the cyclic group C₄ may be written in three equivalent ways--denoted here by , , and --by permuting the symbols used for the group elements (Cotton 1990, p. 11).

The first such table can be written as follows.

	1
1	1
				1
			1
		1

The multiplication table for a second representation may be obtained from by interchanging and .

	1
1	1
		1
				1
			1

And finally, a multiplication table for the third representation can be obtained from by interchanging and .

	1
1	1
			1
		1
				1