Multiplication (often denoted by the , by the mid-line dot operator , by

^{2}) is called a

_{''m''}; if , the product is an

^{3}, a two with a superscript three. In this example, the number two is the base, and three is the exponent. In general, the exponent (or superscript) indicates how many times the base appears in the expression, so that the expression
:$a^n\; =\; \backslash underbrace\_n$
indicates that ''n'' copies of the base ''a'' are to be multiplied together. This notation can be used whenever multiplication is known to be Power associativity, power associative.

Multiplication

an

Arithmetic Operations In Various Number Systems

at cut-the-knot

Modern Chinese Multiplication Techniques on an Abacus

{{Authority control Multiplication, Elementary arithmetic Mathematical notation Articles containing proofs

juxtaposition
Juxtaposition is an act or instance of placing two elements close together or side by side. This is often done in order to compare/contrast the two, to show similarities or differences, etc.
Speech
Juxtaposition in literary terms is the showing ...

, or, on computer
A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic sets of operations known as Computer program, programs. These ...

s, by an asterisk
The asterisk , from Late Latin , from Ancient Greek , ''asteriskos'', "little star", is a Typography, typographical symbol. It is so called because it resembles a conventional image of a star (heraldry), star.
Computer scientists and mathem ...

) is one of the four elementary
In computational complexity theory, the complexity class ELEMENTARY of elementary recursive functions is the union of the classes
: \begin
\mathsf & = \bigcup_ k\mathsf \\
& = \mathsf\left(2^n\right)\cup\mathsf\left(2^\right)\ ...

mathematical operations of arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...

, with the other ones being addition
Addition (usually signified by the plus symbol
The plus and minus signs, and , are mathematical symbol
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object
A mathematical object is an ...

, subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

, and division
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
*Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting o ...

. The result of a multiplication operation is called a product.
The multiplication of whole numbers may be thought of as repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the ''multiplicand'', as the quantity of the other one, the ''multiplier''. Both numbers can be referred to as ''factors''.
:$a\backslash times\; b\; =\; \backslash underbrace\_$
For example, 4 multiplied by 3, often written as $3\; \backslash times\; 4$ and spoken as "3 times 4", can be calculated by adding 3 copies of 4 together:
:$3\; \backslash times\; 4\; =\; 4\; +\; 4\; +\; 4\; =\; 12$
Here, 3 (the ''multiplier'') and 4 (the ''multiplicand'') are the ''factors'', and 12 is the ''product''.
One of the main properties
Property (''latin: Res Privata'') in the abstract is what belongs to or with something, whether as an attribute or as a component of said thing. In the context of this article, it is one or more components (rather than attributes), whether phys ...

of multiplication is the commutative property
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, which states in this case that adding 3 copies of 4 gives the same result as adding 4 copies of 3:
:$4\; \backslash times\; 3\; =\; 3\; +\; 3\; +\; 3\; +\; 3\; =\; 12$
Thus the designation of multiplier and multiplicand does not affect the result of the multiplication.
Systematic generalizations of this basic definition define the multiplication of integers (including negative numbers), rational numbers (fractions), and real numbers.
Multiplication can also be visualized as counting
Counting is the process of determining the number of Element (mathematics), elements of a finite set of objects, i.e., determining the size (mathematics), size of a set. The traditional way of counting consists of continually increasing a (mental ...

objects arranged in a rectangle
In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

(for whole numbers) or as finding the area
Area is the quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

of a rectangle whose sides have some given length
Length is a measure of distance
Distance is a numerical measurement
'
Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be us ...

s. The area of a rectangle does not depend on which side is measured first—a consequence of the commutative property.
The product of two measurements is a new type of measurement. For example, multiplying the lengths of the two sides of a rectangle gives its area. Such a product is the subject of dimensional analysis
In engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range ...

.
The inverse operation of multiplication is division
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
*Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting o ...

. For example, since 4 multiplied by 3 equals 12, 12 divided by 3 equals 4. Indeed, multiplication by 3, followed by division by 3, yields the original number. The division of a number other than 0 by itself equals 1.
Multiplication is also defined for other types of numbers, such as complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s, and for more abstract constructs, like matrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...

. For some of these more abstract constructs, the order in which the operands are multiplied together matters. A listing of the many different kinds of products used in mathematics is given in Product (mathematics)
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

.
Notation and terminology

Inarithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...

, multiplication is often written using the multiplication sign
The multiplication sign, also known as the times sign or the dimension sign, is the symbol , used in mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related ...

