View source for Multiplication

You do not have permission to edit this page, for the following reasons:

This IP address has been blocked from editing Wikipedia.
This does not affect your ability to read Wikipedia pages.
Most people who see this message have done nothing wrong. Some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism, and can sometimes affect other users who are unrelated to that abuse. Review the information below for assistance if you do not believe that you have done anything wrong.

The IP address or range 35.175.0.0/16 has been blocked by ‪JJMC89‬ for the following reason(s):

The IP address that you are currently using has been blocked because it is believed to be a web host provider or colocation provider. To prevent abuse, web hosts and colocation providers may be blocked from editing Wikipedia.
You will not be able to edit Wikipedia using a web host or colocation provider because it hides your IP address, much like a proxy or VPN.
We recommend that you attempt to use another connection to edit. For example, if you use a proxy or VPN to connect to the internet, turn it off when editing Wikipedia. If you edit using a mobile connection, try using a Wi-Fi connection, and vice versa. If you are using a corporate internet connection, switch to a different Wi-Fi network. If you have a Wikipedia account, please log in.
If you do not have any other way to edit Wikipedia, you will need to request an IP block exemption.

How to appeal if you are confident that your connection does not use a colocation provider's IP address:
If you are confident that you are not using a web host, you may appeal this block by adding the following text on your talk page: {{unblock|reason=Caught by a colocation web host block but this host or IP is not a web host. My IP address is _______. Place any further information here. ~~~~}}. You must fill in the blank with your IP address for this block to be investigated. Your IP address can be determined here. Alternatively, if you wish to keep your IP address private you can use the unblock ticket request system. There are several reasons you might be editing using the IP address of a web host or colocation provider (such as if you are using VPN software or a business network); please use this method of appeal only if you think your IP address is in fact not a web host or colocation provider.

Administrators: The IP block exemption user right should only be applied to allow users to edit using web host in exceptional circumstances, and requests should usually be directed to the functionaries team via email. If you intend to give the IPBE user right, a CheckUser needs to take a look at the account. This can be requested most easily at SPI Quick Checkuser Requests. Unblocking an IP or IP range with this template is highly discouraged without at least contacting the blocking administrator.

This block will expire on 05:20, 2 March 2025. Your current IP address is 35.175.117.137.

Even when blocked, you will usually still be able to edit your user talk page, as well as email administrators and other editors.

For information on how to proceed, please read the FAQ for blocked users and the guideline on block appeals. The guide to appealing blocks may also be helpful.

Other useful links: Blocking policy · Help:I have been blocked
This IP address range has been globally blocked.
This does not affect your ability to read Wikipedia pages.
Most people who see this message have done nothing wrong. Some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism, and can sometimes affect other users who are unrelated to that abuse. Review the information below for assistance if you do not believe that you have done anything wrong.

This block affects editing on all Wikimedia wikis.
The IP address or range 35.175.0.0/16 has been globally blocked by ‪Jon Kolbert‬ for the following reason(s):

Open proxy/Webhost: See the help page if you are affected

This block will expire on 05:38, 21 February 2029. Your current IP address is 35.175.117.137.

Even while globally blocked, you will usually still be able to edit pages on Meta-Wiki.

If you believe you were blocked by mistake, you can find additional information and instructions in the No open proxies global policy. Otherwise, to discuss the block please post a request for review on Meta-Wiki. You could also send an email to the stewards VRT queue at stewards@wikimedia.org including all above details.

Other useful links: Global blocks · Help:I have been blocked

You can view and copy the source of this page:

{{Short description|Arithmetical operation}}
{{About|the mathematical operation}}
{{redirect|⋅|the symbol|Interpunct#In mathematics and science}}
{{More citations needed|date=April 2012}}
{{Use dmy dates|date=September 2023|cs1-dates=y}}
[[File:Multiply 4 bags 3 marbles.svg|thumb|right|Four bags with three marbles per bag gives twelve marbles (4 × 3 = 12).]]
[[File:Multiply scaling.svg|thumb|right|Multiplication can also be thought of as [[Scale factor|scaling]]. Here, 2 is being multiplied by 3 using scaling, giving 6 as a result.]]
[[File:Multiplication as scaling integers.gif|thumb|Animation for the multiplication 2 × 3 = 6]]
[[File:Multiplication scheme 4 by 5.jpg|thumb|right|4 × 5 = 20. The large rectangle is made up of 20 squares, each 1 unit by 1 unit.]]
[[File:Multiply field fract.svg|thumb|right|Area of a cloth {{nowrap|1=4.5m × 2.5m = 11.25m<sup>2</sup>}}; {{nowrap|1=4{{sfrac|1|2}} × 2{{sfrac|1|2}} = 11{{sfrac|1|4}}}}]]
{{Arithmetic operations}}

'''Multiplication''' (often denoted by the [[multiplication sign|cross symbol]] {{char|'''×'''}}, by the mid-line [[Multiplication#Notation|dot operator]] {{char|'''⋅'''}}, by [[Juxtaposition#Mathematics|juxtaposition]], or, on [[computer]]s, by an [[asterisk]] {{char|'''*'''}}) is one of the four [[Elementary arithmetic|elementary]] [[Operation (mathematics)|mathematical operations]] of [[arithmetic]], with the other ones being [[addition]], [[subtraction]], and [[division (mathematics)|division]]. The result of a multiplication operation is called a ''[[product (mathematics)|product]]''.

The multiplication of [[Natural number|whole numbers]] may be thought of as [[multiplication and repeated addition|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'''''.
:<math>a\times b = \underbrace{b + \cdots + b}_{a \text{ times}}  .</math>

For example, 4 multiplied by 3, often written as <math> 3 \times 4 </math> and spoken as "3 times 4", can be calculated by adding 3 copies of 4 together:
:<math>3 \times 4 = 4 + 4 + 4 = 12.</math>
Here, 3 (the ''multiplier'') and 4 (the ''multiplicand'') are the ''factors'', and 12 is the ''product''.

One of the main [[#Properties|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:
:<math>4 \times 3 = 3 + 3 + 3 + 3 = 12.</math>

Thus, the designation of multiplier and multiplicand does not affect the result of the multiplication.<ref name="Devlin">{{cite web |last=Devlin |first=Keith |url=http://www.maa.org/external_archive/devlin/devlin_01_11.html |title=What Exactly is Multiplication? |author-link=Keith Devlin |publisher=[[Mathematical Association of America]] |date=January 2011 |quote=With multiplication you have a multiplicand (written second) multiplied by a multiplier (written first) |access-date=May 14, 2017 |archive-url=https://web.archive.org/web/20170527070801/http://www.maa.org/external_archive/devlin/devlin_01_11.html |archive-date=May 27, 2017 |url-status=live }}</ref>

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 [[length]]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 (or [[physical quantities]]) is a new type of measurement, usually with a [[derived unit|derived]] [[unit of measurement|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 (mathematics)|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 number]]s, and [[matrix (mathematics)|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_(mathematics)|sign]] of complex numbers.

