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 addition, subtraction, 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 x b=\underbrace{b+ … +b}a.
For example, 4 multiplied by 3, often written as
3 x 4
3 x 4=4+4+4=12.
One of the main properties of multiplication is the commutative property, which states in this case that adding 3 copies of 4 gives the same result as adding 4 copies of 3:
4 x 3=3+3+3+3=12.
Thus, the designation of multiplier and multiplicand does not affect the result of the multiplication.[1]
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 (for whole numbers) or as finding the area of a rectangle whose sides have some given lengths. 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 (or physical quantities) is a new type of measurement, usually with a derived unit. For example, multiplying the lengths (in meters or feet) of the two sides of a rectangle gives its area (in square meters or square feet). Such a product is the subject of dimensional analysis.
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.
Several mathematical concepts expand upon the fundamental idea of multiplication. The product of a sequence, vector multiplication, complex numbers, and matrices are all examples where this can be seen. These more advanced constructs tend to affect the basic properties in their own ways, such as becoming noncommutative in matrices and some forms of vector multiplication or changing the sign of complex numbers.
Multiplication signs | |
Sign: | × ⋅ |
Unicode: | |
Different From: |
See main article: Multiplication sign.
See also: Multiplier (linguistics). In arithmetic, multiplication is often written using the multiplication sign (either or) between the terms (that is, in infix notation). For example,
2 x 3=6,
3 x 4=12,
2 x 3 x 5=6 x 5=30,
2 x 2 x 2 x 2 x 2=32.
There are other mathematical notations for multiplication:
5 ⋅ 2
The middle dot notation or dot operator, 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 (·) is used. In other countries that use a comma 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,[2] and the International System of Units (SI) standard has since been widely adopted, this usage is now found only in the more traditional journals such as The Lancet.[3]
xy
x
y
5x
x
5(2)
(5)2
(5)(2)
In computer programming, the asterisk (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 sets (such as ASCII and EBCDIC) that lacked a multiplication sign (such as ⋅
or ×
), while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language.[4]
The numbers to be multiplied are generally called the "factors" (as in factorization). 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;[1] however, sometimes the first factor is the multiplicand and the second the multiplier.[5] 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".[6] In algebra, a number that is the multiplier of a variable or expression (e.g., the 3 in
3xy2
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 x \pi
\pi
5133 x 486 x \pi
The product of two numbers or the multiplication between two numbers can be defined for common special cases: natural numbers, integers, rational numbers, real numbers, complex numbers, and quaternions.
The product of two natural numbers
r,s\inN
An integer can be either zero, a nonzero natural number, or minus a nonzero natural number. The product of zero and another integer is always zero. The product of two nonzero integers is determined by the product of their positive amounts, combined with the sign derived from the following rule:
c c |
In words:
Two fractions can be multiplied by multiplying their numerators and denominators:
which is defined when
n,n' ≠ 0
There are several equivalent ways to define formally the real numbers; see Construction of the real numbers. The definition of multiplication is a part of all these definitions.
A fundamental aspect of these definitions is that every real number can be approximated to any accuracy by rational numbers. A standard way for expressing this is that every real number is the least upper bound of a set of rational numbers. In particular, every positive real number is the least upper bound of the truncations of its infinite decimal representation; for example,
\pi
\{3, 3.1, 3.14, 3.141,\ldots\}.
A fundamental property of real numbers is that rational approximations are compatible with arithmetic operations, and, in particular, with multiplication. This means that, if and are positive real numbers such that
a=\supx\inx
b=\supy\iny,
a ⋅ b=\supx\inx ⋅ y.
As changing the signs transforms least upper bounds into greatest lower bounds, the simplest way to deal with a multiplication involving one or two negative numbers, is to use the rule of signs described above in . The construction of the real numbers through Cauchy sequences is often preferred in order to avoid consideration of the four possible sign configurations.
