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<title>Addition</title>

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Addition

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</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="Addition_(disambiguation)" class="mw-disambig" title="Addition (disambiguation)">Addition (disambiguation)</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">"Add" redirects here. For other uses, see <a href="ADD_(disambiguation)" class="mw-disambig" title="ADD (disambiguation)">ADD (disambiguation)</a>.</div>

Addition (usually signified by the <a href="Plus_and_minus_signs#Plus_sign" title="Plus and minus signs">plus symbol</a>, +) is one of the four basic <a href="Operation_(mathematics)" title="Operation (mathematics)">operations</a> of <a href="Arithmetic" title="Arithmetic">arithmetic</a>, the other three being <a href="Subtraction" title="Subtraction">subtraction</a>, <a href="Multiplication" title="Multiplication">multiplication</a>, and <a href="Division_(mathematics)" title="Division (mathematics)">division</a>. The addition of two <a href="Natural_number" title="Natural number">whole numbers</a> results in the total or <a href="Summation" title="Summation">sum</a> of those values combined. For example, the adjacent image shows two columns of apples, one with three apples and the other with two apples, totaling to five apples. This observation is expressed as "3 + 2 = 5", which is read as "three plus two <a href="Equality_(mathematics)" title="Equality (mathematics)">equals</a> five".

Besides <a href="Counting" title="Counting">counting</a> items, addition can also be defined and executed without referring to <a href="Concrete_object" class="mw-redirect" title="Concrete object">concrete objects</a>, using abstractions called <a href="Number" title="Number">numbers</a> instead, such as <a href="Integer" title="Integer">integers</a>, <a href="Real_number" title="Real number">real numbers</a>, and <a href="Complex_number" title="Complex number">complex numbers</a>. Addition belongs to arithmetic, a branch of <a href="Mathematics" title="Mathematics">mathematics</a>. In <a href="Algebra" title="Algebra">algebra</a>, another area of mathematics, addition can also be performed on abstract objects such as <a href="Euclidean_vector" title="Euclidean vector">vectors</a>, <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>, and elements of <a href="Additive_group" title="Additive group">additive groups</a>.

Addition has several important properties. It is <a href="Commutative_property" title="Commutative property">commutative</a>, meaning that the order of the <a href="Operand" title="Operand">numbers being added</a> does not matter, so 3 + 2 = 2 + 3, and it is <a href="Associativity" class="mw-redirect" title="Associativity">associative</a>, meaning that when one adds more than two numbers, the order in which addition is performed does not matter. Repeated addition of <a href="1_(number)" class="mw-redirect" title="1 (number)">1</a> is the same as counting (see <a href="Successor_function" title="Successor function">Successor function</a>). Addition of <a href="0_(number)" class="mw-redirect" title="0 (number)">0</a> does not change a number. Addition also obeys rules concerning related operations such as subtraction and multiplication.

Performing addition is one of the simplest numerical tasks to perform. Addition of very small numbers is accessible to toddlers; the most basic task, 1 + 1, can be performed by infants as young as five months, and even some members of other animal species. In <a href="Primary_education" title="Primary education">primary education</a>, students are taught to add numbers in the <a href="Decimal" title="Decimal">decimal</a> system, beginning with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient <a href="Abacus" title="Abacus">abacus</a> to the modern <a href="Computer" title="Computer">computer</a>, where research on the most efficient implementations of addition continues to this day.

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</style><table class="sidebar nomobile nowraplinks"><tbody><tr><td class="sidebar-above" style="background:var(--background-color-neutral, #efefef);">

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

<mrow>

<mtr>

<mtd>

</mrow>

</mrow>

</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

<mtext>summand</mtext>

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<mtext>summand</mtext>

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</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

<mtext>addend</mtext>

</mrow>

<mtext>addend</mtext>

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</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

<mtext>augend</mtext>

</mrow>

<mtext>addend</mtext>

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</mstyle>

</mtd>

</mtr>

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</mrow>

</mrow>

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</mrow>

<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;">

</mrow>

</mstyle>

</mrow>

<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;">

<mrow>

<mtr>

<mtd>

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</mtd>

</mtr>

<mtr>

<mtd>

<mtext>minuend</mtext>

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<mtext>subtrahend</mtext>

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</mtd>

</mtr>

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</mrow>

<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>

</mstyle>

</mrow>

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

</semantics>

</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"><a href="Multiplication" title="Multiplication">Multiplication</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>

<mtext>factor</mtext>

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<mtext>factor</mtext>

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</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

<mtext>multiplier</mtext>

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<mtext>multiplicand</mtext>

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</mstyle>

</mtd>

</mtr>

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</mrow>

</mrow>

</mstyle>

</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>

</semantics>

</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;">

<mrow>

<mtr>

<mtd>

<mfrac>

<mtext>dividend</mtext>

</mrow>

</mstyle>

<mtext>divisor</mtext>

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</mstyle>

</mfrac>

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</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mtext>numerator</mtext>

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</mstyle>

<mtext>denominator</mtext>

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</mfrac>

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<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>

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

Addition is written using the <a href="Plus_and_minus_signs" title="Plus and minus signs">plus sign</a> "+" <a href="Infix_notation" title="Infix notation">between the terms</a>, and the result is expressed with an <a href="Equals_sign" title="Equals sign">equals sign</a>. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+2=3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1+2=3}</annotation>

</semantics>

</math><img src="./43f7e9a80913b9d5ec8d169140265d3e2fff5b4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 1+2=3}" loading="lazy"> reads "one plus two equals three".<a href="#cite_note-FOOTNOTEMusserPetersonBurger2013[httpsbooksgooglecombooksid8jh7DwAAQBAJpgPA87_87]-2">[2]</a> Nonetheless, some situations where addition is "understood", even though no symbol appears: a whole number followed immediately by a <a href="Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">fraction</a> indicates the sum of the two, called a mixed number, with an example,<a href="#cite_note-FOOTNOTEDevineOlsonOlson1991263-3">[3]</a><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3{\frac {1}{2}}=3+{\frac {1}{2}}=3.5.}">

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3{\frac {1}{2}}=3+{\frac {1}{2}}=3.5.}</annotation>

</semantics>

</math> This notation can cause confusion, since in most other contexts, <a href="Juxtaposition#Mathematics" title="Juxtaposition">juxtaposition</a> denotes <a href="Multiplication" title="Multiplication">multiplication</a> instead.<a href="#cite_note-FOOTNOTEMazur2014161-4">[4]</a>

The numbers or the objects to be added in general addition are collectively referred to as the terms,<a href="#cite_note-FOOTNOTEDepartment_of_the_Army1961[httpsarchiveorgdetailsTM11-684page16mode1upviewtheater_Section_5.1]-5">[5]</a> the addends or the summands.<a href="#cite_note-FOOTNOTEMusserPetersonBurger2013[httpsbooksgooglecombooksid8jh7DwAAQBAJpgPA87_87]-2">[2]</a> This terminology carries over to the summation of multiple terms.

This is to be distinguished from factors, which are <a href="Multiplication" title="Multiplication">multiplied</a>.

Some authors call the first addend the augend.<a href="#cite_note-FOOTNOTEShmerkoYanushkevichLyshevski200980Schmid1974Schmid1983-6">[6]</a> In fact, during the <a href="Renaissance" title="Renaissance">Renaissance</a>, many authors did not consider the first addend an "addend" at all. Today, due to the <a href="Commutative_property" title="Commutative property">commutative property</a> of addition, "augend" is rarely used, and both terms are generally called addends.<a href="#cite_note-FOOTNOTESchwartzman199419-7">[7]</a>

All of the above terminology derives from <a href="Latin" title="Latin">Latin</a>. "<a href="https://en.wiktionary.org/wiki/addition" class="extiw external" title="wikt:addition">Addition</a>" and "<a href="https://en.wiktionary.org/wiki/add" class="extiw external" title="wikt:add">add</a>" are <a href="English_language" title="English language">English</a> words derived from the Latin <a href="Verb" title="Verb">verb</a> addere, which is in turn a <a href="Compound_(linguistics)" title="Compound (linguistics)">compound</a> of ad "to" and dare "to give", from the <a href="Proto-Indo-European_root" title="Proto-Indo-European root">Proto-Indo-European root</a> *deh₃- "to give"; thus to add is to give to.<a href="#cite_note-FOOTNOTESchwartzman199419-7">[7]</a> Using the <a href="Gerundive" title="Gerundive">gerundive</a> <a href="Affix" title="Affix">suffix</a> -nd results in "addend", "thing to be added".<a href="#cite_note-8">[a]</a> Likewise from augere "to increase", one gets "augend", "thing to be increased".<a href="#cite_note-9">[8]</a>

"Sum" and "summand" derive from the Latin <a href="Noun" title="Noun">noun</a> summa "the highest" or "the top", used in Medieval Latin phrase summa linea ("top line") meaning the sum of a column of numerical quantities, following the <a href="Ancient_Greece" title="Ancient Greece">ancient Greek</a> and <a href="Ancient_Rome" title="Ancient Rome">Roman</a> practice of putting the sum at the top of a column.<a href="#cite_note-FOOTNOTESchwartzman1994212-11">[10]</a>

Addere and summare date back at least to <a href="Anicius_Manlius_Severinus_Boethius" class="mw-redirect" title="Anicius Manlius Severinus Boethius">Boethius</a>, if not to earlier Roman writers such as <a href="Vitruvius" title="Vitruvius">Vitruvius</a> and <a href="Sextus_Julius_Frontinus" class="mw-redirect" title="Sextus Julius Frontinus">Frontinus</a>; Boethius also used several other terms for the addition operation. The later <a href="Middle_English" title="Middle English">Middle English</a> terms "adden" and "adding" were popularized by <a href="Geoffrey_Chaucer" title="Geoffrey Chaucer">Chaucer</a>.<a href="#cite_note-FOOTNOTEKarpinski1925150–153-12">[11]</a>

<div class="mw-heading mw-heading2"><h2 id="Definition_and_interpretations">Definition and interpretations</h2></div>

Addition is one of the four basic <a href="Operation_(mathematics)" title="Operation (mathematics)">operations</a> of <a href="Arithmetic" title="Arithmetic">arithmetic</a>, with the other three being <a href="Subtraction" title="Subtraction">subtraction</a>, <a href="Multiplication" title="Multiplication">multiplication</a>, and <a href="Division_(mathematics)" title="Division (mathematics)">division</a>. This operation works by adding two or more terms.<a href="#cite_note-FOOTNOTELewis19741-13">[12]</a> An arbitrary of many operation of additions is called the <a href="Summation" title="Summation">summation</a>.<a href="#cite_note-FOOTNOTEMartin200349-14">[13]</a> An infinite summation is a delicate procedure known as a <a href="Series_(mathematics)" title="Series (mathematics)">series</a>,<a href="#cite_note-FOOTNOTEStewart19998-15">[14]</a> and it can be expressed through <a href="Capital_sigma_notation" class="mw-redirect" title="Capital sigma notation">capital sigma notation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8e2b0b7618be940f4e8c0d27f05ab75fbc13e83c.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="{\textstyle \sum }" loading="lazy">, which compactly denotes <a href="Iteration" title="Iteration">iteration</a> of the operation of addition based on the given indexes.<a href="#cite_note-FOOTNOTEApostol196737-16">[15]</a> For example,

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{5}k^{2}=1^{2}+2^{2}+3^{2}+4^{2}+5^{2}=55.}</annotation>

</semantics>

</math>

Addition is used to model many physical processes. Even for the simple case of adding <a href="Natural_number" title="Natural number">natural numbers</a>, there are many possible interpretations and even more visual representations.