(either or) between the terms (that is, in infix notation
Infix notation is the notation commonly used in arithmetical and logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ) ...

). For example,
:$2\backslash times\; 3\; =\; 6$ ("two times three six")
:$3\backslash times\; 4\; =\; 12$
:$2\backslash times\; 3\backslash times\; 5\; =\; 6\backslash times\; 5\; =\; 30$
:$2\backslash times\; 2\backslash times\; 2\backslash times\; 2\backslash times\; 2\; =\; 32$
There are other mathematical notation
Mathematical notation is a system of symbol
A symbol is a mark, sign, or word
In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or prag ...

s for multiplication:
* To reduce confusion between the multiplication sign × and the common variable , multiplication is also denoted by dot signs, usually a middle-position dot (rarely period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in musical composition
* Period, a descriptor for a historical or period drama ...

):
: or
:The middle dot notation, encoded in Unicode as , is now standard in the United States and other countries where the period is used as a decimal point
A decimal separator is a symbol used to separate the integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spok ...

. When the dot operator character is not accessible, the interpunct
An interpunct, , also known as an interpoint, middle dot, middot and centered dot or centred dot, is a punctuation mark consisting of a vertically centered dot used for interword separation in ancient Latin alphabet, Latin script. (Word-separati ...

(·) is used. In other countries that use a comma
The comma is a punctuation
Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...

as a decimal mark, either the period or a middle dot is used for multiplication.
:Historically, in the United Kingdom and Ireland, the middle dot was sometimes used for the decimal to prevent it from disappearing in the ruled line, and the period/full stop was used for multiplication. However, since the Ministry of Technology
The Minister of Technology was a position in the government of the United Kingdom
The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain,Usage is mixed. The Guardian' and Telegraph' ...

ruled to use the period as the decimal point in 1968, and the SI standard has since been widely adopted, this usage is now found only in the more traditional journals such as ''The Lancet
''The Lancet'' is a weekly peer-reviewed
Peer review is the evaluation of work by one or more people with similar competencies as the producers of the work ( peers). It functions as a form of self-regulation by qualified members of a prof ...

''.
* In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

, multiplication involving variables is often written as a juxtaposition
Juxtaposition is an act or instance of placing two elements close together or side by side. This is often done in order to compare/contrast the two, to show similarities or differences, etc.
Speech
Juxtaposition in literary terms is the showing ...

(e.g., ''xy'' for ''x'' times ''y'' or 5''x'' for five times ''x''), also called ''implied multiplication''. The notation can also be used for quantities that are surrounded by parentheses
A bracket is either of two tall fore- or back-facing punctuation
Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding ...

(e.g., 5(2) or (5)(2) for five times two). This implicit usage of multiplication can cause ambiguity when the concatenated variables happen to match the name of another variable, when a variable name in front of a parenthesis can be confused with a function name, or in the correct determination of the order of operations
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

.
* In vector multiplication, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a cross product
In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

of two vectors
Vector may refer to:
Biology
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector
*Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...

, yielding a vector as its result, while the dot denotes taking the dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

of two vectors, resulting in a scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
*Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...

.
In computer programming
Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a particular task. Programming involves tasks such as analysis, generating algorithms, Profilin ...

, the asterisk
The asterisk , from Late Latin , from Ancient Greek , ''asteriskos'', "little star", is a Typography, typographical symbol. It is so called because it resembles a conventional image of a star (heraldry), star.
Computer scientists and mathem ...

(as in `5*2`

) is still the most common notation. This is due to the fact that most computers historically were limited to small character set
Character encoding is the process of assigning numbers to Graphics, graphical character (computing), characters, especially the written characters of Language, human language, allowing them to be Data storage, stored, Data communication, transmit ...

s (such as ASCII
ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding
Character encoding is the process of assigning numbers to graphical
Graphics (from Greek
Greek may refer to:
Greece
Anything of, ...

and EBCDIC
Extended Binary Coded Decimal Interchange Code (EBCDIC; ) is an eight-bit character encoding used mainly on IBM mainframe and IBM midrange computer operating systems. It descended from the code used with punched cards and the corresponding six-b ...

