Multiplication table

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In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system.

Contents 1 In basic arithmetic 2 Traditional use 3 Patterns in the tables 4 In abstract algebra

[edit] In basic arithmetic

A multiplication table ("times table", as used to teach schoolchildren multiplication) is 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. Traditionally, the heading for the first row and first column contains the symbol for the multiplication operator.

×	1	2	3	4	5	6	7	8	9	10	11	12
1	1	2	3	4	5	6	7	8	9	10	11	12
2	2	4	6	8	10	12	14	16	18	20	22	24
3	3	6	9	12	15	18	21	24	27	30	33	36
4	4	8	12	16	20	24	28	32	36	40	44	48
5	5	10	15	20	25	30	35	40	45	50	55	60
6	6	12	18	24	30	36	42	48	54	60	66	72
7	7	14	21	28	35	42	49	56	63	70	77	84
8	8	16	24	32	40	48	56	64	72	80	88	96
9	9	18	27	36	45	54	63	72	81	90	99	108
10	10	20	30	40	50	60	70	80	90	100	110	120
11	11	22	33	44	55	66	77	88	99	110	121	132
12	12	24	36	48	60	72	84	96	108	120	132	144

So, for example, 3×6=18 by looking up where 3 and 6 intersect. This table does not give the zeros. That is because any real number times zero is zero. Multiplication tables vary from country to country. They may have ranges from 1×1 to 10×10, from 2×1 to 9×9, or from 1×1 to 12×12 to quote a few examples. The layout, however, follows the left-to-right, top-to-bottom movement of Western culture's reading. Since Descartes' promotion of the rectangular coordinate system, the presentation of the multiplication facts in the first quadrant makes pedagogical sense for the math learner. The XY Chart shown below has taken advantage of this.

The XY Chart multiplication table shows the facts in the first quadrant. Under this arrangement as multiplication learners navigate from the factors to products they follow the same steps as plotting points by following the two coordinates. Curiously, the layout of the XY Chart corresponds closely to the definition of Cartesian products.

[edit] Traditional use

The traditional rote learning of multiplication was based on memorisation of columns in the table, in a form like

1 × 7 = 7

2 × 7 = 14

3 × 7 = 21

4 × 7 = 28

5 × 7 = 35

6 × 7 = 42

7 × 7 = 49

8 × 7 = 56

9 × 7 = 63

[edit] Patterns in the tables

For example, for multiplication by 6 a pattern emerges:

  2 × 6 = 12
  4 × 6 = 24
  6 × 6 = 36
  8 × 6 = 48
 10 × 6 = 60

 number × 6 = half_of_number_times_10  + number 

The rule is convenient for even numbers, but also true for odd ones:

 1 × 6 = 05 +  1 =  6
 2 × 6 = 10 +  2 = 12
 3 × 6 = 15 +  3 = 18
 4 × 6 = 20 +  4 = 24
 5 × 6 = 25 +  5 = 30
 6 × 6 = 30 +  6 = 36
 7 × 6 = 35 +  7 = 42
 8 × 6 = 40 +  8 = 48
 9 × 6 = 45 +  9 = 54
10 × 6 = 50 + 10 = 60

[edit] In abstract algebra

Multiplication tables can also define binary operations on groups, fields, rings, and other algebraic systems. In such contexts they can be called Cayley tables. For an example, see octonion.