<div class="mw-heading mw-heading3"><h3 id="Combining_sets">Combining sets</h3></div>

Possibly the most basic interpretation of addition lies in combining <a href="Set_(mathematics)" title="Set (mathematics)">sets</a>, that is:<a href="#cite_note-FOOTNOTEMusserPetersonBurger2013[httpsbooksgooglecombooksid8jh7DwAAQBAJpgPA87_87]-2">[2]</a>

/* start https://en.wikipedia.org/ */

.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}

/* end https://en.wikipedia.org/ */

</style><blockquote class="templatequote">When two or more disjoint collections are combined into a single collection, the number of objects in the single collection is the sum of the numbers of objects in the original collections.</blockquote>

This interpretation is easy to visualize, with little danger of ambiguity. It is also useful in higher mathematics (for the rigorous definition it inspires, see <a href="#Natural_numbers">§ Natural numbers</a> below). However, it is not obvious how one should extend this interpretation to include fractional or negative numbers.<a href="#cite_note-17">[16]</a>

One possibility is to consider collections of objects that can be easily divided, such as pies or, still better, segmented rods. Rather than solely combining collections of segments, rods can be joined end-to-end, which illustrates another conception of addition: adding not the rods but the lengths of the rods.<a href="#cite_note-FOOTNOTENational_Research_Council2001[httpbooksgooglecombooksidpvI7uDPo0-YCpgPA74_74]-18">[17]</a>

<div class="mw-heading mw-heading3"><h3 id="Extending_a_length">Extending a length</h3></div>

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.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}

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</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:392px;max-width:392px"><div class="trow"><div class="tsingle" style="width:390px;max-width:390px"><div class="thumbimage" style="height:106px;overflow:hidden"></div><div class="thumbcaption">A number-line visualization of the algebraic addition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2+4=6}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2+4=6}</annotation>

</semantics>

</math><img src="./2d24e36ee185b418693ed8cd88c8bed6b6b24006.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 2+4=6}" loading="lazy">. A "jump" that has a distance of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./901fc910c19990d0dbaaefe4726ceb1a4e217a0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}" loading="lazy"> followed by another that is as long as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./295b4bf1de7cd3500e740e0f4f0635db22d87b42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 4}" loading="lazy">, is the same as a translation by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./39d81124420a058a7474dfeda48228fb6ee1e253.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 6}" loading="lazy">.</div></div></div><div class="trow"><div class="tsingle" style="width:390px;max-width:390px"><div class="thumbimage" style="height:68px;overflow:hidden"></div><div class="thumbcaption">A number-line visualization of the unary addition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2+4=6}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2+4=6}</annotation>

</semantics>

</math><img src="./2d24e36ee185b418693ed8cd88c8bed6b6b24006.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 2+4=6}" loading="lazy">. A translation by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./295b4bf1de7cd3500e740e0f4f0635db22d87b42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 4}" loading="lazy"> is equivalent to four translations by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">.</div></div></div></div></div>

A second interpretation of addition comes from extending an initial length by a given length:<a href="#cite_note-FOOTNOTEMosley2001[httpbooksgooglecombooksidI-_WcWjemUCpgPA8_8]-19">[18]</a>

<blockquote class="templatequote">When an original length is extended by a given amount, the final length is the sum of the original length and the length of the extension.</blockquote>

The sum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a2391acf09244b9dba74eb940e871a6be7e7973a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a+b}" loading="lazy"> can be interpreted as a <a href="Binary_operation" title="Binary operation">binary operation</a> that combines <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> algebraically, or it can be interpreted as the addition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> more units to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">. Under the latter interpretation, the parts of a sum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a2391acf09244b9dba74eb940e871a6be7e7973a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a+b}" loading="lazy"> play asymmetric roles, and the operation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a2391acf09244b9dba74eb940e871a6be7e7973a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a+b}" loading="lazy"> is viewed as applying the <a href="Unary_operation" title="Unary operation">unary operation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dd9afaf716c7ad1ccd064f65fc135cbb7eac4ec9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.806ex; height:2.343ex;" alt="{\displaystyle +b}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">.<a href="#cite_note-FOOTNOTELiLappan2014204-20">[19]</a> Instead of calling both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> addends, it is more appropriate to call <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> the "augend" in this case, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> plays a passive role. The unary view is also useful when discussing <a href="Subtraction" title="Subtraction">subtraction</a>, because each unary addition operation has an inverse unary subtraction operation, and vice versa.

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

<div class="mw-heading mw-heading3"><h3 id="Commutativity">Commutativity</h3></div>

Addition is <a href="Commutative" class="mw-redirect" title="Commutative">commutative</a>, meaning that one can change the order of the terms in a sum, but still get the same result. Symbolically, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> are any two numbers, then:<a href="#cite_note-FOOTNOTEMusserPetersonBurger2013[httpsbooksgooglecombooksid8jh7DwAAQBAJpgPA89_89]-21">[20]</a>

</mstyle>

</mrow>

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

</semantics>

</math>

The fact that addition is commutative is known as the "commutative law of addition"<a href="#cite_note-FOOTNOTEBerg1967[httpsbooksgooglecombooksidaGXFCUaFCW0CpgPA14_14]-22">[21]</a> or "commutative property of addition".<a href="#cite_note-FOOTNOTEBehrJungst1971[httpsbooksgooglecombooksidGJXOBQAAQBAJpgPA59_59]-23">[22]</a> Some other <a href="Binary_operation" title="Binary operation">binary operations</a> are commutative too as in <a href="Multiplication" title="Multiplication">multiplication</a>,<a href="#cite_note-FOOTNOTERosen2013See_the_[httpsbooksgooglecombooksid-oVvEAAAQBAJpgSL1-PA1_Appendix_I]-24">[23]</a> but others are not as in <a href="Subtraction" title="Subtraction">subtraction</a> and <a href="Division_(mathematics)" title="Division (mathematics)">division</a>.<a href="#cite_note-FOOTNOTEPosamentierFarberGermain-WilliamsParis2013[httpsbooksgooglecombooksidVfCgAQAAQBAJpgPA71_71]-25">[24]</a>

<div class="mw-heading mw-heading3"><h3 id="Associativity">Associativity</h3></div>

Addition is <a href="Associativity" class="mw-redirect" title="Associativity">associative</a>, which means that when three or more numbers are added together, the <a href="Order_of_operations" title="Order of operations">order of operations</a> does not change the result. For any three numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy">, it is true that:<a href="#cite_note-FOOTNOTEMusserPetersonBurger2013[httpsbooksgooglecombooksid8jh7DwAAQBAJpgPA90_90]-26">[25]</a>

</mstyle>

</mrow>

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

</semantics>

</math>

For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+2)+3=1+(2+3)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1+2)+3=1+(2+3)}</annotation>

</semantics>

</math><img src="./c5d0a74a478cd69181525209c16dcec52eee9d99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.053ex; height:2.843ex;" alt="{\displaystyle (1+2)+3=1+(2+3)}" loading="lazy">.

When addition is used together with other operations, the <a href="Order_of_operations" title="Order of operations">order of operations</a> becomes important. In the standard order of operations, addition is a lower priority than <a href="Exponentiation" title="Exponentiation">exponentiation</a>, <a href="Nth_root" title="Nth root">nth roots</a>, multiplication and division, but is given equal priority to subtraction.<a href="#cite_note-FOOTNOTEBronsteinSemendjajew1987-27">[26]</a>

<div class="mw-heading mw-heading3"><h3 id="Identity_element">Identity element</h3></div>

Adding <a href="0_(number)" class="mw-redirect" title="0 (number)">zero</a> to any number does not change the number. In other words, zero is the <a href="Identity_element" title="Identity element">identity element</a> for addition, and is also known as the <a href="Additive_identity" title="Additive identity">additive identity</a>. In symbols, for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">, one has:<a href="#cite_note-FOOTNOTEMusserPetersonBurger2013[httpsbooksgooglecombooksid8jh7DwAAQBAJpgPA90_90]-26">[25]</a>

</mstyle>

</mrow>

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

</semantics>

</math>

This law was first identified in <a href="Brahmagupta" title="Brahmagupta">Brahmagupta</a>'s <a href="Brahmasphutasiddhanta" class="mw-redirect" title="Brahmasphutasiddhanta">Brahmasphutasiddhanta</a> in 628 AD, although he wrote it as three separate laws, depending on whether <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> is negative, positive, or zero itself, and he used words rather than algebraic symbols. Later <a href="Indian_mathematicians" class="mw-redirect" title="Indian mathematicians">Indian mathematicians</a> refined the concept; around the year 830, <a href="Mahavira_(mathematician)" class="mw-redirect" title="Mahavira (mathematician)">Mahavira</a> wrote, "zero becomes the same as what is added to it", corresponding to the unary statement <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0+a=a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ba565116cf2f2d984f7b8365b054b70eb8f89308.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.561ex; height:2.343ex;" alt="{\displaystyle 0+a=a}" loading="lazy">. In the 12th century, <a href="Bh%C4%81skara_II" title="Bhāskara II">Bhaskara</a> wrote, "In the addition of cipher, or subtraction of it, the quantity, positive or negative, remains the same", corresponding to the unary statement <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+0=a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4564e28f0f8274644ca4e58664c0593ed48de541.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.561ex; height:2.343ex;" alt="{\displaystyle a+0=a}" loading="lazy">.<a href="#cite_note-FOOTNOTEKaplan200069–71-28">[27]</a>

<div class="mw-heading mw-heading3"><h3 id="Successor">Successor</h3></div>

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

Within the context of integers, addition of <a href="1_(number)" class="mw-redirect" title="1 (number)">one</a> also plays a special role: for any integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">, the integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8028f23cfdcb108712e2bc53369305574afe820b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.233ex; height:2.343ex;" alt="{\displaystyle a+1}" loading="lazy"> is the least integer greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">, also known as the <a href="Successor_function" title="Successor function">successor</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">. For instance, 3 is the successor of 2, and 7 is the successor of 6. Because of this succession, the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a2391acf09244b9dba74eb940e871a6be7e7973a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a+b}" loading="lazy"> can also be seen as the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy">-th successor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">, making addition an iterated succession. For example, 6 + 2 is 8, because 8 is the successor of 7, which is the successor of 6, making 8 the second successor of 6.<a href="#cite_note-FOOTNOTEHempel2001[httpbooksgooglecombooksidyTY9La4P2n8CpgPA7_7]-29">[28]</a>

<div class="mw-heading mw-heading3"><h3 id="Units">Units</h3></div>

To numerically add physical quantities with <a href="Units_of_measurement" class="mw-redirect" title="Units of measurement">units</a>, they must be expressed with common units.<a href="#cite_note-FOOTNOTEFierro2012Section_2.3-30">[29]</a> For example, adding 50 milliliters to 150 milliliters gives 200 milliliters. However, if a measure of 5 feet is extended by 2 inches, the sum is 62 inches, since 60 inches is synonymous with 5 feet. On the other hand, it is usually meaningless to try to add 3 meters and 4 square meters, since those units are incomparable; this sort of consideration is fundamental in <a href="Dimensional_analysis" title="Dimensional analysis">dimensional analysis</a>.<a href="#cite_note-31">[30]</a>

<div class="mw-heading mw-heading2"><h2 id="Performing_addition">Performing addition</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Innate_ability">Innate ability</h3></div>