) that lacked a multiplication sign (such as `⋅`

or `×`

), while the asterisk appeared on every keyboard. This usage originated in the FORTRAN
Fortran (; formerly FORTRAN) is a general-purpose, compiled language, compiled imperative programming, imperative programming language that is especially suited to numerical analysis, numeric computation and computational science, scientific com ...

programming language.
The numbers to be multiplied are generally called the "". The number to be multiplied is the "multiplicand", and the number by which it is multiplied is the "multiplier". Usually, the multiplier is placed first and the multiplicand is placed second; however sometimes the first factor is the multiplicand and the second the multiplier. Also, as the result of multiplication does not depend on the order of the factors, the distinction between "multiplicand" and "multiplier" is useful only at a very elementary level and in some multiplication algorithm
A multiplication algorithm is an algorithm (or method) to multiplication, multiply two numbers. Depending on the size of the numbers, different algorithms are used. Efficient multiplication algorithms have existed since the advent of the decimal sy ...

s, such as the long multiplication
A multiplication algorithm is an algorithm
of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers ''a'' and ''b'' in locations named A and B. The algorithm proceeds by successive subtractions in ...

. Therefore, in some sources, the term "multiplicand" is regarded as a synonym for "factor". In algebra, a number that is the multiplier of a variable or expression (e.g., the 3 in 3''xy''coefficient
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

.
The result of a multiplication is called a product. When one factor is an integer, the product is a multiple of the other or of the product of the others. Thus is a multiple of , as is . A product of integers is a multiple of each factor; for example, 15 is the product of 3 and 5 and is both a multiple of 3 and a multiple of 5.
Computation

200px, The Educated Monkey – a For_example:_set_the_monkey's_feet_to_4_and_9,_and_get_the_product_–_36_–_in_its_hands..html" ;"title="tin toy dated 1918, used as a multiplication "calculator". For example: set the monkey's feet to 4 and 9, and get the product – 36 – in its hands.">tin toy dated 1918, used as a multiplication "calculator". For example: set the monkey's feet to 4 and 9, and get the product – 36 – in its hands. Many common methods for multiplying numbers using pencil and paper require amultiplication table
In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication binary operation, operation for an algebraic system.
The decimal multiplication table was traditionally taug ...

of memorized or consulted products of small numbers (typically any two numbers from 0 to 9). However, one method, the peasant multiplication algorithm, does not. The example below illustrates "long multiplication" (the "standard algorithm", "grade-school multiplication"):
23958233
× 5830
———————————————
00000000 ( = 23,958,233 × 0)
71874699 ( = 23,958,233 × 30)
191665864 ( = 23,958,233 × 800)
+ 119791165 ( = 23,958,233 × 5,000)
———————————————
139676498390 ( = 139,676,498,390 )
Multiplying numbers to more than a couple of decimal places by hand is tedious and error-prone. Common logarithm
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s were invented to simplify such calculations, since adding logarithms is equivalent to multiplying. The slide rule
The slide rule is a mechanical . The slide rule is used primarily for and and for functions such as , , s, and . They are not designed for addition or subtraction which was usually performed manually, with used to keep track of the magnitude ...

allowed numbers to be quickly multiplied to about three places of accuracy. Beginning in the early 20th century, mechanical calculator
An electronic calculator is typically a portable device used to perform s, ranging from basic to complex .
The first calculator was created in the early 1960s. Pocket-sized devices became available in the 1970s, especially after the , the f ...

s, such as the MarchantMarchant is a surname. Notable people with the surname include:
* Chesten Marchant (died 1676), last monoglot Cornish speaker
* David R. Marchant, glacial geologist
* Edward Dalton Marchant (1806–1887), American artist
* George Marchant (1857–1 ...

, automated multiplication of up to 10-digit numbers. Modern electronic computer
A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic sets of operations known as Computer program, programs. These ...

s and calculators have greatly reduced the need for multiplication by hand.
Historical algorithms

Methods of multiplication were documented in the writings ofancient Egyptian
Ancient Egypt was a civilization of Ancient history, ancient North Africa, concentrated along the lower reaches of the Nile, Nile River, situated in the place that is now the country Egypt. Ancient Egyptian civilization followed prehistori ...

, , and Chinese
Chinese can refer to:
* Something related to China
China, officially the People's Republic of China (PRC), is a country in East Asia. It is the List of countries and dependencies by population, world's most populous country, with a populat ...

civilizations.
The Ishango bone
The Ishango bone, discovered at the "Fisherman Settlement" of Ishango in the Democratic Republic of Congo, is a bone toolIn archaeology
Archaeology or archeology is the study of human activity through the recovery and analysis of material c ...