==Notation==
{{Infobox symbol
|name= Multiplication signs
|sign=× ⋅
|unicode={{unichar|00D7|Multiplication sign|html=}}<br />{{unichar|22C5|Dot operator|html=}} 
|see also = 
|different from= {{unichar|00B7|Middle dot|nlink=Interpunct}}<br /> {{unichar|002E|nlink=Full stop}}
}}
{{main|Multiplication sign}}
{{See also|Multiplier (linguistics)}}
In [[arithmetic]], multiplication is often written using the [[multiplication sign]] (either {{char|&times;}} or {{char|<math>\times</math>}}) between the terms (that is, in [[infix notation]]).<ref>{{Citation |last=Khan Academy |title=Intro to multiplication {{!}} Multiplication and division {{!}} Arithmetic {{!}} Khan Academy |date=2015-08-14 |url=https://www.youtube.com/watch?v=RNxwasijbAo |access-date=2017-03-07 |archive-url=https://web.archive.org/web/20170324175113/https://www.youtube.com/watch?v=RNxwasijbAo |archive-date=2017-03-24 |url-status=live }}</ref> For example,
:<math>2\times 3 = 6,</math> ("two times three [[equals sign|equals]] six")
:<math>3\times 4 = 12  ,</math>
:<math>2\times 3\times 5 = 6\times  5 = 30,</math>
:<math>2\times 2\times 2\times 2 \times 2 = 32.</math>

There are other [[mathematical notation]]s for multiplication:
* To reduce confusion between the multiplication sign × and the common variable {{mvar|x}}, multiplication is also denoted by dot signs,<ref>{{Citation |last=Khan Academy |title=Why aren't we using the multiplication sign? {{!}} Introduction to algebra {{!}} Algebra I {{!}} Khan Academy |date=2012-09-06 |url=https://www.youtube.com/watch?v=vDaIKB19TvY |access-date=2017-03-07 |archive-url=https://web.archive.org/web/20170327163705/https://www.youtube.com/watch?v=vDaIKB19TvY |archive-date=2017-03-27 |url-status=live }}</ref> usually a middle-position dot (rarely [[full stop|period]]): <math>5 \cdot 2</math>.
:The middle dot notation or '''dot operator''', encoded in Unicode as {{unichar|22C5|dot operator}}, is now standard in the United States and other countries where the period is used as a [[decimal separator|decimal point]]. When the dot operator character is not accessible, the [[interpunct]]&nbsp;(·) is used. In other countries that use a [[Comma (punctuation)|comma]] as a decimal mark, either the period or a middle dot is used for multiplication.{{citation needed|date=August 2011}}
: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,<ref>{{Cite journal |doi=10.1038/218111c0 |title=Victory on Points |journal=Nature |volume=218 |issue = 5137 |page=111 |date=1968 |bibcode=1968Natur.218S.111. |doi-access=free}}</ref> 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]]''.<ref>{{cite web |title=The Lancet – Formatting guidelines for electronic submission of manuscripts |url=http://download.thelancet.com/pb/assets/raw/Lancet/authors/artwork-guidelines.pdf |access-date=2017-04-25}}</ref>
* {{anchor|Implicit|Explicit}}In [[algebra]], multiplication involving [[Variable (mathematics)|variables]] is often written as a [[Juxtaposition#Mathematics|juxtaposition]] (e.g., <math>xy</math> for <math>x</math> times <math>y</math> or <math>5x</math> for five times <math>x</math>), also called ''implied multiplication''.<ref name="TI_1982"/> The notation can also be used for quantities that are surrounded by [[parentheses]] (e.g., <math>5(2)</math>, <math>(5)2</math> or <math>(5)(2)</math> 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]].<ref name="Peterson_2019"/><ref name="Peterson_2023"/>
* In [[vector multiplication]], there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a [[cross product]] of two [[vector (mathematics)|vectors]], yielding a vector as its result, while the dot denotes taking the [[dot product]] of two vectors, resulting in a [[scalar (mathematics)|scalar]].

In [[computer programming]], the [[asterisk]] (as in <code>5*2</code>) is still the most common notation. This is due to the fact that most computers historically were limited to small [[character set]]s (such as [[ASCII]] and [[EBCDIC]]) that lacked a multiplication sign (such as <code>⋅</code> or <code>×</code>), while the asterisk appeared on every keyboard.{{citation needed|date=February 2023}} This usage originated in the [[Fortran|FORTRAN]] programming language.<ref name="fortran">{{cite book
|last = Fuller |first = William R.
|title = FORTRAN Programming: A Supplement for Calculus Courses
|series = Universitext
|date = 1977
|url = https://books.google.com/books?id=mnLjBwAAQBAJ&pg=PA10
|page = 10
|publisher = Springer
|doi = 10.1007/978-1-4612-9938-7
|isbn = 978-0-387-90283-8
}}</ref>

{{anchor|Terminology|multiplier}}
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;<ref name="Devlin"/> however, sometimes the first factor is the multiplicand and the second the multiplier.<ref>{{cite web
|last=Ramone |first=Crewton |url=http://www.crewtonramoneshouseofmath.com/multiplicand-and-multiplier.html
|title=Multiplicand and Multiplier
|access-date=10 November 2015
|publisher=Crewton Ramone's House of Math
|archive-url=https://web.archive.org/web/20151026161239/http://www.crewtonramoneshouseofmath.com/multiplicand-and-multiplier.html |archive-date=26 October 2015 |url-status=live}}.</ref>
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]]s, such as the [[long multiplication]]. Therefore, in some sources, the term "multiplicand" is regarded as a synonym for "factor".<ref>{{cite book |last=Litvin |first=Chester |url=https://books.google.com/books?id=-ULmPYAA8voC&q=Can+the+multiplicand+be+the+first+number&pg=PA6 |title=Advance Brain Stimulation by Psychoconduction |date=2012 |publisher=Trafford |isbn=978-1-4669-0152-0 |pages=2–3, 5–6 |via=[[Google Book Search]]}}</ref>
In algebra, a number that is the multiplier of a variable or expression (e.g., the 3 in <math>3xy^2</math>) is called a [[coefficient]].