Studies on mathematical development starting around the 1980s have exploited the phenomenon of <a href="Habituation" title="Habituation">habituation</a>: <a href="Infant" title="Infant">infants</a> look longer at situations that are unexpected.<a href="#cite_note-FOOTNOTEWynn19985-32">[31]</a> A seminal experiment by <a href="Karen_Wynn" title="Karen Wynn">Karen Wynn</a> in 1992 involving <a href="Mickey_Mouse" title="Mickey Mouse">Mickey Mouse</a> dolls manipulated behind a screen demonstrated that five-month-old infants expect 1 + 1 to be 2, and they are comparatively surprised when a physical situation seems to imply that 1 + 1 is either 1 or 3. This finding has since been affirmed by a variety of laboratories using different methodologies.<a href="#cite_note-FOOTNOTEWynn199815-33">[32]</a> Another 1992 experiment with older <a href="Toddler" title="Toddler">toddlers</a>, between 18 and 35 months, exploited their development of motor control by allowing them to retrieve <a href="Ping-pong" class="mw-redirect" title="Ping-pong">ping-pong</a> balls from a box; the youngest responded well for small numbers, while older subjects were able to compute sums up to 5.<a href="#cite_note-FOOTNOTEWynn199817-34">[33]</a>

Even some nonhuman animals show a limited ability to add, particularly <a href="Primate" title="Primate">primates</a>. In a 1995 experiment imitating Wynn's 1992 result (but using <a href="Eggplant" title="Eggplant">eggplants</a> instead of dolls), <a href="Rhesus_macaque" title="Rhesus macaque">rhesus macaque</a> and <a href="Cottontop_tamarin" class="mw-redirect" title="Cottontop tamarin">cottontop tamarin</a> monkeys performed similarly to human infants. More dramatically, after being taught the meanings of the <a href="Arabic_numerals" title="Arabic numerals">Arabic numerals</a> 0 through 4, one <a href="Common_Chimpanzee" class="mw-redirect" title="Common Chimpanzee">chimpanzee</a> was able to compute the sum of two numerals without further training.<a href="#cite_note-FOOTNOTEWynn199819-35">[34]</a> More recently, <a href="Asian_elephant" title="Asian elephant">Asian elephants</a> have demonstrated an ability to perform basic arithmetic.<a href="#cite_note-36">[35]</a>

<div class="mw-heading mw-heading3"><h3 id="Addition_by_counting">Addition by counting</h3></div>

Typically, children first master <a href="Counting" title="Counting">counting</a>. When given a problem that requires that two items and three items be combined, young children model the situation with physical objects, often fingers or a drawing, and then count the total. As they gain experience, they learn or discover the strategy of "counting-on": asked to find two plus three, children count three past two, saying "three, four, five" (usually ticking off fingers), and arriving at five. This strategy seems almost universal; children can easily pick it up from peers or teachers.<a href="#cite_note-FOOTNOTESmith2002130-37">[36]</a> Most discover it independently. With additional experience, children learn to add more quickly by exploiting the commutativity of addition by counting up from the larger number, in this case, starting with three and counting "four, five." Eventually children begin to recall certain addition facts ("<a href="Number_bond" title="Number bond">number bonds</a>"), either through experience or rote memorization. Once some facts are committed to memory, children begin to derive unknown facts from known ones. For example, a child asked to add six and seven may know that 6 + 6 = 12 and then reason that 6 + 7 is one more, or 13.<a href="#cite_note-38">[37]</a> Such derived facts can be found very quickly and most elementary school students eventually rely on a mixture of memorized and derived facts to add fluently.<a href="#cite_note-Henry-39">[38]</a>

Different nations introduce whole numbers and arithmetic at different ages, with many countries teaching addition in pre-school.<a href="#cite_note-40">[39]</a> However, throughout the world, addition is taught by the end of the first year of elementary school.<a href="#cite_note-41">[40]</a>

<div class="mw-heading mw-heading3"><h3 id="Single-digit_addition">Single-digit addition </h3></div>

An ability to add a pair of single digits (numbers from 0 to 9) is a prerequisite for addition of arbitrary numbers in the <a href="Decimal" title="Decimal">decimal</a> system. With 10 choices for each of the two digits to be added, this makes 100 single-digit "addition facts", which can be presented in an addition table.

<th>+

</th>

<th>9

</th></tr>

<tr>

<th scope="row">0

</th>

<td>9

</td></tr>

<tr>

<th scope="row">1

</th>

<td>10

</td></tr>

<tr>

<th scope="row">2

</th>

<td>11

</td></tr>

<tr>

<th scope="row">3

</th>

<td>12

</td></tr>

<tr>

<th scope="row">4

</th>

<td>13

</td></tr>

<tr>

<th scope="row">5

</th>

<td>14

</td></tr>

<tr>

<th scope="row">6

</th>

<td>15

</td></tr>

<tr>

<th scope="row">7

</th>

<td>16

</td></tr>

<tr>

<th scope="row">8

</th>

<td>17

</td></tr>

<tr>

<th scope="row">9

</th>

<td>18

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

Learning to fluently and accurately compute single-digit additions is a major focus of early schooling in arithmetic. Sometimes students are encouraged to memorize the full addition table by <a href="Rote_learning" title="Rote learning">rote</a>, but pattern-based strategies are typically more enlightening and, for most people, more efficient:<a href="#cite_note-FOOTNOTEFosnotDolk200199-42">[41]</a>

<ul><li>Commutative property: Mentioned above, using the pattern <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b=b+a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./684f43b5094501674e8314be5e24a80ee64682e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.234ex; height:2.343ex;" alt="{\displaystyle a+b=b+a}" loading="lazy"> reduces the number of "addition facts" from 100 to 55.</li>

<li>One or two more: Adding 1 or 2 is a basic task, and it can be accomplished through counting on or, ultimately, <a href="Intuition_(knowledge)" class="mw-redirect" title="Intuition (knowledge)">intuition</a>.<a href="#cite_note-FOOTNOTEFosnotDolk200199-42">[41]</a></li>

<li>Zero: Since zero is the additive identity, adding zero is trivial. Nonetheless, in the teaching of arithmetic, some students are introduced to addition as a process that always increases the addends; <a href="Word_problem_(mathematics_education)" title="Word problem (mathematics education)">word problems</a> may help rationalize the "exception" of zero.<a href="#cite_note-FOOTNOTEFosnotDolk200199-42">[41]</a></li>

<li>Doubles: Adding a number to itself is related to counting by two and to <a href="Multiplication" title="Multiplication">multiplication</a>. Doubles facts form a backbone for many related facts, and students find them relatively easy to grasp.<a href="#cite_note-FOOTNOTEFosnotDolk200199-42">[41]</a></li>

<li>Near-doubles: Sums such as 6 + 7 = 13 can be quickly derived from the doubles fact 6 + 6 = 12 by adding one more, or from 7 + 7 = 14 but subtracting one.<a href="#cite_note-FOOTNOTEFosnotDolk200199-42">[41]</a></li>

<li>Five and ten: Sums of the form 5 + x and 10 + x are usually memorized early and can be used for deriving other facts. For example, 6 + 7 = 13 can be derived from 5 + 7 = 12 by adding one more.<a href="#cite_note-FOOTNOTEFosnotDolk200199-42">[41]</a></li>

<li>Making ten: An advanced strategy uses 10 as an intermediate for sums involving 8 or 9; for example, 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14.<a href="#cite_note-FOOTNOTEFosnotDolk200199-42">[41]</a></li></ul>

As students grow older, they commit more facts to memory and learn to derive other facts rapidly and fluently. Many students never commit all the facts to memory, but can still find any basic fact quickly.<a href="#cite_note-Henry-39">[38]</a>

<div class="mw-heading mw-heading3"><h3 id="Carry">Carry</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Carry_(arithmetic)" title="Carry (arithmetic)">Carry (arithmetic)</a></div>

The standard algorithm for adding multidigit numbers is to align the addends vertically and add the columns by using the above addition table, starting from the ones column on the right. If the result of a column exceeds nine, the extra digit is "<a href="Carry_(arithmetic)" title="Carry (arithmetic)">carried</a>" into the next column. For example, in the following image, the ones in the addition of 59 + 27 is 9 + 7 = 16, and the digit 1 is the carry.<a href="#cite_note-43">[42]</a> An alternate strategy starts adding from the most significant digit on the left; this route makes carrying a little clumsier, but it is faster at getting a rough estimate of the sum.<a href="#cite_note-45">[b]</a>

<div class="mw-heading mw-heading3"><h3 id="Decimal_fractions">Decimal fractions</h3></div>

<a href="Decimal_fractions" class="mw-redirect" title="Decimal fractions">Decimal fractions</a> can be added by a simple modification of the above process. One aligns two decimal fractions above each other, with the decimal point in the same location. If necessary, one can add trailing zeros to a shorter decimal to make it the same length as the longer decimal. Finally, one performs the same addition process as above, except the decimal point is placed in the answer, exactly where it was placed in the summands.<a href="#cite_note-FOOTNOTEWingard-Nelson2014[httpsarchiveorgdetailsdecimalsfraction0000wing_v8b6page40mode1upqdecimalviewtheater_40]-46">[44]</a> As an example, 45.1 + 4.34 can be solved as follows:

<pre> 4 5 . 1 0

+ 0 4 . 3 4

————————————

4 9 . 4 4

</pre>

<div class="mw-heading mw-heading3"><h3 id="Scientific_notation">Scientific notation</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Scientific_notation#Basic_operations" title="Scientific notation">Scientific notation § Basic operations</a></div>

In <a href="Scientific_notation" title="Scientific notation">scientific notation</a>, numbers are written in the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a\times 10^{b}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=a\times 10^{b}}</annotation>

</semantics>

</math><img src="./7eea4aa2cfd766558bc1a03f0297b9e34521c293.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.761ex; height:2.676ex;" alt="{\displaystyle x=a\times 10^{b}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> is the <a href="Significand" title="Significand">significand</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{b}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 10^{b}}</annotation>

</semantics>

</math><img src="./d27d5434463417d4d6b12bf6c2835d270bd6b814.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.263ex; height:2.676ex;" alt="{\displaystyle 10^{b}}" loading="lazy"> is the exponential part. To add numbers in scientific notation, they should be expressed with the same exponent, so that the two significands can simply be added.<a href="#cite_note-47">[45]</a>

For example:

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&2.34\times 10^{-5}+5.67\times 10^{-6}\\&\quad =2.34\times 10^{-5}+0.567\times 10^{-5}\\&\quad =2.907\times 10^{-5}.\end{aligned}}}</annotation>

</semantics>

</math><img src="./e75fab741fc87cb367a83ec40b3806b76d7cb5bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:33.44ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}&2.34\times 10^{-5}+5.67\times 10^{-6}\\&\quad =2.34\times 10^{-5}+0.567\times 10^{-5}\\&\quad =2.907\times 10^{-5}.\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Non-decimal">Non-decimal</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Binary_addition" class="mw-redirect" title="Binary addition">Binary addition</a></div>

Addition in other bases is very similar to decimal addition. As an example, one can consider addition in binary.<a href="#cite_note-48">[46]</a> Adding two single-digit binary numbers is relatively simple, using a form of carrying:

<dd>1 + 1 → 0, carry 1 (since 1 + 1 = 2 = 0 + (1 × 21))</dd></dl>

Adding two "1" digits produces a digit "0", while 1 must be added to the next column. This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix (10), the digit to the left is incremented:

<dl><dd>5 + 5 → 0, carry 1 (since 5 + 5 = 10 = 0 + (1 × 101))</dd>

<dd>7 + 9 → 6, carry 1 (since 7 + 9 = 16 = 6 + (1 × 101))</dd></dl>

This is known as carrying.<a href="#cite_note-49">[47]</a> When the result of an addition exceeds the value of a digit, the procedure is to "carry" the excess amount divided by the radix (that is, 10/10) to the left, adding it to the next positional value. This is correct since the next position has a weight that is higher by a factor equal to the radix. Carrying works the same way in binary:

<pre> 1 1 1 1 1 (carried digits)

0 1 1 0 1

+ 1 0 1 1 1

—————————————

1 0 0 1 0 0 = 36

</pre>

In this example, two numerals are being added together: 011012 (1310) and 101112 (2310). The top row shows the carry bits used. Starting in the rightmost column, 1 + 1 = 102. The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. The second column from the right is added: 1 + 0 + 1 = 102 again; the 1 is carried, and 0 is written at the bottom. The third column: 1 + 1 + 1 = 112. This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 1001002 (3610).

