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{{Short description|Arithmetical operation}}{{About|the mathematical operation}}{{redirect|⋅|the symbol|interpunct}}{{More citations needed|date=April 2012}}{{Use dmy dates|date=September 2023|cs1-dates=y}}{{Arithmetic operations}}(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 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 = 11.25m2}}; {{nowrap|1=4{{sfrac|1|2}} × 2{{sfrac|1|2}} = 11{{sfrac|1|4}}}})Multiplication (often denoted by the cross symbol {{char|×}}, by the mid-line dot operator {{char|⋅}}, by juxtaposition, or, on computers, by an asterisk {{char|*}}) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a product.The multiplication of whole numbers may be thought of as repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the multiplicand, as the quantity of the other one, the multiplier; both numbers can be referred to as factors. For example, 4 multiplied by 3, often written as 3 times 4 and spoken as "3 times 4", can be calculated by adding 3 copies of 4 together: Here, 3 (the multiplier) and 4 (the multiplicand) are the factors, and 12 is the product.One of the main properties of multiplication is the commutative property, which states in this case that adding 3 copies of 4 gives the same result as adding 4 copies of 3: Thus, the designation of multiplier and multiplicand does not affect the result of the multiplication.WEB, Devlin, Keith,weblink What Exactly is Multiplication?, Keith Devlin, Mathematical Association of America, January 2011, With multiplication you have a multiplicand (written second) multiplied by a multiplier (written first), May 14, 2017,weblink" title="web.archive.org/web/20170527070801weblink">weblink May 27, 2017, live, Systematic generalizations of this basic definition define the multiplication of integers (including negative numbers), rational numbers (fractions), and real numbers.Multiplication can also be visualized as counting objects arranged in a rectangle (for whole numbers) or as finding the area of a rectangle whose sides have some given lengths. The area of a rectangle does not depend on which side is measured first—a consequence of the commutative property.The product of two measurements (or physical quantities) is a new type of measurement, usually with a derived unit. For example, multiplying the lengths (in meters or feet) of the two sides of a rectangle gives its area (in square meters or square feet). Such a product is the subject of dimensional analysis.The inverse operation of multiplication is division. For example, since 4 multiplied by 3 equals 12, 12 divided by 3 equals 4. Indeed, multiplication by 3, followed by division by 3, yields the original number. The division of a number other than 0 by itself equals 1.Several mathematical concepts expand upon the fundamental idea of multiplication. The product of a sequence, vector multiplication, complex numbers, and matrices are all examples where this can be seen. These more advanced constructs tend to affect the basic properties in their own ways, such as becoming noncommutative in matrices and some forms of vector multiplication or changing the sign of complex numbers.

Notation

factoids

{{unicharDot operator|html=}} |see also = 00B7nlink=Interpunct}} {{unicharnlink=Full stop}}}}{{See also|Multiplier (linguistics)}}In arithmetic, multiplication is often written using the multiplication sign (either {{char|×}} or {{char|times}}) between the terms (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/20170324175113weblink |archive-date=2017-03-24 |url-status=live }} For example, 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/20170327163705weblink |archive-date=2017-03-27 |url-status=live }} usually a middle-position dot (rarely period): 5 cdot 2.

• {{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 due to the fact that most computers historically were limited to small character sets (such as ASCII and EBCDIC) that lacked a multiplication sign (such as ⋅ or ×), while the asterisk appeared on every keyboard.{{citation needed|date=February 2023}} This usage originated in the FORTRAN programming language.FULLER >FIRST = WILLIAM R., FORTRAN Programming: A Supplement for Calculus Courses, Universitext, 1977,weblink 10, Springer, 10.1007/978-1-4612-9938-7, 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; however, sometimes the first factor is the multiplicand and the second the multiplier.RAMONE >FIRST=CREWTON,weblink Multiplicand and Multiplier, 10 November 2015, Crewton Ramone's House of Mathweblink >archive-date=26 October 2015, live, .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".BOOK, Litvin, Chester,weblink Advance Brain Stimulation by Psychoconduction, 2012, Trafford, 978-1-4669-0152-0, 2–3, 5–6, 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, 2times pi is a multiple of pi, as is 5133 times 486 times pi. A product of integers is a multiple of each factor; for example, 15 is the product of 3 and 5 and is both a multiple of 3 and a multiple of 5.

Definitions

{{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 = yessection}}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 r,sinmathbb{N} is defined as:

Product of two integers

An integer can be either zero, a positive, or a negative 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:begin{array}{|c|c c|} end{array}(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:

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 numbers. A standard way for expressing this is that every real number is the least upper bound of a set of rational numbers. In particular, every positive real number is the least upper bound of the truncations of its infinite decimal representation; for example, pi is the least upper bound of {3,; 3.1,; 3.14,; 3,141,ldots}.A fundamental property of real numbers is that rational approximations are compatible with arithmetic operations, and, in particular, with multiplication. This means that, if {{mvar|a}} and {{mvar|b}} are positive real numbers such that a=sup_{xin A} x and b=sup_{yin B} y, then acdot b=sup_{xin A, yin B}xcdot 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.

