multiplication table
In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system.
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Contents
- 1 In basic arithmetic
- 2 Traditional use
- 3 Patterns in the tables
- 4 In abstract algebra
- 5 External links
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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.
| × |
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.
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
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
In general:
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
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.
External links
- Printable Tables in Black and White or Color
- Learn the Multiplication Table then Test Yourself
- Printable worksheets at kwizNET Learning System
- Arithmetic Operations In Various Number Systems, Abacus In Various Number Systems, Soroban In Various Number Systems and Suan Pan In Various Number Systems at cut-the-knot
- Actual XY Charts at Multiplication Tables
Category: Elementary arithmetic |