The result of a multiplication is called a [[product (mathematics)|product]]. When one factor is an integer, the product is a [[multiple (mathematics)|''multiple'']] of the other or of the product of the others. Thus, <math>2\times \pi</math> is a multiple of <math>\pi</math>, as is <math>5133 \times 486 \times \pi</math>. 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==

{{Expert needed|Mathematics
| talk = Merging new section with "Multiplication of Different Kinds of Numbers"
| reason = defining multiplication is not straightforward and different proposals have been made over the centuries, with competing ideas (e.g. recursive vs. non-recursive definitions)
| section = yes
| date = September 2023|section
}}

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.

===Product of two natural numbers===
[[File:Three by Four.svg|thumb|3 by 4 is 12.]]

The product of two natural numbers <math>r,s\in\mathbb{N}</math> is defined as:

<math display="block"> r \cdot s \equiv \sum_{i=1}^s r = \underbrace{ r+r+\cdots+r }_{s\text{ times}} \equiv \sum_{j=1}^r s = \underbrace{ s+s+\cdots+s }_{r\text{ times}} . </math>

===Product of two integers===
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 [[absolute value|positive amounts]], combined with the sign derived from the following rule:

<math display="block">\begin{array}{|c|c c|}
                   \hline
  \times & +  &  - \\ \hline
    +  & + & - \\ 
    -  & - & + \\ \hline
\end{array}</math>
(This rule is a consequence of the [[distributivity]] of multiplication over addition, and is not an ''additional rule''.)

In words:

* A positive number multiplied by a positive number is positive (product of natural numbers),
* A positive number multiplied by a negative number is negative,
* A negative number multiplied by a positive number is negative,
* A negative number multiplied by a negative number is positive.

===Product of two fractions===
Two fractions can be multiplied by multiplying their numerators and denominators:

:<math display="block"> \frac{z}{n} \cdot \frac{z'}{n'} = \frac{z\cdot z'}{n\cdot n'} , </math>
:which is defined when <math> n,n'\neq  0 </math>.

=== Product of two real numbers ===

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 number]]s. 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 [[truncation]]s of its infinite [[decimal representation]]; for example, <math>\pi</math> is the least upper bound of <math>\{3,\; 3.1,\; 3.14,\; 3.141,\ldots\}.</math>

A fundamental property of real numbers is that rational approximations are compatible with [[arithmetic operation]]s, and, in particular, with multiplication. This means that, if {{mvar|a}} and {{mvar|b}} are positive real numbers such that <math>a=\sup_{x\in A} x</math> and <math>b=\sup_{y\in B} y,</math> then <math>a\cdot b=\sup_{x\in A, y\in B}x\cdot y.</math> In particular, the product of two positive real numbers is the least upper bound of the term-by-term products of the [[sequence]]s of their decimal representations.

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 {{slink|#Product of two integers}}. The construction of the real numbers through [[Cauchy sequence]]s is often preferred in order to avoid consideration of the four possible sign configurations.

===Product of two complex numbers===
Two complex numbers can be multiplied by the distributive law and the fact that <math> i^2=-1</math>, as follows:
:<math>\begin{align}
  (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{align}</math>

[[File:Komplexe zahlenebene.svg|thumb|upright=1.25|A complex number in polar coordinates]]

The geometric meaning of complex multiplication can be understood by rewriting complex numbers in [[polar coordinates]]:

:<math>a + b\, i = r \cdot ( \cos(\varphi) +  i \sin(\varphi) ) = r \cdot  e ^{ i \varphi} </math>

Furthermore,
:<math>c + d\, i = s \cdot ( \cos(\psi) +  i\sin(\psi) ) = s \cdot e^{i\psi},</math>

from which one obtains
:<math>(a \cdot c - b \cdot d) + (a \cdot d + b \cdot c) i = r \cdot s \cdot e^{i(\varphi + \psi)}.</math>

The geometric meaning is that the magnitudes are multiplied and the arguments are added.

===Product of two quaternions===
The product of two [[quaternion]]s can be found in the article on [[quaternions]]. Note, in this case, that <matH>a \cdot b</math> and <math>b \cdot a</matH> are in general different.

==Computation==
{{Main|Multiplication algorithm}}
[[file:צעצוע מכני משנת 1918 לחישובי לוח הכפל The Educated Monkey.jpg|upright|right|thumb|The Educated Monkey—a [[tin toy]] dated 1918, used as a multiplication "calculator". <small>For example: set the monkey's feet to 4 and 9, and get the product—36—in its hands.</small>]]

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 [[Ancient Egyptian multiplication|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]], the above multiplication is depicted similarly but with the original product kept horizontal and computation starting with the first digit of the multiplier:<ref>{{Cite web |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
Multiplying numbers to more than a couple of decimal places by hand is tedious and error-prone. [[Common logarithm]]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]]s, such as the [[Marchant Calculator|Marchant]], automated multiplication of up to 10-digit numbers. Modern electronic [[computer]]s and calculators have greatly reduced the need for multiplication by hand.

===Historical algorithms===
Methods of multiplication were documented in the writings of [[ancient Egypt]]ian, {{Citation needed span|text=Greek, Indian,|date=December 2021|reason=This claim is not sourced in the subsections below.}} and [[History of China#Ancient China|Chinese]] civilizations.

The [[Ishango bone]], dated to about 18,000 to 20,000&nbsp;BC, may hint at a knowledge of multiplication in the [[Upper Paleolithic]] era in [[Central Africa]], but this is speculative.<ref>{{cite arXiv|last=Pletser|first=Vladimir|date=2012-04-04|title=Does the Ishango Bone Indicate Knowledge of the Base 12? An Interpretation of a Prehistoric Discovery, the First Mathematical Tool of Humankind|class=math.HO|eprint=1204.1019}}</ref>{{Verification needed|date=December 2021}}

====Egyptians====
{{Main|Ancient Egyptian multiplication}}
The Egyptian method of multiplication of integers and fractions, which is documented in the [[Rhind Mathematical Papyrus]], was by successive additions and doubling. For instance, to find the product of 13 and 21 one had to double 21 three times, obtaining {{nowrap|1=2 × 21 = 42}}, {{nowrap|1=4 × 21 = 2 × 42 = 84}}, {{nowrap|1=8 × 21 =  2 × 84 = 168}}. The full product could then be found by adding the appropriate terms found in the doubling sequence:<ref>{{Cite web |title=Peasant Multiplication |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]] used a [[sexagesimal]] [[positional number system]], analogous to the modern-day [[decimal expansion|decimal system]]. Thus, Babylonian multiplication was very similar to modern decimal multiplication. Because of the relative difficulty of remembering {{nowrap|60 × 60}} different products, Babylonian mathematicians employed [[multiplication table]]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.{{Citation needed|date=December 2021}}