Product of two complex numbers

Two complex numbers can be multiplied by the distributive law and the fact that i^2=-1, as follows: end{align}(File:Komplexe zahlenebene.svg|thumb|upright=1.25|A complex number in polar coordinates)Geometric meaning of complex multiplication can be understood rewriting complex numbers in polar coordinates: Furthermore, from which one obtains The geometric meaning is that the magnitudes are multiplied and the arguments are added.

Product of two quaternions

The product of two quaternions can be found in the article on quaternions. Note, in this case, that a cdot b and b cdot a are in general different.

Computation

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". 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"): 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:WEB, Multiplication,weblink 2022-03-15, mathematische-basteleien.de, 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.

Historical algorithms

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.ARXIV, Pletser, Vladimir, 2012-04-04, Does the Ishango Bone Indicate Knowledge of the Base 12? An Interpretation of a Prehistoric Discovery, the First Mathematical Tool of Humankind, math.HO, 1204.1019, {{Verification needed|date=December 2021}}

Egyptians

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:WEB, Peasant Multiplication,weblink 2021-12-29, cut-the-knot.org,

Babylonians

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}}

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 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.JOURNAL,weblink Ancient times table hidden in Chinese bamboo strips, Nature, Jane, Qiu, Jane Qiu, 7 January 2014, 22 January 2014, 10.1038/nature.2014.14482, 130132289,weblink" title="web.archive.org/web/20140122064930weblink">weblink 22 January 2014, live, free,

Modern methods

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: 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.WEB, Bernhard, Adrienne, How modern mathematics emerged from a lost Islamic library,weblink 2022-04-22, bbc.com, en,

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: ! scope="col" width="40pt" | ×! scope="col" width="120pt" | 30! scope="col" width="40pt" | 4! scope="row" | 10|300|40! scope="row" | 3|90|12and then add the entries.

Computer algorithms

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.JOURNAL, Harvey, David, van der Hoeven, Joris, Lecerf, Grégoire, Even faster integer multiplication, 2016, Journal of Complexity, 36, 1–30, 10.1016/j.jco.2016.03.001, 0885-064X, 1407.3360, 205861906, In March 2019, David Harvey and Joris van der Hoeven submitted a paper presenting an integer multiplication algorithm with a complexity of O(nlog n).David Harvey, Joris Van Der Hoeven (2019). Integer multiplication in time O(n log n) {{Webarchive|url=https://web.archive.org/web/20190408180939weblink |date=2019-04-08 }} The algorithm, also based on the fast Fourier transform, is conjectured to be asymptotically optimal.WEB,weblink Mathematicians Discover the Perfect Way to Multiply, Hartnett, Kevin, Quanta Magazine, 11 April 2019, en, 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).WEB,weblink Multiplication Hits the Speed Limit, Klarreich, Erica, cacm.acm.org, en, 2020-01-25,weblink 2020-10-31, live,

Products of measurements

One can only meaningfully add or subtract quantities of the same type, but quantities of different types can be multiplied or divided without problems. For example, four bags with three marbles each can be thought of as: 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: In this case, the hour units cancel out, leaving the product with only kilometer units.Other examples of multiplication involving units include:

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 textstyle prod, which derives from the capital letter Π (pi) in the Greek alphabet (much like the same way the summation symbol textstyle sum is derived from the Greek letter Σ (sigma)).WEB, Weisstein, Eric W., Product,weblink 2020-08-16, mathworld.wolfram.com, en, WEB, Summation and Product Notation,weblink 2020-08-16, math.illinoisstate.edu, The meaning of this notation is given by which results in 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 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 xm; if {{nowrap|m > n}}, the product is an empty product whose value is 1—regardless of the expression for the factors.

Properties of capital pi notation

By definition, If all factors are identical, a product of {{mvar|n}} factors is equivalent to exponentiation: Associativity and commutativity of multiplication imply if {{mvar|a}} is a non-negative integer, or if all x_i are positive real numbers, and if all a_i are non-negative integers, or if {{mvar|x}} is a positive real number.

Infinite products

One may also consider products of infinitely many terms; these are called infinite products. Notationally, this consists in replacing n above by the infinity symbol ∞. The product of such an infinite sequence is defined as the limit of the product of the first n terms, as n grows without bound. That is, One can similarly replace m with negative infinity, and define: provided both limits exist.{{Citation needed|date=December 2021}}

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.WEB, Weisstein, Eric W., Exponentiation,weblink 2021-12-29, mathworld.wolfram.com, en, In general, the exponent (or superscript) indicates how many times the base appears in the expression, so that the expression 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.

Properties

Image:Multiplication chart.svg|thumb|right|upright 1.0|Multiplication of numbers 0–10. Line labels = multiplicand. X axis = multiplier. Y axis = product.Extension of this pattern into other quadrants gives the reason why a negative number times a negative number yields a positive number.Note also how multiplication by zero causes a reduction in dimensionality, as does multiplication by a singular matrix where the determinantdeterminantFor 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:WEB, Multiplication, Encyclopedia of Mathematics,weblink 2021-12-29, BOOK, Biggs, Norman L., Discrete Mathematics, Oxford University Press, 2002, 978-0-19-871369-2, 25, en,

x cdot y = y cdot x.