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In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system.
The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation for arithmetic operations with baseten numbers. Many educators believe it is necessary to memorize the table up to 9 × 9.^{[1]}
The oldest known multiplication tables were used by the Babylonians about 4000 years ago. They used base 60.^{[2]} The oldest known tables using base 10 are the decimal multiplication table on bamboo strips dating to about 305 BC, found in China.^{[2]}
The table is sometimes attributed to Pythagoras. It is also called the Table of Pythagoras in many languages (for example French, Italian and apparently Russian long ago), sometimes in English.^{[4]}
In 493 A.D., Victorius of Aquitaine wrote a 98column multiplication table which gave (in Roman numerals) the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then the fractions down to 1/144" (Maher & Makowski 2001, p. 383)
In his 1820 book The Philosophy of Arithmetic,^{[5]} mathematician John Leslie published a multiplication table up to 99 × 99, which allows numbers to be multiplied in pairs of digits at a time. Leslie also recommended that young pupils memorize the multiplication table up to 25 × 25.
0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20 

1  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20 
2  2  4  6  8  10  12  14  16  18  20  22  24  26  28  30  32  34  36  38  40 
3  3  6  9  12  15  18  21  24  27  30  33  36  39  42  45  48  51  54  57  60 
4  4  8  12  16  20  24  28  32  36  40  44  48  52  56  60  64  68  72  76  80 
5  5  10  15  20  25  30  35  40  45  50  55  60  65  70  75  80  85  90  95  100 
6  6  12  18  24  30  36  42  48  54  60  66  72  78  84  90  96  102  108  114  120 
7  7  14  21  28  35  42  49  56  63  70  77  84  91  98  105  112  119  126  133  140 
8  8  16  24  32  40  48  56  64  72  80  88  96  104  112  120  128  136  144  152  160 
9  9  18  27  36  45  54  63  72  81  90  99  108  117  126  135  144  153  162  171  180 
10  10  20  30  40  50  60  70  80  90  100  110  120  130  140  150  160  170  180  190  200 
11  11  22  33  44  55  66  77  88  99  110  121  132  143  154  165  176  187  198  209  220 
12  12  24  36  48  60  72  84  96  108  120  132  144  156  168  180  192  204  216  228  240 
13  13  26  39  52  65  78  91  104  117  130  143  156  169  182  195  208  221  234  247  260 
14  14  28  42  56  70  84  98  112  126  140  154  168  182  196  210  224  238  252  266  280 
15  15  30  45  60  75  90  105  120  135  150  165  180  195  210  225  240  255  270  285  300 
16  16  32  48  64  80  96  112  128  144  160  176  192  208  224  240  256  272  288  304  320 
17  17  34  51  68  85  102  119  136  153  170  187  204  221  238  255  272  289  306  323  340 
18  18  36  54  72  90  108  126  144  162  180  198  216  234  252  270  288  306  324  342  360 
19  19  38  57  76  95  114  133  152  171  190  209  228  247  266  285  304  323  342  361  380 
20  20  40  60  80  100  120  140  160  180  200  220  240  260  280  300  320  340  360  380  400 
The traditional rote learning of multiplication was based on memorization of columns in the table, in a form like
1 × 10 = 10
2 × 10 = 20
3 × 10 = 30
4 × 10 = 40
5 × 10 = 50
6 × 10 = 60
7 × 10 = 70
8 × 10 = 80
9 × 10 = 90
This form of writing the multiplication table in columns with complete number sentences is still used in some countries instead of the modern grid above.
There is a pattern in the multiplication table that can help people to memorize the table more easily. It uses the figures below:
→ → 1 2 3 2 4 ↑ 4 5 6 ↓ ↑ ↓ 7 8 9 6 8 ← ← 0 0 Fig. 1 Fig. 2
For example, to memorize all the multiples of 7:
Figure 1 is used for multiples of 1, 3, 7, and 9. Figure 2 is used for the multiples of 2, 4, 6, and 8. These patterns can be used to memorize the multiples of any number from 0 to 10, except 5. As you would start on the number you are multiplying, when you multiply by 0, you stay on 0 (0 is external and so the arrows have no effect on 0, otherwise 0 is used as a link to create a perpetual cycle). The pattern also works with multiples of 10, by starting at 1 and simply adding 0, giving you 10, then just apply every number in the pattern to the "tens" unit as you would normally do as usual to the "ones" unit.
Tables can also define binary operations on groups, fields, rings, and other algebraic systems. In such contexts they can be called Cayley tables. Here are the addition and multiplication tables for the finite field Z_{5}.


For other examples, see group, and octonion.
The Chinese multiplication table consists of eightyone sentences with five Chinese characters per sentence, making it is easy for children to learn by heart. A shorter version of the table consists of only fortyfive sentences, as terms such as "nine eights beget seventytwo" are identical to "eight nines beget seventytwo" so there is no need to learn them twice.
A bundle of 21 bamboo slips dated 305 B.C. in the Warring States period in the Tsinghua Bamboo Slips (清华简) collection is the world's earliest known example of a decimal multiplication table.^{[6]}
In 1989, the National Council of Teachers of Mathematics (NCTM) developed new standards which were based on the belief that all students should learn higherorder thinking skills, and which recommended reduced emphasis on the teaching of traditional methods that relied on rote memorization, such as multiplication tables. Widely adopted texts such as Investigations in Numbers, Data, and Space (widely known as TERC after its producer, Technical Education Research Centers) omitted aids such as multiplication tables in early editions. NCTM made it clear in their 2006 Focal Points that basic mathematics facts must be learned, though there is no consensus on whether rote memorization is the best method.