The multiplication table is sometimes attributed to the ancient Greek mathematician Pythagoras. It is also called the Table of Pythagoras in many languages, sometimes in English. The Greco-Roman mathematician Nichomachus, a follower of Neopythagoreanism, included a multiplication table in his Introduction to Arithmetic, whereas the oldest surviving Greek multiplication table is on a wax tablet dated to the 1st century AD and currently housed in the British Museum. In 493 AD, Victorius of Aquitaine wrote a 98-column multiplication table which gave 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." In his 1820 book The Philosophy of Arithmetic, 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 50 × 50. The illustration below shows a table up to 12 × 12, which is a size commonly used in schools.
×
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
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, such as Bosnia and Herzegovina, instead of the modern grid above.
Patterns in the tables
There is a pattern in the multiplication table that can help people to memorize the table more easily. It uses the figures below: 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. 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. For example, to recall all the multiples of 7:
Look at the 7 in the first picture and follow the arrow.
The next number in the direction of the arrow is 4. So think of the next number after 7 that ends with 4, which is 14.
The next number in the direction of the arrow is 1. So think of the next number after 14 that ends with 1, which is 21.
After coming to the top of this column, start with the bottom of the next column, and travel in the same direction. The number is 8. So think of the next number after 21 that ends with 8, which is 28.
Proceed in the same way until the last number, 3, corresponding to 63.
Next, use the 0 at the bottom. It corresponds to 70.
Then, start again with the 7. This time it will correspond to 77.
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 fieldZ5. For every natural number n, there are also addition and multiplication tables for the ringZn.
The Chinese multiplication table consists of eighty-one sentences with four or five Chinese characters per sentence, making it easy for children to learn by heart. A shorter version of the table consists of only forty-five sentences, as terms such as "nine eights beget seventy-two" are identical to "eight nines beget seventy-two" so there is no need to learn them twice. A minimum version by removing all "one" sentences, consists of only thirty-six sentences, which is most commonly used in schools in China. It's often in this order: 2x2=4, 2x3=6,..., 2x8=16, 2x9=18, 3x3, 3x4,..., 3x9, 4x4,..., 4x9, 5x5,...,9x9
A bundle of 21 bamboo slips dated 305 BC in the Warring States period in the Tsinghua Bamboo Slips collection is the world's earliest known example of a decimal multiplication table.
Standards-based mathematics reform in the US
In 1989, the National Council of Teachers of Mathematics developed new standards which were based on the belief that all students should learn higher-order thinking skills, 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 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.