, dated to about 18,000 to 20,000 BC, may hint at a knowledge of multiplication in the Upper Paleolithic
The Upper Paleolithic (or Upper Palaeolithic) also called the is the third and last subdivision of the or Old . Very broadly, it dates to between 50,000 and years ago (the beginning of the ), according to some theories coinciding with the ...

era in Central Africa
Central Africa is a subregion
A subregion is a part of a larger region
In geography
Geography (from Greek: , ''geographia'', literally "earth description") is a field of science devoted to the study of the lands, features, inhab ...

, but this is speculative.
Egyptians

The Egyptian method of multiplication of integers and fractions, which is documented in theRhind Mathematical Papyrus
The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum
The British Museum, in the Bloomsbury
Bloomsbury is a district in the West End of London
The West End of London (commonly referred to as the West End ...

, was by successive additions and doubling. For instance, to find the product of 13 and 21 one had to double 21 three times, obtaining , , . The full product could then be found by adding the appropriate terms found in the doubling sequence:
:13 × 21 = (1 + 4 + 8) × 21 = (1 × 21) + (4 × 21) + (8 × 21) = 21 + 84 + 168 = 273.
Babylonians

TheBabylonians
Babylonia () was an ancient
Ancient history is the aggregate of past eventsWordNet Search – ...

used a sexagesimal
Sexagesimal, also known as base 60 or sexagenary, is a numeral system
A numeral system (or system of numeration) is a writing system
A writing system is a method of visually representing verbal communication
Communication (from Latin ...

positional number system
Positional notation (or place-value notation, or positional numeral system) denotes usually the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...

, analogous to the modern-day decimal systemDecimal system may refer to:
* Decimal (base ten) number system, used in mathematics for writing numbers and performing arithmetic
* Dewey Decimal Classification, Dewey Decimal System, a subject classification system used in libraries
* Decimal curr ...

. Thus, Babylonian multiplication was very similar to modern decimal multiplication. Because of the relative difficulty of remembering different products, Babylonian mathematicians employed multiplication table
In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication binary operation, operation for an algebraic system.
The decimal multiplication table was traditionally taug ...

s. These tables consisted of a list of the first twenty multiples of a certain ''principal number'' ''n'': ''n'', 2''n'', ..., 20''n''; followed by the multiples of 10''n'': 30''n'' 40''n'', and 50''n''. Then to compute any sexagesimal product, say 53''n'', one only needed to add 50''n'' and 3''n'' computed from the table.
Chinese

In the mathematical text ''Zhoubi Suanjing
The ''Zhoubi Suanjing'' () is one of the oldest Chinese mathematics, Chinese mathematical texts. "Zhou" refers to the ancient Zhou dynasty (1046–256 BCE); "Bi" means thigh and according to the book, it refers to the gnomon of the sundial. The ...

'', dated prior to 300 BC, and the ''Nine Chapters on the Mathematical Art
''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained () ...

'', multiplication calculations were written out in words, although the early Chinese mathematicians employed Rod calculus involving place value addition, subtraction, multiplication, and division. The Chinese were already using a by the end of the 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#REDIRECT Spring and Autumn period
The Spri ...

period.
Modern methods

The modern method of multiplication based on theHindu–Arabic numeral system
The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Arabic numeral system or Hindu numeral system) is a positional notation, positional decimal numeral system, and is t ...

was first described by Brahmagupta
Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, doc ...

. Brahmagupta gave rules for addition, subtraction, multiplication, and division. Henry Burchard Fine
Henry Burchard Fine (September 14, 1858 – December 22, 1928) was an American university dean and mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) inc ...

, then a professor of mathematics at Princeton University
Princeton University is a private
Private or privates may refer to:
Music
* "In Private
"In Private" was the third single in a row to be a charting success for United Kingdom, British singer Dusty Springfield, after an absence of nearly tw ...

, wrote the following:
:''The Indians are the inventors not only of the positional decimal system itself, but of most of the processes involved in elementary reckoning with the system. Addition and subtraction they performed quite as they are performed nowadays; multiplication they effected in many ways, ours among them, but division they did cumbrously.''
These place value decimal arithmetic algorithms were introduced to Arab countries by Al Khwarizmi in the early 9th century and popularized in the Western world by Fibonacci
Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician
A mathematician is someone who uses an extensive knowledge of mathem ...

in the 13th century.
Grid method

Grid method multiplication
The grid method (also known as the box method) of multiplication is an introductory approach to multi-digit multiplication calculations that involve numbers larger than ten. Because it is often taught in mathematics education
In contemporary educa ...