Two complex numbers can be multiplied by the distributive law and the fact that
i2=-1
\begin{align} (a+bi) ⋅ (c+di)&=a ⋅ c+a ⋅ di+bi ⋅ c+b ⋅ d ⋅ i2\\ &=(a ⋅ c-b ⋅ d)+(a ⋅ d+b ⋅ c)i \end{align}
Geometric meaning of complex multiplication can be understood rewriting complex numbers in polar coordinates:
a+bi=r ⋅ (\cos(\varphi)+i\sin(\varphi))=r ⋅ e
Furthermore,
c+di=s ⋅ (\cos(\psi)+i\sin(\psi))=s ⋅ ei\psi,
from which one obtains
(a ⋅ c-b ⋅ d)+(a ⋅ d+b ⋅ c)i=r ⋅ s ⋅ ei(\varphi.
The geometric meaning is that the magnitudes are multiplied and the arguments are added.
The product of two quaternions can be found in the article on quaternions. Note, in this case, that and
b ⋅ a</matH>areingeneraldifferent. ==Computation== {{Main|Multiplicationalgorithm}} [[file:צעצועמכנימשנת1918לחישובילוחהכפלTheEducatedMonkey.jpg|upright|right|thumb|TheEducatedMonkey—a[[tintoy]]dated1918,usedasamultiplication"calculator".<small>Forexample:setthemonkey'sfeetto4and9,andgettheproduct—36—initshands.</small>]] Manycommonmethodsformultiplyingnumbersusingpencilandpaperrequirea[[multiplicationtable]]ofmemorizedorconsultedproductsofsmallnumbers(typicallyanytwonumbersfrom0to9).However,onemethod,the[[AncientEgyptianmultiplication|peasantmultiplication]]algorithm,doesnot.Theexamplebelowillustrates"longmultiplication"(the"standardalgorithm","grade-schoolmultiplication"): 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) Insomecountriessuchas[[Germany]],theabovemultiplicationisdepictedsimilarlybutwiththeoriginalproductkepthorizontalandcomputationstartingwiththefirstdigitofthemultiplier:<ref>{{Citeweb|title=Multiplication|url=http://www.mathematische-basteleien.de/multiplication.htm|access-date=2022-03-15|website=mathematische-basteleien.de}}</ref> 23958233·5830 ——————————————— 119791165 191665864 71874699 00000000——————————————— 139676498390 Multiplyingnumberstomorethanacoupleofdecimalplacesbyhandistediousanderror-prone.[[Commonlogarithm]]swereinventedtosimplifysuchcalculations,sinceaddinglogarithmsisequivalenttomultiplying.The[[sliderule]]allowednumberstobequicklymultipliedtoaboutthreeplacesofaccuracy.Beginningintheearly20thcentury,mechanical[[calculator]]s,suchasthe[[MarchantCalculator|Marchant]],automatedmultiplicationofupto10-digitnumbers.Modernelectronic[[computer]]sandcalculatorshavegreatlyreducedtheneedformultiplicationbyhand. ===Historicalalgorithms=== Methodsofmultiplicationweredocumentedinthewritingsof[[ancientEgypt]]ian,{{Citationneededspan|text=Greek,Indian,|date=December2021|reason=Thisclaimisnotsourcedinthesubsectionsbelow.}}and[[HistoryofChina#AncientChina|Chinese]]civilizations. The[[Ishangobone]],datedtoabout18,000to20,000 BC,mayhintataknowledgeofmultiplicationinthe[[UpperPaleolithic]]erain[[CentralAfrica]],butthisisspeculative.