HOME

TheInfoList



OR:

Multiplication (often denoted by the cross symbol , by the mid-line
dot operator 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 addi ...
, by
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 showin ...
, or, on
computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations ( computation) automatically. Modern digital electronic computers can perform generic sets of operations known as programs. These prog ...
s, by an
asterisk The asterisk ( ), from Late Latin , from Ancient Greek , ''asteriskos'', "little star", is a typographical symbol. It is so called because it resembles a conventional image of a heraldic star. Computer scientists and mathematicians often voc ...
) is one of the four
elementary Elementary may refer to: Arts, entertainment, and media Music * ''Elementary'' (Cindy Morgan album), 2001 * ''Elementary'' (The End album), 2007 * ''Elementary'', a Melvin "Wah-Wah Watson" Ragin album, 1977 Other uses in arts, entertainment, a ...
mathematical operations In mathematics, an operation is a function which takes zero or more input values (also called "''operands''" or "arguments") to a well-defined output value. The number of operands is the arity of the operation. The most commonly studied operati ...
of
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
, with the other ones being
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
,
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. 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\times b = \underbrace_ For example, 4 multiplied by 3, often written as 3 \times 4 and spoken as "3 times 4", can be calculated by adding 3 copies of 4 together: :3 \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 is the ownership of land, resources, improvements or other tangible objects, or intellectual property. Property may also refer to: Mathematics * Property (mathematics) Philosophy and science * Property (philosophy), in philosophy an ...
of multiplication is the
commutative property 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 of ...
, which states in this case that adding 3 copies of 4 gives the same result as adding 4 copies of 3: :4 \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 objects arranged in a
rectangle In 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 parallelogram contain ...
(for whole numbers) or as finding the
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
of a rectangle whose sides have some given
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...
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 and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as mi ...
. The inverse operation of multiplication is division. 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 extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, and for more abstract constructs, like
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
. 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, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called ''factors''. For example, 30 is the product of 6 and 5 (the result of multiplication), and x\cd ...
.


Notation and terminology

In
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
, multiplication is often written using the multiplication sign (either or ) between the terms (that is, in
infix notation Infix notation is the notation commonly used in arithmetical and logical formulae and statements. It is characterized by the placement of operators between operands—" infixed operators"—such as the plus sign in . Usage Binary relations are ...
). For example, :2\times 3 = 6 ("two times three equals six") :3\times 4 = 12 :2\times 3\times 5 = 6\times 5 = 30 :2\times 2\times 2\times 2\times 2 = 32 There are other
mathematical notation Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathem ...
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): :5 \cdot 2 or 5\,.\,3 :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. 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 script. (Word-separating spaces did n ...
 (·) is used. In other countries that use a
comma The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
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 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 general medical journal and one of the oldest of its kind. It is also the world's highest-impact academic journal. It was founded in England in 1823. The journal publishes original research articles ...
''. * In
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
, 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 showin ...
(e.g., xy for x times y or 5x 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 marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
(e.g., 5(2), (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 and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression. For examp ...
. * 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 mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
of two vectors, 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'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
of two vectors, resulting in a scalar. In
computer programming Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as anal ...
, the
asterisk The asterisk ( ), from Late Latin , from Ancient Greek , ''asteriskos'', "little star", is a typographical symbol. It is so called because it resembles a conventional image of a heraldic star. Computer scientists and mathematicians often voc ...
(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 graphical characters, especially the written characters of human language, allowing them to be stored, transmitted, and transformed using digital computers. The numerical values tha ...
s (such as
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because ...
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 ...
) that lacked a multiplication sign (such as or ×), while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language. The numbers to be multiplied are generally called the " factors". 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 algorithms, such as the long multiplication. 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 3xy^2) is called a
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
. 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 2\times \pi is a multiple of \pi, as is 5133 \times 486 \times \pi. 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.


Definitions

The product of two numbers or the multiplication between two numbers can be defined for common special cases: integers, natural numbers, fractions, real numbers, complex numbers, and quaternions.


Product of two natural numbers

Placing several stones into a rectangular pattern with r rows and s columns gives : r \cdot s = \sum_^s r = \underbrace_= \sum_^r s = \underbrace_ stones.


Product of two integers

Integers allow positive and negative numbers. Their product is determined by the product of their positive amounts, combined with the sign derived from the following rule: :\begin \hline \times & - & + \\ \hline - & + & - \\ + & - & + \\ \hline \end (This rule is a necessary consequence of demanding
distributivity 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 arithmeti ...
of multiplication over addition, and is not an ''additional rule''.) In words, we have: * A negative number multiplied by a negative number is positive, * A negative number multiplied by a positive number is negative, * A positive number multiplied by a negative number is negative, * A positive number multiplied by a positive number is positive.