, or the box method, is used in primary schools in England and Wales and in some areas of the United States to help teach an understanding of how multiple digit multiplication works. An example of multiplying 34 by 13 would be to lay the numbers out in a grid as follows:
:
and then add the entries.
Computer algorithms

The classical method of multiplying two -digit numbers requires digit multiplications.Multiplication algorithm
A multiplication algorithm is an algorithm (or method) to multiplication, multiply two numbers. Depending on the size of the numbers, different algorithms are used. Efficient multiplication algorithms have existed since the advent of the decimal sy ...

s have been designed that reduce the computation time considerably when multiplying large numbers. Methods based on the discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discret ...

reduce the computational complexity to . In 2016, the factor was replaced by a function that increases much slower, though still not constant. In March 2019, David Harvey and Joris van der Hoeven submitted a paper presenting an integer multiplication algorithm with a complexity of $O(n\backslash log\; n).$ The algorithm, also based on the fast Fourier transform, is conjectured to be asymptotically optimal. The algorithm is not practically useful, as it only becomes faster for multiplying extremely large numbers (having more than bits).
Products of measurements

One can only meaningfully add or subtract quantities of the same type, but quantities of different types can be multiplied or divided without problems. For example, four bags with three marbles each can be thought of as: :× marbles per bag= 12 marbles. When two measurements are multiplied together, the product is of a type depending on the types of measurements. The general theory is given bydimensional analysis
In engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range ...

. This analysis is routinely applied in physics, but it also has applications in finance and other applied fields.
A common example in physics is the fact that multiplying speed
In everyday use and in kinematics
Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, bodies (objects), and systems of bodies (groups of objects) without considerin ...

by time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

gives distance
Distance is a numerical measurement
'
Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

. For example:
:50 kilometers per hour × 3 hours = 150 kilometers.
In this case, the hour units cancel out, leaving the product with only kilometer units.
Other examples of multiplication involving units include:
:2.5 meters × 4.5 meters = 11.25 square meters
:11 meters/seconds × 9 seconds = 99 meters
:4.5 residents per house × 20 houses = 90 residents
Product of a sequence

Capital pi notation

The product of a sequence of factors can be written with the product symbol, which derives from the capital letter $\backslash textstyle\; \backslash prod$ (pi) in theGreek alphabet
The Greek alphabet has been used to write the Greek language since the late ninth or early eighth century BC. It is derived from the earlier Phoenician alphabet, and was the first alphabetic script in history to have distinct letters for vowels ...

(much like the same way the capital letter $\backslash textstyle\; \backslash sum$ (sigma) is used in the context of summation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

). Unicode position contains a glyph for denoting such a product, distinct from , the letter. The meaning of this notation is given by:
:$\backslash prod\_^4\; i\; =\; 1\backslash cdot\; 2\backslash cdot\; 3\backslash cdot\; 4,$
that is
:$\backslash prod\_^4\; i\; =\; 24.$
The subscript gives the symbol for a bound variable
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

(''i'' in this case), called the "index of multiplication", together with its lower bound (''1''), whereas the superscript (here ''4'') gives its upper bound. The lower and upper bound are expressions denoting integers. The factors of the product are obtained by taking the expression following the product operator, with successive integer values substituted for the index of multiplication, starting from the lower bound and incremented by 1 up to (and including) the upper bound. For example:
:$\backslash prod\_^6\; i\; =\; 1\backslash cdot\; 2\backslash cdot\; 3\backslash cdot\; 4\backslash cdot\; 5\; \backslash cdot\; 6\; =\; 720.$
More generally, the notation is defined as
:$\backslash prod\_^n\; x\_i\; =\; x\_m\; \backslash cdot\; x\_\; \backslash cdot\; x\_\; \backslash cdot\; \backslash ,\backslash ,\backslash cdots\backslash ,\backslash ,\; \backslash cdot\; x\_\; \backslash cdot\; x\_n,$
where ''m'' and ''n'' are integers or expressions that evaluate to integers. In the case where , the value of the product is the same as that of the single factor ''x''empty product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

whose value is 1—regardless of the expression for the factors.
Properties of capital pi notation

By definition, :$\backslash prod\_^x\_i=x\_1\backslash cdot\; x\_2\backslash cdot\backslash ldots\backslash cdot\; x\_n.$ If all factors are identical, a product of factors is equivalent toexponentiation
Exponentiation is a mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...