====Chinese====
{{see also|Chinese multiplication table}}
[[File:Multiplication algorithm.GIF|thumb|right|upright 1.0|{{nowrap|1=38 × 76 = 2888}}]]
In the mathematical text ''[[Zhoubi Suanjing]]'', dated prior to 300&nbsp;BC, and the ''[[Nine Chapters on the Mathematical Art]]'', 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 [[Chinese multiplication table|decimal multiplication table]] by the end of the [[Warring States]] period.<ref name="Nature">{{cite journal | url =http://www.nature.com/news/ancient-times-table-hidden-in-chinese-bamboo-strips-1.14482 | title =Ancient times table hidden in Chinese bamboo strips | journal =Nature | first =Jane |last=Qiu |author-link=Jane Qiu| date =7 January 2014 | access-date =22 January 2014 | 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 =22 January 2014 | url-status =live | doi-access =free }}</ref>

===Modern methods===
[[Image:Gelosia multiplication 45 256.png|right|upright 1.0|thumb|Product of 45 and 256. Note the order of the numerals in 45 is reversed down the left column. The carry step of the multiplication can be performed at the final stage of the calculation (in bold), returning the final product of {{nowrap|1=45 × 256 = 11520}}. This is a variant of [[Lattice multiplication]].]]
The modern method of multiplication based on the [[Hindu–Arabic numeral system]] was first described by [[Brahmagupta]]. Brahmagupta gave rules for addition, subtraction, multiplication, and division. [[Henry Burchard Fine]], then a professor of mathematics at [[Princeton University]], 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.''<ref>{{cite book |last=Fine |first=Henry B. |author-link=Henry Burchard Fine |title=The Number System of Algebra – Treated Theoretically and Historically |edition=2nd |date=1907 |page=90 |url=https://archive.org/download/numbersystemofal00fineuoft/numbersystemofal00fineuoft.pdf}}</ref>
These place value decimal arithmetic algorithms were introduced to Arab countries by [[Al Khwarizmi]] in the early 9th&nbsp;century and popularized in the Western world by [[Fibonacci]] in the 13th century.<ref>{{Cite web |last=Bernhard |first=Adrienne |title=How modern mathematics emerged from a lost Islamic library |url=https://www.bbc.com/future/article/20201204-lost-islamic-library-maths |access-date=2022-04-22 |website=bbc.com |language=en}}</ref>

====Grid method====
[[Grid method multiplication]], or the box method, is used in primary schools in England and Wales and in some areas{{Which|date=December 2021}} 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:

:{| 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'''
|}

and then add the entries.

===Computer algorithms===
{{Main|Multiplication algorithm#Fast multiplication algorithms for large inputs}}
The classical method of multiplying two {{math|''n''}}-digit numbers requires {{math|''n''<sup>2</sup>}} digit multiplications. [[Multiplication algorithm]]s have been designed that reduce the computation time considerably when multiplying large numbers. Methods based on the [[Discrete Fourier transform#Multiplication of large integers|discrete Fourier transform]] reduce the [[computational complexity]] to {{math|''O''(''n'' log ''n'' log log ''n'')}}. In 2016, the factor {{math|log log ''n''}} was replaced by a function that increases much slower, though still not constant.<ref>{{Cite journal|last1=Harvey|first1=David|last2=van der Hoeven|first2=Joris|last3=Lecerf|first3=Grégoire|title=Even faster integer multiplication|date=2016|journal=Journal of Complexity|volume=36|pages=1–30|doi=10.1016/j.jco.2016.03.001|issn=0885-064X|arxiv=1407.3360|s2cid=205861906}}</ref> In March 2019, David Harvey and Joris van der Hoeven submitted a paper presenting an integer multiplication algorithm with a complexity of <math>O(n\log n).</math><ref>David Harvey, Joris Van Der Hoeven (2019). [https://hal.archives-ouvertes.fr/hal-02070778 Integer multiplication in time O(n log n)] {{Webarchive|url=https://web.archive.org/web/20190408180939/https://hal.archives-ouvertes.fr/hal-02070778 |date=2019-04-08 }}</ref> The algorithm, also based on the fast Fourier transform, is conjectured to be asymptotically optimal.<ref>{{Cite web|url=https://www.quantamagazine.org/mathematicians-discover-the-perfect-way-to-multiply-20190411/|title=Mathematicians Discover the Perfect Way to Multiply|last=Hartnett|first=Kevin|website=Quanta Magazine|date=11 April 2019|language=en|access-date=2020-01-25}}</ref> The algorithm is not practically useful, as it only becomes faster for multiplying extremely large numbers (having more than {{math|2<sup>1729<sup>12</sup></sup>}} bits).<ref>{{Cite web|url=https://cacm.acm.org/magazines/2020/1/241707-multiplication-hits-the-speed-limit/fulltext|title=Multiplication Hits the Speed Limit|last=Klarreich|first=Erica|website=cacm.acm.org|language=en|access-date=2020-01-25|archive-url=https://archive.today/20201031123457/https://cacm.acm.org/magazines/2020/1/241707-multiplication-hits-the-speed-limit/fulltext|archive-date=2020-10-31|url-status=live}}</ref>

==Products of measurements==
{{Main|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:<ref name="Devlin"/>
:[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 in physics|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

==Product of a sequence{{anchor|Product of sequences|Products of sequences}}==<!--linked from below-->
===Capital pi notation{{Anchor|Capital Pi notation}}===<!--This section is linked from [[Pi (letter)]], [[Capital Pi notation]], [[Capital pi notation]]-->
{{Further information|Iterated binary operation#Notation}}

The product of a sequence of factors can be written with the product symbol <math>\textstyle \prod</math>, which derives from the capital letter Π (pi) in the [[Greek alphabet]] (much like the same way the [[summation symbol]] <math>\textstyle \sum</math> is derived from the Greek letter Σ (sigma)).<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Product|url=https://mathworld.wolfram.com/Product.html|access-date=2020-08-16|website=mathworld.wolfram.com|language=en}}</ref><ref>{{Cite web|title=Summation and Product Notation|url=https://math.illinoisstate.edu/day/courses/old/305/contentsummationnotation.html|access-date=2020-08-16|website=math.illinoisstate.edu}}</ref> The meaning of this notation is given by
:<math>\prod_{i=1}^4 (i+1) = (1+1)\,(2+1)\,(3+1)\, (4+1),</math>
which results in
:<math>\prod_{i=1}^4 (i+1) = 120.</math>

In such a notation, the [[variable (mathematics)|variable]] {{mvar|i}} represents a varying [[integer]], called the multiplication index, that runs from the lower value {{math|1}} indicated in the subscript to the upper value {{math|4}} 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
:<math>\prod_{i=m}^n x_i = x_m \cdot x_{m+1} \cdot x_{m+2} \cdot \,\,\cdots\,\, \cdot x_{n-1} \cdot x_n,</math>
where ''m'' and ''n'' are integers or expressions that evaluate to integers. In the case where {{nowrap|1=''m'' = ''n''}}, the value of the product is the same as that of the single factor ''x''<sub>''m''</sub>; if {{nowrap|''m'' > ''n''}}, the product is an [[empty product]] whose value is&nbsp;1—regardless of the expression for the factors.