Product of two fractions

Two fractions can be multiplied by multiplying their numerators and denominators: : \frac \cdot \frac = \frac


Product of two real numbers

The rigorous definition of the product of two real numbers is a byproduct of the
Construction of the real numbers In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ...
. This construction implies that, for every real number there is a set of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
such that is the least upper bound of the elements of : :a=\sup_ x. If is another real number that is the least upper bound of , the product a\cdot b is defined as :a\cdot b=\sup_x\cdot y. This definition does not depend of a particular choice of and . That is, if they are changed without changing their least upper bound, then the least upper bound defining a\cdot b is not changed.


Product of two complex numbers

Two complex numbers can be multiplied by the distributive law and the fact that i^2=-1, as follows: :\begin (a + b\, i) \cdot (c + d\, i) &= a \cdot c + a \cdot d\, i + b \, i \cdot c + b \cdot d \cdot i^2\\ &= (a \cdot c - b \cdot d) + (a \cdot d + b \cdot c) \, i \end Geometric meaning of complex multiplication can be understood rewriting complex numbers in
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
: :a + b\, i = r \cdot ( \cos(\varphi) + i \sin(\varphi) ) = r \cdot e ^ Furthermore, :c + d\, i = s \cdot ( \cos(\psi) + i\sin(\psi) ) = s \cdot e^, from which one obtains :(a \cdot c - b \cdot d) + (a \cdot d + b \cdot c) i = r \cdot s \cdot e^. The geometric meaning is that the magnitudes are multiplied and the arguments are added.


Product of two quaternions

The product of two
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s can be found in the article on
quaternions In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
. Note, in this case, that a \cdot b and b \cdot a are in general different.


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 a multiplication table 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 ) In some countries such as
Germany Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated betwee ...
, the above multiplication is depicted similarly but with the original product kept horizontal and computation starting with the first digit of the multiplier: 23958233 · 5830 ——————————————— 119791165 191665864 71874699 00000000 ——————————————— 139676498390 Multiplying numbers to more than a couple of decimal places by hand is tedious and error-prone.
Common logarithm In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered ...
s were invented to simplify such calculations, since adding logarithms is equivalent to multiplying. The slide rule 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 electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...
s, such as the
Marchant Marchant is a surname. Notable people with the surname include: * Adio Marchant (born 1987), English singer and songwriter known professionally as Bipolar Sunshine * Alison Marchant, Australian politician * Chesten Marchant (died 1676), last monogl ...
, automated multiplication of up to 10-digit numbers. Modern electronic
computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations ( computation) automatically. Modern digital electronic computers can perform generic sets of operations known as programs. These prog ...
s and calculators have greatly reduced the need for multiplication by hand.


Historical algorithms

Methods of multiplication were documented in the writings of ancient Egyptian, and Chinese civilizations. The Ishango bone, 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) is the third and last subdivision of the Paleolithic or Old Stone Age. Very broadly, it dates to between 50,000 and 12,000 years ago (the beginning of the Holocene), according to some theories coin ...
era in
Central Africa Central Africa is a subregion of the African continent comprising various countries according to different definitions. Angola, Burundi, the Central African Republic, Chad, the Democratic Republic of the Congo, the Republic of the Co ...
, but this is speculative.


Egyptians

The Egyptian method of multiplication of integers and fractions, which is documented in the
Rhind Mathematical Papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scotland, Scottish antiquarian, who ...
, 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

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 ...
used a
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
positional number system, analogous to the modern-day decimal system. Thus, Babylonian multiplication was very similar to modern decimal multiplication. Because of the relative difficulty of remembering different products, Babylonian mathematicians employed multiplication tables. 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'', dated prior to 300 BC, and the ''
Nine Chapters on the Mathematical Art ''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest sur ...
'', 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 decimal multiplication table 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 and concluded with the Qin wars of conquest ...
period.


Modern methods

The modern method of multiplication based on the
Hindu–Arabic numeral system The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Hindu numeral system or Arabic numeral system) is a positional decimal numeral system, and is the most common syste ...
was first described by
Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the '' Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical tr ...
. Brahmagupta gave rules for addition, subtraction, multiplication, and division. Henry Burchard Fine, then a professor of mathematics at
Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...
, 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 Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronom ...
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 from the Republic of Pisa, considered to be "the most talented Wester ...
in the 13th century.