<div class="mw-heading mw-heading3"><h3 id="Computers">Computers</h3></div>

<a href="Analog_computer" title="Analog computer">Analog computers</a> work directly with physical quantities, so their addition mechanisms depend on the form of the addends. A mechanical adder might represent two addends as the positions of sliding blocks, in which case they can be added with an <a href="Arithmetic_mean" title="Arithmetic mean">averaging</a> <a href="Lever" title="Lever">lever</a>. If the addends are the rotation speeds of two <a href="Axle" title="Axle">shafts</a>, they can be added with a <a href="Differential_(mechanics)" class="mw-redirect" title="Differential (mechanics)">differential</a>. A hydraulic adder can add the <a href="Pressure" title="Pressure">pressures</a> in two chambers by exploiting <a href="Newton's_laws_of_motion" title="Newton's laws of motion">Newton's second law</a> to balance forces on an assembly of <a href="Piston" title="Piston">pistons</a>. The most common situation for a general-purpose analog computer is to add two <a href="Voltage" title="Voltage">voltages</a> (referenced to <a href="Ground_(electricity)" title="Ground (electricity)">ground</a>); this can be accomplished roughly with a <a href="Resistor" title="Resistor">resistor</a> <a href="Electronic_circuit" title="Electronic circuit">network</a>, but a better design exploits an <a href="Operational_amplifier" title="Operational amplifier">operational amplifier</a>.<a href="#cite_note-FOOTNOTETruittRogers19601,_44–49,_2,_77–78-50">[48]</a>

Addition is also fundamental to the operation of <a href="Computer" title="Computer">digital computers</a>, where the efficiency of addition, in particular the <a href="Carry_(arithmetic)" title="Carry (arithmetic)">carry</a> mechanism, is an important limitation to overall performance.<a href="#cite_note-FOOTNOTEGschwindMcCluskey1975[httpbooksgooglecombooksidVLmrCAAAQBAJpgPA233_233]-51">[49]</a>

The <a href="Abacus" title="Abacus">abacus</a>, also called a counting frame, is a calculating tool that was in use centuries before the adoption of the written modern numeral system and is still widely used by merchants, traders and clerks in <a href="Asia" title="Asia">Asia</a>, <a href="Africa" title="Africa">Africa</a>, and elsewhere; it dates back to at least 2700–2300 BC, when it was used in <a href="Sumer" title="Sumer">Sumer</a>.<a href="#cite_note-52">[50]</a>

<a href="Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a> invented the mechanical calculator in 1642;<a href="#cite_note-53">[51]</a> it was the first operational <a href="Adding_machine" title="Adding machine">adding machine</a>. <a href="Pascal's_calculator" class="mw-redirect" title="Pascal's calculator">Pascal's calculator</a> was limited by its gravity-assisted carry mechanism, which forced its wheels to only turn one way so it could add. To subtract, the operator had to use the <a href="Method_of_complements" title="Method of complements">Pascal's calculator's complement</a>, which required as many steps as an addition.<a href="#cite_note-54">[52]</a> <a href="Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a> built the <a href="Stepped_reckoner" title="Stepped reckoner">stepped reckoner</a>, another mechanical calculator, finished in 1694, and <a href="Giovanni_Poleni" title="Giovanni Poleni">Giovanni Poleni</a> improved on the design in 1709 with a calculating clock made of wood that could perform all four arithmetical operations. These early attempts were not commercially successful but inspired later mechanical calculators of the 19th century.<a href="#cite_note-55">[53]</a>

<a href="Adder_(electronics)" title="Adder (electronics)">Adders</a> execute integer addition in electronic digital computers, usually using <a href="Binary_arithmetic" class="mw-redirect" title="Binary arithmetic">binary arithmetic</a>. The simplest architecture is the ripple carry adder, which follows the standard multi-digit algorithm. One slight improvement is the <a href="Carry_bypass_adder" class="mw-redirect" title="Carry bypass adder">carry skip</a> design, again following human intuition; one does not perform all the carries in computing 999 + 1, but one bypasses the group of 9s and skips to the answer.<a href="#cite_note-FOOTNOTEFlynnOberman20012,_8-56">[54]</a>

In practice, computational addition may be achieved via <a href="Exclusive_or" title="Exclusive or">XOR</a> and <a href="Bitwise_operation#AND" title="Bitwise operation">AND</a> bitwise logical operations in conjunction with bitshift operations. Both XOR and AND gates are straightforward to realize in digital logic, allowing the realization of <a href="Adder_(electronics)" title="Adder (electronics)">full adder</a> circuits, which in turn may be combined into more complex logical operations. In modern digital computers, integer addition is typically the fastest arithmetic instruction, yet it has the largest impact on performance since it underlies all <a href="Floating-point_arithmetic" title="Floating-point arithmetic">floating-point operations</a> as well as such basic tasks as <a href="Memory_address" title="Memory address">address</a> generation during <a href="Memory_(computers)" class="mw-redirect" title="Memory (computers)">memory</a> access and fetching <a href="Instruction_(computer_science)" class="mw-redirect" title="Instruction (computer science)">instructions</a> during <a href="Control_flow" title="Control flow">branching</a>. To increase speed, modern designs calculate digits in <a href="Parallel_algorithm" title="Parallel algorithm">parallel</a>; these schemes go by such names as carry select, <a href="Carry_lookahead_adder" class="mw-redirect" title="Carry lookahead adder">carry lookahead</a>, and the <a href="Ling_adder" title="Ling adder">Ling</a> pseudocarry. Many implementations are, in fact, hybrids of these last three designs.<a href="#cite_note-FOOTNOTEFlynnOberman20011–9LiuTanSongChen2010194-57">[55]</a>

Some decimal computers in the late 1950s and early 1960s used add tables instead of adders, e.g., RCA 301,<a href="#cite_note-58">[56]</a> <a href="IBM_1620" title="IBM 1620">IBM 1620</a>.<a href="#cite_note-ModOne.1620-59">[57]</a>

Arithmetic implemented on a computer can deviate from the mathematical ideal in various ways. For example, if the result of an addition is too large for a computer to store, an <a href="Arithmetic_overflow" class="mw-redirect" title="Arithmetic overflow">arithmetic overflow</a> occurs, resulting in an error message and/or an incorrect answer. Unanticipated arithmetic overflow is a fairly common cause of <a href="Software_bug" title="Software bug">program errors</a>. Such overflow bugs may be hard to discover and diagnose because they may manifest themselves only for very large input data sets, which are less likely to be used in validation tests.<a href="#cite_note-60">[58]</a> The <a href="Year_2000_problem" title="Year 2000 problem">Year 2000 problem</a> was a series of bugs where overflow errors occurred due to the use of a 2-digit format for years.<a href="#cite_note-FOOTNOTENeumann1987-61">[59]</a>

Computers have another way of representing numbers, called <a href="Floating-point_arithmetic" title="Floating-point arithmetic">floating-point arithmetic</a>, which is similar to the scientific notation described above and which reduces the overflow problem. Each floating point number has two parts, an exponent and a mantissa. To add two floating-point numbers, the exponents must match, which typically means shifting the mantissa of the smaller number. If the disparity between the larger and smaller numbers is too great, a loss of precision may result. If many smaller numbers are to be added to a large number, it is best to add the smaller numbers together first and then add the total to the larger number, rather than adding small numbers to the large number one at a time. This makes floating-point addition non-associative in general.<a href="#cite_note-goldberg-62">[60]</a>

<div class="mw-heading mw-heading2"><h2 id="Addition_of_numbers">Addition of numbers</h2></div>

To prove the usual properties of addition, one must first define addition for the context in question. Addition is first defined on the <a href="Natural_number" title="Natural number">natural numbers</a>. In <a href="Set_theory" title="Set theory">set theory</a>, addition is then extended to progressively larger sets that include the natural numbers: the <a href="Integer" title="Integer">integers</a>, the <a href="Rational_number" title="Rational number">rational numbers</a>, and the <a href="Real_number" title="Real number">real numbers</a>.<a href="#cite_note-63">[61]</a> In <a href="Mathematics_education" title="Mathematics education">mathematics education</a>,<a href="#cite_note-65">[c]</a> positive fractions are added before negative numbers are even considered; this is also the historical route.<a href="#cite_note-66">[63]</a>

<div class="mw-heading mw-heading3"><h3 id="Natural_numbers">Natural numbers</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Natural_number" title="Natural number">Natural number</a></div>

There are two popular ways to define the sum of two natural numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy">. If one defines natural numbers to be the <a href="Cardinal_number" title="Cardinal number">cardinalities</a> of finite sets (the cardinality of a set is the number of elements in the set), then it is appropriate to define their sum as follows:<a href="#cite_note-FOOTNOTEBegle197549Johnson1975120DevineOlsonOlson199175-67">[64]</a>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3e3ea80ce89011c04b5ef106c35f78179384c6da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.372ex; height:2.843ex;" alt="{\displaystyle N(S)}" loading="lazy"> be the cardinality of a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">. Take two disjoint sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy">, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(A)=a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1f786657989e5bb121b90403f14d680499bf1ba5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.944ex; height:2.843ex;" alt="{\displaystyle N(A)=a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(B)=b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dccd11bf537e4cac12bcc89982ebab6b3d717eec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.733ex; height:2.843ex;" alt="{\displaystyle N(B)=b}" loading="lazy">. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a2391acf09244b9dba74eb940e871a6be7e7973a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a+b}" loading="lazy"> is defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(A\cup B)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(A\cup B)}</annotation>

</semantics>

</math><img src="./3facc2fbd8e65e33a64e9da652db03fcbda1fca7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.963ex; height:2.843ex;" alt="{\displaystyle N(A\cup B)}" loading="lazy">.

</blockquote>

Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cup B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cdb575990bcfbcdf616aa6fd76e8b30bf7fd2169.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\cup B}" loading="lazy"> means the <a href="Union_(set_theory)" title="Union (set theory)">union</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy">. An alternate version of this definition allows <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> to possibly overlap and then takes their <a href="Disjoint_union" title="Disjoint union">disjoint union</a>, a mechanism that allows common elements to be separated out and therefore counted twice.

The other popular definition is recursive:<a href="#cite_note-FOOTNOTEEnderton1977[httpbooksgooglecombooksidJlR-Ehk35XkCpgPA79_79]-68">[65]</a>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7524dec7439727959673a9272c5884243276e491.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.906ex; height:2.509ex;" alt="{\displaystyle n^{+}}" loading="lazy"> be the successor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, that is the number following <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> in the natural numbers, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{+}=1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0^{+}=1}</annotation>

</semantics>

</math><img src="./3142e49a7d964a02bdc03fc569bdc387f57dc135.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.934ex; height:2.509ex;" alt="{\displaystyle 0^{+}=1}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1^{+}=2}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./db96fe89d90013f33e3df10bb2228b39b66cec4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.934ex; height:2.509ex;" alt="{\displaystyle 1^{+}=2}" loading="lazy">. Define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+0=a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4564e28f0f8274644ca4e58664c0593ed48de541.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.561ex; height:2.343ex;" alt="{\displaystyle a+0=a}" loading="lazy">. Define the general sum recursively by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b^{+}=(a+b)^{+}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a+b^{+}=(a+b)^{+}}</annotation>

</semantics>

</math><img src="./88a9b8d0d1ea91e54afa590279fd0ebbd57864e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.065ex; height:3.009ex;" alt="{\displaystyle a+b^{+}=(a+b)^{+}}" loading="lazy">. Hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+1=1+0^{+}=(1+0)^{+}=1^{+}=2}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1+1=1+0^{+}=(1+0)^{+}=1^{+}=2}</annotation>

</semantics>

</math><img src="./09ce13d673fbdd65a499bd62083f0c287fe4ae5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.556ex; height:3.009ex;" alt="{\displaystyle 1+1=1+0^{+}=(1+0)^{+}=1^{+}=2}" loading="lazy">.