==== Properties of capital pi notation====
By definition, 
:<math>\prod_{i=1}^{n}x_i=x_1\cdot x_2\cdot\ldots\cdot x_n.</math>

If all factors are identical, a product of {{mvar|n}} factors is equivalent to [[exponentiation]]:
:<math>\prod_{i=1}^{n}x=x\cdot x\cdot\ldots\cdot x=x^n.</math>

[[Associativity]] and [[commutativity]] of multiplication imply
:<math>\prod_{i=1}^{n}{x_iy_i} =\left(\prod_{i=1}^{n}x_i\right)\left(\prod_{i=1}^{n}y_i\right)</math> and
:<math>\left(\prod_{i=1}^{n}x_i\right)^a =\prod_{i=1}^{n}x_i^a</math>
if {{mvar|a}} is a non-negative integer, or if all <math>x_i</math> are positive [[real number]]s, and
:<math>\prod_{i=1}^{n}x^{a_i} =x^{\sum_{i=1}^{n}a_i}</math>
if all <math>a_i</math> are non-negative integers, or if {{mvar|x}} is a positive real number.

===Infinite products===
{{Main|Infinite product}}
One may also consider products of infinitely many terms; these are called [[infinite product]]s. Notationally, this consists in replacing ''n'' above by the [[infinity symbol]] ∞. The product of such an infinite sequence is defined as the [[limit of a sequence|limit]] of the product of the first ''n'' terms, as ''n'' grows without bound. That is,
:<math>\prod_{i=m}^\infty x_i = \lim_{n\to\infty} \prod_{i=m}^n x_i.</math>

One can similarly replace ''m'' with negative infinity, and define:
:<math>\prod_{i=-\infty}^\infty x_i = \left(\lim_{m\to-\infty}\prod_{i=m}^0 x_i\right) \cdot \left(\lim_{n\to\infty} \prod_{i=1}^n x_i\right),</math>
provided both limits exist.{{Citation needed|date=December 2021}}

==Exponentiation==
{{Main|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 2<sup>3</sup>, a two with a [[superscript]] three. In this example, the number two is the ''base'', and three is the ''exponent''.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Exponentiation|url=https://mathworld.wolfram.com/Exponentiation.html|access-date=2021-12-29|website=mathworld.wolfram.com|language=en}}</ref> In general, the exponent (or superscript) indicates how many times the base appears in the expression, so that the expression
:<math>a^n = \underbrace{a\times a \times \cdots  \times a}_n = \prod_{i=1}^{n}a </math>
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]].

==Properties==
[[Image:Multiplication chart.svg|thumb|right|upright 1.0|Multiplication of numbers 0–10. Line labels = multiplicand. ''X''&nbsp;axis = multiplier. ''Y''&nbsp;axis = product.<br>Extension of this pattern into other quadrants gives the reason why a negative number times a negative number yields a positive number.<br>Note also how multiplication by zero causes a reduction in dimensionality, as does multiplication by a [[singular matrix]] where the [[determinant]] is 0. In this process, information is lost and cannot be regained.]]
For [[real number|real]] and [[complex number|complex]] numbers, which includes, for example, [[natural number]]s, [[integer]]s, and [[rational number|fractions]], multiplication has certain properties:

;[[Commutative property]]
:The order in which two numbers are multiplied does not matter:<ref name=":0">{{Cite web |title=Multiplication |website=Encyclopedia of Mathematics |url=https://encyclopediaofmath.org/wiki/Multiplication |access-date=2021-12-29}}</ref><ref name=":1">{{Cite book |last=Biggs |first=Norman L. |title=Discrete Mathematics |publisher=Oxford University Press |date=2002 |isbn=978-0-19-871369-2 |pages=25 |language=en}}</ref>
::<math>x \cdot y = y \cdot x.</math>

;[[Associative property]]
:Expressions solely involving multiplication or addition are invariant with respect to the [[order of operations]]:<ref name=":0"/><ref name=":1"/>
::<math>(x \cdot y) \cdot z = x \cdot (y \cdot z).</math>

;[[Distributive property]]
:Holds with respect to multiplication over addition. This identity is of prime importance in simplifying algebraic expressions:<ref name=":0"/><ref name=":1"/>
::<math>x \cdot(y + z) = x \cdot y + x \cdot z.</math>

;[[Identity element]]
:The multiplicative identity is 1; anything multiplied by 1 is itself. This feature of 1 is known as the '''identity property''':<ref name=":0"/><ref name=":1"/>
::<math>x \cdot 1 = x.</math>

;[[Absorbing element|Property of 0]]
:Any number multiplied by 0 is 0. This is known as the '''zero property''' of multiplication:<ref name=":0"/>
::<math>x \cdot 0 = 0.</math>

;[[Additive inverse|Negation]]
:−1 times any number is equal to the '''[[additive inverse]]''' of that number:
::<math>(-1) \cdot x = (-x)</math>, where <math>(-x) + x = 0.</math>

:−1 times −1 is 1:
::<math>(-1) \cdot (-1) = 1.</math>

;[[Inverse element]]
:Every number ''x'', [[division by zero|except 0]], has a '''[[multiplicative inverse]]''', <math>\frac{1}{x}</math>, such that <math>x \cdot \left(\frac{1}{x}\right) = 1</math>.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Multiplicative Inverse |url=https://mathworld.wolfram.com/ |access-date=2022-04-19 |website=Wolfram MathWorld |language=en}}</ref>