Grid method

Grid method multiplication, 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 algorithms 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 samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
reduce the
computational complexity In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) ...
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\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: : bags× 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 by
dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as mi ...
. 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, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quant ...
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 or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
. 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 \textstyle \prod, which derives from the capital letter Π (pi) in the
Greek alphabet The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as ...
(much like the same way the summation symbol \textstyle \sum is derived from the Greek letter Σ (sigma). The meaning of this notation is given by :\prod_^4 (i+1) = (1+1)\,(2+1)\,(3+1)\, (4+1), which results in :\prod_^4 (i+1) = 120. In such a notation, the variable represents a varying
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
, called the multiplication index, that runs from the lower value indicated in the subscript to the upper value given by the superscript. The product is obtained by multiplying together all factors obtained by substituting the multiplication index for an integer between the lower and the upper values (the bounds included) in the expression that follows the product operator. More generally, the notation is defined as :\prod_^n x_i = x_m \cdot x_ \cdot x_ \cdot \,\,\cdots\,\, \cdot x_ \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''''m''; if , the product is an empty product whose value is 1—regardless of the expression for the factors.


Properties of capital pi notation

By definition, :\prod_^x_i=x_1\cdot x_2\cdot\ldots\cdot x_n. If all factors are identical, a product of factors is equivalent to
exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...
: :\prod_^x=x\cdot x\cdot\ldots\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 valid rule of replacement ...
and
commutativity 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 of ...
of multiplication imply :\prod_^ =\left(\prod_^x_i\right)\left(\prod_^y_i\right) and :\left(\prod_^x_i\right)^a =\prod_^x_i^a if is a nonnegative integer, or if all x_i are positive
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, and :\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 called
infinite product In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product : \prod_^ a_n = a_1 a_2 a_3 \cdots is defined to be the limit of the partial products ''a''1''a''2...''a'n'' as ''n'' increases without bound. ...
s. Notationally, this consists in replacing ''n'' above by the Infinity symbol ∞. The product of such an infinite sequence is defined as the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of the product of the first ''n'' terms, as ''n'' grows without bound. That is, :\prod_^\infty x_i = \lim_ \prod_^n x_i. One can similarly replace ''m'' with negative infinity, and define: :\prod_^\infty x_i = \left(\lim_\prod_^0 x_i\right) \cdot \left(\lim_ \prod_^n x_i\right), provided both limits exist.


Exponentiation

When multiplication is repeated, the resulting operation is known as ''
exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...
''. For instance, the product of three factors of two (2×2×2) is "two raised to the third power", and is denoted by 23, a two with a
superscript A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, whil ...
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 = \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 associative In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity. Definition An algebra (or more generally a magma) is said to be power-associative if the subalgebra ...
.


Properties

For real and complex 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 ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s,
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s, and
fractions A fraction (from la, fractus, "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 ...
, multiplication has certain properties: ;
Commutative property 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 of ...
:The order in which two numbers are multiplied does not matter: ::x\cdot y = y\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 valid rule of replaceme ...
:Expressions solely involving multiplication or addition are invariant with respect to the
order of operations In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression. For examp ...
: ::(x\cdot y)\cdot z = x\cdot(y\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 arithmet ...
:Holds with respect to multiplication over addition. This identity is of prime importance in simplifying algebraic expressions: ::x\cdot(y + z) = x\cdot y + x\cdot z ;
Identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
:The multiplicative identity is 1; anything multiplied by 1 is itself. This feature of 1 is known as the identity property: ::x\cdot 1 = x ; Property of 0 :Any number multiplied by 0 is 0. This is known as the zero property of multiplication: ::x\cdot 0 = 0 ;
Negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
:−1 times any number is equal to the
additive inverse In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (op ...
of that number. ::(-1)\cdot x = (-x) where (-x)+x=0 :–1 times –1 is 1. ::(-1)\cdot (-1) = 1 ;
Inverse element In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
:Every number ''x'', except 0, has a
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
, \frac, such that x\cdot\left(\frac\right) = 1. ;
Order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
preservation :Multiplication by a positive number preserves the
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
: ::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 extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
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 most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
and
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s.


