Thedecimal multiplication table was traditionally taught as an essential part ofelementary arithmetic around the world, as it lays the foundation for arithmetic operations with base-ten numbers. Many educators believe it is necessary to memorize the table up to 9 × 9.[1]
The multiplication table is sometimes attributed to the ancient Greek mathematicianPythagoras (570–495 BC). It is also called the Table of Pythagoras in many languages (for example French, Italian and Russian), sometimes in English.[4] TheGreco-Roman mathematicianNichomachus (60–120 AD), a follower ofNeopythagoreanism, included a multiplication table in hisIntroduction to Arithmetic, whereas the oldest survivingGreek multiplication table is on a wax tablet dated to the 1st century AD and currently housed in theBritish Museum.[5]
In 493 AD,Victorius of Aquitaine wrote a 98-column multiplication table which gave (inRoman 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."[6]
In his 1820 bookThe Philosophy of Arithmetic,[7] mathematicianJohn Leslie published a table of "quarter-squares" which could be used, with some additional steps, for multiplication up to 1000 × 1000. Leslie also recommended that young pupils memorize the multiplication table up to 50 × 50.
In 1897,August Leopold Crelle publishedCalculating tables giving the products of every two numbers from one to one thousand[8] which is a simple multiplication table for products up to 1000 × 10000.
Tables showing all products of numbers from 1 to 10 or 1 to 12 are the sizes most commonly found in primary schools. The table below shows products up to 12 × 12:
×
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 common multi-digitmultiplication algorithms taught in school break that problem down into a sequence of single-digit multiplication and multi-digit addition problems. Single-digit multiplication can be summarized in a 100-entry table of all products of digits from 0 to 9. Because0 ×a = 0 for any numbera, the rows and columns for multiplication by 0 are typically left out. Multiplication of integers iscommutative,a ×b =b ×a. Therefore, the table is symmetric across its main diagonal, and can be reduced to 45 entries by only showing entriesa ×b wherea ≥b, as shown below. The table could be reduced further (to 36 entries) by leaving off rows and columns for multiplication by 1, themultiplicative identity, which satisfiesa × 1 =a.
1
1
2
2
4
3
3
6
9
4
4
8
12
16
5
5
10
15
20
25
6
6
12
18
24
30
36
7
7
14
21
28
35
42
49
8
8
16
24
32
40
48
56
64
9
9
18
27
36
45
54
63
72
81
×
1
2
3
4
5
6
7
8
9
The traditionalrote learning of multiplication was based on memorization of columns in the table, arranged as follows.
This form of writing the multiplication table in columns with complete number sentences is still used in some countries, such as Colombia, Bosnia and Herzegovina,[citation needed] instead of the modern grids 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
5
0
Figure 1: Odd
Figure 2: Even
Cycles of the unit digit of multiples of integers ending in 1, 3, 7 and 9 (upper row), and 2, 4, 6 and 8 (lower row) on atelephone keypad
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.
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.
For every natural numbern, addition and multiplication inZn, the ring of integers modulon, is described by ann byn table(see:Modular arithmetic). For example, the tables forZ5 are:
The Chinese multiplication table consists of eighty-one terms. It was historically called thenine-nine table, because in ancient times it started with 9 × 9: nine nines beget eighty-one, eight nines beget seventy-two, etc. It was known in China as early as theSpring and Autumn period, and survived through the age of the abacus; pupils in elementary school today still must memorize it. A shorter version of the table consists of only forty-five sentences:
九九乘法口诀表 (The Nine-nine multiplication table)
×
1 一 yī
2 二 èr
3 三 sān
4 四 sì
5 五 wǔ
6 六 liù
7 七 qī
8 八 bā
9 九 jiǔ
1 一 yī
一一 得一
2 二 èr
一二 得二
二二 得四
3 三 sān
一三 得三
二三 得六
三三 得九
4 四 sì
一四 得四
二四 得八
三四 十二
四四 十六
5 五 wǔ
一五 得五
二五 一十
三五 十五
四五 二十
五五 二十五
6 六 liù
一六 得六
二六 十二
三六 十八
四六 二十四
五六 三十
六六 三十六
7 七 qī
一七 得七
二七 十四
三七 二十一
四七 二十八
五七 三十五
六七 四十二
七七 四十九
8 八 bā
一八 得八
二八 十六
三八 二十四
四八 三十二
五八 四十
六八 四十八
七八 五十六
八八 六十四
9 九 jiǔ
一九 得九
二九 十八
三九 二十七
四九 三十六
五九 四十五
六九 五十四
七九 六十三
八九 七十二
九九 八十一
Mokkan discovered atHeijō Palace suggest that the multiplication table may have been introduced to Japan through Chinese mathematical treatises such as theSunzi Suanjing, because their expression of the multiplication table share the character如 in products less than ten.[9] Chinese and Japanese share a similar system of eighty-one short, easily memorable sentences taught to students to help them learn the multiplication table up to 9 × 9. In current usage, the sentences that express products less than ten include an additional particle in both languages. In the case of modern Chinese, this is得 (dé); and in Japanese, this isが (ga). This is useful for those who practice calculation with asuanpan or asoroban, because the sentences remind them to shift one column to the right when inputting a product that does not begin with atens digit. In particular, the Japanese multiplication table uses non-standard pronunciations for numbers in some specific instances (such as the replacement ofsan roku withsaburoku; indicated in bold below).