;[[Order theory|Order]] preservation
:Multiplication by a positive number preserves the [[Order theory|order]]:
::For {{nowrap|''a'' > 0}}, if {{nowrap|''b'' > ''c'',}} then {{nowrap|''ab'' > ''ac''}}.
:Multiplication by a negative number reverses the order:
::For {{nowrap|''a'' < 0}}, if {{nowrap|''b'' > ''c'',}} then {{nowrap|''ab'' < ''ac''}}.
:The [[complex number]]s do not have an ordering that is compatible with both addition and multiplication.<ref>{{Cite web |last=Angell |first=David |title=ORDERING COMPLEX NUMBERS... NOT* |url=https://web.maths.unsw.edu.au/~angell/articles/complexorder.pdf |access-date=29 December 2021 |publisher=UNSW Sydney, School of Mathematics and Statistics}}</ref>

Other mathematical systems that include a multiplication operation may not have all these properties. For example, multiplication is not, in general, commutative for [[Matrix (mathematics)|matrices]] and [[quaternion]]s.<ref name=":0"/> [[Hurwitz's theorem (composition algebras)|Hurwitz's theorem]] shows that for the [[hypercomplex number]]s of [[dimension]] 8 or greater, including the [[octonion]]s, [[sedenion]]s, and [[trigintaduonion]]s, multiplication is generally not associative.<ref>{{cite web | last=Cawagas | first=Raoul E. | last2=Carrascal | first2=Alexander S. | last3=Bautista | first3=Lincoln A. | last4=Maria | first4=John P. Sta. | last5=Urrutia | first5=Jackie D. | last6=Nobles | first6=Bernadeth | title=The Subalgebra Structure of the Cayley-Dickson Algebra of Dimension 32 (trigintaduonion) | arxiv=0907.2047v3 | year=2009 | url=https://arxiv.org/abs/0907.2047v3 | doi=10.48550/arXiv.0907.2047}}</ref>

==Axioms==
{{Main|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:
:<math>x \times 0 = 0</math>
:<math>x \times S(y) = (x \times y) + x</math>

Here ''S''(''y'') represents the [[Successor ordinal|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 [[Mathematical induction|induction]]. For instance, ''S''(0), denoted by 1, is a multiplicative identity because
:<math>x \times 1 = x \times S(0) = (x \times 0) + x = 0 + x = x.</math>

The axioms for [[integer]]s typically define them as equivalence classes of ordered pairs of natural numbers. The model is based on treating (''x'',''y'') as equivalent to {{nowrap|''x'' − ''y''}} 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
:<math>(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).</math>

The rule that −1 × −1 = 1 can then be deduced from
:<math>(0, 1) \times (0, 1) = (0 \times 0 + 1 \times 1,\, 0 \times 1 + 1 \times 0) = (1,0).</math>

Multiplication is extended in a similar way to [[rational number]]s and then to [[real number]]s.{{Citation needed|date=December 2021}}

==Multiplication with set theory==
The product of non-negative integers can be defined with set theory using [[Cardinal number#Cardinal multiplication|cardinal numbers]] or the [[Peano axioms#Arithmetic|Peano axioms]]. See [[#Multiplication of different kinds of numbers|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]].<ref>{{Cite web |date=2018-04-11 |title=10.2: Building the Real Numbers |url=https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_(Boman_and_Rogers)/10%3A_Epilogue_to_Real_Analysis/10.02%3A_Building_the_Real_Numbers |access-date=2023-06-23 |website=Mathematics LibreTexts |language=en}}</ref>

==Multiplication in group theory==<!--linked from below-->
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 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 (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 zero is excluded. This is easily seen by the nonexistence of an inverse for all elements other than 1 and&nbsp;−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''' <math>\cdot</math> '''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 <math>\left( \mathbb{Q}/ \{ 0 \} ,\, \cdot \right)</math>.<ref>{{cite book |last1=Burns |first1=Gerald |title=Introduction to group theory with applications |date=1977 |publisher=Academic Press |location=New York |isbn=9780121457501 }}</ref>

==Multiplication of different kinds of numbers==<!--linked from above-->
Numbers can ''count'' (3&nbsp;apples), ''order'' (the 3rd&nbsp;apple), or ''measure'' (3.5&nbsp;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 [[Matrix (mathematics)|matrices]]) or do not look much like numbers (such as [[quaternion]]s).

; Integers
: <math>N\times M</math> 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
: <math>N\times (-M) = (-N)\times M = - (N\times M)</math> and
: <math>(-N)\times (-M) = N\times M</math>
: The same sign rules apply to rational and real numbers.

; [[Rational number]]s
: Generalization to fractions <math>\frac{A}{B}\times \frac{C}{D}</math> is by multiplying the numerators and denominators, respectively: <math>\frac{A}{B}\times \frac{C}{D} = \frac{(A\times C)}{(B\times D)}</math>. This gives the area of a rectangle <math>\frac{A}{B}</math> high and <math>\frac{C}{D}</math> wide, and is the same as the number of things in an array when the rational numbers happen to be whole numbers.<ref name=":0"/>

; [[Real number]]s
: 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 number]]s
: Considering complex numbers <math>z_1</math> and <math>z_2</math> as ordered pairs of real numbers <math>(a_1, b_1)</math> and <math>(a_2, b_2)</math>, the product <math>z_1\times z_2</math> is <math>(a_1\times a_2 - b_1\times b_2, a_1\times b_2 + a_2\times b_1)</math>. This is the same as for reals <math>a_1\times a_2</math> when the ''imaginary parts'' <math>b_1</math> and <math>b_2</math> are zero.

: Equivalently, denoting <math>\sqrt{-1}</math> as <math>i</math>, <math>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.</math><ref name=":0"/>
: Alternatively, in trigonometric form, if <math>z_1 = r_1(\cos\phi_1+i\sin\phi_1), z_2 = r_2(\cos\phi_2+i\sin\phi_2)</math>, then<math display="inline">z_1z_2 = r_1r_2(\cos(\phi_1 + \phi_2) + i\sin(\phi_1 + \phi_2)).</math><ref name=":0"/>

; 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]] (polynomials can be added and multiplied, but polynomials are not numbers in any usual sense).

