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{{Short description|Arithmetical operation}}
{{About|the mathematical operation}}
{{redirect|⋅|the symbol|
{{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 [[Scale factor|scaling]]. Here, 2 is being multiplied by 3 using scaling, giving 6 as a result.]]
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[[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]]''.
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===Product of two integers===
An integer can be either zero, a
<math display="block">\begin{array}{|c|c c|}
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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
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.
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[[File:Komplexe zahlenebene.svg|thumb|upright=1.25|A complex number in polar coordinates]]
:<math>a + b\, i = r \cdot ( \cos(\varphi) + i \sin(\varphi) ) = r \cdot e ^{ i \varphi} </math>
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