</blockquote>

Again, there are minor variations upon this definition in the literature. Taken literally, the above definition is an application of the <a href="Recursion#The_recursion_theorem" title="Recursion">recursion theorem</a> on the <a href="Partially_ordered_set" title="Partially ordered set">partially ordered set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./578dc62c2bb6d5e2b2624c6b58b02787df469372.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {N} ^{2}}" loading="lazy">.<a href="#cite_note-69">[66]</a> On the other hand, some sources prefer to use a restricted recursion theorem that applies only to the set of natural numbers. One then considers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> to be temporarily "fixed", applies recursion on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> to define a function "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a178498f0589f80da65e17b0aa8bf3bdb4869080.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.038ex; height:2.176ex;" alt="{\displaystyle a+}" loading="lazy">", and pastes these unary operations for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> together to form the full binary operation.<a href="#cite_note-70">[67]</a>

This recursive formulation of addition was developed by Dedekind as early as 1854, and he would expand upon it in the following decades. He proved the associative and commutative properties, among others, through <a href="Mathematical_induction" title="Mathematical induction">mathematical induction</a>.<a href="#cite_note-FOOTNOTEFerreirós1999223-71">[68]</a>

<div class="mw-heading mw-heading3"><h3 id="Integers">Integers</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Integer" title="Integer">Integer</a></div>

The simplest conception of an integer is that it consists of an <a href="Absolute_value" title="Absolute value">absolute value</a> (which is a natural number) and a <a href="Sign_(mathematics)" title="Sign (mathematics)">sign</a> (generally either <a href="Positive_number" class="mw-redirect" title="Positive number">positive</a> or <a href="Negative_numbers" class="mw-redirect" title="Negative numbers">negative</a>). The integer zero is a special third case, being neither positive nor negative. The corresponding definition of addition must proceed by cases:<a href="#cite_note-FOOTNOTESmith1980234SparksRees197966-72">[69]</a>

<blockquote class="templatequote">For an integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |n|}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./35a62139afd28f74d3306e3bf603bebdecefe169.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.688ex; height:2.843ex;" alt="{\displaystyle |n|}" loading="lazy"> be its absolute value. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> be integers. If either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> is zero, treat it as an identity. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> are both positive, define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b=|a|+|b|}">

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./55bf110589aa13db6127d4bbd066d77886589e7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.821ex; height:2.843ex;" alt="{\displaystyle a+b=|a|+|b|}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> are both negative, define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b=-(|a|+|b|)}">

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a+b=-(|a|+|b|)}</annotation>

</semantics>

</math><img src="./7ecd1c8ee24d1badcf4b585283485061d4a9c21a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.439ex; height:2.843ex;" alt="{\displaystyle a+b=-(|a|+|b|)}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> have different signs, define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a2391acf09244b9dba74eb940e871a6be7e7973a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a+b}" loading="lazy"> to be the difference between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a|}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8b61d5baa05004815f3abc52f517ce62b609b9b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.523ex; height:2.843ex;" alt="{\displaystyle |a|}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |b|}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./881f49e94388a46a05d329251551ce20baf4f05d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.291ex; height:2.843ex;" alt="{\displaystyle |b|}" loading="lazy">, with the sign of the term whose absolute value is larger.

</blockquote>

As an example, −6 + 4 = −2; because −6 and 4 have different signs, their absolute values are subtracted, and since the absolute value of the negative term is larger, the answer is negative.

Although this definition can be useful for concrete problems, the number of cases to consider complicates proofs unnecessarily. So the following method is commonly used for defining integers. It is based on the remark that every integer is the difference of two natural integers and that two such differences, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a-b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1b80866c2bf2f1bc1f2e4c97e7937f5663150ea6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a-b}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c-d}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cae892d55d0f7dd8e31a9de69a788fa910411f00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.063ex; height:2.343ex;" alt="{\displaystyle c-d}" loading="lazy"> are equal if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+d=b+c}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./452fe2e66cb2e61da56ebdb3f8b8521557709cf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.229ex; height:2.343ex;" alt="{\displaystyle a+d=b+c}" loading="lazy">. So, one can define formally the integers as the <a href="Equivalence_class" title="Equivalence class">equivalence classes</a> of <a href="Ordered_pair" title="Ordered pair">ordered pairs</a> of natural numbers under the <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)\sim (c,d)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a,b)\sim (c,d)}</annotation>

</semantics>

</math><img src="./f9eed2e08692e7ce42d13803af10157e550d6328.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.235ex; height:2.843ex;" alt="{\displaystyle (a,b)\sim (c,d)}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+d=b+c}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./452fe2e66cb2e61da56ebdb3f8b8521557709cf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.229ex; height:2.343ex;" alt="{\displaystyle a+d=b+c}" loading="lazy">.<a href="#cite_note-FOOTNOTECampbell1970[httpsarchiveorgdetailsstructureofarith00camppage83_83]-73">[70]</a> The equivalence class of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d7e5710198f33b00695903460983021e75860e2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}" loading="lazy"> contains either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a-b,0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a-b,0)}</annotation>

</semantics>

</math><img src="./d338865ad4db49d6cc42d3644f96587e7b9037a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.073ex; height:2.843ex;" alt="{\displaystyle (a-b,0)}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\geq b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ed5d3957d5f94566507526017e4ebb67c02efe81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle a\geq b}" loading="lazy">, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,b-a)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,b-a)}</annotation>

</semantics>

</math><img src="./21f778d142ef66765b84238913ab637e3cef485e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.073ex; height:2.843ex;" alt="{\displaystyle (0,b-a)}" loading="lazy"> if otherwise. Given that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is a natural number, then one can denote <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3b96423f11618625bf7c2f1af4e5fefa8de1e091.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.203ex; height:2.176ex;" alt="{\displaystyle +n}" loading="lazy"> the equivalence class of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n,0)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5fd8a7c3a302914ba5ae7cac4d8df11b59943934.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.4ex; height:2.843ex;" alt="{\displaystyle (n,0)}" loading="lazy">, and by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4f00139753ecf4fe00a10a17bd5620b70a61b29e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.203ex; height:2.176ex;" alt="{\displaystyle -n}" loading="lazy"> the equivalence class of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,n)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./fda9516a58dc3bd99e060e9ec8565620a57a3a9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.4ex; height:2.843ex;" alt="{\displaystyle (0,n)}" loading="lazy">. This allows identifying the natural number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> with the equivalence class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3b96423f11618625bf7c2f1af4e5fefa8de1e091.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.203ex; height:2.176ex;" alt="{\displaystyle +n}" loading="lazy">.

The addition of ordered pairs is done component-wise:<a href="#cite_note-FOOTNOTECampbell1970[httpsarchiveorgdetailsstructureofarith00camppage84_84]-74">[71]</a>

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<annotation encoding="application/x-tex">{\displaystyle (a,b)+(c,d)=(a+c,b+d).}</annotation>

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A straightforward computation shows that the equivalence class of the result depends only on the equivalence classes of the summands, and thus that this defines an addition of equivalence classes, that is, integers.<a href="#cite_note-FOOTNOTEEnderton1977[httpbooksgooglecombooksidJlR-Ehk35XkCpgPA92_92]-75">[72]</a> Another straightforward computation shows that this addition is the same as the above case definition.

<div class="mw-heading mw-heading3"><h3 id="Rational_numbers_(fractions)">Rational numbers (fractions)</h3></div>

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

Addition of <a href="Rational_number" title="Rational number">rational numbers</a> involves the <a href="Fraction" title="Fraction">fractions</a>. The computation can be done by using the <a href="Least_common_denominator" class="mw-redirect" title="Least common denominator">least common denominator</a>, but a conceptually simpler definition involves only integer addition and multiplication:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}.}</annotation>

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As an example, the sum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {3}{4}}+{\frac {1}{8}}={\frac {3\,\times \,8\,+\,4\,\times \,1}{4\times 8}}={\frac {24\,+\,4}{32}}={\frac {28}{32}}={\frac {7}{8}}}">

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<annotation encoding="application/x-tex">{\textstyle {\frac {3}{4}}+{\frac {1}{8}}={\frac {3\,\times \,8\,+\,4\,\times \,1}{4\times 8}}={\frac {24\,+\,4}{32}}={\frac {28}{32}}={\frac {7}{8}}}</annotation>

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</math><img src="./65b83081aabf80e50215bdad284e082529766024.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:38.326ex; height:3.843ex;" alt="{\textstyle {\frac {3}{4}}+{\frac {1}{8}}={\frac {3\,\times \,8\,+\,4\,\times \,1}{4\times 8}}={\frac {24\,+\,4}{32}}={\frac {28}{32}}={\frac {7}{8}}}" loading="lazy">.<a href="#cite_note-FOOTNOTECameronCraig201329-76">[73]</a>

Addition of fractions is much simpler when the <a href="Denominator" class="mw-redirect" title="Denominator">denominators</a> are the same; in this case, one can simply add the numerators while leaving the denominator the same:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {a}{c}}+{\frac {b}{c}}={\frac {a+b}{c}},}</annotation>

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so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{4}}+{\frac {2}{4}}={\frac {1\,+\,2}{4}}={\frac {3}{4}}}">

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<annotation encoding="application/x-tex">{\textstyle {\frac {1}{4}}+{\frac {2}{4}}={\frac {1\,+\,2}{4}}={\frac {3}{4}}}</annotation>

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</math><img src="./22dd4c9cc9f8b08b9e56f358a74101befe2ee76e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.544ex; height:3.676ex;" alt="{\textstyle {\frac {1}{4}}+{\frac {2}{4}}={\frac {1\,+\,2}{4}}={\frac {3}{4}}}" loading="lazy">.<a href="#cite_note-FOOTNOTECameronCraig201329-76">[73]</a>

The commutativity and associativity of rational addition are easy consequences of the laws of integer arithmetic.<a href="#cite_note-77">[74]</a>

<div class="mw-heading mw-heading3"><h3 id="Real_numbers">Real numbers</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Construction_of_the_real_numbers" title="Construction of the real numbers">Construction of the real numbers</a></div>

A common construction of the set of real numbers is the Dedekind completion of the set of rational numbers. A real number is defined to be a <a href="Dedekind_cut" title="Dedekind cut">Dedekind cut</a> of rationals: a <a href="Non-empty_set" class="mw-redirect" title="Non-empty set">non-empty set</a> of rationals that is closed downward and has no <a href="Greatest_element" class="mw-redirect" title="Greatest element">greatest element</a>. The sum of real numbers a and b is defined element by element:<a href="#cite_note-FOOTNOTEEnderton1977[httpbooksgooglecombooksidJlR-Ehk35XkCpgPA114_114]-78">[75]</a>

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<annotation encoding="application/x-tex">{\displaystyle a+b=\{q+r\mid q\in a,r\in b\}.}</annotation>

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This definition was first published, in a slightly modified form, by <a href="Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> in 1872.<a href="#cite_note-79">[76]</a>

