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<title>Multiplication</title>

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Multiplication

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the mathematical operation. For other uses, see <a href="Multiplication_(disambiguation)" class="mw-disambig" title="Multiplication (disambiguation)">Multiplication (disambiguation)</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">"⋅" redirects here. For the symbol, see <a href="Interpunct#In_mathematics_and_science" title="Interpunct">Interpunct § In mathematics and science</a>.</div>

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Multiplication is one of the four elementary mathematical operations of <a href="Arithmetic" title="Arithmetic">arithmetic</a>, with the other ones being <a href="Addition" title="Addition">addition</a>, <a href="Subtraction" title="Subtraction">subtraction</a>, and <a href="Division_(mathematics)" title="Division (mathematics)">division</a>. The result of a multiplication operation is called a <a href="Product_(mathematics)" title="Product (mathematics)">product</a>. Multiplication is often denoted by the cross symbol, ×, by the mid-line dot operator, ·, by juxtaposition, or, in programming languages, by an asterisk, *.

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. This is to be distinguished from <a href="Term_(arithmetic)" class="mw-redirect" title="Term (arithmetic)">terms</a>, which are added.

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<mtext> times</mtext>

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<annotation encoding="application/x-tex">{\displaystyle a\times b=\underbrace {b+\cdots +b} _{a{\text{ times}}}.}</annotation>

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</math><img src="./fb333bd4d31edafab90c5762a3294101de5ed902.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:19.212ex; height:5.676ex;" alt="{\displaystyle a\times b=\underbrace {b+\cdots +b} _{a{\text{ times}}}.}" loading="lazy"></dd></dl>

Whether the first factor is the multiplier or the multiplicand may be ambiguous or depend upon context. For example, the expression <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times 4}">

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<annotation encoding="application/x-tex">{\displaystyle 3\times 4}</annotation>

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</math><img src="./7fda443e7a6e78fa880a6dccbf8bdf43a10d9988.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 3\times 4}" loading="lazy">, can be phrased as "3 times 4" and evaluated as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4+4+4}">

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<annotation encoding="application/x-tex">{\displaystyle 4+4+4}</annotation>

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</math><img src="./8a1b250399ee6c333a849320f712e15909839b74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.168ex; height:2.343ex;" alt="{\displaystyle 4+4+4}" loading="lazy">, where 3 is the multiplier, but also as "3 multiplied by 4", in which case 3 becomes the multiplicand.<a href="#cite_note-1">[1]</a> 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.<a href="#cite_note-Devlin-2">[2]</a><a href="#cite_note-3">[3]</a>

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<a href="Arithmetic_operations" class="mw-redirect" title="Arithmetic operations">Arithmetic operations</a></td></tr><tr><td class="sidebar-content" style="font-size:130%;">

<table class="infobox-subbox infobox-3cols-child infobox-table"><tbody><tr><th colspan="4" class="infobox-header"><a href="Addition" title="Addition">Addition</a> (+)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;">

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</mtd>

</mtr>

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<mtd>

<mtext>summand</mtext>

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<mtext>summand</mtext>

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</mtd>

</mtr>

<mtr>

<mtd>

<mtext>addend</mtext>

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<mtext>addend</mtext>

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</mtd>

</mtr>

<mtr>

<mtd>

<mtext>augend</mtext>

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<mtext>addend</mtext>

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<annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}</annotation>

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</math><img src="./ea99a27b5a763ef48889c450ac8157083ea97118.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:20.217ex; height:9.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}" loading="lazy"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;">

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<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{sum}}}</annotation>

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</math><img src="./b8609baca9fdbc4c529f5894884a08122d695dad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.931ex; height:1.343ex;" alt="{\displaystyle \scriptstyle {\text{sum}}}" loading="lazy"></td></tr><tr><th colspan="4" class="infobox-header"><a href="Subtraction" title="Subtraction">Subtraction</a> (−)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;">

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<mtr>

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</mtd>

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<mtr>

<mtd>

<mtext>minuend</mtext>

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<mtext>subtrahend</mtext>

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</mtd>

</mtr>

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<annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}</annotation>

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</math><img src="./2780b756445a5f8f95b16c33e3b924f976958ea0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.356ex; height:4.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}" loading="lazy"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;">

<mtext>difference</mtext>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{difference}}}</annotation>

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</math><img src="./1ac22c4e24eef2036cff5bfea924cc0dbb30c5d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.857ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {\text{difference}}}" loading="lazy"></td></tr><tr><th colspan="4" class="infobox-header"> (×)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;">

<mrow>

<mtr>

<mtd>

<mtext>factor</mtext>

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<mtext>factor</mtext>

</mrow>

</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

<mtext>multiplier</mtext>

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<mtext>multiplicand</mtext>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}</annotation>

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</math><img src="./93f7b476e32221c7b05d356289c8085aef54059b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.176ex; height:4.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}" loading="lazy"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;">

<mtext>product</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{product}}}</annotation>

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</math><img src="./a5c8b7509b8be1043622cb7b1b9a36ca8bfc2616.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.578ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {\text{product}}}" loading="lazy"></a></td></tr><tr><th colspan="4" class="infobox-header"><a href="Division_(mathematics)" title="Division (mathematics)">Division</a> (÷)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;">

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<mtr>

<mtd>

<mfrac>

<mtext>dividend</mtext>

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</mstyle>

<mtext>divisor</mtext>

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</mstyle>

</mfrac>

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</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mtext>numerator</mtext>

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</mstyle>

<mtext>denominator</mtext>

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</mstyle>

</mfrac>

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</mtr>

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</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}</annotation>

</semantics>

</math><img src="./5d5d22ff59234f0d437be740306e8dd905991e1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:14.15ex; height:8.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}" loading="lazy"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;">

<mrow>

<mtr>

<mtd>

<mtext>fraction</mtext>

</mrow>

</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

<mtext>quotient</mtext>

</mrow>

</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

<mtext>ratio</mtext>

</mrow>

</mstyle>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}</annotation>

</semantics>

</math><img src="./2359c3ca6e50e7ae8065baa710440b3c79895023.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:8.197ex; height:7.176ex;" alt="{\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}" loading="lazy"></a></td></tr><tr><th colspan="4" class="infobox-header"><a href="Exponentiation" title="Exponentiation">Exponentiation</a></th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;">

<mrow>

<mtr>

<mtd>

<msup>

</mrow>

<mtext>exponent</mtext>

</mrow>

</msup>

</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

<mtext>power</mtext>

</mrow>

</msup>

</mstyle>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}</annotation>

</semantics>

</math><img src="./ecb107371002b62a60fcbd13e742f4d81f872b67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.618ex; height:4.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}" loading="lazy"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;">

<mtext>power</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{power}}}</annotation>

</semantics>

</math><img src="./b0d0a9fbffb659c0055d5ee6fde3f7f28e96f45c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.297ex; height:1.509ex;" alt="{\displaystyle \scriptstyle {\text{power}}}" loading="lazy"></td></tr><tr><th colspan="4" class="infobox-header"><a href="Nth_root" title="Nth root">nth root</a> (√)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;">

<mroot>

<mtext>radicand</mtext>

</mrow>

</mstyle>

<mtext>degree</mtext>

</mrow>

</mroot>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}</annotation>

</semantics>

</math><img src="./5582d567e7e7fbcdb728291770905e09beb0ea18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.422ex; height:2.676ex;" alt="{\displaystyle \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}" loading="lazy"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{root}}}</annotation>

</semantics>

</math><img src="./2a015c1122190da3f1f1732d88b8bb03a8d7eb91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.928ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {\text{root}}}" loading="lazy"></td></tr><tr><th colspan="4" class="infobox-header"><a href="Logarithm" title="Logarithm">Logarithm</a> (log)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;">

<msub>

</mrow>

</msub>

<mtext>anti-logarithm</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}</annotation>

</semantics>

</math><img src="./2435266fcae4aa91d3d70a74bb91b5b35ef52edd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.454ex; height:2.176ex;" alt="{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}" loading="lazy"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;">

<mtext>logarithm</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{logarithm}}}</annotation>

</semantics>

</math><img src="./fe5d50baa86b950ff6d15760b7a38df1f8d8c868.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.948ex; height:2.009ex;" alt="{\displaystyle \scriptstyle {\text{logarithm}}}" loading="lazy"></td></tr></tbody></table></td>

</tr><tr><td class="sidebar-navbar"></td></tr></tbody></table>

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 <a href="Physical_quantities" class="mw-redirect" title="Physical quantities">physical quantities</a>) is a new type of measurement (or new quantity), usually with a derived <a href="Unit_of_measurement" title="Unit of measurement">unit of measurement</a>. 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 <a href="Dimensional_analysis" title="Dimensional analysis">dimensional analysis</a>.

The <a href="Inverse_operation" class="mw-redirect" title="Inverse operation">inverse operation</a> of multiplication is <a href="Division_(mathematics)" title="Division (mathematics)">division</a>. 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.

<div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Multiplication_sign" title="Multiplication sign">Multiplication sign</a></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Multiplier_(linguistics)" title="Multiplier (linguistics)">Multiplier (linguistics)</a></div>

In <a href="Arithmetic" title="Arithmetic">arithmetic</a>, multiplication is often written using the <a href="Multiplication_sign" title="Multiplication sign">multiplication sign</a> (either × or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \times }</annotation>

</semantics>

</math><img src="./0ffafff1ad26cbe49045f19a67ce532116a32703.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }" loading="lazy">) between the factors (that is, in <a href="Infix_notation" title="Infix notation">infix notation</a>).<a href="#cite_note-mpb-4">[4]</a> For example,

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\times 3=6,}</annotation>

</semantics>

</math><img src="./5a91f8ab41b52a86b48b8d350ba7b3fae3586260.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.073ex; height:2.509ex;" alt="{\displaystyle 2\times 3=6,}" loading="lazy"> ("two times three <a href="Equals_sign" title="Equals sign">equals</a> six")</dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3\times 4=12,}</annotation>

</semantics>

</math><img src="./6aab6cfecb8789154929bbd74562bc63595b6159.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.236ex; height:2.509ex;" alt="{\displaystyle 3\times 4=12,}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\times 3\times 5=6\times 5=30,}</annotation>

</semantics>

</math><img src="./eef7661693aaea5975725ccf780f95c3399917e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.502ex; height:2.509ex;" alt="{\displaystyle 2\times 3\times 5=6\times 5=30,}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\times 2\times 2\times 2\times 2=32.}</annotation>

</semantics>

</math><img src="./77328d358be847c985842557eef888daec60e454.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:23.244ex; height:2.176ex;" alt="{\displaystyle 2\times 2\times 2\times 2\times 2=32.}" loading="lazy"></dd></dl>

There are other <a href="Mathematical_notation" title="Mathematical notation">mathematical notations</a> for multiplication:

<ul><li>To reduce confusion between the multiplication sign × and the common variable x, multiplication is also denoted by dot signs, usually a middle-position dot (rarely <a href="Full_stop" title="Full stop">period</a>): <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\cdot 2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 5\cdot 2}</annotation>