<ref>{{citearXiv|last=Pletser|first=Vladimir|date=2012-04-04|title=DoestheIshangoBoneIndicateKnowledgeoftheBase12?AnInterpretationofaPrehistoricDiscovery,theFirstMathematicalToolofHumankind|class=math.HO|eprint=1204.1019}}</ref>{{Verificationneeded|date=December2021}} ====Egyptians==== {{Main|AncientEgyptianmultiplication}} TheEgyptianmethodofmultiplicationofintegersandfractions,whichisdocumentedinthe[[RhindMathematicalPapyrus]],wasbysuccessiveadditionsanddoubling.Forinstance,tofindtheproductof13and21onehadtodouble21threetimes,obtaining{{nowrap|1=2×21=42}},{{nowrap|1=4×21=2×42=84}},{{nowrap|1=8×21=2×84=168}}.Thefullproductcouldthenbefoundbyaddingtheappropriatetermsfoundinthedoublingsequence:<ref>{{Citeweb|title=PeasantMultiplication|url=http://www.cut-the-knot.org/Curriculum/Algebra/PeasantMultiplication.shtml|access-date=2021-12-29|website=cut-the-knot.org}}</ref> :13×21=(1+4+8)×21=(1×21)+(4×21)+(8×21)=21+84+168=273. ====Babylonians==== The[[Babylonians]]useda[[sexagesimal]][[positionalnumbersystem]],analogoustothemodern-day[[decimalexpansion|decimalsystem]].Thus,Babylonianmultiplicationwasverysimilartomoderndecimalmultiplication.Becauseoftherelativedifficultyofremembering{{nowrap|60×60}}differentproducts,Babylonianmathematiciansemployed[[multiplicationtable]]s.Thesetablesconsistedofalistofthefirsttwentymultiplesofacertain''principalnumber''''n'':''n'',2''n'',...,20''n'';followedbythemultiplesof10''n'':30''n''40''n'',and50''n''.Thentocomputeanysexagesimalproduct,say53''n'',oneonlyneededtoadd50''n''and3''n''computedfromthetable.{{Citationneeded|date=December2021}} ====Chinese==== {{seealso|Chinesemultiplicationtable}} [[File:Multiplicationalgorithm.GIF|thumb|right|upright1.0|{{nowrap|1=38×76=2888}}]] Inthemathematicaltext''[[ZhoubiSuanjing]]'',datedpriorto300 BC,andthe''[[NineChaptersontheMathematicalArt]]'',multiplicationcalculationswerewrittenoutinwords,althoughtheearlyChinesemathematiciansemployed[[Rodcalculus]]involvingplacevalueaddition,subtraction,multiplication,anddivision.TheChinesewerealreadyusinga[[Chinesemultiplicationtable|decimalmultiplicationtable]]bytheendofthe[[WarringStates]]period.<refname="Nature">{{citejournal|url=http://www.nature.com/news/ancient-times-table-hidden-in-chinese-bamboo-strips-1.14482|title=AncienttimestablehiddeninChinesebamboostrips|journal=Nature|first=Jane|last=Qiu|author-link=JaneQiu|date=7January2014|access-date=22January2014|doi=10.1038/nature.2014.14482|s2cid=130132289|archive-url=https://web.archive.org/web/20140122064930/http://www.nature.com/news/ancient-times-table-hidden-in-chinese-bamboo-strips-1.14482|archive-date=22January2014|url-status=live|doi-access=free}}</ref> ===Modernmethods=== [[Image:Gelosiamultiplication45256.png|right|upright1.0|thumb|Productof45and256.Notetheorderofthenumeralsin45isreverseddowntheleftcolumn.Thecarrystepofthemultiplicationcanbeperformedatthefinalstageofthecalculation(inbold),returningthefinalproductof{{nowrap|1=45×256=11520}}.Thisisavariantof[[Latticemultiplication]].]] Themodernmethodofmultiplicationbasedonthe[[Hindu–Arabicnumeralsystem]]wasfirstdescribedby[[Brahmagupta]].Brahmaguptagaverulesforaddition,subtraction,multiplication,anddivision.[[HenryBurchardFine]],thenaprofessorofmathematicsat[[PrincetonUniversity]],wrotethefollowing: :''TheIndiansaretheinventorsnotonlyofthepositionaldecimalsystemitself,butofmostoftheprocessesinvolvedinelementaryreckoningwiththesystem.Additionandsubtractiontheyperformedquiteastheyareperformednowadays;multiplicationtheyeffectedinmanyways,oursamongthem,butdivisiontheydidcumbrously.''