Axioms

In the book '' Arithmetices principia, nova methodo exposita'', Giuseppe Peano proposed axioms for arithmetic based on his axioms for natural numbers. Peano arithmetic has two axioms for multiplication: :x \times 0 = 0 :x \times S(y) = (x \times y) + x Here ''S''(''y'') represents the successor 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. For instance, ''S''(0), denoted by 1, is a multiplicative identity because :x \times 1 = x \times S(0) = (x \times 0) + x = 0 + x = x. The axioms for
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
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,\, x_m) \times (y_p,\, y_m) = (x_p \times y_p + x_m \times y_m,\; x_p \times y_m + x_m \times y_p). The rule that −1 × −1 = 1 can then be deduced from :(0, 1) \times (0, 1) = (0 \times 0 + 1 \times 1,\, 0 \times 1 + 1 \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 of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s and then to
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s.


Multiplication with set theory

The product of non-negative integers can be defined with set theory using cardinal numbers or the
Peano axioms In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...
. 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 *Ernst von Below (1863–1955), German World War I general *Fred Below ...
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 In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ...
.


Multiplication in group theory

There are many sets that, under the operation of multiplication, satisfy the axioms that define 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 In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
. 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 In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
, 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. Here, it is straightforward to verify closure, associativity, and inclusion of identity (the
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
) 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 \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 \left( \mathbb/ \ ,\, \cdot \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 as
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
) or do not look much like numbers (such as
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s). ;Integers :N\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\times (-M) = (-N)\times M = - (N\times M) and :(-N)\times (-M) = N\times M :The same sign rules apply to rational and real numbers. ;
Rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s :Generalization to fractions \frac\times \frac is by multiplying the numerators and denominators respectively: \frac\times \frac = \frac. This gives the area of a rectangle \frac high and \frac wide, and is the same as the number of things in an array when the rational numbers happen to be whole numbers. ;
Real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s :Real numbers and their products can be defined in terms of sequences of rational numbers. ;
Complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s :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\times z_2 is (a_1\times a_2 - b_1\times b_2, a_1\times b_2 + a_2\times b_1). This is the same as for reals a_1\times a_2 when the ''imaginary parts'' b_1 and b_2 are zero. :Equivalently, denoting \sqrt as i, we have z_1 \times z_2 = (a_1+b_1i)(a_2+b_2i)=(a_1 \times a_2)+(a_1\times b_2i)+(b_1\times a_2i)+(b_1\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(\cos\phi_1+i\sin\phi_1), z_2 = r_2(\cos\phi_2+i\sin\phi_2), thenz_1z_2 = r_1r_2(\cos(\phi_1 + \phi_2) + i\sin(\phi_1 + \phi_2)). ;Further generalizations :See Multiplication in group theory, above, and
Multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
, 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. An example of a ring that is not any of the above number systems is a
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
(you can add and multiply polynomials, but polynomials are not numbers in any usual sense.) ;Division :Often division, \frac, is the same as multiplication by an inverse, x\left(\frac\right). Multiplication for some types of "numbers" may have corresponding division, without inverses; in an
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
''x'' may have no inverse "\frac" but \frac may be defined. In a
division ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
there are inverses, but \frac may be ambiguous in non-commutative rings since x\left(\frac\right) need not be the same as \left(\frac\right)x.


See also

*
Dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as mi ...
* Multiplication algorithm ** Karatsuba algorithm, for large numbers ** Toom–Cook multiplication, for very large numbers **
Schönhage–Strassen algorithm The Schönhage–Strassen algorithm is an asymptotically fast multiplication algorithm for large integers. It was developed by Arnold Schönhage and Volker Strassen in 1971.A. Schönhage and V. Strassen,Schnelle Multiplikation großer Zahlen, ...
, for huge numbers * Multiplication table * Binary multiplier, how computers multiply ** Booth's multiplication algorithm **
Floating-point arithmetic In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be ...
**
Fused multiply–add Fuse or FUSE may refer to: Devices * Fuse (electrical), a device used in electrical systems to protect against excessive current ** Fuse (automotive), a class of fuses for vehicles * Fuse (hydraulic), a device used in hydraulic systems to prot ...
** Multiply–accumulate ** Wallace tree *
Multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
, reciprocal *
Factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \ ...
* Genaille–Lucas rulers * Lunar arithmetic * Napier's bones * Peasant multiplication *
Product (mathematics) In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called ''factors''. For example, 30 is the product of 6 and 5 (the result of multiplication), and x\cd ...
, for generalizations * Slide rule


Notes


References

*


External links


Multiplication
an
Arithmetic Operations In Various Number Systems
at
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

Modern Chinese Multiplication Techniques on an Abacus
{{Authority control Elementary arithmetic Mathematical notation Articles containing proofs