</semantics>

</math><img src="./494e592f1901abf068780eddb93af3c52a021c5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.004ex; height:2.176ex;" alt="{\displaystyle 5\cdot 2}" loading="lazy">.<a href="#cite_note-mpb-4">[4]</a> The middle dot notation or dot operator is now standard in the United States<a href="#cite_note-mpb-4">[4]</a><a href="#cite_note-5">[5]</a> and other countries.<a href="#cite_note-humez-6">[6]</a> When the dot operator character is not accessible, the <a href="Interpunct" title="Interpunct">interpunct</a> (·) is used.<a href="#cite_note-humez-6">[6]</a> In most European and other countries that use a <a href="Comma_(punctuation)" class="mw-redirect" title="Comma (punctuation)">comma</a> as a <a href="Decimal_point" class="mw-redirect" title="Decimal point">decimal point</a> (and a period as a <a href="Thousands_separator" class="mw-redirect" title="Thousands separator">thousands separator</a>), the multiplication sign or a middle dot is used to indicate multiplication. Historically, in the United Kingdom and Ireland, the middle dot was sometimes used for the decimal point to prevent it from disappearing in the ruled line, and the full stop (period) was used for multiplication. However, since the <a href="Ministry_of_Technology" title="Ministry of Technology">Ministry of Technology</a> ruled in 1968 that the period be used as the decimal point,<a href="#cite_note-7">[7]</a> and the <a href="International_System_of_Units" title="International System of Units">International System of Units</a> (SI) standard has since been widely adopted, this usage is now found only in the more traditional journals such as <a href="The_Lancet" title="The Lancet">The Lancet</a>.<a href="#cite_note-8">[8]</a></li>

<li>In <a href="Algebra" title="Algebra">algebra</a>, multiplication involving <a href="Variable_(mathematics)" title="Variable (mathematics)">variables</a> is often written as a <a href="Juxtaposition#Mathematics" title="Juxtaposition">juxtaposition</a> (e.g., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle xy}</annotation>

</semantics>

</math><img src="./c72eb345e496513fb8b2fa4aa8c4d89b855f9a01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> times <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 5x}</annotation>

</semantics>

</math><img src="./5adce1d349cd87969996dd19d6f1c75a30926372.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.492ex; height:2.176ex;" alt="{\displaystyle 5x}" loading="lazy"> for five times <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">), also called implied multiplication. The notation can also be used for quantities that are surrounded by <a href="Parentheses" class="mw-redirect" title="Parentheses">parentheses</a> (e.g., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5(2)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 5(2)}</annotation>

</semantics>

</math><img src="./2a7ec534cae3d03d91782f05db0228f8fa7e1fb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle 5(2)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (5)2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (5)2}</annotation>

</semantics>

</math><img src="./e28154db4fc89a6d84dfb9d6fb2d15ddede538de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle (5)2}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (5)(2)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (5)(2)}</annotation>

</semantics>

</math><img src="./c2a9191aab090aa424333c85aa878258d675f945.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.944ex; height:2.843ex;" alt="{\displaystyle (5)(2)}" loading="lazy"> for five times two). <a href="#cite_note-9">[9]</a>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 <a href="Order_of_operations" title="Order of operations">order of operations</a>.<a href="#cite_note-Peterson_2019-10">[10]</a><a href="#cite_note-Peterson_2023-11">[11]</a></li>

<li>In <a href="Vector_multiplication" title="Vector multiplication">vector multiplication</a>, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a <a href="Cross_product" title="Cross product">cross product</a> of two <a href="Vector_(mathematics)" class="mw-redirect" title="Vector (mathematics)">vectors</a>, yielding a vector as its result, while the dot denotes taking the <a href="Dot_product" title="Dot product">dot product</a> of two vectors, resulting in a <a href="Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a>.</li></ul>

In <a href="Computer_programming" title="Computer programming">computer programming</a>, the <a href="Asterisk" title="Asterisk">asterisk</a> (as in <code>5*2</code>) is still the most common notation. This is because most computers historically were limited to small <a href="Character_set" class="mw-redirect" title="Character set">character sets</a> (such as <a href="ASCII" title="ASCII">ASCII</a> and <a href="EBCDIC" title="EBCDIC">EBCDIC</a>) that lacked a multiplication sign (such as <code>⋅</code> or <code>×</code>), while the asterisk appeared on every keyboard.<a href="#cite_note-12">[12]</a> This usage originated in the <a href="Fortran" title="Fortran">FORTRAN</a> programming language.<a href="#cite_note-fortran-13">[13]</a>

The numbers to be multiplied are generally called the "factors" (as in <a href="Factorization" title="Factorization">factorization</a>). 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;<a href="#cite_note-multiplicand_on_Britannica-14">[14]</a><a href="#cite_note-multiplicand_via_Wolfram_Mathworld-15">[15]</a> 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 <a href="Multiplication_algorithm" title="Multiplication algorithm">multiplication algorithms</a>, such as the <a href="Long_multiplication" class="mw-redirect" title="Long multiplication">long multiplication</a>. Therefore, in some sources, the term "multiplicand" is regarded as a synonym for "factor".<a href="#cite_note-16">[16]</a>

In algebra, a number that is the multiplier of a variable or expression (e.g., the 3 in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3xy^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3xy^{2}}</annotation>

</semantics>

</math><img src="./2d78903022b95f5d3dc883053c53461a9d61d969.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.707ex; height:3.009ex;" alt="{\displaystyle 3xy^{2}}" loading="lazy">) is called a <a href="Coefficient" title="Coefficient">coefficient</a>.

The result of a multiplication is called a <a href="Product_(mathematics)" title="Product (mathematics)">product</a>. When one factor is an integer, the product is a <a href="Multiple_(mathematics)" title="Multiple (mathematics)">multiple</a> of the other or of the product of the others. Thus, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times \pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\times \pi }</annotation>

</semantics>

</math><img src="./9dcd6d0dfb48b0a89f47825221da1843c87cdcb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.335ex; height:2.176ex;" alt="{\displaystyle 2\times \pi }" loading="lazy"> is a multiple of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

</semantics>

</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy">, as is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5133\times 486\times \pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 5133\times 486\times \pi }</annotation>

</semantics>

</math><img src="./bb435b96d4ca46315f757ed9a7d2f42767941ebe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.15ex; height:2.176ex;" alt="{\displaystyle 5133\times 486\times \pi }" loading="lazy">. 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.

<div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2></div>

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.

<div class="mw-heading mw-heading3"><h3 id="Product_of_two_natural_numbers">Product of two natural numbers</h3></div>

The product of two natural numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r,s\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r,s\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./aa499cab93f2b4a9532a856ee701c75c8e59a870.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.692ex; height:2.509ex;" alt="{\displaystyle r,s\in \mathbb {N} }" loading="lazy"> is defined as:

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

</munder>

</mrow>

<mtext> times</mtext>

</mrow>

</munder>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

</munder>

</mrow>

<mtext> times</mtext>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 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}}}.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Product_of_two_integers">Product of two integers</h3></div>

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 <a href="Absolute_value" title="Absolute value">positive amounts</a>, combined with the sign derived from the following rule:

<th style="padding:0.2em 1em;">×

</th>

<th style="padding:0.2em 1em;">+

</th>

<th style="padding:0.2em 1em;">−

</th></tr>

<tr>

<th style="padding:0.2em 1em;">+

</th>

<td>−

</td></tr>

<tr>

<th style="padding:0.2em 1em;">−

</th>

<td>+

</td></tr></tbody></table>

(This rule is a consequence of the <a href="Distributivity" class="mw-redirect" title="Distributivity">distributivity</a> of multiplication over addition, and is not an additional rule.)

In words:

<ul><li>A positive number multiplied by a positive number is positive (product of natural numbers),</li>

<li>A positive number multiplied by a negative number is negative,</li>

<li>A negative number multiplied by a positive number is negative,</li>

<li>A negative number multiplied by a negative number is positive.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Product_of_two_fractions">Product of two fractions</h3></div>

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

<mfrac>

</mfrac>

</mrow>

<mfrac>

<msup>

</msup>

<msup>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</msup>

</mrow>

<mrow>

<msup>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {z}{n}}\cdot {\frac {z'}{n'}}={\frac {z\cdot z'}{n\cdot n'}},}</annotation>

</semantics>

</math></dd>

<dd>which is defined when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,n'\neq 0}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n,n'\neq 0}</annotation>

</semantics>

</math><img src="./817c78364f24a013fb14ae19e432bd0a46667e88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.769ex; height:3.009ex;" alt="{\displaystyle n,n'\neq 0}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Product_of_two_real_numbers">Product of two real numbers</h3></div>

There are several equivalent ways to define formally the real numbers; see <a href="Construction_of_the_real_numbers" title="Construction of the real numbers">Construction of the real numbers</a>. 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 <a href="Rational_number" title="Rational number">rational numbers</a>. A standard way for expressing this is that every real number is the <a href="Least_upper_bound" class="mw-redirect" title="Least upper bound">least upper bound</a> of a set of rational numbers. In particular, every positive real number is the least upper bound of the <a href="Truncation" title="Truncation">truncations</a> of its infinite <a href="Decimal_representation" title="Decimal representation">decimal representation</a>; for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

</semantics>

</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> is the least upper bound of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{3,\;3.1,\;3.14,\;3.141,\ldots \}.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{3,\;3.1,\;3.14,\;3.141,\ldots \}.}</annotation>

</semantics>

</math><img src="./a4943b7b770c34dcef0b9a8ab8eb0e3d20078adc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.331ex; height:2.843ex;" alt="{\displaystyle \{3,\;3.1,\;3.14,\;3.141,\ldots \}.}" loading="lazy">

A fundamental property of real numbers is that rational approximations are compatible with <a href="Arithmetic_operation" class="mw-redirect" title="Arithmetic operation">arithmetic operations</a>, and, in particular, with multiplication. This means that, if a and b are positive real numbers such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=\sup _{x\in A}x}">

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=\sup _{x\in A}x}</annotation>

</semantics>

</math><img src="./fd5fb5e1bb0804b8708ecbb89d9138eaac561c94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.546ex; height:3.843ex;" alt="{\displaystyle a=\sup _{x\in A}x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=\sup _{y\in B}y,}">

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=\sup _{y\in B}y,}</annotation>

</semantics>

</math><img src="./5e9e7f38c026d918b9d6f627326f5e219f3f5c35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:9.787ex; height:4.676ex;" alt="{\displaystyle b=\sup _{y\in B}y,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b=\sup _{x\in A,y\in B}x\cdot y.}">

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\cdot b=\sup _{x\in A,y\in B}x\cdot y.}</annotation>

</semantics>

</math><img src="./eaebfd1c11652605dd39029bc7bd76a9c74ef2bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.09ex; height:4.676ex;" alt="{\displaystyle a\cdot b=\sup _{x\in A,y\in B}x\cdot y.}" loading="lazy"> In particular, the product of two positive real numbers is the least upper bound of the term-by-term products of the <a href="Sequence" title="Sequence">sequences</a> 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 <a href="#Product_of_two_integers">§ Product of two integers</a>. The construction of the real numbers through <a href="Cauchy_sequence" title="Cauchy sequence">Cauchy sequences</a> is often preferred in order to avoid consideration of the four possible sign configurations.

