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Multiplication Table

A multiplication table is an array showing the result of applying a binary operator to elements of a given set S. 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 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 C_4^((1)), C_4^((2)), and C_4^((3))--by permuting the symbols used for the group elements (Cotton 1990, p. 11).

The first such table can be written as follows.

C_4^((1)) 1 A B C
1 1 A B C
A A B C 1
B B C 1 A
C C 1 A B

The multiplication table for a second representation C_4^((2)) may be obtained from C_4^((1)) by interchanging A and B.

C_4^((2)) 1 A B C
1 1 A B C
A A 1 C B
B B C A 1
C C B 1 A

And finally, a multiplication table for the third representation C_4^((3)) can be obtained from C_4^((1)) by interchanging A and C.

C_4^((3)) 1 A B C
1 1 A B C
A A C 1 B
B B 1 C A
C C B A 1

See also

Abstract Group, Binary Operator, Group, Truth Table

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References

Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, 1990.

Referenced on Wolfram|Alpha

Multiplication Table

Cite this as:

Weisstein, Eric W. "Multiplication Table." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MultiplicationTable.html

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