:
:$\backslash prod\_^x=x\backslash cdot\; x\backslash cdot\backslash ldots\backslash cdot\; x=x^n.$
Associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule ...

and commutativity
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of multiplication imply
:$\backslash prod\_^\; =\backslash left(\backslash prod\_^x\_i\backslash right)\backslash left(\backslash prod\_^y\_i\backslash right)$ and
:$\backslash left(\backslash prod\_^x\_i\backslash right)^a\; =\backslash prod\_^x\_i^a$
if is a nonnegative integer, or if all $x\_i$ are positive real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s, and
:$\backslash prod\_^x^\; =x^$
if all $a\_i$ are nonnegative integers, or if is a positive real number.
Infinite products

One may also consider products of infinitely many terms; these are calledinfinite product In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...

s. Notationally, this consists in replacing ''n'' above by the Infinity symbol
The infinity symbol (\infty, , or ∞) is a mathematical symbol representing the concept of infinity. In algebraic geometry, the figure is called a lemniscate.
History
The shape of a sideways figure eight has a long pedigree; for instance, it ...

∞. The product of such an infinite sequence is defined as the limit
Limit or Limits may refer to:
Arts and media
* Limit (music), a way to characterize harmony
* Limit (song), "Limit" (song), a 2016 single by Luna Sea
* Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...

of the product of the first ''n'' terms, as ''n'' grows without bound. That is,
:$\backslash prod\_^\backslash infty\; x\_i\; =\; \backslash lim\_\; \backslash prod\_^n\; x\_i.$
One can similarly replace ''m'' with negative infinity, and define:
:$\backslash prod\_^\backslash infty\; x\_i\; =\; \backslash left(\backslash lim\_\backslash prod\_^0\; x\_i\backslash right)\; \backslash cdot\; \backslash left(\backslash lim\_\; \backslash prod\_^n\; x\_i\backslash right),$
provided both limits exist.
Properties

Forreal
Real may refer to:
* Reality
Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...

and complex
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...

numbers, which includes, for example, natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s, integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s, and fractions
A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, ...

, multiplication has certain properties:
;Commutative property
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

:The order in which two numbers are multiplied does not matter:
::$x\backslash cdot\; y\; =\; y\backslash cdot\; x.$
;Associative property
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule ...

:Expressions solely involving multiplication or addition are invariant with respect to the order of operations
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

:
::$(x\backslash cdot\; y)\backslash cdot\; z\; =\; x\backslash cdot(y\backslash cdot\; z)$
;Distributive property
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmetic, ...

:Holds with respect to multiplication over addition. This identity is of prime importance in simplifying algebraic expressions:
::$x\backslash cdot(y\; +\; z)\; =\; x\backslash cdot\; y\; +\; x\backslash cdot\; z$
;Identity element
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

:The multiplicative identity is 1; anything multiplied by 1 is itself. This feature of 1 is known as the identity property:
::$x\backslash cdot\; 1\; =\; x$
; Property of 0
:Any number multiplied by 0 is 0. This is known as the zero property of multiplication:
::$x\backslash cdot\; 0\; =\; 0$
;Negation
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, ...

:−1 times any number is equal to the additive inverse
In mathematics, the additive inverse of a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be ...

of that number.
::$(-1)\backslash cdot\; x\; =\; (-x)$ where $(-x)+x=0$
:–1 times –1 is 1.
::$(-1)\backslash cdot\; (-1)\; =\; 1$
;Inverse element
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

:Every number ''x'', except 0, has a multiplicative inverse
Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

, $\backslash frac$, such that $x\backslash cdot\backslash left(\backslash frac\backslash right)\; =\; 1$.
;Order
Order, ORDER or Orders may refer to:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...

preservation
:Multiplication by a positive number preserves the order
Order, ORDER or Orders may refer to:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...

:
::For , if then .
:Multiplication by a negative number reverses the order:
::For , if then .
:The complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s do not have an ordering that is compatible with both addition and multiplication.
Other mathematical systems that include a multiplication operation may not have all these properties. For example, multiplication is not, in general, commutative for matrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...

and quaternion
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s.
Axioms

In the book ''Arithmetices principia, nova methodo exposita
The 1889 treatise ''Arithmetices principia, nova methodo exposita'' (''The principles of arithmetic, presented by a new method''; 1889) by Giuseppe Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician
...

