multiplication#Capital Pi notation

{{Infobox symbol

|name= Multiplication signs

|sign=× ⋅

|unicode={{unichar|00D7|Multiplication sign|html=}}
{{unichar|22C5|Dot operator|html=}}

|see also =

|different from= {{unichar|00B7|Middle dot|nlink=Interpunct}}
{{unichar|002E|nlink=Full stop}}

}}

{{main|Multiplication sign}}

{{See also|Multiplier (linguistics)}}

In arithmetic, multiplication is often written using the multiplication sign (either {{char|×}} or {{char|$\backslash times$}}) between the factors (that is, in infix notation).{{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 }} For example,

:$2\backslash times\; 3\; =\; 6,$ ("two times three equals six")

:$3\backslash times\; 4\; =\; 12\; ,$

:$2\backslash times\; 3\backslash times\; 5\; =\; 6\backslash times\; 5\; =\; 30,$

:$2\backslash times\; 2\backslash times\; 2\backslash times\; 2\; \backslash times\; 2\; =\; 32.$

There are other mathematical notations for multiplication:

• To reduce confusion between the multiplication sign × and the common variable {{mvar|x}}, multiplication is also denoted by dot signs,{{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 }} usually a middle-position dot (rarely period): $5\; \backslash cdot\; 2$.

: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 . 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.{{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,{{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}} 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.{{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}}

• {{anchor|Implicit|Explicit}}In algebra, multiplication involving variables is often written as a juxtaposition (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 (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 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 vectors, yielding a vector as its result, while the dot denotes taking the dot product of two vectors, resulting in a scalar.

In computer programming, the asterisk (as in 5*2) is still the most common notation. This is because 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.{{citation needed|date=February 2023}} This usage originated in the FORTRAN programming language.{{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

}}

{{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;{{cite web |title=Multiplicand {{!}} mathematics {{!}} Britannica |url=https://www.britannica.com/science/multiplicand |website=www.britannica.com |publisher=Encyclopædia Britannica, Inc. |access-date=15 November 2024 |language=en}}{{cite web |last1=Weisstein |first1=Eric W. |title=Multiplicand |url=https://mathworld.wolfram.com/Multiplicand.html |website=mathworld.wolfram.com |publisher=Wolfram Research, Inc. |access-date=15 November 2024 |language=en}} however, sometimes the first factor is considered 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".{{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}}

In algebra, a number that is the multiplier of a variable or expression (e.g., the 3 in $3xy^2$) is called a coefficient.

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, $2\backslash times\; \backslash pi$ is a multiple of $\backslash pi$, as is $5133\; \backslash times\; 486\; \backslash times\; \backslash pi$. 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.

{{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.

File:Three by Four.svg

The product of two natural numbers $r,s\backslash in\backslash mathbb\{N\}$ is defined as:

$$r\; \backslash cdot\; s\; \backslash equiv\; \backslash sum\_\{i=1\}^s\; r\; =\; \backslash underbrace\{\; r+r+\backslash cdots+r\; \}\_\{s\backslash text\{\; times\}\}\; \backslash equiv\; \backslash sum\_\{j=1\}^r\; s\; =\; \backslash underbrace\{\; s+s+\backslash cdots+s\; \}\_\{r\backslash text\{\; times\}\}\; .$$

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:

(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.

Two fractions can be multiplied by multiplying their numerators and denominators:

:$$\backslash frac\{z\}\{n\}\; \backslash cdot\; \backslash frac\{z\text{'}\}\{n\text{'}\}\; =\; \backslash frac\{z\backslash cdot\; z\text{'}\}\{n\backslash cdot\; n\text{'}\}\; ,$$

:which is defined when $n,n\text{'}\backslash neq\; 0$.