The commutativity and associativity of real addition are immediate; defining the real number 0 as the set of negative rationals, it is easily seen as the additive identity. Probably the trickiest part of this construction pertaining to addition is the definition of additive inverses.<a href="#cite_note-80">[77]</a>

Unfortunately, dealing with the multiplication of Dedekind cuts is a time-consuming case-by-case process similar to the addition of signed integers.<a href="#cite_note-81">[78]</a> Another approach is the metric completion of the rational numbers. A real number is essentially defined to be the limit of a <a href="Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a> of rationals, lim an. Addition is defined term by term:<a href="#cite_note-82">[79]</a>

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<annotation encoding="application/x-tex">{\displaystyle \lim _{n}a_{n}+\lim _{n}b_{n}=\lim _{n}(a_{n}+b_{n}).}</annotation>

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This definition was first published by <a href="Georg_Cantor" title="Georg Cantor">Georg Cantor</a>, also in 1872, although his formalism was slightly different.<a href="#cite_note-FOOTNOTEFerreirós1999128-83">[80]</a>

One must prove that this operation is well-defined, dealing with co-Cauchy sequences. Once that task is done, all the properties of real addition follow immediately from the properties of rational numbers. Furthermore, the other arithmetic operations, including multiplication, have straightforward, analogous definitions.<a href="#cite_note-FOOTNOTEBurrill1967140-84">[81]</a>

<div class="mw-heading mw-heading3"><h3 id="Complex_numbers">Complex numbers</h3></div>

<a href="Complex_number" title="Complex number">Complex numbers</a> are added by adding the real and imaginary parts of the summands.<a href="#cite_note-85">[82]</a><a href="#cite_note-86">[83]</a> That is to say:

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<annotation encoding="application/x-tex">{\displaystyle (a+bi)+(c+di)=(a+c)+(b+d)i.}</annotation>

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</math><img src="./e7a659c86589cad841e37da3ce64af337ed5caab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.332ex; height:2.843ex;" alt="{\displaystyle (a+bi)+(c+di)=(a+c)+(b+d)i.}" loading="lazy"></dd></dl>

Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point X obtained by building a <a href="Parallelogram" title="Parallelogram">parallelogram</a> three of whose vertices are O, A and B.<a href="#cite_note-FOOTNOTEÖzhan2022[httpbooksgooglecombooksidIS51EAAAQBAJpgPA10_10]-87">[84]</a>

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

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Group_(mathematics)" title="Group (mathematics)">Group (mathematics)</a>, <a href="Ring_(mathematics)" title="Ring (mathematics)">Ring (mathematics)</a>, <a href="Module_(mathematics)" title="Module (mathematics)">Module (mathematics)</a>, <a href="Vector_space" title="Vector space">Vector space</a>, <a href="Field_(mathematics)" title="Field (mathematics)">Field (mathematics)</a>, and <a href="Algebra_over_a_field" title="Algebra over a field">Algebra over a field</a></div>

Many binary operations can be viewed as generalizations of the addition operation on the real numbers. The field of algebra is centrally concerned with such generalized operations, and they also appear in <a href="Set_theory" title="Set theory">set theory</a> and <a href="Category_theory" title="Category theory">category theory</a>.

<div class="mw-heading mw-heading3"><h3 id="Abelian_group">Abelian group</h3></div>

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

In <a href="Group_theory" title="Group theory">group theory</a>, a <a href="Group_(mathematics)" title="Group (mathematics)">Group</a> is an algebraic structure that allows for composing any two elements.

In the special case where the order does not matter, the composition operator is sometimes called addition. Such groups are referred to as Abelian or commutative; the composition operator is often written as "+".

<div class="mw-heading mw-heading3"><h3 id="Linear_algebra">Linear algebra</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Vector_addition" class="mw-redirect" title="Vector addition">Vector addition</a>, <a href="Matrix_addition" title="Matrix addition">Matrix addition</a>, <a href="Modular_arithmetic" title="Modular arithmetic">Modular arithmetic</a>, and <a href="Linear_combination" title="Linear combination">Linear combination</a></div>

In <a href="Linear_algebra" title="Linear algebra">linear algebra</a>, a <a href="Vector_space" title="Vector space">vector space</a> is an algebraic structure that allows for adding any two <a href="Coordinate_vector" title="Coordinate vector">vectors</a> and for scaling vectors. A familiar vector space is the set of all ordered pairs of real numbers; the ordered pair <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}">

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</math><img src="./d7e5710198f33b00695903460983021e75860e2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}" loading="lazy"> is interpreted as a vector from the origin in the Euclidean plane to the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}">

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

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</math><img src="./d7e5710198f33b00695903460983021e75860e2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}" loading="lazy"> in the plane. The sum of two vectors is obtained by adding their individual coordinates:

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<annotation encoding="application/x-tex">{\displaystyle (a,b)+(c,d)=(a+c,b+d).}</annotation>

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This addition operation is central to <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>, in which <a href="Velocity" title="Velocity">velocities</a>, <a href="Acceleration" title="Acceleration">accelerations</a> and <a href="Force" title="Force">forces</a> are all represented by vectors.<a href="#cite_note-FOOTNOTEGbur20111-88">[85]</a>

<a href="Matrix_addition" title="Matrix addition">Matrix addition</a> is defined for two matrices of the same dimensions. The sum of two m × n (pronounced "m by n") matrices A and B, denoted by A + B, is again an m × n matrix computed by adding corresponding elements:<a href="#cite_note-89">[86]</a><a href="#cite_note-90">[87]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {A} +\mathbf {B} &={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\\\end{bmatrix}}+{\begin{bmatrix}b_{11}&b_{12}&\cdots &b_{1n}\\b_{21}&b_{22}&\cdots &b_{2n}\\\vdots &\vdots &\ddots &\vdots \\b_{m1}&b_{m2}&\cdots &b_{mn}\\\end{bmatrix}}\\[8mu]&={\begin{bmatrix}a_{11}+b_{11}&a_{12}+b_{12}&\cdots &a_{1n}+b_{1n}\\a_{21}+b_{21}&a_{22}+b_{22}&\cdots &a_{2n}+b_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}+b_{m1}&a_{m2}+b_{m2}&\cdots &a_{mn}+b_{mn}\\\end{bmatrix}}\\\end{aligned}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\begin{bmatrix}1&3\\1&0\\1&2\end{bmatrix}}+{\begin{bmatrix}0&0\\7&5\\2&1\end{bmatrix}}&={\begin{bmatrix}1+0&3+0\\1+7&0+5\\1+2&2+1\end{bmatrix}}\\[8mu]&={\begin{bmatrix}1&3\\8&5\\3&3\end{bmatrix}}\end{aligned}}}</annotation>

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</math><img src="./dddc9dbb423d947a48a13619bf1c0d8cf527fe5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.338ex; width:40.195ex; height:19.843ex;" alt="{\displaystyle {\begin{aligned}{\begin{bmatrix}1&3\\1&0\\1&2\end{bmatrix}}+{\begin{bmatrix}0&0\\7&5\\2&1\end{bmatrix}}&={\begin{bmatrix}1+0&3+0\\1+7&0+5\\1+2&2+1\end{bmatrix}}\\[8mu]&={\begin{bmatrix}1&3\\8&5\\3&3\end{bmatrix}}\end{aligned}}}" loading="lazy"></dd></dl>

In <a href="Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a>, the set of available numbers is restricted to a finite subset of the integers, and addition "wraps around" when reaching a certain value, called the modulus.<a href="#cite_note-FOOTNOTEOmondi2020[httpbooksgooglecombooksid_m7NDwAAQBAJpgPA142_142]-91">[88]</a> For example, the set of integers modulo 12 has twelve elements; it inherits an addition operation from the integers that is central to <a href="Set_theory_(music)" title="Set theory (music)">musical set theory</a>.<a href="#cite_note-FOOTNOTEPrinceton_University_Press2008[httpbooksgooglecombooksidZOfUsvemJDMCpgPA938_938]-92">[89]</a> The set of integers modulo 2 has just two elements; the addition operation it inherits is known in <a href="Boolean_logic" class="mw-redirect" title="Boolean logic">Boolean logic</a> as the "<a href="Exclusive_or" title="Exclusive or">exclusive or</a>" function.<a href="#cite_note-FOOTNOTEPratt2017[httpbooksgooglecombooksidGbM-DgAAQBAJpgPA314_314]-93">[90]</a> A similar "wrap around" operation arises in <a href="Geometry" title="Geometry">geometry</a>, where the sum of two <a href="Angle" title="Angle">angle measures</a> is often taken to be their sum as real numbers modulo 2π. This amounts to an addition operation on the <a href="Circle" title="Circle">circle</a>, which in turn generalizes to the operations of higher-dimensional <a href="Lie_group" title="Lie group">Lie groups</a>.<a href="#cite_note-94">[91]</a>

The general theory of <a href="Abstract_algebra" title="Abstract algebra">abstract algebra</a> allows an "addition" operation to be any <a href="Associative" class="mw-redirect" title="Associative">associative</a> and <a href="Commutative" class="mw-redirect" title="Commutative">commutative</a> operation on a set. Basic <a href="Algebraic_structure" title="Algebraic structure">algebraic structures</a> with such an addition operation include <a href="Commutative_monoid" class="mw-redirect" title="Commutative monoid">commutative monoids</a> and <a href="Abelian_group" title="Abelian group">abelian groups</a>.<a href="#cite_note-FOOTNOTENicholson2012[httpbooksgooglecombooksidKNufK-P-6WoCpgPA70_70]BhattacharyaJainNagpaul1994[httpsbooksgooglecombooksidhiQ8e0b48swCpgPA159_159]-95">[92]</a>

<a href="Linear_combination" title="Linear combination">Linear combinations</a> combine multiplication and summation; they are sums in which each term has a multiplier, usually a <a href="Real_numbers" class="mw-redirect" title="Real numbers">real</a> or <a href="Complex_numbers" class="mw-redirect" title="Complex numbers">complex</a> number. Linear combinations are especially useful in contexts where straightforward addition would violate some normalization rule, such as <a href="Mixed_strategy" class="mw-redirect" title="Mixed strategy">mixing</a> of <a href="Strategy_(game_theory)" title="Strategy (game theory)">strategies</a> in <a href="Game_theory" title="Game theory">game theory</a> or <a href="Quantum_superposition" title="Quantum superposition">superposition</a> of <a href="Quantum_state" title="Quantum state">states</a> in <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>.<a href="#cite_note-FOOTNOTERieffelPolak201116-96">[93]</a>

<div class="mw-heading mw-heading3"><h3 id="Set_theory_and_category_theory">Set theory and category theory</h3></div>

A far-reaching generalization of the addition of natural numbers is the addition of <a href="Ordinal_number" title="Ordinal number">ordinal numbers</a> and <a href="Cardinal_number" title="Cardinal number">cardinal numbers</a> in set theory. These give two different generalizations of the addition of natural numbers to the <a href="Transfinite_number" title="Transfinite number">transfinite</a>. Unlike most addition operations, the addition of ordinal numbers is not commutative.<a href="#cite_note-FOOTNOTECheng2017124–132-97">[94]</a> Addition of cardinal numbers, however, is a commutative operation closely related to the <a href="Disjoint_union" title="Disjoint union">disjoint union</a> operation.<a href="#cite_note-FOOTNOTESchindler201434-98">[95]</a>