'', Giuseppe Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

proposed axioms for arithmetic based on his axioms for natural numbers. Peano arithmetic has two axioms for multiplication:
:$x\; \backslash times\; 0\; =\; 0$
:$x\; \backslash times\; S(y)\; =\; (x\; \backslash times\; y)\; +\; x$
Here ''S''(''y'') represents the successor
Successor is someone who, or something which succeeds or comes after (see success (disambiguation), success and Succession (disambiguation), succession)
Film and TV
* The Successor (film), ''The Successor'' (film), a 1996 film including Laura Girli ...

of ''y''; i.e., the natural number that follows ''y''. The various properties like associativity can be proved from these and the other axioms of Peano arithmetic, including induction
Induction may refer to:
Philosophy
* Inductive reasoning, in logic, inferences from particular cases to the general case
Biology and chemistry
* Labor induction (birth/pregnancy)
* Induction chemotherapy, in medicine
* Induction period, the t ...

. For instance, ''S''(0), denoted by 1, is a multiplicative identity because
:$x\; \backslash times\; 1\; =\; x\; \backslash times\; S(0)\; =\; (x\; \backslash times\; 0)\; +\; x\; =\; 0\; +\; x\; =\; x.$
The axioms for integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s typically define them as equivalence classes of ordered pairs of natural numbers. The model is based on treating (''x'',''y'') as equivalent to when ''x'' and ''y'' are treated as integers. Thus both (0,1) and (1,2) are equivalent to −1. The multiplication axiom for integers defined this way is
:$(x\_p,\backslash ,\; x\_m)\; \backslash times\; (y\_p,\backslash ,\; y\_m)\; =\; (x\_p\; \backslash times\; y\_p\; +\; x\_m\; \backslash times\; y\_m,\backslash ;\; x\_p\; \backslash times\; y\_m\; +\; x\_m\; \backslash times\; y\_p).$
The rule that −1 × −1 = 1 can then be deduced from
:$(0,\; 1)\; \backslash times\; (0,\; 1)\; =\; (0\; \backslash times\; 0\; +\; 1\; \backslash times\; 1,\backslash ,\; 0\; \backslash times\; 1\; +\; 1\; \backslash times\; 0)\; =\; (1,0).$
Multiplication is extended in a similar way to rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

s and then to real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s.
Multiplication with set theory

The product of non-negative integers can be defined with set theory usingcardinal numbers
150px, Aleph null, the smallest infinite cardinal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...

or the Peano axioms
In mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...

. See below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Fred Below (1926–1988), American blues drummer
*Fritz von Below (1853 ...

how to extend this to multiplying arbitrary integers, and then arbitrary rational numbers. The product of real numbers is defined in terms of products of rational numbers; see construction of the real numbers.
Multiplication in group theory

There are many sets that, under the operation of multiplication, satisfy the axioms that define group (mathematics), group structure. These axioms are closure, associativity, and the inclusion of an identity element and inverses. A simple example is the set of non-zero rational numbers. Here we have identity 1, as opposed to groups under addition where the identity is typically 0. Note that with the rationals, we must exclude zero because, under multiplication, it does not have an inverse: there is no rational number that can be multiplied by zero to result in 1. In this example, we have an abelian group, but that is not always the case. To see this, consider the set of invertible square matrices of a given dimension over a given field (mathematics), field. Here, it is straightforward to verify closure, associativity, and inclusion of identity (the identity matrix) and inverses. However, matrix multiplication is not commutative, which shows that this group is non-abelian. Another fact worth noticing is that the integers under multiplication do not form a group—even if we exclude zero. This is easily seen by the nonexistence of an inverse for all elements other than 1 and −1. Multiplication in group theory is typically notated either by a dot or by juxtaposition (the omission of an operation symbol between elements). So multiplying element a by element b could be notated as a $\backslash cdot$ b or ab. When referring to a group via the indication of the set and operation, the dot is used. For example, our first example could be indicated by $\backslash left(\; \backslash mathbb/\; \backslash \; ,\backslash ,\; \backslash cdot\; \backslash right)$.Multiplication of different kinds of numbers

Numbers can ''count'' (3 apples), ''order'' (the 3rd apple), or ''measure'' (3.5 feet high); as the history of mathematics has progressed from counting on our fingers to modelling quantum mechanics, multiplication has been generalized to more complicated and abstract types of numbers, and to things that are not numbers (such asmatrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...