<div class="mw-heading mw-heading3"><h3 id="Product_of_two_complex_numbers">Product of two complex numbers</h3></div>

Two complex numbers can be multiplied by the distributive law and the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{2}=-1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i^{2}=-1}</annotation>

</semantics>

</math><img src="./88e98a401d352e5037d5043028e2d7f449e83fa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.926ex; height:2.843ex;" alt="{\displaystyle i^{2}=-1}" loading="lazy">, as follows:

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(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{aligned}}}</annotation>

</semantics>

</math><img src="./4a13d5e1f0ba272ad74c3981769e9db0d07ae052.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.71ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}(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{aligned}}}" loading="lazy"></dd></dl>

The geometric meaning of complex multiplication can be understood by rewriting complex numbers in <a href="Polar_coordinates" class="mw-redirect" title="Polar coordinates">polar coordinates</a>:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a+b\,i=r\cdot (\cos(\varphi )+i\sin(\varphi ))=r\cdot e^{i\varphi }}</annotation>

</semantics>

</math><img src="./9c901ed651f73bb284b601eb3a3f6acf59366b80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.333ex; height:3.176ex;" alt="{\displaystyle a+b\,i=r\cdot (\cos(\varphi )+i\sin(\varphi ))=r\cdot e^{i\varphi }}" loading="lazy"></dd></dl>

Furthermore,

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c+d\,i=s\cdot (\cos(\psi )+i\sin(\psi ))=s\cdot e^{i\psi },}</annotation>

</semantics>

</math><img src="./d21ffe9c6e3bbfb2e2f3168c58bccdec627e9b8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.04ex; height:3.176ex;" alt="{\displaystyle c+d\,i=s\cdot (\cos(\psi )+i\sin(\psi ))=s\cdot e^{i\psi },}" loading="lazy"></dd></dl>

from which one obtains

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a\cdot c-b\cdot d)+(a\cdot d+b\cdot c)i=r\cdot s\cdot e^{i(\varphi +\psi )}.}</annotation>

</semantics>

</math><img src="./6fb11ff643384aacc5f1cbc59b8d047cbfb00884.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.387ex; height:3.343ex;" alt="{\displaystyle (a\cdot c-b\cdot d)+(a\cdot d+b\cdot c)i=r\cdot s\cdot e^{i(\varphi +\psi )}.}" loading="lazy"></dd></dl>

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

<div class="mw-heading mw-heading3"><h3 id="Product_of_two_quaternions">Product of two quaternions</h3></div>

The product of two <a href="Quaternion" title="Quaternion">quaternions</a> can be found in the article on <a href="Quaternions" class="mw-redirect" title="Quaternions">quaternions</a>. Note, in this case, that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\cdot b}</annotation>

</semantics>

</math><img src="./620419d3ed53abc98659a5fc0f3a5eb6177830ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.906ex; height:2.176ex;" alt="{\displaystyle a\cdot b}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\cdot a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b\cdot a}</annotation>

</semantics>

</math><img src="./86cfc2b64ea5a17c1fe61677e219b10446590c20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.906ex; height:2.176ex;" alt="{\displaystyle b\cdot a}" loading="lazy"> are in general different.

<div class="mw-heading mw-heading2"><h2 id="Computation">Computation</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Multiplication_algorithm" title="Multiplication algorithm">Multiplication algorithm</a></div>

Many common methods for multiplying numbers using pencil and paper require a <a href="Multiplication_table" title="Multiplication table">multiplication table</a> of memorized or consulted products of small numbers (typically any two numbers from 0 to 9). However, one method, the <a href="Ancient_Egyptian_multiplication" title="Ancient Egyptian multiplication">peasant multiplication</a> algorithm, does not. The example below illustrates "long multiplication" (the "standard algorithm", "grade-school multiplication"):

<pre> 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 )

</pre>

In some countries such as <a href="Germany" title="Germany">Germany</a>, 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:<a href="#cite_note-17">[17]</a>

<pre>23958233 · 5830

———————————————

119791165

191665864

71874699

00000000

———————————————

139676498390

</pre>

Multiplying numbers to more than a couple of decimal places by hand is tedious and error-prone. <a href="Common_logarithm" title="Common logarithm">Common logarithms</a> were invented to simplify such calculations, since adding logarithms is equivalent to multiplying. The <a href="Slide_rule" title="Slide rule">slide rule</a> allowed numbers to be quickly multiplied to about three places of accuracy. Beginning in the early 20th century, mechanical <a href="Calculator" title="Calculator">calculators</a>, such as the <a href="Marchant_Calculator" class="mw-redirect" title="Marchant Calculator">Marchant</a>, automated multiplication of up to 10-digit numbers. Modern electronic <a href="Computer" title="Computer">computers</a> and calculators have greatly reduced the need for multiplication by hand.

<div class="mw-heading mw-heading3"><h3 id="Historical_algorithms">Historical algorithms</h3></div>

Methods of multiplication were documented in the writings of <a href="Ancient_Egypt" title="Ancient Egypt">ancient Egyptian</a>, Greek, Indian, and <a href="History_of_China#Ancient_China" title="History of China">Chinese</a> civilizations.

The <a href="Ishango_bone" title="Ishango bone">Ishango bone</a>, dated to about 18,000 to 20,000 BC, may hint at a knowledge of multiplication in the <a href="Upper_Paleolithic" title="Upper Paleolithic">Upper Paleolithic</a> era in <a href="Central_Africa" title="Central Africa">Central Africa</a>, but this is speculative.<a href="#cite_note-18">[18]</a>

<div class="mw-heading mw-heading4"><h4 id="Egyptians">Egyptians</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Ancient_Egyptian_multiplication" title="Ancient Egyptian multiplication">Ancient Egyptian multiplication</a></div>

The Egyptian method of multiplication of integers and fractions, which is documented in the <a href="Rhind_Mathematical_Papyrus" title="Rhind Mathematical Papyrus">Rhind Mathematical Papyrus</a>, was by successive additions and doubling. For instance, to find the product of 13 and 21 one had to double 21 three times, obtaining 2 × 21 = 42, 4 × 21 = 2 × 42 = 84, 8 × 21 = 2 × 84 = 168. The full product could then be found by adding the appropriate terms found in the doubling sequence:<a href="#cite_note-19">[19]</a>

<div class="mw-heading mw-heading4"><h4 id="Babylonians">Babylonians</h4></div>

The <a href="Babylonians" class="mw-redirect" title="Babylonians">Babylonians</a> used a <a href="Sexagesimal" title="Sexagesimal">sexagesimal</a> <a href="Positional_number_system" class="mw-redirect" title="Positional number system">positional number system</a>, analogous to the modern-day <a href="Decimal_expansion" class="mw-redirect" title="Decimal expansion">decimal system</a>. Thus, Babylonian multiplication was very similar to modern decimal multiplication. Because of the relative difficulty of remembering 60 × 60 different products, Babylonian mathematicians employed <a href="Multiplication_table" title="Multiplication table">multiplication tables</a>. 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.

<div class="mw-heading mw-heading4"><h4 id="Chinese">Chinese</h4></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Chinese_multiplication_table" title="Chinese multiplication table">Chinese multiplication table</a></div>

In the mathematical text <a href="Zhoubi_Suanjing" title="Zhoubi Suanjing">Zhoubi Suanjing</a>, dated prior to 300 BC, and the <a href="Nine_Chapters_on_the_Mathematical_Art" class="mw-redirect" title="Nine Chapters on the Mathematical Art">Nine Chapters on the Mathematical Art</a>, multiplication calculations were written out in words, although the early Chinese mathematicians employed <a href="Rod_calculus" title="Rod calculus">Rod calculus</a> involving place value addition, subtraction, multiplication, and division. The Chinese were already using a <a href="Chinese_multiplication_table" title="Chinese multiplication table">decimal multiplication table</a> by the end of the <a href="Warring_States" class="mw-redirect" title="Warring States">Warring States</a> period.<a href="#cite_note-Nature-20">[20]</a>

<div class="mw-heading mw-heading3"><h3 id="Modern_methods">Modern methods</h3></div>

The modern method of multiplication based on the <a href="Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a> was first described by <a href="Brahmagupta" title="Brahmagupta">Brahmagupta</a>. Brahmagupta gave rules for addition, subtraction, multiplication, and division. <a href="Henry_Burchard_Fine" title="Henry Burchard Fine">Henry Burchard Fine</a>, then a professor of mathematics at <a href="Princeton_University" title="Princeton University">Princeton University</a>, wrote the following:

<dl><dd>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.<a href="#cite_note-21">[21]</a></dd></dl>

These place value decimal arithmetic algorithms were introduced to Arab countries by <a href="Al_Khwarizmi" class="mw-redirect" title="Al Khwarizmi">Al Khwarizmi</a> in the early 9th century and popularized in the Western world by <a href="Fibonacci" title="Fibonacci">Fibonacci</a> in the 13th century.<a href="#cite_note-22">[22]</a>

<div class="mw-heading mw-heading4"><h4 id="Grid_method">Grid method</h4></div>

<a href="Grid_method_multiplication" title="Grid method multiplication">Grid method multiplication</a>, or the box method, is used in primary schools in England and Wales and in some areas 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:

<th scope="col">×

</th>

<th scope="col">30

</th>

<th scope="col">4

</th></tr>

<tr>

<th scope="row">10

</th>

<td>300

</td>

<td>40

</td></tr>

<tr>

<th scope="row">3

</th>

<td>90

</td>

<td>12

</td></tr></tbody></table></dd></dl>

and then add the entries.