In <a href="Category_theory" title="Category theory">category theory</a>, disjoint union is seen as a particular case of the <a href="Coproduct" title="Coproduct">coproduct</a> operation,<a href="#cite_note-FOOTNOTERiehl2016100-99">[96]</a> and general coproducts are perhaps the most abstract of all the generalizations of addition. The coproduct such as <a href="Direct_sum" title="Direct sum">direct sum</a> is named to evoke their connection with addition.<a href="#cite_note-FOOTNOTEBhattacharyaJainNagpaul1994[httpsbooksgooglecombooksidhiQ8e0b48swCpgPA196_196]-100">[97]</a>

<div class="mw-heading mw-heading2"><h2 id="Related_operations">Related operations</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Arithmetic">Arithmetic</h3></div>

<a href="Subtraction" title="Subtraction">Subtraction</a> can be thought of as a kind of addition—that is, the addition of an <a href="Additive_inverse" title="Additive inverse">additive inverse</a>. Subtraction is itself a sort of inverse to addition, in that adding <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"> and subtracting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

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</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"> are <a href="Inverse_function" title="Inverse function">inverse functions</a>.<a href="#cite_note-FOOTNOTEKay2021[httpsbooksgooglecombooksidaw81EAAAQBAJpgPA44_44]-101">[98]</a> Given a set with an addition operation, one cannot always define a corresponding subtraction operation on that set; the set of natural numbers is a simple example. On the other hand, a subtraction operation uniquely determines an addition operation, an additive inverse operation, and an additive identity; for this reason, an additive group can be described as a set that is closed under subtraction.<a href="#cite_note-102">[99]</a>

<a href="Multiplication" title="Multiplication">Multiplication</a> can be thought of as <a href="Multiplication_and_repeated_addition" title="Multiplication and repeated addition">repeated addition</a>. If a single term x appears in a sum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> times, then the sum is the <a href="Product_(mathematics)" title="Product (mathematics)">product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> and x. Nonetheless, this works only for <a href="Natural_number" title="Natural number">natural numbers</a>.<a href="#cite_note-FOOTNOTEMusserPetersonBurger2013[httpsbooksgooglecombooksid8jh7DwAAQBAJpgPA101_101]-103">[100]</a> By the definition in general, multiplication is the operation between two numbers, called the multiplier and the multiplicand, that are combined into a single number called the product.<a href="#cite_note-FOOTNOTEIsodaOlfosNoine2021[httpbooksgooglecombooksidowcLEAAAQBAJpgPA163_163&ndash;164]-104">[101]</a>

In the real and complex numbers, addition and multiplication can be interchanged by the <a href="Exponential_function" title="Exponential function">exponential function</a>:<a href="#cite_note-FOOTNOTERudin1976178-105">[102]</a>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle e^{a+b}=e^{a}e^{b}.}</annotation>

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</math>

This identity allows multiplication to be carried out by consulting a <a href="Mathematical_table" title="Mathematical table">table</a> of <a href="Logarithm" title="Logarithm">logarithms</a> and computing addition by hand; it also enables multiplication on a <a href="Slide_rule" title="Slide rule">slide rule</a>. The formula is still a good first-order approximation in the broad context of <a href="Lie_group" title="Lie group">Lie groups</a>, where it relates multiplication of infinitesimal group elements with addition of vectors in the associated <a href="Lie_algebra" title="Lie algebra">Lie algebra</a>.<a href="#cite_note-FOOTNOTELee2003526Proposition_20.9-106">[103]</a>

There are even more generalizations of multiplication than addition.<a href="#cite_note-107">[104]</a> In general, multiplication operations always <a href="Distributivity" class="mw-redirect" title="Distributivity">distribute</a> over addition; this requirement is formalized in the definition of a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a>. In some contexts, integers, distributivity over addition, and the existence of a multiplicative identity are enough to determine the multiplication operation uniquely. The distributive property also provides information about the addition operation; by expanding the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+1)(a+b)}">

</mstyle>

</mrow>

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

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</math><img src="./5acba069a646bb109088be3153796966c2bf0fcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.852ex; height:2.843ex;" alt="{\displaystyle (1+1)(a+b)}" loading="lazy"> in both ways, one concludes that addition is forced to be commutative. For this reason, ring addition is commutative in general.<a href="#cite_note-108">[105]</a>

<a href="Division_(mathematics)" title="Division (mathematics)">Division</a> is an arithmetic operation remotely related to addition. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a/b=ab^{-1}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a/b=ab^{-1}}</annotation>

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</math><img src="./012d02f448f2eccf0983a017c00b38479ba8a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.048ex; height:3.176ex;" alt="{\displaystyle a/b=ab^{-1}}" loading="lazy">, division is right distributive over addition: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+b)/c=a/c+b/c}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a+b)/c=a/c+b/c}</annotation>

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</math><img src="./1c7b02f39b3ed42244933d9d447aba8771633423.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.551ex; height:2.843ex;" alt="{\displaystyle (a+b)/c=a/c+b/c}" loading="lazy">.<a href="#cite_note-109">[106]</a> However, division is not left distributive over addition, such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/(2+2)}">

</mrow>

</mstyle>

</mrow>

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

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</math><img src="./06d829bdb1cf286b8e62a861992bc086aae414d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.3ex; height:2.843ex;" alt="{\displaystyle 1/(2+2)}" loading="lazy"> is not the same as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/2+1/2}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

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</math><img src="./f9394dbcd47b8907463ef1537d379490ca7497bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.815ex; height:2.843ex;" alt="{\displaystyle 1/2+1/2}" loading="lazy">.<a href="#cite_note-110">[107]</a>

<div class="mw-heading mw-heading3"><h3 id="Ordering">Ordering</h3></div>

The maximum operation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max(a,b)}">

</mstyle>

</mrow>

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

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</math><img src="./0603f0969a6eb143615dc0339002987aa0dd9daa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.396ex; height:2.843ex;" alt="{\displaystyle \max(a,b)}" loading="lazy"> is a binary operation similar to addition. In fact, if two nonnegative numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> are of different <a href="Orders_of_magnitude" class="mw-redirect" title="Orders of magnitude">orders of magnitude</a>, their sum is approximately equal to their maximum. This approximation is extremely useful in the applications of mathematics, for example, in truncating <a href="Taylor_series" title="Taylor series">Taylor series</a>. However, it presents a perpetual difficulty in <a href="Numerical_analysis" title="Numerical analysis">numerical analysis</a>, essentially since "max" is not invertible. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> is much greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">, then a straightforward calculation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+b)-b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e611eb45467f3b1ec65b85fe38e7d7e6339ec780.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.715ex; height:2.843ex;" alt="{\displaystyle (a+b)-b}" loading="lazy"> can accumulate an unacceptable <a href="Round-off_error" title="Round-off error">round-off error</a>, perhaps even returning zero. See also <a href="Loss_of_significance" class="mw-redirect" title="Loss of significance">Loss of significance</a>.<a href="#cite_note-goldberg-62">[60]</a>

The approximation becomes exact in a kind of infinite limit; if either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> is an infinite <a href="Cardinal_number" title="Cardinal number">cardinal number</a>, their cardinal sum is exactly equal to the greater of the two.<a href="#cite_note-112">[d]</a> Accordingly, there is no subtraction operation for infinite cardinals.<a href="#cite_note-FOOTNOTEEnderton1977[httpbooksgooglecombooksidJlR-Ehk35XkCpgPA164_164]-113">[109]</a>

Maximization is commutative and associative, like addition. Furthermore, since addition preserves the ordering of real numbers, addition distributes over "max" in the same way that multiplication distributes over addition:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a+\max(b,c)=\max(a+b,a+c).}</annotation>

</semantics>

</math>

For these reasons, in <a href="Tropical_geometry" title="Tropical geometry">tropical geometry</a> one replaces multiplication with addition and addition with maximization. In this context, addition is called "tropical multiplication", maximization is called "tropical addition", and the tropical "additive identity" is <a href="Extended_real_number_line" title="Extended real number line">negative infinity</a>.<a href="#cite_note-FOOTNOTEMikhalkin20061-114">[110]</a> Some authors prefer to replace addition with minimization; then the additive identity is positive infinity.<a href="#cite_note-FOOTNOTEAkianBapatGaubert20054-115">[111]</a>

Tying these observations together, tropical addition is approximately related to regular addition through the <a href="Logarithm" title="Logarithm">logarithm</a>:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log(a+b)\approx \max(\log a,\log b),}</annotation>

</semantics>

</math>

which becomes more accurate as the base of the logarithm increases.<a href="#cite_note-FOOTNOTEMikhalkin20062-116">[112]</a> The approximation can be made exact by extracting a constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy">, named by analogy with the <a href="Planck_constant" title="Planck constant">Planck constant</a> from <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>,<a href="#cite_note-FOOTNOTELitvinovMaslovSobolevskii19993-117">[113]</a> and taking the "<a href="Classical_limit" title="Classical limit">classical limit</a>" as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> tends to zero:

</mrow>

</munder>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \max(a,b)=\lim _{h\to 0}h\log(e^{a/h}+e^{b/h}).}</annotation>

</semantics>

</math>

In this sense, the maximum operation is a dequantized version of addition.<a href="#cite_note-FOOTNOTEViro20014-118">[114]</a>

<div class="mw-heading mw-heading3"><h3 id="In_probability_theory">In probability theory</h3></div>

<a href="Convolution" title="Convolution">Convolution</a> is used to add two independent <a href="Random_variable" title="Random variable">random variables</a> defined by <a href="Probability_distribution" title="Probability distribution">distribution functions</a>. Its usual definition combines integration, subtraction, and multiplication.<a href="#cite_note-FOOTNOTEGbur2011300-119">[115]</a>

<ul><li><a href="Lunar_arithmetic" title="Lunar arithmetic">Lunar arithmetic</a>, a version of arithmetic with addition and multiplication replaced by digit-by-digit max and min</li>

<li><a href="Mental_calculation" title="Mental calculation">Mental arithmetic</a>, methods for performing addition without mechanical or written aid</li>

<li><a href="Minkowski_sum" class="mw-redirect" title="Minkowski sum">Minkowski sum</a>, an addition operation on geometric shapes</li>

<li><a href="Parallel_addition_(mathematics)" class="mw-redirect" title="Parallel addition (mathematics)">Parallel addition (mathematics)</a>, the reciprocal value of a sum of reciprocal values</li>

<li><a href="Prefix_sum" title="Prefix sum">Prefix sum</a>, computational problem of finding running totals</li>

<li><a href="Pythagorean_addition" title="Pythagorean addition">Pythagorean addition</a>, combining two side lengths of a right triangle to produce the length of the hypotenuse</li>

<li><a href="Verbal_arithmetic" title="Verbal arithmetic">Verbal arithmetic</a> (also known as cryptarithms), puzzles involving addition</li>

<li><a href="Velocity-addition_formula" title="Velocity-addition formula">Velocity-addition formula</a> for adding relativistic velocities</li></ul>

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

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> "Addend" is not a Latin word; in Latin it must be further conjugated, as in numerus addendus "the number to be added".

</li>

<li id="cite_note-45"><a href="#cite_ref-45">^</a> For example, <a href="Al-Khwarizmi" title="Al-Khwarizmi">al-Khwarizmi</a> performed multi-digit addition in this way from left to right.<a href="#cite_note-FOOTNOTECrossleyHenry1990-44">[43]</a>

</li>

<li id="cite_note-65"><a href="#cite_ref-65">^</a> This is according to a survey of the nations with highest TIMSS mathematics test scores.<a href="#cite_note-FOOTNOTESchmidtHouangCogan20024-64">[62]</a>

</li>

<li id="cite_note-112"><a href="#cite_ref-112">^</a> Enderton calls this statement the "Absorption Law of Cardinal Arithmetic"; it depends on the comparability of cardinals and therefore on the <a href="Axiom_of_Choice" class="mw-redirect" title="Axiom of Choice">Axiom of Choice</a>.