; Division
: Often division, <math>\frac{x}{y}</math>, is the same as multiplication by an inverse, <math>x\left(\frac{1}{y}\right)</math>. Multiplication for some types of "numbers" may have corresponding division, without inverses; in an [[integral domain]] ''x'' may have no inverse "<math>\frac{1}{x}</math>" but <math>\frac{x}{y}</math> may be defined. In a [[division ring]] there are inverses, but <math>\frac{x}{y}</math> may be ambiguous in non-commutative rings since <math>x\left(\frac{1}{y}\right)</math> need not be the same as <math>\left(\frac{1}{y}\right)x</math>.{{Citation needed|date=December 2021}}

==See also==
{{div col|colwidth=35em}}
* [[Dimensional analysis]]
* [[Multiplication algorithm]]
** [[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 arithmetic]]
** [[Multiply–accumulate operation]]
*** [[Fused multiply–add]]
** [[Wallace tree]]
* [[Multiplicative inverse]], reciprocal
* [[Factorial]]
* [[Genaille–Lucas rulers]]
* [[Lunar arithmetic]]
* [[Napier's bones]]
* [[Peasant multiplication]]
* [[Product (mathematics)]], for generalizations
* [[Slide rule]]
{{div col end}}

==References==
{{Reflist|refs=
<ref name="TI_1982">{{cite book |title=Announcing the TI Programmable 88! |publisher=[[Texas Instruments]] |date=1982<!--or 1983--> |url=http://www.datamath.net/Leaflets/TI-88_Announcement.pdf |access-date=2017-08-03 |url-status=live |archive-url=https://web.archive.org/web/20170803091337/http://www.datamath.net/Leaflets/TI-88_Announcement.pdf |archive-date=2017-08-03 |quote=Now, implied multiplication is recognized by the [[Algebraic Operating System|AOS]] and the square root, logarithmic, and trigonometric functions can be followed by their arguments as when working with pencil and paper.}} (NB. The TI-88 only existed as a prototype and was never released to the public.)</ref>
<ref name="Peterson_2019">{{cite web |title=Order of Operations: Implicit Multiplication? |publisher=The Math Doctors |date=2019-10-14 |department=Algebra / PEMDAS |author-first=Dave |author-last=Peterson |url=https://www.themathdoctors.org/order-of-operations-implicit-multiplication/ |access-date=2023-09-25 |url-status=live |archive-url=https://web.archive.org/web/20230924215608/https://www.themathdoctors.org/order-of-operations-implicit-multiplication/ |archive-date=2023-09-24}}</ref>
<ref name="Peterson_2023">{{cite web |title=Implied Multiplication 1: Not as Bad as You Think |publisher=The Math Doctors |date=2023-08-18 |department=Algebra / Ambiguity, PEMDAS |author-first=Dave |author-last=Peterson |url=https://www.themathdoctors.org/implied-multiplication-1-not-as-bad-as-you-think/ |access-date=2023-09-25 |url-status=live |archive-url=https://web.archive.org/web/20230924222451/https://www.themathdoctors.org/implied-multiplication-1-not-as-bad-as-you-think/ |archive-date=2023-09-24 |postscript=none}}; {{cite web |title=Implied Multiplication 2: Is There a Standard? |publisher=The Math Doctors |date=2023-08-25 |department=Algebra, Arithmetic / Ambiguity, PEMDAS |first=Dave |last=Peterson |url=https://www.themathdoctors.org/implied-multiplication-2-is-there-a-standard/ |access-date=2023-09-25 |url-status=live |archive-url=https://web.archive.org/web/20230924222706/https://www.themathdoctors.org/implied-multiplication-2-is-there-a-standard/ |archive-date=2023-09-24 |postscript=none}}; {{cite web |title=Implied Multiplication 3: You Can't Prove It |publisher=The Math Doctors |date=2023-09-01 |department=Algebra / PEMDAS |author-first=Dave |author-last=Peterson |url=https://www.themathdoctors.org/implied-multiplication-3-you-cant-prove-it/ |access-date=2023-09-25 |url-status=live |archive-url=https://web.archive.org/web/20230924222904/https://www.themathdoctors.org/implied-multiplication-3-you-cant-prove-it/ |archive-date=2023-09-24}}</ref>
}}

==Further reading==
* {{cite book |author=[[Carl Boyer|Boyer, Carl B.]] (revised by [[Uta Merzbach|Merzbach, Uta C.]]) |title=History of Mathematics |publisher=John Wiley and Sons, Inc. |date=1991 |isbn=978-0-471-54397-8 |url-access=registration |url=https://archive.org/details/historyofmathema00boye }}

==External links==
* [https://www.cut-the-knot.org/do_you_know/multiplication.shtml Multiplication] and [http://www.cut-the-knot.org/blue/SysTable.shtml Arithmetic Operations In Various Number Systems] at [[cut-the-knot]]
* [https://web.archive.org/web/20120719043305/http://webhome.idirect.com/~totton/suanpan/mod_mult/ Modern Chinese Multiplication Techniques on an Abacus]

{{Elementary arithmetic}}
{{Hyperoperations}}
{{Authority control}}

[[Category:Multiplication| ]]
[[Category:Elementary arithmetic]]
[[Category:Mathematical notation]]
[[Category:Articles containing proofs]]

Pages transcluded onto the current version of this page (help):