<ref>{{citebook|last=Fine|first=HenryB.|author-link=HenryBurchardFine|title=TheNumberSystemofAlgebra–TreatedTheoreticallyandHistorically|edition=2nd|date=1907|page=90|url=https://archive.org/download/numbersystemofal00fineuoft/numbersystemofal00fineuoft.pdf}}</ref> TheseplacevaluedecimalarithmeticalgorithmswereintroducedtoArabcountriesby[[AlKhwarizmi]]intheearly9th centuryandpopularizedintheWesternworldby[[Fibonacci]]inthe13thcentury.<ref>{{Citeweb|last=Bernhard|first=Adrienne|title=HowmodernmathematicsemergedfromalostIslamiclibrary|url=https://www.bbc.com/future/article/20201204-lost-islamic-library-maths|access-date=2022-04-22|website=bbc.com|language=en}}</ref> ====Gridmethod==== [[Gridmethodmultiplication]],ortheboxmethod,isusedinprimaryschoolsinEnglandandWalesandinsomeareas{{Which|date=December2021}}oftheUnitedStatestohelpteachanunderstandingofhowmultipledigitmultiplicationworks.Anexampleofmultiplying34by13wouldbetolaythenumbersoutinagridasfollows: :{|class="wikitable"style="text-align:center;" !scope="col"width="40pt"|× !scope="col"width="120pt"|30 !scope="col"width="40pt"|4 |- !scope="row"|10 |'''300''' |'''40''' |- !scope="row"|3 |'''90''' |'''12''' |} andthenaddtheentries. ===Computeralgorithms=== {{Main|Multiplicationalgorithm#Fastmultiplicationalgorithmsforlargeinputs}} Theclassicalmethodofmultiplyingtwo{{math|''n''}}-digitnumbersrequires{{math|''n''<sup>2</sup>}}digitmultiplications.[[Multiplicationalgorithm]]shavebeendesignedthatreducethecomputationtimeconsiderablywhenmultiplyinglargenumbers.Methodsbasedonthe[[DiscreteFouriertransform#Multiplicationoflargeintegers|discreteFouriertransform]]reducethe[[computationalcomplexity]]to{{math|''O''(''n''log''n''loglog''n'')}}.In2016,thefactor{{math|loglog''n''}}wasreplacedbyafunctionthatincreasesmuchslower,thoughstillnotconstant.<ref>{{Citejournal|last1=Harvey|first1=David|last2=vanderHoeven|first2=Joris|last3=Lecerf|first3=Grégoire|title=Evenfasterintegermultiplication|date=2016|journal=JournalofComplexity|volume=36|pages=1–30|doi=10.1016/j.jco.2016.03.001|issn=0885-064X|arxiv=1407.3360|s2cid=205861906}}</ref>InMarch2019,DavidHarveyandJorisvanderHoevensubmittedapaperpresentinganintegermultiplicationalgorithmwithacomplexityof<math>O(nlogn).
See main article: Dimensional analysis. 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:[1]
[4 bags] × [3 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. 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 by time gives distance. 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
The product of a sequence of factors can be written with the product symbol
style\prod
style\sum
4 | |
\prod | |
i=1 |
(i+1)=(1+1)(2+1)(3+1)(4+1),
4 | |
\prod | |
i=1 |
(i+1)=120.
In such a notation, the variable represents a varying integer, 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
n | |
\prod | |
i=m |
xi=xm ⋅ xm+1 ⋅ xm+2 ⋅ … ⋅ xn-1 ⋅ xn,
By definition,
n | |
\prod | |
i=1 |
xi=x1 ⋅ x2 ⋅ \ldots ⋅ xn.
If all factors are identical, a product of factors is equivalent to exponentiation:
n | |
\prod | |
i=1 |
x=x ⋅ x ⋅ \ldots ⋅ x=xn.