<div class="mw-heading mw-heading3"><h3 id="Computer_algorithms">Computer algorithms</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Multiplication_algorithm#Fast_multiplication_algorithms_for_large_inputs" title="Multiplication algorithm">Multiplication algorithm § Fast multiplication algorithms for large inputs</a></div>

The classical method of multiplying two n-digit numbers requires n2 digit multiplications. <a href="Multiplication_algorithm" title="Multiplication algorithm">Multiplication algorithms</a> have been designed that reduce the computation time considerably when multiplying large numbers. Methods based on the <a href="Discrete_Fourier_transform#Multiplication_of_large_integers" title="Discrete Fourier transform">discrete Fourier transform</a> reduce the <a href="Computational_complexity" title="Computational complexity">computational complexity</a> to O(n log n log log n). In 2016, the factor log log n was replaced by a function that increases much slower, though still not constant.<a href="#cite_note-23">[23]</a> In March 2019, David Harvey and Joris van der Hoeven submitted a paper presenting an integer multiplication algorithm with a complexity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n\log n).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(n\log n).}</annotation>

</semantics>

</math><img src="./d33777001ea38436a3f2b44b2c1fa99268a339c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.765ex; height:2.843ex;" alt="{\displaystyle O(n\log n).}" loading="lazy"><a href="#cite_note-24">[24]</a> The algorithm, also based on the fast Fourier transform, is conjectured to be asymptotically optimal.<a href="#cite_note-25">[25]</a> The algorithm is not practically useful, as it only becomes faster for multiplying extremely large numbers (having more than 2172912 bits).<a href="#cite_note-26">[26]</a>

<div class="mw-heading mw-heading2"><h2 id="Products_of_measurements">Products of measurements</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Dimensional_analysis" title="Dimensional analysis">Dimensional analysis</a></div>

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:<a href="#cite_note-Devlin-2">[2]</a>

<dl><dd>[4 bags] × [3 marbles per bag] = 12 marbles.</dd></dl>

When two measurements are multiplied together, the product is of a type depending on the types of measurements. The general theory is given by <a href="Dimensional_analysis" title="Dimensional analysis">dimensional analysis</a>. 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 <a href="Speed" title="Speed">speed</a> by <a href="Time_in_physics" title="Time in physics">time</a> gives <a href="Distance" title="Distance">distance</a>. For example:

<dl><dd>50 kilometers per hour × 3 hours = 150 kilometers.</dd></dl>

In this case, the hour units cancel out, leaving the product with only kilometer units.

Other examples of multiplication involving units include:

<dl><dd>2.5 meters × 4.5 meters = 11.25 square meters</dd>

<dd>11 meters/seconds × 9 seconds = 99 meters</dd>

<dd>4.5 residents per house × 20 houses = 90 residents</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Product_of_a_sequence">Product of a sequence</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Capital_pi_notation">Capital pi notation</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Iterated_binary_operation#Notation" title="Iterated binary operation">Iterated binary operation § Notation</a></div>

The product of a sequence of factors can be written with the product symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \prod }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \prod }</annotation>

</semantics>

</math><img src="./423a3226e80f549c55e3873aecbf57af9296e0fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.194ex; height:2.843ex;" alt="{\displaystyle \textstyle \prod }" loading="lazy">, which derives from the capital letter Π (pi) in the <a href="Greek_alphabet" title="Greek alphabet">Greek alphabet</a> (much like the same way the <a href="Summation_symbol" class="mw-redirect" title="Summation symbol">summation symbol</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \sum }</annotation>

</semantics>

</math><img src="./92795a77657ae4e5746d1e5d8aa40151e176e723.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.454ex; height:2.843ex;" alt="{\displaystyle \textstyle \sum }" loading="lazy"> is derived from the Greek letter Σ (sigma)).<a href="#cite_note-27">[27]</a><a href="#cite_note-28">[28]</a> The meaning of this notation is given by

</mrow>

</mrow>

</munderover>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \prod _{i=1}^{4}(i+1)=(1+1)\,(2+1)\,(3+1)\,(4+1),}</annotation>

</semantics>

</math><img src="./1fd1bb0514745c6d1b398dcff281b03eb6841ca7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.389ex; height:7.343ex;" alt="{\displaystyle \prod _{i=1}^{4}(i+1)=(1+1)\,(2+1)\,(3+1)\,(4+1),}" loading="lazy"></dd></dl>

which results in

</mrow>

</mrow>

</munderover>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \prod _{i=1}^{4}(i+1)=120.}</annotation>

</semantics>

</math><img src="./88a3dfc66c3ab59bbb6c8d1b7b6d70f8072e5d90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.817ex; height:7.343ex;" alt="{\displaystyle \prod _{i=1}^{4}(i+1)=120.}" loading="lazy"></dd></dl>

In such a notation, the <a href="Variable_(mathematics)" title="Variable (mathematics)">variable</a> i represents a varying <a href="Integer" title="Integer">integer</a>, called the multiplication index, that runs from the lower value 1 indicated in the subscript to the upper value 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

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle \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},}</annotation>

</semantics>

</math><img src="./cd6a5719f5cda293cd79836a3c90f59e7447557e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.63ex; height:6.843ex;" alt="{\displaystyle \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},}" loading="lazy"></dd></dl>

where m and n are integers or expressions that evaluate to integers. In the case where m = n, the value of the product is the same as that of the single factor xm; if m > n, the product is an <a href="Empty_product" title="Empty product">empty product</a> whose value is 1—regardless of the expression for the factors.

<div class="mw-heading mw-heading4"><h4 id="Properties_of_capital_pi_notation">Properties of capital pi notation</h4></div>

By definition,

</mrow>

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</munderover>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \prod _{i=1}^{n}x_{i}=x_{1}\cdot x_{2}\cdot \ldots \cdot x_{n}.}</annotation>

</semantics>

</math><img src="./30a3a2eef2f68451d914a6ea33cdb641516ea124.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.308ex; height:6.843ex;" alt="{\displaystyle \prod _{i=1}^{n}x_{i}=x_{1}\cdot x_{2}\cdot \ldots \cdot x_{n}.}" loading="lazy"></dd></dl>

If all factors are identical, a product of n factors is equivalent to <a href="Exponentiation" title="Exponentiation">exponentiation</a>:

</mrow>

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</munderover>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \prod _{i=1}^{n}x=x\cdot x\cdot \ldots \cdot x=x^{n}.}</annotation>

</semantics>

</math><img src="./7bc5f2a7154bbff0562cc8f7d1a317caa9fb56ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.828ex; height:6.843ex;" alt="{\displaystyle \prod _{i=1}^{n}x=x\cdot x\cdot \ldots \cdot x=x^{n}.}" loading="lazy"></dd></dl>

<a href="Associativity" class="mw-redirect" title="Associativity">Associativity</a> and <a href="Commutativity" class="mw-redirect" title="Commutativity">commutativity</a> of multiplication imply

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \prod _{i=1}^{n}{x_{i}y_{i}}=\left(\prod _{i=1}^{n}x_{i}\right)\left(\prod _{i=1}^{n}y_{i}\right)}</annotation>

</semantics>

</math><img src="./42afdc3f58ec71d5bbea42d733d5fb3e9405320c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.054ex; height:7.509ex;" alt="{\displaystyle \prod _{i=1}^{n}{x_{i}y_{i}}=\left(\prod _{i=1}^{n}x_{i}\right)\left(\prod _{i=1}^{n}y_{i}\right)}" loading="lazy"> and</dd>

<msup>

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<mrow>

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</munderover>

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</msubsup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(\prod _{i=1}^{n}x_{i}\right)^{a}=\prod _{i=1}^{n}x_{i}^{a}}</annotation>

</semantics>

</math><img src="./b8b345e62727e9f991a1f6214cfc175e90e66620.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.156ex; height:7.509ex;" alt="{\displaystyle \left(\prod _{i=1}^{n}x_{i}\right)^{a}=\prod _{i=1}^{n}x_{i}^{a}}" loading="lazy"></dd></dl>

if a is a non-negative integer, or if all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> are positive <a href="Real_number" title="Real number">real numbers</a>, and

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<annotation encoding="application/x-tex">{\displaystyle \prod _{i=1}^{n}x^{a_{i}}=x^{\sum _{i=1}^{n}a_{i}}}</annotation>

</semantics>

</math><img src="./5f053c2374cb00e7144cab0ed3a67eedacfbdb26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.02ex; height:6.843ex;" alt="{\displaystyle \prod _{i=1}^{n}x^{a_{i}}=x^{\sum _{i=1}^{n}a_{i}}}" loading="lazy"></dd></dl>

if all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation>

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</math><img src="./0bc77764b2e74e64a63341054fa90f3e07db275f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}" loading="lazy"> are non-negative integers, or if x is a positive real number.

<div class="mw-heading mw-heading3"><h3 id="Infinite_products">Infinite products</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Infinite_product" title="Infinite product">Infinite product</a></div>

One may also consider products of infinitely many factors; these are called <a href="Infinite_product" title="Infinite product">infinite products</a>. Notationally, this consists in replacing n above by the <a href="Infinity_symbol" title="Infinity symbol">infinity symbol</a> ∞. The product of such an infinite sequence is defined as the <a href="Limit_of_a_sequence" title="Limit of a sequence">limit</a> of the product of the first n factors, as n grows without bound. That is,

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</munder>

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<annotation encoding="application/x-tex">{\displaystyle \prod _{i=m}^{\infty }x_{i}=\lim _{n\to \infty }\prod _{i=m}^{n}x_{i}.}</annotation>

</semantics>

</math><img src="./54769034d506aa9194ee35c7175118c3c359f92d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.015ex; height:6.843ex;" alt="{\displaystyle \prod _{i=m}^{\infty }x_{i}=\lim _{n\to \infty }\prod _{i=m}^{n}x_{i}.}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle \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),}</annotation>

</semantics>

</math><img src="./6c9de50a83cd6f6a8b1396ef59c2555631584fe9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.804ex; height:7.509ex;" alt="{\displaystyle \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),}" loading="lazy"></dd></dl>

provided both limits exist.

<div class="mw-heading mw-heading2"><h2 id="Exponentiation">Exponentiation</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Exponentiation" title="Exponentiation">Exponentiation</a></div>

When multiplication is repeated, the resulting operation is known as <a href="Exponentiation" title="Exponentiation">exponentiation</a>. 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 <a href="Superscript" class="mw-redirect" title="Superscript">superscript</a> three. In this example, the number two is the base, and three is the exponent.<a href="#cite_note-29">[29]</a> In general, the exponent (or superscript) indicates how many times the base appears in the expression, so that the expression

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<annotation encoding="application/x-tex">{\displaystyle a^{n}=\underbrace {a\times a\times \cdots \times a} _{n}=\prod _{i=1}^{n}a}</annotation>

</semantics>

</math><img src="./8f173e636557f9f2eb2f43a25442e68f5dda1884.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:28.165ex; height:7.509ex;" alt="{\displaystyle a^{n}=\underbrace {a\times a\times \cdots \times a} _{n}=\prod _{i=1}^{n}a}" loading="lazy"></dd></dl>

indicates that n copies of the base a are to be multiplied together. This notation can be used whenever multiplication is known to be <a href="Power_associativity" title="Power associativity">power associative</a>.