⋅ (edit)
Template:About (view source) (template editor protected)
Template:Ambox (view source) (template editor protected)
Template:Anchor (view source) (template editor protected)
Template:Arithmetic operations (edit)
Template:Authority control (view source) (template editor protected)
Template:C (view source) (template editor protected)
Template:Category handler (view source) (protected)
Template:Category link without namespace (view source) (template editor protected)
Template:Char (view source) (semi-protected)
Template:Citation (view source) (protected)
Template:Citation needed (view source) (protected)
Template:Citation needed span (view source) (template editor protected)
Template:Cite arXiv (view source) (template editor protected)
Template:Cite book (view source) (protected)
Template:Cite journal (view source) (protected)
Template:Cite web (view source) (protected)
Template:DMCA (view source) (template editor protected)
Template:Dated maintenance category (view source) (template editor protected)
Template:Dated maintenance category (articles) (view source) (template editor protected)
Template:Delink (view source) (protected)
Template:Div col (view source) (template editor protected)
Template:Div col/styles.css (view source) (template editor protected)
Template:Div col end (view source) (template editor protected)
Template:Elementary arithmetic (edit)
Template:Expert needed (view source) (template editor protected)
Template:Expert needed/catcheck (view source) (template editor protected)
Template:FULLROOTPAGENAME (view source) (template editor protected)
Template:Find sources mainspace (view source) (template editor protected)
Template:Fix (view source) (protected)
Template:Fix-span (view source) (template editor protected)
Template:Fix/category (view source) (protected)
Template:Further (view source) (template editor protected)
Template:Further information (view source) (template editor protected)
Template:Hex2dec (view source) (semi-protected)
Template:Hlist/styles.css (view source) (protected)
Template:Hyperoperations (edit)
Template:Icon (view source) (template editor protected)
Template:Ifsubst (view source) (template editor protected)
Template:Infobox (view source) (template editor protected)
Template:Infobox symbol (edit)
Template:Main (view source) (template editor protected)
Template:Main other (view source) (protected)
Template:Math (view source) (template editor protected)
Template:Mono (view source) (template editor protected)
Template:Mono/styles.css (view source) (template editor protected)
Template:More citations needed (view source) (template editor protected)
Template:Mvar (view source) (template editor protected)
Template:Navbar (view source) (protected)
Template:Navbox (view source) (template editor protected)
Template:Navbox with columns (view source) (template editor protected)
Template:Nowrap (view source) (protected)
Template:Ns has subpages (view source) (protected)
Template:Pagetype (view source) (protected)
Template:Redirect (view source) (template editor protected)
Template:Reflist (view source) (protected)
Template:Reflist/styles.css (view source) (protected)
Template:Resize (view source) (template editor protected)
Template:Resolve category redirect (view source) (template editor protected)
Template:SDcat (view source) (protected)
Template:Section link (view source) (template editor protected)
Template:See also (view source) (template editor protected)
Template:Sfrac (view source) (extended confirmed protected)
Template:Sfrac/styles.css (view source) (extended confirmed protected)
Template:Short description (view source) (protected)
Template:Short description/lowercasecheck (view source) (protected)
Template:Slink (view source) (extended confirmed protected)
Template:Template other (view source) (protected)
Template:Tooltip (view source) (template editor protected)
Template:Tooltip/styles.css (view source) (template editor protected)
Template:Trim (view source) (protected)
Template:Unichar (view source) (semi-protected)
Template:Unichar/gc (view source) (semi-protected)
Template:Unichar/glyph (view source) (semi-protected)
Template:Unichar/hexformat (view source) (semi-protected)
Template:Unichar/main (view source) (semi-protected)
Template:Unichar/name (view source) (semi-protected)
Template:Unichar/notes (view source) (semi-protected)
Template:Unichar/ulink (view source) (semi-protected)
Template:Use dmy dates (view source) (template editor protected)
Template:Verification needed (view source) (semi-protected)
Template:Verify source (view source) (template editor protected)
Template:Webarchive (view source) (template editor protected)
Template:Which (view source) (template editor protected)
Template:Yesno (view source) (protected)
Template:Yesno-no (view source) (template editor protected)
Template:Yesno-yes (view source) (template editor protected)
Category:Mathematics articles needing expert attention (edit)
Module:About (view source) (template editor protected)
Module:Anchor (view source) (template editor protected)
Module:Arguments (view source) (protected)
Module:Authority control (view source) (template editor protected)
Module:Authority control/auxiliary (view source) (template editor protected)
Module:Authority control/config (view source) (template editor protected)
Module:BaseConvert (view source) (template editor protected)
Module:Category handler (view source) (protected)
Module:Category handler/blacklist (view source) (protected)
Module:Category handler/config (view source) (protected)
Module:Category handler/data (view source) (protected)
Module:Category handler/shared (view source) (protected)
Module:Check for unknown parameters (view source) (protected)
Module:Citation/CS1 (view source) (protected)
Module:Citation/CS1/COinS (view source) (protected)
Module:Citation/CS1/Configuration (view source) (protected)
Module:Citation/CS1/Date validation (view source) (protected)
Module:Citation/CS1/Identifiers (view source) (protected)
Module:Citation/CS1/Utilities (view source) (protected)
Module:Citation/CS1/Whitelist (view source) (protected)
Module:Citation/CS1/styles.css (view source) (protected)
Module:DecodeEncode (view source) (template editor protected)
Module:Delink (view source) (protected)
Module:Disambiguation/templates (view source) (protected)
Module:EditAtWikidata (view source) (protected)
Module:Find sources (view source) (template editor protected)
Module:Find sources/config (view source) (template editor protected)
Module:Find sources/links (view source) (template editor protected)
Module:Find sources/templates/Find sources mainspace (view source) (template editor protected)
Module:Format link (view source) (template editor protected)
Module:Hatnote (view source) (template editor protected)
Module:Hatnote/styles.css (view source) (template editor protected)
Module:Hatnote list (view source) (template editor protected)
Module:Icon (view source) (template editor protected)
Module:Icon/data (view source) (template editor protected)
Module:Infobox (view source) (template editor protected)
Module:Infobox/styles.css (view source) (template editor protected)
Module:Infobox3cols (view source) (template editor protected)
Module:InfoboxImage (view source) (template editor protected)
Module:Labelled list hatnote (view source) (template editor protected)
Module:LoadData (view source) (semi-protected)
Module:Message box (view source) (protected)
Module:Message box/ambox.css (view source) (protected)
Module:Message box/configuration (view source) (protected)
Module:Namespace detect/config (view source) (protected)
Module:Namespace detect/data (view source) (protected)
Module:Navbar (view source) (protected)
Module:Navbar/configuration (view source) (protected)
Module:Navbar/styles.css (view source) (protected)
Module:Navbox (view source) (template editor protected)
Module:Navbox/configuration (view source) (template editor protected)
Module:Navbox/styles.css (view source) (template editor protected)
Module:Navbox with columns (view source) (template editor protected)
Module:Navbox with columns/configuration (view source) (template editor protected)
Module:Ns has subpages (view source) (protected)
Module:Numcr2namecr (view source) (semi-protected)
Module:Pagetype (view source) (protected)
Module:Pagetype/config (view source) (protected)
Module:Pagetype/disambiguation (view source) (protected)
Module:Pagetype/rfd (view source) (protected)
Module:Pagetype/setindex (view source) (protected)
Module:Pagetype/softredirect (view source) (protected)
Module:Redirect hatnote (view source) (template editor protected)
Module:Resolve category redirect (view source) (template editor protected)
Module:SDcat (view source) (protected)
Module:Section link (view source) (template editor protected)
Module:Sidebar (view source) (template editor protected)
Module:Sidebar/configuration (view source) (template editor protected)
Module:Sidebar/styles.css (view source) (template editor protected)
Module:String (view source) (protected)
Module:TableTools (view source) (protected)
Module:Template redirect regex (view source) (template editor protected)
Module:Unicode data (view source) (template editor protected)
Module:Unicode data/names/000 (view source) (semi-protected)
Module:Unicode data/names/002 (view source) (semi-protected)
Module:Unsubst (view source) (protected)
Module:Webarchive (view source) (template editor protected)
Module:Webarchive/data (view source) (template editor protected)
Module:Wikitext Parsing (view source) (protected)
Module:Yesno (view source) (protected)

Return to Multiplication.

Retrieved from "https://en.wikipedia.org/wiki/Multiplication"