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, $\backslash pi$ is the least upper bound of $\backslash \{3,\backslash ;\; 3.1,\backslash ;\; 3.14,\backslash ;\; 3.141,\backslash ldots\backslash \}.$

A fundamental property of real numbers is that rational approximations are compatible with arithmetic operations, and, in particular, with multiplication. This means that, if {{mvar|a}} and {{mvar|b}} are positive real numbers such that $a=\backslash sup\_\{x\backslash in\; A\}\; x$ and $b=\backslash sup\_\{y\backslash in\; B\}\; y,$ then $a\backslash cdot\; b=\backslash sup\_\{x\backslash in\; A,\; y\backslash in\; B\}x\backslash cdot\; y.$ In particular, the product of two positive real numbers is the least upper bound of the term-by-term products of the sequences 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 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 $i^2=-1$, as follows:

:$\backslash begin\{align\}$

(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}

File:Komplexe zahlenebene.svg

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

:$a\; +\; b\backslash ,\; i\; =\; r\; \backslash cdot\; (\; \backslash cos(\backslash varphi)\; +\; i\; \backslash sin(\backslash varphi)\; )\; =\; r\; \backslash cdot\; e\; ^\{\; i\; \backslash varphi\}$

Furthermore,

:$c\; +\; d\backslash ,\; i\; =\; s\; \backslash cdot\; (\; \backslash cos(\backslash psi)\; +\; i\backslash sin(\backslash psi)\; )\; =\; s\; \backslash cdot\; e^\{i\backslash psi\},$

from which one obtains

:$(a\; \backslash cdot\; c\; -\; b\; \backslash cdot\; d)\; +\; (a\; \backslash cdot\; d\; +\; b\; \backslash cdot\; c)\; i\; =\; r\; \backslash cdot\; s\; \backslash cdot\; e^\{i(\backslash varphi\; +\; \backslash psi)\}.$

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 $a\; \backslash cdot\; b$ and $b\; \backslash cdot\; a$ are in general different.

{{Main|Multiplication algorithm}}

file:צעצוע מכני משנת 1918 לחישובי לוח הכפל The Educated Monkey.jpg 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, the multiplication above is depicted similarly but with the original problem written on a single line and computation starting with the first digit of the multiplier:{{Cite web |title=Multiplication |url=http://www.mathematische-basteleien.de/multiplication.htm |access-date=2022-03-15 |website=mathematische-basteleien.de}}

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 logarithms 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 calculators, such as the Marchant, automated multiplication of up to 10-digit numbers. Modern electronic computers and calculators have greatly reduced the need for multiplication by hand.

Methods of multiplication were documented in the writings of ancient Egyptian, {{Citation needed span|text=Greek, Indian,|date=December 2021|reason=This claim is not sourced in the subsections below.}} 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 era in Central Africa, but this is speculative.{{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}}{{Verification needed|date=December 2021}}

{{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:{{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}}

:13 × 21 = (1 + 4 + 8) × 21 = (1 × 21) + (4 × 21) + (8 × 21) = 21 + 84 + 168 = 273.

The Babylonians used a sexagesimal 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 {{nowrap|60 × 60}} 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, 2n, ..., 20n; followed by the multiples of 10n: 30n 40n, and 50n. Then to compute any sexagesimal product, say 53n, one only needed to add 50n and 3n computed from the table.{{Citation needed|date=December 2021}}

{{see also|Chinese multiplication table}}

File:Multiplication algorithm.GIF

In the mathematical text Zhoubi Suanjing, dated prior to 300 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 decimal multiplication table by the end of the Warring States period.{{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 }}

Image:Gelosia multiplication 45 256.png.]]

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.{{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}}

These place value decimal arithmetic algorithms were introduced to Arab countries by Al Khwarizmi in the early 9th century and popularized in the Western world by Fibonacci in the 13th century.{{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}}

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:

:

and then add the entries.