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

For <a href="Real_number" title="Real number">real</a> and <a href="Complex_number" title="Complex number">complex</a> numbers, which includes, for example, <a href="Natural_number" title="Natural number">natural numbers</a>, <a href="Integer" title="Integer">integers</a>, and <a href="Rational_number" title="Rational number">fractions</a>, multiplication has certain properties:

<dl><dt><a href="Commutative_property" title="Commutative property">Commutative property</a></dt>

<dd>The order in which two numbers are multiplied does not matter:<a href="#cite_note-:0-30">[30]</a><a href="#cite_note-:1-31">[31]</a>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x\cdot y=y\cdot x.}</annotation>

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</math><img src="./f21580dd12734adc7e6eee7e7b31d0f9f3f9dcc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.074ex; height:2.009ex;" alt="{\displaystyle x\cdot y=y\cdot x.}" loading="lazy"></dd></dl></dd></dl>

<dl><dt><a href="Associative_property" title="Associative property">Associative property</a></dt>

<dd>Expressions solely involving multiplication or addition are invariant with respect to the <a href="Order_of_operations" title="Order of operations">order of operations</a>:<a href="#cite_note-:0-30">[30]</a><a href="#cite_note-:1-31">[31]</a>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (x\cdot y)\cdot z=x\cdot (y\cdot z).}</annotation>

</semantics>

</math><img src="./86e9f4a061a2bd043c888316d8a7507fd65c0b79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.227ex; height:2.843ex;" alt="{\displaystyle (x\cdot y)\cdot z=x\cdot (y\cdot z).}" loading="lazy"></dd></dl></dd></dl>

<dl><dt><a href="Distributive_property" title="Distributive property">Distributive property</a></dt>

<dd>Holds with respect to multiplication over addition. This identity is of prime importance in simplifying algebraic expressions:<a href="#cite_note-:0-30">[30]</a><a href="#cite_note-:1-31">[31]</a>

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<annotation encoding="application/x-tex">{\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z.}</annotation>

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</math><img src="./aa8d1f2ca9f31fc5ab031af8800412fb06c97a6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.749ex; height:2.843ex;" alt="{\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z.}" loading="lazy"></dd></dl></dd></dl>

<dl><dt><a href="Identity_element" title="Identity element">Identity element</a></dt>

<dd>The multiplicative identity is 1; anything multiplied by 1 is itself. This feature of 1 is known as the identity property:<a href="#cite_note-:0-30">[30]</a><a href="#cite_note-:1-31">[31]</a>

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<annotation encoding="application/x-tex">{\displaystyle x\cdot 1=x.}</annotation>

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</math><img src="./adbae0b8dd959593c8a554f7c0ae8ef44dd16a29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.246ex; height:2.176ex;" alt="{\displaystyle x\cdot 1=x.}" loading="lazy"></dd></dl></dd></dl>

<dl><dt><a href="Absorbing_element" title="Absorbing element">Property of 0</a></dt>

<dd>Any number multiplied by 0 is 0. This is known as the zero property of multiplication:<a href="#cite_note-:0-30">[30]</a>

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<annotation encoding="application/x-tex">{\displaystyle x\cdot 0=0.}</annotation>

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</math><img src="./ad22b9680e3200627c854a842be78fa346f6b672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.079ex; height:2.176ex;" alt="{\displaystyle x\cdot 0=0.}" loading="lazy"></dd></dl></dd></dl>

<dl><dt><a href="Additive_inverse" title="Additive inverse">Negation</a></dt>

<dd>−1 times any number is equal to the <a href="Additive_inverse" title="Additive inverse">additive inverse</a> of that number:

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<annotation encoding="application/x-tex">{\displaystyle (-1)\cdot x=(-x)}</annotation>

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</math><img src="./248d74c79bb9e2e4ed765acb885d7f2d18294711.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.834ex; height:2.843ex;" alt="{\displaystyle (-1)\cdot x=(-x)}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-x)+x=0.}">

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<annotation encoding="application/x-tex">{\displaystyle (-x)+x=0.}</annotation>

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</math><img src="./fcf3de93528dfca77af9f4fb0ad39d40b661f6b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.025ex; height:2.843ex;" alt="{\displaystyle (-x)+x=0.}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>−1 times −1 is 1:

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<annotation encoding="application/x-tex">{\displaystyle (-1)\cdot (-1)=1.}</annotation>

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</math><img src="./ec8a91eaba3b81f6f8af33d6036c622abb922322.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.147ex; height:2.843ex;" alt="{\displaystyle (-1)\cdot (-1)=1.}" loading="lazy"></dd></dl></dd></dl>

<dl><dt><a href="Inverse_element" title="Inverse element">Inverse element</a></dt>

<dd>Every number x, <a href="Division_by_zero" title="Division by zero">except 0</a>, has a <a href="Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{x}}}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{x}}}</annotation>

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</math><img src="./68f89eaf83a3811c69adb4bf1119bafd661a4c08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.166ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{x}}}" loading="lazy">, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot \left({\frac {1}{x}}\right)=1}">

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<annotation encoding="application/x-tex">{\displaystyle x\cdot \left({\frac {1}{x}}\right)=1}</annotation>

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</math><img src="./1dcf6f625e02d33dcac6b78c2e0448d73accd07f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.857ex; height:6.176ex;" alt="{\displaystyle x\cdot \left({\frac {1}{x}}\right)=1}" loading="lazy">.<a href="#cite_note-32">[32]</a></dd></dl>

<dl><dt><a href="Order_theory" title="Order theory">Order</a> preservation</dt>

<dd>Multiplication by a positive number preserves the <a href="Order_theory" title="Order theory">order</a>:

<dd>Multiplication by a negative number reverses the order:

<dd>The <a href="Complex_number" title="Complex number">complex numbers</a> do not have an ordering that is compatible with both addition and multiplication.<a href="#cite_note-33">[33]</a></dd></dl>

Other mathematical systems that include a multiplication operation may not have all these properties. For example, multiplication is not, in general, commutative for <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> and <a href="Quaternion" title="Quaternion">quaternions</a>.<a href="#cite_note-:0-30">[30]</a> <a href="Hurwitz's_theorem_(composition_algebras)" title="Hurwitz's theorem (composition algebras)">Hurwitz's theorem</a> shows that for the <a href="Hypercomplex_number" title="Hypercomplex number">hypercomplex numbers</a> of <a href="Dimension" title="Dimension">dimension</a> 8 or greater, including the <a href="Octonion" title="Octonion">octonions</a>, <a href="Sedenion" title="Sedenion">sedenions</a>, and <a href="Trigintaduonion" title="Trigintaduonion">trigintaduonions</a>, multiplication is generally not associative.<a href="#cite_note-34">[34]</a>

<div class="mw-heading mw-heading2"><h2 id="Axioms">Axioms</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Peano_axioms" title="Peano axioms">Peano axioms</a></div>

In the book <a href="Arithmetices_principia%2C_nova_methodo_exposita" title="Arithmetices principia, nova methodo exposita">Arithmetices principia, nova methodo exposita</a>, <a href="Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a> proposed axioms for arithmetic based on his axioms for natural numbers. Peano arithmetic has two axioms for multiplication:

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x\times 0=0}</annotation>

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</math><img src="./7c53848975b79c30253c1a29c94c2039626593e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.593ex; height:2.176ex;" alt="{\displaystyle x\times 0=0}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle x\times S(y)=(x\times y)+x}</annotation>

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</math><img src="./52009f37ff7a2abb73a860ec211bdf292b111363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.038ex; height:2.843ex;" alt="{\displaystyle x\times S(y)=(x\times y)+x}" loading="lazy"></dd></dl>

Here S(y) represents the <a href="Successor_ordinal" title="Successor ordinal">successor</a> 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 <a href="Mathematical_induction" title="Mathematical induction">induction</a>. For instance, S(0), denoted by 1, is a multiplicative identity because

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<annotation encoding="application/x-tex">{\displaystyle x\times 1=x\times S(0)=(x\times 0)+x=0+x=x.}</annotation>

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</math><img src="./a12b719027307239746f7de98730f473f83b3376.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.988ex; height:2.843ex;" alt="{\displaystyle x\times 1=x\times S(0)=(x\times 0)+x=0+x=x.}" loading="lazy"></dd></dl>

The axioms for <a href="Integer" title="Integer">integers</a> typically define them as equivalence classes of ordered pairs of natural numbers. The model is based on treating (x,y) as equivalent to 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

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<annotation encoding="application/x-tex">{\displaystyle (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}).}</annotation>

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</math><img src="./665406be617cc190f2199937ac79eb00637bb9ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:64.796ex; height:3.009ex;" alt="{\displaystyle (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}).}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle (0,1)\times (0,1)=(0\times 0+1\times 1,\,0\times 1+1\times 0)=(1,0).}</annotation>

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</math><img src="./cf8d62bafe97a241d1df058cad686e4b6cf25997.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.761ex; height:2.843ex;" alt="{\displaystyle (0,1)\times (0,1)=(0\times 0+1\times 1,\,0\times 1+1\times 0)=(1,0).}" loading="lazy"></dd></dl>

Multiplication is extended in a similar way to <a href="Rational_number" title="Rational number">rational numbers</a> and then to <a href="Real_number" title="Real number">real numbers</a>.

<div class="mw-heading mw-heading2"><h2 id="Multiplication_with_set_theory">Multiplication with set theory</h2></div>

The product of non-negative integers can be defined with set theory using <a href="Cardinal_number#Cardinal_multiplication" title="Cardinal number">cardinal numbers</a> or the <a href="Peano_axioms#Arithmetic" title="Peano axioms">Peano axioms</a>. See <a href="#Multiplication_of_different_kinds_of_numbers">below</a> 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 <a href="Construction_of_the_real_numbers" title="Construction of the real numbers">construction of the real numbers</a>.<a href="#cite_note-35">[35]</a>

<div class="mw-heading mw-heading2"><h2 id="Multiplication_in_group_theory">Multiplication in group theory</h2></div>

There are many sets that, under the operation of multiplication, satisfy the axioms that define <a href="Group_(mathematics)" title="Group (mathematics)">group</a> structure. These axioms are closure, associativity, and the inclusion of an identity element and inverses.

A simple example is the set of non-zero <a href="Rational_numbers" class="mw-redirect" title="Rational numbers">rational numbers</a>. 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 <a href="Abelian_group" title="Abelian group">abelian group</a> 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 <a href="Field_(mathematics)" title="Field (mathematics)">field</a>. Here, it is straightforward to verify closure, associativity, and inclusion of identity (the <a href="Identity_matrix" title="Identity matrix">identity matrix</a>) 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }">

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<annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation>

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</math><img src="./ba2c023bad1bd39ed49080f729cbf26bc448c9ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" loading="lazy"> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbb {Q} /\{0\},\,\cdot \right)}">

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<annotation encoding="application/x-tex">{\displaystyle \left(\mathbb {Q} /\{0\},\,\cdot \right)}</annotation>

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</math><img src="./bd9e40385175e571c9f7b46301e2aa2d11621454.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.335ex; height:2.843ex;" alt="{\displaystyle \left(\mathbb {Q} /\{0\},\,\cdot \right)}" loading="lazy">.<a href="#cite_note-36">[36]</a>

<div class="mw-heading mw-heading2"><h2 id="Multiplication_of_different_kinds_of_numbers">Multiplication of different kinds of numbers</h2></div>

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 <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>) or do not look much like numbers (such as <a href="Quaternion" title="Quaternion">quaternions</a>).

