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 base-ten numbers. Many educators believe it is necessary to memorize the table up to 9 × 9..
History
Pre-modern times
The oldest known multiplication tables were used by the Babylonians about 4000 years ago. However, they used a base of 60. The oldest known tables using a base of 10 are the Chinese decimal multiplication table on bamboo strips dating to about 305 BC, during China's Warring States period.
The multiplication table is sometimes attributed to the ancient Greek mathematician Pythagoras (570–495 BC). It is also called the Table of Pythagoras in many languages (for example French, Italian and Russian), sometimes in English.for example in An Elementary Treatise on Arithmetic by John Farrar The Greco-Roman mathematician Nichomachus (60–120 AD), 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.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), , pp. 58, 129.
In 493 AD, Victorius of Aquitaine wrote a 98-column 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."David W. Maher and John F. Makowski. "Literary evidence for Roman arithmetic with fractions". Classical Philology, 96/4 (October 2001), p. 383.
Modern times In his 1820 book The Philosophy of Arithmetic, mathematician John 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 published Calculating tables giving the products of every two numbers from one to one thousand 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:
×1234567891011121 1 2 3 4 5 6 7 8 9 10 11 122 2 4 6 8 10 12 14 16 18 20 22 243 3 6 9 12 15 18 21 24 27 30 33 364 4 8 12 16 20 24 28 32 36 40 44 485 5 10 15 20 25 30 35 40 45 50 55 606 6 12 18 24 30 36 42 48 54 60 66 727 7 14 21 28 35 42 49 56 63 70 77 848 8 16 24 32 40 48 56 64 72 80 88 969 9 18 27 36 45 54 63 72 81 90 99 10810 10 20 30 40 50 60 70 80 90 100 110 12011 11 22 33 44 55 66 77 88 99 110 121 13212 12 24 36 48 60 72 84 96 108 120 132 144
The common multi-digit multiplication 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. Because for any number , the rows and columns for multiplication by 0 are typically left out. Multiplication of integers is commutative, . Therefore, the table is symmetric across its main diagonal, and can be reduced to 45 entries by only showing entries where , as shown below. The table could be reduced further (to 36 entries) by leaving off rows and columns for multiplication by 1, the multiplicative identity, which satisfies .
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 81×123456789
The traditional rote learning of multiplication was based on memorization of columns in the table, arranged as follows.
1 × 1 = 1 2 × 1 = 2 3 × 1 = 3 4 × 1 = 4 5 × 1 = 5 6 × 1 = 6 7 × 1 = 7 8 × 1 = 8 9 × 1 = 9 10 × 1 = 10 11 × 1 = 11 12 × 1 = 121 × 2 = 2 2 × 2 = 4 3 × 2 = 6 4 × 2 = 8 5 × 2 = 10 6 × 2 = 12 7 × 2 = 14 8 × 2 = 16 9 × 2 = 18 10 × 2 = 20 11 × 2 = 22 12 × 2 = 241 × 3 = 3 2 × 3 = 6 3 × 3 = 9 4 × 3 = 12 5 × 3 = 15 6 × 3 = 18 7 × 3 = 21 8 × 3 = 24 9 × 3 = 27 10 × 3 = 30 11 × 3 = 33 12 × 3 = 361 × 4 = 4 2 × 4 = 8 3 × 4 = 12 4 × 4 = 16 5 × 4 = 20 6 × 4 = 24 7 × 4 = 28 8 × 4 = 32 9 × 4 = 36 10 × 4 = 40 11 × 4 = 44 12 × 4 = 481 × 5 = 5 2 × 5 = 10 3 × 5 = 15 4 × 5 = 20 5 × 5 = 25 6 × 5 = 30 7 × 5 = 35 8 × 5 = 40 9 × 5 = 45 10 × 5 = 50 11 × 5 = 55 12 × 5 = 601 × 6 = 6 2 × 6 = 12 3 × 6 = 18 4 × 6 = 24 5 × 6 = 30 6 × 6 = 36 7 × 6 = 42 8 × 6 = 48 9 × 6 = 54 10 × 6 = 60 11 × 6 = 66 12 × 6 = 721 × 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 10 × 7 = 70 11 × 7 = 77 12 × 7 = 841 × 8 = 8 2 × 8 = 16 3 × 8 = 24 4 × 8 = 32 5 × 8 = 40 6 × 8 = 48 7 × 8 = 56 8 × 8 = 64 9 × 8 = 72 10 × 8 = 80 11 × 8 = 88 12 × 8 = 961 × 9 = 9 2 × 9 = 18 3 × 9 = 27 4 × 9 = 36 5 × 9 = 45 6 × 9 = 54 7 × 9 = 63 8 × 9 = 72 9 × 9 = 81 10 × 9 = 90 11 × 9 = 99 12 × 9 = 1081 × 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 10 × 10 = 100 11 × 10 = 110 12 × 10 = 1201 × 11 = 11 2 × 11 = 22 3 × 11 = 33 4 × 11 = 44 5 × 11 = 55 6 × 11 = 66 7 × 11 = 77 8 × 11 = 88 9 × 11 = 99 10 × 11 = 110 11 × 11 = 121 12 × 11 = 1321 × 12 = 12 2 × 12 = 24 3 × 12 = 36 4 × 12 = 48 5 × 12 = 60 6 × 12 = 72 7 × 12 = 84 8 × 12 = 96 9 × 12 = 108 10 × 12 = 120 11 × 12 = 132 12 × 12 = 144
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, instead of the modern grids 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: → →↑1 23↓↑2 4↓ 4 5 6 7 896 8←← 0 5 0 Figure 1: OddFigure 2: Even
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. Continue like this.