{{Main|Multiplication algorithm#Fast multiplication algorithms for large inputs}}

The classical method of multiplying two {{math|n}}-digit numbers requires {{math|n²}} digit multiplications. Multiplication algorithms have been designed that reduce the computation time considerably when multiplying large numbers. Methods based on the 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.{{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}} In March 2019, David Harvey and Joris van der Hoeven submitted a paper presenting an integer multiplication algorithm with a complexity of $O(n\backslash log\; n).$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 }} The algorithm, also based on the fast Fourier transform, is conjectured to be asymptotically optimal.{{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}} The algorithm is not practically useful, as it only becomes faster for multiplying extremely large numbers (having more than {{math|2¹⁷²⁹¹²}} bits).{{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|date=January 2020 |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}}

{{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:

:[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

==Product of a sequence{{anchor|Product of sequences|Products of sequences}}==

===Capital pi notation{{Anchor|Capital Pi notation}}===

{{Further information|Iterated binary operation#Notation}}

The product of a sequence of factors can be written with the product symbol $\backslash textstyle\; \backslash prod$, which derives from the capital letter Π (pi) in the Greek alphabet (much like the same way the summation symbol $\backslash textstyle\; \backslash sum$ is derived from the Greek letter Σ (sigma)).{{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}}{{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}} The meaning of this notation is given by

:$\backslash prod\_\{i=1\}^4\; (i+1)\; =\; (1+1)\backslash ,(2+1)\backslash ,(3+1)\backslash ,\; (4+1),$

which results in

:$\backslash prod\_\{i=1\}^4\; (i+1)\; =\; 120.$

In such a notation, the 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

:$\backslash prod\_\{i=m\}^n\; x\_i\; =\; x\_m\; \backslash cdot\; x\_\{m+1\}\; \backslash cdot\; x\_\{m+2\}\; \backslash cdot\; \backslash ,\backslash ,\backslash cdots\backslash ,\backslash ,\; \backslash cdot\; x\_\{n-1\}\; \backslash cdot\; x\_n,$

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_m; if {{nowrap|m > n}}, the product is an empty product whose value is 1—regardless of the expression for the factors.

By definition,

:$\backslash prod\_\{i=1\}^\{n\}x\_i=x\_1\backslash cdot\; x\_2\backslash cdot\backslash ldots\backslash cdot\; x\_n.$

If all factors are identical, a product of {{mvar|n}} factors is equivalent to exponentiation:

:$\backslash prod\_\{i=1\}^\{n\}x=x\backslash cdot\; x\backslash cdot\backslash ldots\backslash cdot\; x=x^n.$

Associativity and commutativity of multiplication imply

:$\backslash prod\_\{i=1\}^\{n\}\{x\_iy\_i\}\; =\backslash left(\backslash prod\_\{i=1\}^\{n\}x\_i\backslash right)\backslash left(\backslash prod\_\{i=1\}^\{n\}y\_i\backslash right)$ and

:$\backslash left(\backslash prod\_\{i=1\}^\{n\}x\_i\backslash right)^a\; =\backslash prod\_\{i=1\}^\{n\}x\_i^a$

if {{mvar|a}} is a non-negative integer, or if all $x\_i$ are positive real numbers, and

:$\backslash prod\_\{i=1\}^\{n\}x^\{a\_i\}\; =x^\{\backslash sum\_\{i=1\}^\{n\}a\_i\}$

if all $a\_i$ are non-negative integers, or if {{mvar|x}} is a positive real number.

{{Main|Infinite product}}

One may also consider products of infinitely many factors; 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 factors, as n grows without bound. That is,

:$\backslash prod\_\{i=m\}^\backslash infty\; x\_i\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\; \backslash prod\_\{i=m\}^n\; x\_i.$

One can similarly replace m with negative infinity, and define:

:$\backslash prod\_\{i=-\backslash infty\}^\backslash infty\; x\_i\; =\; \backslash left(\backslash lim\_\{m\backslash to-\backslash infty\}\backslash prod\_\{i=m\}^0\; x\_i\backslash right)\; \backslash cdot\; \backslash left(\backslash lim\_\{n\backslash to\backslash infty\}\; \backslash prod\_\{i=1\}^n\; x\_i\backslash right),$

provided both limits exist.{{Citation needed|date=December 2021}}

{{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 2³, a two with a superscript three. In this example, the number two is the base, and three is the exponent.{{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}} In general, the exponent (or superscript) indicates how many times the base appears in the expression, so that the expression

:$a^n\; =\; \backslash underbrace\{a\backslash times\; a\; \backslash times\; \backslash cdots\; \backslash times\; a\}\_n\; =\; \backslash prod\_\{i=1\}^\{n\}a$

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.