Associativity and commutativity of multiplication imply
n | |
\prod | |
i=1 |
{xiyi}
n | |
=\left(\prod | |
i=1 |
xi\right)\left(\prod
n | |
i=1 |
yi\right)
n | |
\left(\prod | |
i=1 |
a | |
x | |
i\right) |
n | |
=\prod | |
i=1 |
a | |
x | |
i |
xi
n | |
\prod | |
i=1 |
ai | |
x |
| ||||||||||
=x |
ai
See main article: Infinite product. One may also consider products of infinitely many terms; these are called infinite products. Notationally, this consists in replacing n above by the infinity symbol ∞. The product of such an infinite sequence is defined as the limit of the product of the first n terms, as n grows without bound. That is,
infty | |
\prod | |
i=m |
xi=\limn\toinfty
n | |
\prod | |
i=m |
xi.
One can similarly replace m with negative infinity, and define:
infty | |
\prod | |
i=-infty |
xi=\left(\limm\to-infty
0 | |
\prod | |
i=m |
xi\right) ⋅ \left(\limn\toinfty
n | |
\prod | |
i=1 |
xi\right),
See main article: Exponentiation. When multiplication is repeated, the resulting operation is known as exponentiation. 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 three. In this example, the number two is the base, and three is the exponent.[12] In general, the exponent (or superscript) indicates how many times the base appears in the expression, so that the expression
an=\underbrace{a x a x … x a}n=
n | |
\prod | |
i=1 |
a
For real and complex numbers, which includes, for example, natural numbers, integers, and fractions, multiplication has certain properties:
x ⋅ y=y ⋅ x.
(x ⋅ y) ⋅ z=x ⋅ (y ⋅ z).
x ⋅ (y+z)=x ⋅ y+x ⋅ z.
x ⋅ 1=x.
x ⋅ 0=0.
(-1) ⋅ x=(-x)
(-x)+x=0.
−1 times −1 is 1:
(-1) ⋅ (-1)=1.
1 | |
x |
x ⋅ \left(
1 | |
x |
\right)=1
Other mathematical systems that include a multiplication operation may not have all these properties. For example, multiplication is not, in general, commutative for matrices and quaternions.[13]
See main article: Peano 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 x 0=0
x x S(y)=(x x 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 x 1=x x S(0)=(x x 0)+x=0+x=x.
The axioms for integers 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
(xp,xm) x (yp,ym)=(xp x yp+xm x ym, xp x ym+xm x yp).
The rule that −1 × −1 = 1 can then be deduced from
(0,1) x (0,1)=(0 x 0+1 x 1,0 x 1+1 x 0)=(1,0).
Multiplication is extended in a similar way to rational numbers and then to real numbers.
The product of non-negative integers can be defined with set theory using cardinal numbers or the Peano axioms. See 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.[17]
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. Here identity 1 is had, as opposed to groups under addition where the identity is typically 0. Note that with the rationals, zero must be excluded 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, an abelian group is had, 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) 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 zero is excluded. 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
⋅
\left(Q/\{0\}, ⋅ \right)
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) or do not look much like numbers (such as quaternions).
N x M
N x (-M)=(-N) x M=-(N x M)
(-N) x (-M)=N x M
The same sign rules apply to rational and real numbers.
A | |
B |
x
C | |
D |
A | |
B |
x
C | |
D |
=
(A x C) | |
(B x D) |
A | |
B |
C | |
D |
z1
z2
(a1,b1)
(a2,b2)
z1 x z2
(a1 x a2-b1 x b2,a1 x b2+a2 x b1)
a1 x a2
b1
b2
Equivalently, denoting
\sqrt{-1}
i
z1 x z2=(a1+b1i)(a2+b2i)=(a1 x a2)+(a1 x b2i)+(b1 x a2i)+(b1 x
2)=(a | |
b | |
1a |
2-b1b2)+(a1b2+b1a2)i.
Alternatively, in trigonometric form, if
z1=r1(\cos\phi1+i\sin\phi1),z2=r2(\cos\phi2+i\sin\phi2)
x | |
y |
x\left( | 1 |
y |
\right)
1 | |
x |
x | |
y |
x | |
y |
x\left( | 1 |
y |
\right)
\left( | 1 |
y |
\right)x