<dl><dt>Integers</dt>

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<annotation encoding="application/x-tex">{\displaystyle N\times M}</annotation>

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</math><img src="./5f4c0be393d026f1ee4c2baa3e84a774013914a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.346ex; height:2.176ex;" alt="{\displaystyle N\times M}" loading="lazy"> 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</dd>

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<annotation encoding="application/x-tex">{\displaystyle N\times (-M)=(-N)\times M=-(N\times M)}</annotation>

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</math><img src="./55ff14b1171cce6a308b982e0cd6019b3d4c2c6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.088ex; height:2.843ex;" alt="{\displaystyle N\times (-M)=(-N)\times M=-(N\times M)}" loading="lazy"> and</dd>

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<annotation encoding="application/x-tex">{\displaystyle (-N)\times (-M)=N\times M}</annotation>

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</math><img src="./dee1d0390840523adbd93a78832e2c05bc4dc48b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.026ex; height:2.843ex;" alt="{\displaystyle (-N)\times (-M)=N\times M}" loading="lazy"></dd>

<dd>The same sign rules apply to rational and real numbers.</dd></dl>

<dl><dt><a href="Rational_number" title="Rational number">Rational numbers</a></dt>

<dd>Generalization to fractions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {A}{B}}\times {\frac {C}{D}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {A}{B}}\times {\frac {C}{D}}}</annotation>

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</math><img src="./1e096995e6dd2346db181c01dcef0a6eae03b33b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.201ex; height:5.343ex;" alt="{\displaystyle {\frac {A}{B}}\times {\frac {C}{D}}}" loading="lazy"> is by multiplying the numerators and denominators, respectively: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {A}{B}}\times {\frac {C}{D}}={\frac {(A\times C)}{(B\times D)}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {A}{B}}\times {\frac {C}{D}}={\frac {(A\times C)}{(B\times D)}}}</annotation>

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</math><img src="./81c86d8b1575d08a100a155dc1694e975e2cc6d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.473ex; height:6.509ex;" alt="{\displaystyle {\frac {A}{B}}\times {\frac {C}{D}}={\frac {(A\times C)}{(B\times D)}}}" loading="lazy">. This gives the area of a rectangle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {A}{B}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {A}{B}}}</annotation>

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</math><img src="./4a1926bc23f1f3411122430184cee4d8b61890d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.6ex; height:5.343ex;" alt="{\displaystyle {\frac {A}{B}}}" loading="lazy"> high and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {C}{D}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {C}{D}}}</annotation>

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</math><img src="./adda89edc64a39ff1f56e37d0897f25241aff4e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.76ex; height:5.343ex;" alt="{\displaystyle {\frac {C}{D}}}" loading="lazy"> wide, and is the same as the number of things in an array when the rational numbers happen to be whole numbers.<a href="#cite_note-:0-30">[30]</a></dd></dl>

<dl><dt><a href="Real_number" title="Real number">Real numbers</a></dt>

<dd>Real numbers and their products <a href="Construction_of_the_real_numbers#Construction_from_Cauchy_sequences" title="Construction of the real numbers">can be defined in terms of sequences of rational numbers</a>.</dd></dl>

<dl><dt><a href="Complex_number" title="Complex number">Complex numbers</a></dt>

<dd>Considering complex numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle z_{1}}</annotation>

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</math><img src="./c3621e468231ab352b7caa30bcf0ce9b452241a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle z_{2}}</annotation>

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</math><img src="./5abf655fa14f7ea44ad0ca781b59ff59c5f49117.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{2}}" loading="lazy"> as ordered pairs of real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},b_{1})}">

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<annotation encoding="application/x-tex">{\displaystyle (a_{1},b_{1})}</annotation>

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</math><img src="./4169cf3c7bae0f4515e37aa3095f85d47636d56b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.179ex; height:2.843ex;" alt="{\displaystyle (a_{1},b_{1})}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{2},b_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle (a_{2},b_{2})}</annotation>

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</math><img src="./8eebf25545e61dd0b2385710f8ea0373a1904be3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.179ex; height:2.843ex;" alt="{\displaystyle (a_{2},b_{2})}" loading="lazy">, the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1}\times z_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle z_{1}\times z_{2}}</annotation>

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</math><img src="./b0dc5823066e7b39dc7997b6ca049ac67d5c413a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.111ex; height:2.009ex;" alt="{\displaystyle z_{1}\times z_{2}}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1}\times a_{2}-b_{1}\times b_{2},a_{1}\times b_{2}+a_{2}\times b_{1})}">

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<annotation encoding="application/x-tex">{\displaystyle (a_{1}\times a_{2}-b_{1}\times b_{2},a_{1}\times b_{2}+a_{2}\times b_{1})}</annotation>

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</math><img src="./fc8ff838c80a8d60ebfefda07846183b88c9db7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.229ex; height:2.843ex;" alt="{\displaystyle (a_{1}\times a_{2}-b_{1}\times b_{2},a_{1}\times b_{2}+a_{2}\times b_{1})}" loading="lazy">. This is the same as for reals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}\times a_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{1}\times a_{2}}</annotation>

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</math><img src="./719f8414ce021210971518e99e745faf0ee2a574.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.408ex; height:2.009ex;" alt="{\displaystyle a_{1}\times a_{2}}" loading="lazy"> when the imaginary parts <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle b_{1}}</annotation>

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</math><img src="./9af2720c91be489f57ecde4bb651b95e113d0144.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle b_{2}}</annotation>

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</math><img src="./2530a260ad35bf21ee61f1f4d6493ae0474f6068.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{2}}" loading="lazy"> are zero.</dd></dl>

<dl><dd>Equivalently, denoting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-1}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {-1}}}</annotation>

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</math><img src="./4ea1ea9ac61e6e1e84ac39130f78143c18865719.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-1}}}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1}\times z_{2}=(a_{1}+b_{1}i)(a_{2}+b_{2}i)=(a_{1}\times a_{2})+(a_{1}\times b_{2}i)+(b_{1}\times a_{2}i)+(b_{1}\times b_{2}i^{2})=(a_{1}a_{2}-b_{1}b_{2})+(a_{1}b_{2}+b_{1}a_{2})i.}">

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<annotation encoding="application/x-tex">{\displaystyle z_{1}\times z_{2}=(a_{1}+b_{1}i)(a_{2}+b_{2}i)=(a_{1}\times a_{2})+(a_{1}\times b_{2}i)+(b_{1}\times a_{2}i)+(b_{1}\times b_{2}i^{2})=(a_{1}a_{2}-b_{1}b_{2})+(a_{1}b_{2}+b_{1}a_{2})i.}</annotation>

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</math><img src="./d1d83413496f24893795cf11870291ee96b5d1db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:114.84ex; height:3.176ex;" alt="{\displaystyle z_{1}\times z_{2}=(a_{1}+b_{1}i)(a_{2}+b_{2}i)=(a_{1}\times a_{2})+(a_{1}\times b_{2}i)+(b_{1}\times a_{2}i)+(b_{1}\times b_{2}i^{2})=(a_{1}a_{2}-b_{1}b_{2})+(a_{1}b_{2}+b_{1}a_{2})i.}" loading="lazy"><a href="#cite_note-:0-30">[30]</a></dd>

<dd>Alternatively, in trigonometric form, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1}=r_{1}(\cos \phi _{1}+i\sin \phi _{1}),z_{2}=r_{2}(\cos \phi _{2}+i\sin \phi _{2})}">

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<annotation encoding="application/x-tex">{\displaystyle z_{1}=r_{1}(\cos \phi _{1}+i\sin \phi _{1}),z_{2}=r_{2}(\cos \phi _{2}+i\sin \phi _{2})}</annotation>

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</math><img src="./842a7f22ba3ce2bb5e9b68b9ecd0a2397beb1126.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.626ex; height:2.843ex;" alt="{\displaystyle z_{1}=r_{1}(\cos \phi _{1}+i\sin \phi _{1}),z_{2}=r_{2}(\cos \phi _{2}+i\sin \phi _{2})}" loading="lazy">, then<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle z_{1}z_{2}=r_{1}r_{2}(\cos(\phi _{1}+\phi _{2})+i\sin(\phi _{1}+\phi _{2})).}">

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<annotation encoding="application/x-tex">{\textstyle z_{1}z_{2}=r_{1}r_{2}(\cos(\phi _{1}+\phi _{2})+i\sin(\phi _{1}+\phi _{2})).}</annotation>

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</math><img src="./0376405bd9c910d46d9f1ea07ae1b685760ca049.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.086ex; height:2.843ex;" alt="{\textstyle z_{1}z_{2}=r_{1}r_{2}(\cos(\phi _{1}+\phi _{2})+i\sin(\phi _{1}+\phi _{2})).}" loading="lazy"><a href="#cite_note-:0-30">[30]</a></dd></dl>

<dl><dt>Further generalizations</dt>

<dd>See <a href="#Multiplication_in_group_theory">Multiplication in group theory</a>, above, and <a href="Multiplicative_group" title="Multiplicative group">multiplicative group</a>, which for example includes matrix multiplication. A very general, and abstract, concept of multiplication is as the "multiplicatively denoted" (second) binary operation in a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a>. An example of a ring that is not any of the number systems above is a <a href="Polynomial_ring" title="Polynomial ring">polynomial ring</a> (polynomials can be added and multiplied, but polynomials are not numbers in any usual sense).</dd></dl>

<dl><dt>Division</dt>

<dd>Often division, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x}{y}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {x}{y}}}</annotation>

</semantics>

</math><img src="./187d36a20476c4ccaaae85662613087bc4c67d35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.166ex; height:5.176ex;" alt="{\displaystyle {\frac {x}{y}}}" loading="lazy">, is the same as multiplication by an inverse, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\left({\frac {1}{y}}\right)}">

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\left({\frac {1}{y}}\right)}</annotation>

</semantics>

</math><img src="./818601c5133f84a324953e979bfe2b0289f1763c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.137ex; height:6.176ex;" alt="{\displaystyle x\left({\frac {1}{y}}\right)}" loading="lazy">. Multiplication for some types of "numbers" may have corresponding division, without inverses; in an <a href="Integral_domain" title="Integral domain">integral domain</a> x may have no inverse "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{x}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{x}}}</annotation>

</semantics>

</math><img src="./68f89eaf83a3811c69adb4bf1119bafd661a4c08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.166ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{x}}}" loading="lazy">" but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x}{y}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {x}{y}}}</annotation>

</semantics>

</math><img src="./187d36a20476c4ccaaae85662613087bc4c67d35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.166ex; height:5.176ex;" alt="{\displaystyle {\frac {x}{y}}}" loading="lazy"> may be defined. In a <a href="Division_ring" title="Division ring">division ring</a> there are inverses, but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x}{y}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {x}{y}}}</annotation>

</semantics>

</math><img src="./187d36a20476c4ccaaae85662613087bc4c67d35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.166ex; height:5.176ex;" alt="{\displaystyle {\frac {x}{y}}}" loading="lazy"> may be ambiguous in non-commutative rings since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\left({\frac {1}{y}}\right)}">

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\left({\frac {1}{y}}\right)}</annotation>

</semantics>

</math><img src="./818601c5133f84a324953e979bfe2b0289f1763c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.137ex; height:6.176ex;" alt="{\displaystyle x\left({\frac {1}{y}}\right)}" loading="lazy"> need not be the same as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {1}{y}}\right)x}">