In abstract algebra Tables can also define binary operations on groups, fields, rings, and other algebraic systems. In such contexts they are called Cayley tables.
For every natural number n, addition and multiplication in Zn, the ring of integers modulo n, is described by an n by n table . For example, the tables for Z5 are:
+012340 0 1 2 3 41 1 2 3 4 02 2 3 4 0 13 3 4 0 1 24 4 0 1 2 3
×012340 0 0 0 0 01 0 1 2 3 42 0 2 4 1 33 0 3 1 4 24 0 4 3 2 1
For other examples, see group.
Hypercomplex numbers
Hypercomplex number multiplication tables show the non-commutative results of multiplying two hypercomplex imaginary units. The simplest example is that of the quaternion multiplication table :
{|class="wikitable" |+Quaternion multiplication table |- !width=15 nowrap|↓ × → !width=15| !width=15| !width=15| !width=15| |- !
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Chinese and Japanese multiplication tables
The Chinese multiplication table consists of eighty-one terms. It was historically called the nine-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 the Spring 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 yī2 二 èr3 三 sān4 四 sì5 五 wǔ6 六 liù7 七 qī8 八 bā9 九 jiǔ1 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 at Heijō Palace suggest that the multiplication table may have been introduced to Japan through Chinese mathematical treatises such as the Sunzi Suanjing, because their expression of the multiplication table share the character in products less than ten. 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 (); and in Japanese, this is (). This is useful for those who practice calculation with a suanpan or a soroban, because the sentences remind them to shift one column to the right when inputting a product that does not begin with a tens digit. In particular, the Japanese multiplication table uses non-standard pronunciations for numbers in some specific instances (such as the replacement of san roku with saburoku; indicated in bold below).
+ The Japanese multiplication table× 1 ichi2 ni3 san4 shi5 go6 roku7 shichi8 ha9 ku1 inin'ichi ga ichiinni ga niinsan ga saninshi ga shiingo ga goinroku ga rokuinshichi ga shichiinhachi ga hachiinku ga ku2 nini ichi ga nini ga shini san ga rokuni shi ga hachini go jūni roku jūnini shichi jūshini hachi jūrokuni ku jūhachi3 sansan ichi ga sansan ni ga roku ga kusan shi jūnisan go jūgo jūhachisan shichi nijūichi nijūshisan ku nijūshichi4 shishi ichi ga shishi ni ga hachishi san jūnishi shi jūrokushi go nijūshi roku nijūshishi shichi nijūhachishi ha sanjūnishi ku sanjūroku5 gogo ichi ga gogo ni jūgo san jūgogo shi nijūgo go nijūgogo roku sanjūgo shichi sanjūgogo ha shijū shijūgo6 rokuroku ichi ga rokuroku ni jūniroku san jūhachiroku shi nijūshiroku go sanjūroku roku sanjūrokuroku shichi shijūniroku ha shijūhachi gojūshi7 shichishichi ichi ga shichishichi ni jūshishichi san nijūichishichi shi nijūhachishichi go sanjūgoshichi roku shijūnishichi shichi shijūkushichi ha gojūrokushichi ku rokujūsan8 hachihachi ichi ga hachihachi ni jūrokuhachi san nijūshihachi shi sanjūnihachi go shijūhachi roku shijūhachihachi shichi gojūroku rokujūshi shichijūni9 kuku ichi ga kuku ni jūhachiku san nijūshichiku shi sanjūrokuku go shijūgoku roku gojūshiku shichi rokujūsanku ha shichijūniku ku hachijūichi
Warring States decimal multiplication bamboo slips A bundle of 21 bamboo slips dated 305 BC in the Warring States period in the Tsinghua Bamboo Slips (清華簡) collection is the world's earliest known example of a decimal multiplication table.Nature article The 2,300-year-old matrix is the world's oldest decimal multiplication table