Image:Multiplication chart.svg where the determinant is 0. In this process, information is lost and cannot be regained.]]

For real and complex numbers, which includes, for example, natural numbers, integers, and fractions, multiplication has certain properties:

;Commutative property

:The order in which two numbers are multiplied does not matter:{{Cite web |title=Multiplication |website=Encyclopedia of Mathematics |url=https://encyclopediaofmath.org/wiki/Multiplication |access-date=2021-12-29}}{{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}}

::$x\; \backslash cdot\; y\; =\; y\; \backslash cdot\; x.$

;Associative property

:Expressions solely involving multiplication or addition are invariant with respect to the order of operations:

::$(x\; \backslash cdot\; y)\; \backslash cdot\; z\; =\; x\; \backslash cdot\; (y\; \backslash cdot\; z).$

;Distributive property

:Holds with respect to multiplication over addition. This identity is of prime importance in simplifying algebraic expressions:

::$x\; \backslash cdot(y\; +\; z)\; =\; x\; \backslash cdot\; y\; +\; x\; \backslash cdot\; z.$

;Identity element

:The multiplicative identity is 1; anything multiplied by 1 is itself. This feature of 1 is known as the identity property:

::$x\; \backslash cdot\; 1\; =\; x.$

;Property of 0

:Any number multiplied by 0 is 0. This is known as the zero property of multiplication:

::$x\; \backslash cdot\; 0\; =\; 0.$

;Negation

:−1 times any number is equal to the additive inverse of that number:

::$(-1)\; \backslash cdot\; x\; =\; (-x)$, where $(-x)\; +\; x\; =\; 0.$

:−1 times −1 is 1:

::$(-1)\; \backslash cdot\; (-1)\; =\; 1.$

;Inverse element

:Every number x, except 0, has a multiplicative inverse, $\backslash frac\{1\}\{x\}$, such that $x\; \backslash cdot\; \backslash left(\backslash frac\{1\}\{x\}\backslash right)\; =\; 1$.{{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}}

;Order preservation

:Multiplication by a positive number preserves the 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 numbers do not have an ordering that is compatible with both addition and multiplication.{{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}}

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. Hurwitz's theorem shows that for the hypercomplex numbers of dimension 8 or greater, including the octonions, sedenions, and trigintaduonions, multiplication is generally not associative.{{cite arXiv | last1=Cawagas | first1=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) | date=2009 | class=math.RA | eprint=0907.2047v3}}

{{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:

:$x\; \backslash times\; 0\; =\; 0$

:$x\; \backslash times\; S(y)\; =\; (x\; \backslash times\; y)\; +\; x$

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

:$x\; \backslash times\; 1\; =\; x\; \backslash times\; S(0)\; =\; (x\; \backslash times\; 0)\; +\; x\; =\; 0\; +\; x\; =\; x.$

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 {{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

:$(x\_p,\backslash ,\; x\_m)\; \backslash times\; (y\_p,\backslash ,\; y\_m)\; =\; (x\_p\; \backslash times\; y\_p\; +\; x\_m\; \backslash times\; y\_m,\backslash ;\; x\_p\; \backslash times\; y\_m\; +\; x\_m\; \backslash times\; y\_p).$

The rule that −1 × −1 = 1 can then be deduced from

:$(0,\; 1)\; \backslash times\; (0,\; 1)\; =\; (0\; \backslash times\; 0\; +\; 1\; \backslash times\; 1,\backslash ,\; 0\; \backslash times\; 1\; +\; 1\; \backslash times\; 0)\; =\; (1,0).$