) or do not look much like numbers (such as quaternion
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s).
;Integers
:$N\backslash times\; M$ is the sum of ''N'' copies of ''M'' when ''N'' and ''M'' are positive whole numbers. This gives the number of things in an array ''N'' wide and ''M'' high. Generalization to negative numbers can be done by
:$N\backslash times\; (-M)\; =\; (-N)\backslash times\; M\; =\; -\; (N\backslash times\; M)$ and
:$(-N)\backslash times\; (-M)\; =\; N\backslash times\; M$
:The same sign rules apply to rational and real numbers.
;Rational numbers
:Generalization to fractions $\backslash frac\backslash times\; \backslash frac$ is by multiplying the numerators and denominators respectively: $\backslash frac\backslash times\; \backslash frac\; =\; \backslash frac$. This gives the area of a rectangle $\backslash frac$ high and $\backslash frac$ wide, and is the same as the number of things in an array when the rational numbers happen to be whole numbers.
;Real numbers
:Real numbers and their products Construction of the real numbers#Construction from Cauchy sequences, can be defined in terms of sequences of rational numbers.
;Complex numbers
:Considering complex numbers $z\_1$ and $z\_2$ as ordered pairs of real numbers $(a\_1,\; b\_1)$ and $(a\_2,\; b\_2)$, the product $z\_1\backslash times\; z\_2$ is $(a\_1\backslash times\; a\_2\; -\; b\_1\backslash times\; b\_2,\; a\_1\backslash times\; b\_2\; +\; a\_2\backslash times\; b\_1)$. This is the same as for reals $a\_1\backslash times\; a\_2$ when the ''imaginary parts'' $b\_1$ and $b\_2$ are zero.
:Equivalently, denoting $\backslash sqrt$ as $i$, we have $z\_1\; \backslash times\; z\_2\; =\; (a\_1+b\_1i)(a\_2+b\_2i)=(a\_1\; \backslash times\; a\_2)+(a\_1\backslash times\; b\_2i)+(b\_1\backslash times\; a\_2i)+(b\_1\backslash times\; b\_2i^2)=(a\_1a\_2-b\_1b\_2)+(a\_1b\_2+b\_1a\_2)i.$
:Alternatively, in trigonometric form, if $z\_1\; =\; r\_1(\backslash cos\backslash phi\_1+i\backslash sin\backslash phi\_1),\; z\_2\; =\; r\_2(\backslash cos\backslash phi\_2+i\backslash sin\backslash phi\_2)$, then$z\_1z\_2\; =\; r\_1r\_2(\backslash cos(\backslash phi\_1\; +\; \backslash phi\_2)\; +\; i\backslash sin(\backslash phi\_1\; +\; \backslash phi\_2)).$
;Further generalizations
:See #Multiplication in group theory, Multiplication in group theory, above, and Multiplicative group, which for example includes matrix multiplication. A very general, and abstract, concept of multiplication is as the "multiplicatively denoted" (second) binary operation in a Ring (mathematics), ring. An example of a ring that is not any of the above number systems is a polynomial ring (you can add and multiply polynomials, but polynomials are not numbers in any usual sense.)
;Division
:Often division, $\backslash frac$, is the same as multiplication by an inverse, $x\backslash left(\backslash frac\backslash right)$. Multiplication for some types of "numbers" may have corresponding division, without inverses; in an integral domain ''x'' may have no inverse "$\backslash frac$" but $\backslash frac$ may be defined. In a division ring there are inverses, but $\backslash frac$ may be ambiguous in non-commutative rings since $x\backslash left(\backslash frac\backslash right)$ need not be the same as $\backslash left(\backslash frac\backslash right)x$.
Exponentiation

When multiplication is repeated, the resulting operation is known asexponentiation
Exponentiation is a mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...

. For instance, the product of three factors of two (2×2×2) is "two raised to the third power", and is denoted by 2See also

* Dimensional analysis *Multiplication algorithm
A multiplication algorithm is an algorithm (or method) to multiplication, multiply two numbers. Depending on the size of the numbers, different algorithms are used. Efficient multiplication algorithms have existed since the advent of the decimal sy ...

** Karatsuba algorithm, for large numbers
** Toom–Cook multiplication, for very large numbers
** Schönhage–Strassen algorithm, for huge numbers
* Multiplication table
* Binary multiplier, how computers multiply
** Booth's multiplication algorithm
** Floating point
** Fused multiply–add
** Multiply–accumulate
** Wallace tree
* Multiplicative inverse, reciprocal
* Factorial
* Genaille–Lucas rulers
* Lunar arithmetic
* Napier's bones
* Peasant multiplication
* Product (mathematics)
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

, for generalizations
* Slide rule
Notes

References

*External links

Multiplication

an

Arithmetic Operations In Various Number Systems

at cut-the-knot

Modern Chinese Multiplication Techniques on an Abacus

{{Authority control Multiplication, Elementary arithmetic Mathematical notation Articles containing proofs