mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a

multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...

operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Ma ...

for an algebraic system. The

decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...

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

In pre-modern time

The oldest known multiplication tables were used by the

Babylonians Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c. 1 ...

about 4000 years ago. However, they used a base of 60. The oldest known tables using a base of 10 are the

Chinese Chinese can refer to: * Something related to China * Chinese people, people of Chinese nationality, citizenship, and/or ethnicity **''Zhonghua minzu'', the supra-ethnic concept of the Chinese nation ** List of ethnic groups in China, people of ...

decimal multiplication table on bamboo strips dating to about 305 BC, during China's

Warring States The Warring States period () was an era in ancient Chinese history characterized by warfare, as well as bureaucratic and military reforms and consolidation. It followed the Spring and Autumn period and concluded with the Qin wars of conquest ...

period.

PSM V26 D467 Table of pythagoras on slats

The multiplication table is sometimes attributed to the ancient Greek mathematician

Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...

(570–495 BC). It is also called the Table of Pythagoras in many languages (for example French, Italian and Russian), sometimes in English. The Greco-Roman mathematician

Nichomachus Nichomachus ( grc-gre, Νιχόμαχος ) was a playwright who lived in Athens in the 5th century BC. He was a younger contemporary of Sophocles. Only the following titles and associated fragments of Nichomachus's plays have survived: ''Alcmae ...

(60–120 AD), a follower of

Neopythagoreanism Neopythagoreanism (or neo-Pythagoreanism) was a school of Hellenistic philosophy which revived Pythagorean doctrines. Neopythagoreanism was influenced by middle Platonism and in turn influenced Neoplatonism. It originated in the 1st century BC ...

, included a multiplication table in his ''

Introduction to Arithmetic The book ''Introduction to Arithmetic'' ( grc-gre, Ἀριθμητικὴ εἰσαγωγή, ''Arithmetike eisagoge'') is the only extant work on mathematics by Nicomachus (60–120 AD). Summary The work contains both philosophical prose an ...

'', whereas the oldest surviving

Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

multiplication table is on a wax tablet dated to the 1st century AD and currently housed in the

British Museum The British Museum is a public museum dedicated to human history, art and culture located in the Bloomsbury area of London. Its permanent collection of eight million works is among the largest and most comprehensive in existence. It docum ...

. In 493 AD,

Victorius of Aquitaine Victorius of Aquitaine, a countryman of Prosper of Aquitaine and also working in Rome, produced in AD 457 an Easter Cycle, which was based on the consular list provided by Prosper's Chronicle. This dependency caused scholars to think that Prosper ...

wrote a 98-column multiplication table which gave (in

Roman numerals Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, eac ...

) 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 modern time

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 nowadays in English-world schools. In China, however, because multiplication of integers is

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...

, many schools use a smaller table as below. Some schools even remove the first column since 1 is the multiplicative identity.

The traditional

rote learning Rote learning is a memorization technique based on repetition. The method rests on the premise that the recall of repeated material becomes faster the more one repeats it. Some of the alternatives to rote learning include meaningful 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 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: Multiplication_mnemonic

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

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

s, and other algebraic systems. In such contexts they are called

Cayley table Named after the 19th century British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplic ...

s. Here are the addition and multiplication tables for the

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...

Z₅: *for every natural number ''n'', there are also addition and multiplication tables for the ring Z_''n''. For other examples, see

, and

octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...

Chinese and Japanese multiplication tables

Mokkan are wooden tablets found at Japanese archaeological sites. Most of the tablets date from the mid-7th to mid-8th century, but some are as late as the early modern period. They have been found in sites across Japan, but mostly around the old capita ...

discovered at

Heijō Palace was the imperial residence in the Japanese capital city Heijō-kyō (today's Nara), during most of the Nara period. The palace, which served as the imperial residence and the administrative centre of for most of the Nara period from 710 to 794 ...

suggest that the multiplication table may have been introduced to Japan through Chinese mathematical treatises such as the

Sunzi Suanjing ''Sunzi Suanjing'' () was a mathematical treatise written during 3rd to 5th centuries AD which was listed as one of the Ten Computational Canons during the Tang dynasty. The specific identity of its author Sunzi (lit. "Master Sun") is still ...

, 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 The suanpan (), also spelled suan pan or souanpan) is an abacus of Chinese origin first described in a 190 CE book of the Eastern Han Dynasty, namely ''Supplementary Notes on the Art of Figures'' written by Xu Yue. However, the exact design ...

or a

soroban The is an abacus developed in Japan. It is derived from the ancient Chinese suanpan, imported to Japan in the 14th century. Like the suanpan, the soroban is still used today, despite the proliferation of practical and affordable pocket electr ...

, 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'').

Warring States decimal multiplication bamboo slips

A bundle of 21 bamboo slips dated 305 BC in the

period in the

Tsinghua Bamboo Slips The Tsinghua Bamboo Strips () are a collection of Chinese texts dating to the Warring States period and written in ink on strips of bamboo, that were acquired in 2008 by Tsinghua University, China. The texts were obtained by illegal excavation, pr ...

(清華簡) collection is the world's earliest known example of a decimal multiplication table.''Nature'' articl
The 2,300-year-old matrix is the world's oldest decimal multiplication table
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Standards-based mathematics reform in the US

In 1989, the

National Council of Teachers of Mathematics Founded in 1920, The National Council of Teachers of Mathematics (NCTM) is a professional organization for schoolteachers of mathematics in the United States. One of its goals is to improve the standards of mathematics in education. NCTM holds an ...

(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 as

Investigations in Numbers, Data, and Space Investigations in Numbers, Data, and Space is a K–5 mathematics curriculum, developed at TERC in Cambridge, Massachusetts, United States. The curriculum is often referred to as ''Investigations'' or simply ''TERC''. Patterned after the NCTM stan ...

(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. 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.

References

{{Authority control Multiplication Mathematics education