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

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.{{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}}

==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. 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 $\backslash cdot$ b or ab. When referring to a group via the indication of the set and operation, the dot is used. For example, our first example could be indicated by $\backslash left(\; \backslash mathbb\{Q\}/\; \backslash \{\; 0\; \backslash \}\; ,\backslash ,\; \backslash cdot\; \backslash right)$.{{cite book |last1=Burns |first1=Gerald |title=Introduction to group theory with applications |date=1977 |publisher=Academic Press |location=New York |isbn=9780121457501 }}

==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) or do not look much like numbers (such as quaternions).

; Integers

: $N\backslash times\; M$ is the sum of N copies of M when N and M are positive whole numbers. This gives the number of things in an array N wide and M high. Generalization to negative numbers can be done by

: $N\backslash times\; (-M)\; =\; (-N)\backslash times\; M\; =\; -\; (N\backslash times\; M)$ and

: $(-N)\backslash times\; (-M)\; =\; N\backslash times\; M$

: The same sign rules apply to rational and real numbers.

; Rational numbers

: Generalization to fractions $\backslash frac\{A\}\{B\}\backslash times\; \backslash frac\{C\}\{D\}$ is by multiplying the numerators and denominators, respectively: $\backslash frac\{A\}\{B\}\backslash times\; \backslash frac\{C\}\{D\}\; =\; \backslash frac\{(A\backslash times\; C)\}\{(B\backslash times\; D)\}$. This gives the area of a rectangle $\backslash frac\{A\}\{B\}$ high and $\backslash frac\{C\}\{D\}$ wide, and is the same as the number of things in an array when the rational numbers happen to be whole numbers.

; Real numbers

: Real numbers and their products can be defined in terms of sequences of rational numbers.

; Complex numbers

: Considering complex numbers $z\_1$ and $z\_2$ as ordered pairs of real numbers $(a\_1,\; b\_1)$ and $(a\_2,\; b\_2)$, the product $z\_1\backslash times\; z\_2$ is $(a\_1\backslash times\; a\_2\; -\; b\_1\backslash times\; b\_2,\; a\_1\backslash times\; b\_2\; +\; a\_2\backslash times\; b\_1)$. This is the same as for reals $a\_1\backslash times\; a\_2$ when the imaginary parts $b\_1$ and $b\_2$ are zero.

: Equivalently, denoting $\backslash sqrt\{-1\}$ as $i$, $z\_1\; \backslash times\; z\_2\; =\; (a\_1+b\_1i)(a\_2+b\_2i)=(a\_1\; \backslash times\; a\_2)+(a\_1\backslash times\; b\_2i)+(b\_1\backslash times\; a\_2i)+(b\_1\backslash times\; b\_2i^2)=(a\_1a\_2-b\_1b\_2)+(a\_1b\_2+b\_1a\_2)i.$

: Alternatively, in trigonometric form, if $z\_1\; =\; r\_1(\backslash cos\backslash phi\_1+i\backslash sin\backslash phi\_1),\; z\_2\; =\; r\_2(\backslash cos\backslash phi\_2+i\backslash sin\backslash phi\_2)$, then$z\_1z\_2\; =\; r\_1r\_2(\backslash cos(\backslash phi\_1\; +\; \backslash phi\_2)\; +\; i\backslash sin(\backslash phi\_1\; +\; \backslash phi\_2)).$

; Further generalizations

: See Multiplication in group theory, 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. An example of a ring that is not any of the number systems above is a polynomial ring (polynomials can be added and multiplied, but polynomials are not numbers in any usual sense).

; Division

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

{{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}}

{{Reflist|refs=

{{cite book |title=Announcing the TI Programmable 88! |publisher=Texas Instruments |date=1982 |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 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.)

{{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}}

{{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}}

}}

• {{cite book |author=Boyer, Carl B. (revised by 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 }}

• [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]

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