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left({\frac {1}{y}}\right)x}</annotation>

</semantics>

</math><img src="./457d1d15a880bc767d849fad69db5107dd459d43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.137ex; height:6.176ex;" alt="{\displaystyle \left({\frac {1}{y}}\right)x}" loading="lazy">.</dd></dl>

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</style><div class="div-col" style="column-width: 35em;">

<ul><li><a href="Dimensional_analysis" title="Dimensional analysis">Dimensional analysis</a></li>

<li><a href="Multiplication_algorithm" title="Multiplication algorithm">Multiplication algorithm</a>

<ul><li><a href="Karatsuba_algorithm" title="Karatsuba algorithm">Karatsuba algorithm</a>, for large numbers</li>

<li><a href="Toom%E2%80%93Cook_multiplication" title="Toom–Cook multiplication">Toom–Cook multiplication</a>, for very large numbers</li>

<li><a href="Sch%C3%B6nhage%E2%80%93Strassen_algorithm" title="Schönhage–Strassen algorithm">Schönhage–Strassen algorithm</a>, for huge numbers</li></ul></li>

<li><a href="Multiplication_table" title="Multiplication table">Multiplication table</a></li>

<li><a href="Binary_multiplier" title="Binary multiplier">Binary multiplier</a>, how computers multiply

<ul><li><a href="Booth's_multiplication_algorithm" title="Booth's multiplication algorithm">Booth's multiplication algorithm</a></li>

<li><a href="Floating-point_arithmetic" title="Floating-point arithmetic">Floating-point arithmetic</a></li>

<li><a href="Multiply%E2%80%93accumulate_operation" title="Multiply–accumulate operation">Multiply–accumulate operation</a>

<ul><li><a href="Fused_multiply%E2%80%93add" class="mw-redirect" title="Fused multiply–add">Fused multiply–add</a></li></ul></li>

<li><a href="Wallace_tree" title="Wallace tree">Wallace tree</a></li></ul></li>

<li><a href="Multiplicative_inverse" title="Multiplicative inverse">Multiplicative inverse</a>, reciprocal</li>

<li><a href="Factorial" title="Factorial">Factorial</a></li>

<li><a href="Genaille%E2%80%93Lucas_rulers" title="Genaille–Lucas rulers">Genaille–Lucas rulers</a></li>

<li><a href="Lunar_arithmetic" title="Lunar arithmetic">Lunar arithmetic</a></li>

<li><a href="Napier's_bones" title="Napier's bones">Napier's bones</a></li>

<li><a href="Peasant_multiplication" class="mw-redirect" title="Peasant multiplication">Peasant multiplication</a></li>

<li><a href="Product_(mathematics)" title="Product (mathematics)">Product (mathematics)</a>, for generalizations</li>

<li><a href="Slide_rule" title="Slide rule">Slide rule</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFFoster2022" class="citation journal cs1">Foster, Colin (2022-03-01). <a rel="nofollow" class="external text" href="https://hdl.handle.net/2134/19299752.v1">"Getting multiplication the right way round"</a>. Mathematics in School. 51 (2). <a href="Mathematical_Association" title="Mathematical Association">The Mathematical Association</a> (UK): 16–17. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2134%2F19299752.v1">2134/19299752.v1</a>. Retrieved 2025-05-15.</cite>

</li>

<li id="cite_note-Devlin-2">^ <a href="#cite_ref-Devlin_2-0">a</a> <a href="#cite_ref-Devlin_2-1">b</a> <cite id="CITEREFDevlin2011" class="citation web cs1"><a href="Keith_Devlin" title="Keith Devlin">Devlin, Keith</a> (January 2011). <a rel="nofollow" class="external text" href="http://www.maa.org/external_archive/devlin/devlin_01_11.html">"What Exactly is Multiplication?"</a>. <a href="Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170527070801/http://www.maa.org/external_archive/devlin/devlin_01_11.html">Archived</a> from the original on 2017-05-27. Retrieved 2017-05-14. <q>With multiplication you have a multiplicand (written second) multiplied by a multiplier (written first)</q></cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFDevlin2011" class="citation web cs1">Devlin, Keith (January 2011). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241212000314/https://profkeithdevlin.org/devlins-angle/2011-posts/#jan11">"What exactly is multiplication?"</a>. profkeithdevlin.org. Archived from <a rel="nofollow" class="external text" href="https://profkeithdevlin.org/devlins-angle/2011-posts/#jan11">the original</a> on 2024-12-12. Retrieved 2024-12-12.</cite>

</li>

<li id="cite_note-mpb-4">^ <a href="#cite_ref-mpb_4-0">a</a> <a href="#cite_ref-mpb_4-1">b</a> <a href="#cite_ref-mpb_4-2">c</a> <cite id="CITEREFMusserPetersonBurger2013" class="citation book cs1">Musser, Gary L.; Peterson, Blake E.; Burger, William F. (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA101">Mathematics for Elementary Teachers: A Contemporary Approach</a>. <a href="John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. p. 101. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-118-48700-6</bdi>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFKlose1966" class="citation book cs1">Klose, Orval (1966). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rG7iBQAAQBAJ&pg=PA39">The Number Systems and Operations of Arithmetic</a>. Pergamon Press. p. 39. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4831-3709-4</bdi>.</cite>

</li>

<li id="cite_note-humez-6">^ <a href="#cite_ref-humez_6-0">a</a> <a href="#cite_ref-humez_6-1">b</a> <cite id="CITEREFHumezHumez2008" class="citation book cs1">Humez, Alexander; Humez, Nicholas (2008-10-02). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RaWdkz9NCYYC&pg=PT115">On the Dot: The Speck That Changed the World</a>. <a href="Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. p. 103. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-971718-7</bdi>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite class="citation journal cs1"><a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F218111c0">"Victory on Points"</a>. Nature. 218 (5137): 111. 1968. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1968Natur.218S.111.">1968Natur.218S.111.</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F218111c0">10.1038/218111c0</a>.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://download.thelancet.com/pb/assets/raw/Lancet/authors/artwork-guidelines.pdf">"The Lancet – Formatting guidelines for electronic submission of manuscripts"</a> (PDF). Retrieved 2017-04-25.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFTall1983" class="citation journal cs1">Tall, David (1983). "Introducing Algebra on the Computer: Today and Tomorrow". Mathematics in School. 12 (5): 37–40. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/30213874">30213874</a>.</cite>

</li>

<li id="cite_note-Peterson_2019-10"><a href="#cite_ref-Peterson_2019_10-0">^</a> <cite id="CITEREFPeterson2019" class="citation web cs1">Peterson, Dave (2019-10-14). <a rel="nofollow" class="external text" href="https://www.themathdoctors.org/order-of-operations-implicit-multiplication/">"Order of Operations: Implicit Multiplication?"</a>. Algebra / PEMDAS. The Math Doctors. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230924215608/https://www.themathdoctors.org/order-of-operations-implicit-multiplication/">Archived</a> from the original on 2023-09-24. Retrieved 2023-09-25.</cite>

</li>

<li id="cite_note-Peterson_2023-11"><a href="#cite_ref-Peterson_2023_11-0">^</a> <cite id="CITEREFPeterson2023" class="citation web cs1">Peterson, Dave (2023-08-18). <a rel="nofollow" class="external text" href="https://www.themathdoctors.org/implied-multiplication-1-not-as-bad-as-you-think/">"Implied Multiplication 1: Not as Bad as You Think"</a>. Algebra / Ambiguity, PEMDAS. The Math Doctors. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230924222451/https://www.themathdoctors.org/implied-multiplication-1-not-as-bad-as-you-think/">Archived</a> from the original on 2023-09-24. Retrieved 2023-09-25</cite>; <cite id="CITEREFPeterson2023" class="citation web cs1">Peterson, Dave (2023-08-25). <a rel="nofollow" class="external text" href="https://www.themathdoctors.org/implied-multiplication-2-is-there-a-standard/">"Implied Multiplication 2: Is There a Standard?"</a>. Algebra, Arithmetic / Ambiguity, PEMDAS. The Math Doctors. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230924222706/https://www.themathdoctors.org/implied-multiplication-2-is-there-a-standard/">Archived</a> from the original on 2023-09-24. Retrieved 2023-09-25</cite>; <cite id="CITEREFPeterson2023" class="citation web cs1">Peterson, Dave (2023-09-01). <a rel="nofollow" class="external text" href="https://www.themathdoctors.org/implied-multiplication-3-you-cant-prove-it/">"Implied Multiplication 3: You Can't Prove It"</a>. Algebra / PEMDAS. The Math Doctors. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230924222904/https://www.themathdoctors.org/implied-multiplication-3-you-cant-prove-it/">Archived</a> from the original on 2023-09-24. Retrieved 2023-09-25.</cite>

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite id="CITEREFGookin2004" class="citation book cs1">Gookin, Dan (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ruPPRwBYdjIC&pg=PA88">C For Dummies</a> (2nd ed.). Wiley. p. 88. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7645-7392-7</bdi>.</cite>

</li>

<li id="cite_note-fortran-13"><a href="#cite_ref-fortran_13-0">^</a> <cite id="CITEREFFuller1977" class="citation book cs1">Fuller, William R. (1977). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mnLjBwAAQBAJ&pg=PA10">FORTRAN Programming: A Supplement for Calculus Courses</a>. Universitext. Springer. p. 10. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4612-9938-7">10.1007/978-1-4612-9938-7</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90283-8</bdi>.</cite>

</li>

<li id="cite_note-multiplicand_on_Britannica-14"><a href="#cite_ref-multiplicand_on_Britannica_14-0">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/science/multiplicand">"Multiplicand | mathematics | Britannica"</a>. www.britannica.com. Encyclopædia Britannica, Inc. Retrieved 2024-11-15.</cite>

</li>

<li id="cite_note-multiplicand_via_Wolfram_Mathworld-15"><a href="#cite_ref-multiplicand_via_Wolfram_Mathworld_15-0">^</a> <cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Multiplicand.html">"Multiplicand"</a>. mathworld.wolfram.com. Wolfram Research, Inc. Retrieved 2024-11-15.</cite>

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <cite id="CITEREFLitvin2012" class="citation book cs1">Litvin, Chester (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-ULmPYAA8voC&q=Can+the+multiplicand+be+the+first+number&pg=PA6">Advance Brain Stimulation by Psychoconduction</a>. Trafford. pp. 2–3, 5–6. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4669-0152-0</bdi> – via <a href="Google_Book_Search" class="mw-redirect" title="Google Book Search">Google Book Search</a>.</cite>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mathematische-basteleien.de/multiplication.htm">"Multiplication"</a>. mathematische-basteleien.de. Retrieved 2022-03-15.</cite>

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFPletser2012" class="citation arxiv cs1">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". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1204.1019">1204.1019</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.HO">math.HO</a>].</cite>