The Japanese multiplication table
×
1ichi
2ni
3san
4shi
5go
6roku
7shichi
8ha
9ku
1 in
in'ichi ga ichi
inni ga ni
insan ga san
inshi ga shi
ingo ga go
inroku ga roku
inshichi ga shichi
inhachi ga hachi
inku ga ku
2 ni
ni ichi ga ni
ninin ga shi
ni san ga roku
ni shi ga hachi
ni go jū
ni roku jūni
ni shichi jūshi
ni hachi jūroku
ni ku jūhachi
3 san
san ichi ga san
san ni ga roku
sazan ga ku
san shi jūni
san go jūgo
saburoku jūhachi
san shichi nijūichi
sanpa nijūshi
san ku nijūshichi
4 shi
shi ichi ga shi
shi ni ga hachi
shi san jūni
shi shi jūroku
shi go nijū
shi roku nijūshi
shi shichi nijūhachi
shi ha sanjūni
shi ku sanjūroku
5 go
go ichi ga go
go ni jū
go san jūgo
go shi nijū
go go nijūgo
go roku sanjū
go shichi sanjūgo
go ha shijū
gokku shijūgo
6 roku
roku ichi ga roku
roku ni jūni
roku san jūhachi
roku shi nijūshi
roku go sanjū
roku roku sanjūroku
roku shichi shijūni
roku ha shijūhachi
rokku gojūshi
7 shichi
shichi ichi ga shichi
shichi ni jūshi
shichi san nijūichi
shichi shi nijūhachi
shichi go sanjūgo
shichi roku shijūni
shichi shichi shijūku
shichi ha gojūroku
shichi ku rokujūsan
8 hachi
hachi ichi ga hachi
hachi ni jūroku
hachi san nijūshi
hachi shi sanjūni
hachi go shijū
hachi roku shijūhachi
hachi shichi gojūroku
rokku gojūshi
hakku shichijūni
9 ku
ku ichi ga ku
ku ni jūhachi
ku san nijūshichi
ku shi sanjūroku
ku go shijūgo
ku roku gojūshi
ku shichi rokujūsan
ku ha shichijūni
ku ku hachijūichi
Warring States decimal multiplication bamboo slips
A bundle of 21 bamboo slips dated 305 BC in theWarring States period in theTsinghua Bamboo Slips (清華簡) collection is the world's earliest known example of a decimal multiplication table.[10]
A modern representation of the Warring States decimal multiplication table used to calculate 12 × 34.5
In 1989, theNational Council of Teachers of Mathematics (NCTM) 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 asInvestigations in Numbers, Data, and Space (widely known asTERC after its producer, Technical Education Research Centers) omitted aids such as multiplication tables in early editions. NCTM made it clear in their 2006Focal Points that basic mathematics facts must be learned, though there is no consensus on whether rote memorization is the best method. In recent years, a number of nontraditional methods have been devised to help children learn multiplication facts, including video-game style apps and books that aim to teach times tables through character-based stories.
In 2024, the recommendation to learn the multiplication table was removed from the California Mathematics Curriculum Framework.[11]
^David E. Smith (1958),History of Mathematics, Volume I: General Survey of the History of Elementary Mathematics. New York: Dover Publications (a reprint of the 1951 publication),ISBN0-486-20429-4, pp. 58, 129.
^David W. Maher and John F. Makowski. "Literary evidence for Roman arithmetic with fractions".Classical Philology, 96/4 (October 2001), p. 383.
^Leslie, John (1820).The Philosophy of Arithmetic; Exhibiting a Progressive View of the Theory and Practice of Calculation, with Tables for the Multiplication of Numbers as Far as One Thousand. Edinburgh: Abernethy & Walker.