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/Curriculum/Algebra/PeasantMultiplication.shtml">"Peasant Multiplication"</a>. cut-the-knot.org. Retrieved 2021-12-29.</cite>

</li>

<li id="cite_note-Nature-20"><a href="#cite_ref-Nature_20-0">^</a> <cite id="CITEREFQiu2014" class="citation journal cs1"><a href="Jane_Qiu" title="Jane Qiu">Qiu, Jane</a> (2014-01-07). <a rel="nofollow" class="external text" href="http://www.nature.com/news/ancient-times-table-hidden-in-chinese-bamboo-strips-1.14482">"Ancient times table hidden in Chinese bamboo strips"</a>. Nature. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnature.2014.14482">10.1038/nature.2014.14482</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:130132289">130132289</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140122064930/http://www.nature.com/news/ancient-times-table-hidden-in-chinese-bamboo-strips-1.14482">Archived</a> from the original on 2014-01-22. Retrieved 2014-01-22.</cite>

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> <cite id="CITEREFFine1907" class="citation book cs1"><a href="Henry_Burchard_Fine" title="Henry Burchard Fine">Fine, Henry B.</a> (1907). <a rel="nofollow" class="external text" href="https://archive.org/download/numbersystemofal00fineuoft/numbersystemofal00fineuoft.pdf">The Number System of Algebra – Treated Theoretically and Historically</a> (PDF) (2nd ed.). p. 90.</cite>

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> <cite id="CITEREFBernhard" class="citation web cs1">Bernhard, Adrienne. <a rel="nofollow" class="external text" href="https://www.bbc.com/future/article/20201204-lost-islamic-library-maths">"How modern mathematics emerged from a lost Islamic library"</a>. bbc.com. Retrieved 2022-04-22.</cite>

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> <cite id="CITEREFHarveyvan_der_HoevenLecerf2016" class="citation journal cs1">Harvey, David; van der Hoeven, Joris; Lecerf, Grégoire (2016). "Even faster integer multiplication". Journal of Complexity. 36: 1–30. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1407.3360">1407.3360</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jco.2016.03.001">10.1016/j.jco.2016.03.001</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0885-064X">0885-064X</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:205861906">205861906</a>.</cite>

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> David Harvey, Joris Van Der Hoeven (2019). <a rel="nofollow" class="external text" href="https://hal.archives-ouvertes.fr/hal-02070778">Integer multiplication in time O(n log n)</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190408180939/https://hal.archives-ouvertes.fr/hal-02070778">Archived</a> 2019-04-08 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> <cite id="CITEREFHartnett2019" class="citation web cs1">Hartnett, Kevin (2019-04-11). <a rel="nofollow" class="external text" href="https://www.quantamagazine.org/mathematicians-discover-the-perfect-way-to-multiply-20190411/">"Mathematicians Discover the Perfect Way to Multiply"</a>. Quanta Magazine. Retrieved 2020-01-25.</cite>

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> <cite id="CITEREFKlarreich2020" class="citation web cs1">Klarreich, Erica (January 2020). <a rel="nofollow" class="external text" href="https://cacm.acm.org/magazines/2020/1/241707-multiplication-hits-the-speed-limit/fulltext">"Multiplication Hits the Speed Limit"</a>. cacm.acm.org. <a rel="nofollow" class="external text" href="https://archive.today/20201031123457/https://cacm.acm.org/magazines/2020/1/241707-multiplication-hits-the-speed-limit/fulltext">Archived</a> from the original on 2020-10-31. Retrieved 2020-01-25.</cite>

</li>

<li id="cite_note-27"><a href="#cite_ref-27">^</a> <cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Product.html">"Product"</a>. mathworld.wolfram.com. Retrieved 2020-08-16.</cite>

</li>

<li id="cite_note-28"><a href="#cite_ref-28">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.illinoisstate.edu/day/courses/old/305/contentsummationnotation.html">"Summation and Product Notation"</a>. math.illinoisstate.edu. Retrieved 2020-08-16.</cite>

</li>

<li id="cite_note-29"><a href="#cite_ref-29">^</a> <cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Exponentiation.html">"Exponentiation"</a>. mathworld.wolfram.com. Retrieved 2021-12-29.</cite>

</li>

<li id="cite_note-:0-30">^ <a href="#cite_ref-:0_30-0">a</a> <a href="#cite_ref-:0_30-1">b</a> <a href="#cite_ref-:0_30-2">c</a> <a href="#cite_ref-:0_30-3">d</a> <a href="#cite_ref-:0_30-4">e</a> <a href="#cite_ref-:0_30-5">f</a> <a href="#cite_ref-:0_30-6">g</a> <a href="#cite_ref-:0_30-7">h</a> <a href="#cite_ref-:0_30-8">i</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Multiplication">"Multiplication"</a>. Encyclopedia of Mathematics. Retrieved 2021-12-29.</cite>

</li>

<li id="cite_note-:1-31">^ <a href="#cite_ref-:1_31-0">a</a> <a href="#cite_ref-:1_31-1">b</a> <a href="#cite_ref-:1_31-2">c</a> <a href="#cite_ref-:1_31-3">d</a> <cite id="CITEREFBiggs2002" class="citation book cs1">Biggs, Norman L. (2002). Discrete Mathematics. Oxford University Press. p. 25. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-871369-2</bdi>.</cite>

</li>

<li id="cite_note-32"><a href="#cite_ref-32">^</a> <cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/">"Multiplicative Inverse"</a>. Wolfram MathWorld. Retrieved 2022-04-19.</cite>

</li>

<li id="cite_note-33"><a href="#cite_ref-33">^</a> <cite id="CITEREFAngell" class="citation web cs1">Angell, David. <a rel="nofollow" class="external text" href="https://web.maths.unsw.edu.au/~angell/articles/complexorder.pdf">"ORDERING COMPLEX NUMBERS... NOT*"</a> (PDF). UNSW Sydney, School of Mathematics and Statistics. Retrieved 2021-12-29.</cite>

</li>

<li id="cite_note-34"><a href="#cite_ref-34">^</a> <cite id="CITEREFCawagasCarrascalBautistaMaria2009" class="citation arxiv cs1">Cawagas, Raoul E.; Carrascal, Alexander S.; Bautista, Lincoln A.; Maria, John P. Sta.; Urrutia, Jackie D.; Nobles, Bernadeth (2009). "The Subalgebra Structure of the Cayley-Dickson Algebra of Dimension 32 (trigintaduonion)". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0907.2047v3">0907.2047v3</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.RA">math.RA</a>].</cite>

</li>

<li id="cite_note-35"><a href="#cite_ref-35">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_(Boman_and_Rogers)/10%3A_Epilogue_to_Real_Analysis/10.02%3A_Building_the_Real_Numbers">"10.2: Building the Real Numbers"</a>. Mathematics LibreTexts. 2018-04-11. Retrieved 2023-06-23.</cite>

</li>

<li id="cite_note-36"><a href="#cite_ref-36">^</a> <cite id="CITEREFBurns1977" class="citation book cs1">Burns, Gerald (1977). Introduction to group theory with applications. New York: Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780121457501</bdi>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFBoyer,_Carl_B._(revised_by_Merzbach,_Uta_C.)1991" class="citation book cs1"><a href="Carl_Boyer" class="mw-redirect" title="Carl Boyer">Boyer, Carl B.</a> (revised by <a href="Uta_Merzbach" title="Uta Merzbach">Merzbach, Uta C.</a>) (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00boye">History of Mathematics</a>. John Wiley and Sons, Inc. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-54397-8</bdi>.</cite><code class="cs1-code">{{cite book}}</code>: CS1 maint: multiple names: authors list (link)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://www.cut-the-knot.org/do_you_know/multiplication.shtml">Multiplication</a> and <a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/blue/SysTable.shtml">Arithmetic Operations In Various Number Systems</a> at <a href="Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120719043305/http://webhome.idirect.com/~totton/suanpan/mod_mult/">Modern Chinese Multiplication Techniques on an Abacus</a></li></ul>

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<a href="Addition" title="Addition">+</a> <a href="Addition" title="Addition">Addition</a> (<a href="Plus_and_minus_signs#Plus_sign" title="Plus and minus signs">+</a>)

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;text-align: center;width:25%;"><div>

<a href="Subtraction" title="Subtraction">−</a> <a href="Subtraction" title="Subtraction">Subtraction</a> (<a href="Plus_and_minus_signs#Minus_sign" title="Plus and minus signs">−</a>)

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(<a href="Multiplication_sign" title="Multiplication sign">×</a> or <a href="Interpunct" title="Interpunct">·</a>)

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<a href="Division_(mathematics)" title="Division (mathematics)">÷</a> <a href="Division_(mathematics)" title="Division (mathematics)">Division</a> (<a href="Division_sign" title="Division sign">÷</a> or <a href="Slash_(punctuation)#Division" title="Slash (punctuation)">∕</a>)

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<ul><li><a href="Successor_function" title="Successor function">Successor (0)</a></li>

<li><a href="Addition" title="Addition">Addition (1)</a></li>

<li><a href="Exponentiation" title="Exponentiation">Exponentiation (3)</a></li>

<li><a href="Tetration" title="Tetration">Tetration (4)</a></li>

<li><a href="Pentation" title="Pentation">Pentation (5)</a></li>

<li><a href="Hexation" class="mw-redirect" title="Hexation">Hexation (6)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Inverse_function" title="Inverse function">Inverse</a> for left argument</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Primitive_recursive_function#Predecessor" title="Primitive recursive function">Predecessor (0)</a></li>

<li><a href="Subtraction" title="Subtraction">Subtraction (1)</a></li>

<li><a href="Division_(mathematics)" title="Division (mathematics)">Division (2)</a></li>

<li><a href="Nth_root" title="Nth root">Root extraction (3)</a></li>

<li><a href="Super-root" class="mw-redirect" title="Super-root">Super-root (4)</a></li></ul>

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<ul><li><a href="Primitive_recursive_function#Predecessor" title="Primitive recursive function">Predecessor (0)</a></li>

<li><a href="Subtraction" title="Subtraction">Subtraction (1)</a></li>

<li><a href="Division_(mathematics)" title="Division (mathematics)">Division (2)</a></li>

<li><a href="Logarithm" title="Logarithm">Logarithm (3)</a></li>

<li><a href="Super-logarithm" class="mw-redirect" title="Super-logarithm">Super-logarithm (4)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related articles</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Ackermann_function" title="Ackermann function">Ackermann function</a></li>

<li><a href="Conway_chained_arrow_notation" title="Conway chained arrow notation">Conway chained arrow notation</a></li>

<li><a href="Grzegorczyk_hierarchy" title="Grzegorczyk hierarchy">Grzegorczyk hierarchy</a></li>

<li><a href="Knuth's_up-arrow_notation" title="Knuth's up-arrow notation">Knuth's up-arrow notation</a></li>

<li><a href="Steinhaus%E2%80%93Moser_notation" title="Steinhaus–Moser notation">Steinhaus–Moser notation</a></li></ul>

</div></td></tr></tbody></table></div>

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