</li>

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

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

<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="#CITEREFEnderton1977">Enderton (1977)</a>, p. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA138">138</a>: "...select two sets K and L with card K = 2 and card L = 3. Sets of fingers are handy; sets of apples are preferred by textbooks."

</li>

<li id="cite_note-FOOTNOTEMusserPetersonBurger2013[httpsbooksgooglecombooksid8jh7DwAAQBAJpgPA87_87]-2">^ <a href="#cite_ref-FOOTNOTEMusserPetersonBurger2013[httpsbooksgooglecombooksid8jh7DwAAQBAJpgPA87_87]_2-0">a</a> <a href="#cite_ref-FOOTNOTEMusserPetersonBurger2013[httpsbooksgooglecombooksid8jh7DwAAQBAJpgPA87_87]_2-1">b</a> <a href="#cite_ref-FOOTNOTEMusserPetersonBurger2013[httpsbooksgooglecombooksid8jh7DwAAQBAJpgPA87_87]_2-2">c</a> <a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger (2013)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA87">87</a>.

</li>

<li id="cite_note-FOOTNOTEDevineOlsonOlson1991263-3"><a href="#cite_ref-FOOTNOTEDevineOlsonOlson1991263_3-0">^</a> <a href="#CITEREFDevineOlsonOlson1991">Devine, Olson & Olson (1991)</a>, p. 263.

</li>

<li id="cite_note-FOOTNOTEMazur2014161-4"><a href="#cite_ref-FOOTNOTEMazur2014161_4-0">^</a> <a href="#CITEREFMazur2014">Mazur (2014)</a>, p. 161.

</li>

<li id="cite_note-FOOTNOTEDepartment_of_the_Army1961[httpsarchiveorgdetailsTM11-684page16mode1upviewtheater_Section_5.1]-5"><a href="#cite_ref-FOOTNOTEDepartment_of_the_Army1961[httpsarchiveorgdetailsTM11-684page16mode1upviewtheater_Section_5.1]_5-0">^</a> <a href="#CITEREFDepartment_of_the_Army1961">Department of the Army (1961)</a>, <a rel="nofollow" class="external text" href="https://archive.org/details/TM11-684/page/16/mode/1up?view=theater">Section 5.1</a>.

</li>

<li id="cite_note-FOOTNOTEShmerkoYanushkevichLyshevski200980Schmid1974Schmid1983-6"><a href="#cite_ref-FOOTNOTEShmerkoYanushkevichLyshevski200980Schmid1974Schmid1983_6-0">^</a> <a href="#CITEREFShmerkoYanushkevichLyshevski2009">Shmerko, Yanushkevich & Lyshevski (2009)</a>, p. 80; <a href="#CITEREFSchmid1974">Schmid (1974)</a>; <a href="#CITEREFSchmid1983">Schmid (1983)</a>.

</li>

<li id="cite_note-FOOTNOTESchwartzman199419-7">^ <a href="#cite_ref-FOOTNOTESchwartzman199419_7-0">a</a> <a href="#cite_ref-FOOTNOTESchwartzman199419_7-1">b</a> <a href="#CITEREFSchwartzman1994">Schwartzman (1994)</a>, p. 19.

</li>

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</style><cite id="CITEREFSchubert1903" class="citation book cs1">Schubert, Hermann (1903). <a rel="nofollow" class="external text" href="https://archive.org/details/cu31924062544915/page/n25/">"Monism in Arithmetic"</a>. Mathematical Essays and Recreations. Chicago: Open Court. p. 10.</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <a href="#CITEREFKarpinski1925">Karpinski (1925)</a>, pp. 56–57, reproduced on p. 104

</li>

<li id="cite_note-FOOTNOTESchwartzman1994212-11"><a href="#cite_ref-FOOTNOTESchwartzman1994212_11-0">^</a> <a href="#CITEREFSchwartzman1994">Schwartzman (1994)</a>, p. 212.

</li>

<li id="cite_note-FOOTNOTEKarpinski1925150–153-12"><a href="#cite_ref-FOOTNOTEKarpinski1925150–153_12-0">^</a> <a href="#CITEREFKarpinski1925">Karpinski (1925)</a>, pp. 150–153.

</li>

<li id="cite_note-FOOTNOTELewis19741-13"><a href="#cite_ref-FOOTNOTELewis19741_13-0">^</a> <a href="#CITEREFLewis1974">Lewis (1974)</a>, p. 1.

</li>

<li id="cite_note-FOOTNOTEMartin200349-14"><a href="#cite_ref-FOOTNOTEMartin200349_14-0">^</a> <a href="#CITEREFMartin2003">Martin (2003)</a>, p. 49.

</li>

<li id="cite_note-FOOTNOTEStewart19998-15"><a href="#cite_ref-FOOTNOTEStewart19998_15-0">^</a> <a href="#CITEREFStewart1999">Stewart (1999)</a>, p. 8.

</li>

<li id="cite_note-FOOTNOTEApostol196737-16"><a href="#cite_ref-FOOTNOTEApostol196737_16-0">^</a> <a href="#CITEREFApostol1967">Apostol (1967)</a>, p. 37.

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> See <a href="#CITEREFViro2001">Viro (2001)</a> for an example of the sophistication involved in adding with sets of "fractional cardinality".

</li>

<li id="cite_note-FOOTNOTENational_Research_Council2001[httpbooksgooglecombooksidpvI7uDPo0-YCpgPA74_74]-18"><a href="#cite_ref-FOOTNOTENational_Research_Council2001[httpbooksgooglecombooksidpvI7uDPo0-YCpgPA74_74]_18-0">^</a> <a href="#CITEREFNational_Research_Council2001">National Research Council (2001)</a>, p. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=pvI7uDPo0-YC&pg=PA74">74</a>.

</li>

<li id="cite_note-FOOTNOTEMosley2001[httpbooksgooglecombooksidI-_WcWjemUCpgPA8_8]-19"><a href="#cite_ref-FOOTNOTEMosley2001[httpbooksgooglecombooksidI-_WcWjemUCpgPA8_8]_19-0">^</a> <a href="#CITEREFMosley2001">Mosley (2001)</a>, p. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=I-__WcWjemUC&pg=PA8">8</a>.

</li>

<li id="cite_note-FOOTNOTELiLappan2014204-20"><a href="#cite_ref-FOOTNOTELiLappan2014204_20-0">^</a> <a href="#CITEREFLiLappan2014">Li & Lappan (2014)</a>, p. 204.

</li>

<li id="cite_note-FOOTNOTEMusserPetersonBurger2013[httpsbooksgooglecombooksid8jh7DwAAQBAJpgPA89_89]-21"><a href="#cite_ref-FOOTNOTEMusserPetersonBurger2013[httpsbooksgooglecombooksid8jh7DwAAQBAJpgPA89_89]_21-0">^</a> <a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger (2013)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA89">89</a>.

</li>

<li id="cite_note-FOOTNOTEBerg1967[httpsbooksgooglecombooksidaGXFCUaFCW0CpgPA14_14]-22"><a href="#cite_ref-FOOTNOTEBerg1967[httpsbooksgooglecombooksidaGXFCUaFCW0CpgPA14_14]_22-0">^</a> <a href="#CITEREFBerg1967">Berg (1967)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aGXFCUaFCW0C&pg=PA14">14</a>.

</li>

<li id="cite_note-FOOTNOTEBehrJungst1971[httpsbooksgooglecombooksidGJXOBQAAQBAJpgPA59_59]-23"><a href="#cite_ref-FOOTNOTEBehrJungst1971[httpsbooksgooglecombooksidGJXOBQAAQBAJpgPA59_59]_23-0">^</a> <a href="#CITEREFBehrJungst1971">Behr & Jungst (1971)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GJXOBQAAQBAJ&pg=PA59">59</a>.

</li>

<li id="cite_note-FOOTNOTERosen2013See_the_[httpsbooksgooglecombooksid-oVvEAAAQBAJpgSL1-PA1_Appendix_I]-24"><a href="#cite_ref-FOOTNOTERosen2013See_the_[httpsbooksgooglecombooksid-oVvEAAAQBAJpgSL1-PA1_Appendix_I]_24-0">^</a> <a href="#CITEREFRosen2013">Rosen (2013)</a>, See the <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-oVvEAAAQBAJ&pg=SL1-PA1">Appendix I</a>.

</li>

<li id="cite_note-FOOTNOTEPosamentierFarberGermain-WilliamsParis2013[httpsbooksgooglecombooksidVfCgAQAAQBAJpgPA71_71]-25"><a href="#cite_ref-FOOTNOTEPosamentierFarberGermain-WilliamsParis2013[httpsbooksgooglecombooksidVfCgAQAAQBAJpgPA71_71]_25-0">^</a> <a href="#CITEREFPosamentierFarberGermain-WilliamsParis2013">Posamentier et al. (2013)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VfCgAQAAQBAJ&pg=PA71">71</a>.

</li>

<li id="cite_note-FOOTNOTEMusserPetersonBurger2013[httpsbooksgooglecombooksid8jh7DwAAQBAJpgPA90_90]-26">^ <a href="#cite_ref-FOOTNOTEMusserPetersonBurger2013[httpsbooksgooglecombooksid8jh7DwAAQBAJpgPA90_90]_26-0">a</a> <a href="#cite_ref-FOOTNOTEMusserPetersonBurger2013[httpsbooksgooglecombooksid8jh7DwAAQBAJpgPA90_90]_26-1">b</a> <a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger (2013)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA90">90</a>.

</li>

<li id="cite_note-FOOTNOTEBronsteinSemendjajew1987-27"><a href="#cite_ref-FOOTNOTEBronsteinSemendjajew1987_27-0">^</a> <a href="#CITEREFBronsteinSemendjajew1987">Bronstein & Semendjajew (1987)</a>.

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<li id="cite_note-FOOTNOTEKaplan200069–71-28"><a href="#cite_ref-FOOTNOTEKaplan200069–71_28-0">^</a> <a href="#CITEREFKaplan2000">Kaplan (2000)</a>, pp. 69–71.

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<li id="cite_note-FOOTNOTEHempel2001[httpbooksgooglecombooksidyTY9La4P2n8CpgPA7_7]-29"><a href="#cite_ref-FOOTNOTEHempel2001[httpbooksgooglecombooksidyTY9La4P2n8CpgPA7_7]_29-0">^</a> <a href="#CITEREFHempel2001">Hempel (2001)</a>, p. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=yTY9La4P2n8C&pg=PA7">7</a>.

</li>

<li id="cite_note-FOOTNOTEFierro2012Section_2.3-30"><a href="#cite_ref-FOOTNOTEFierro2012Section_2.3_30-0">^</a> <a href="#CITEREFFierro2012">Fierro (2012)</a>, Section 2.3.

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<li id="cite_note-31"><a href="#cite_ref-31">^</a> <cite id="CITEREFMoebsLing2022" class="citation book cs1">Moebs, William; et al. (2022). "1.4 Dimensional Analysis". <a rel="nofollow" class="external text" href="https://openstax.org/books/university-physics-volume-1/pages/1-4-dimensional-analysis">University Physics Volume 1</a>. <a href="OpenStax" title="OpenStax">OpenStax</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-947172-20-3</bdi>.</cite>

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<li id="cite_note-FOOTNOTEWynn19985-32"><a href="#cite_ref-FOOTNOTEWynn19985_32-0">^</a> <a href="#CITEREFWynn1998">Wynn (1998)</a>, p. 5.

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<li id="cite_note-FOOTNOTEWynn199815-33"><a href="#cite_ref-FOOTNOTEWynn199815_33-0">^</a> <a href="#CITEREFWynn1998">Wynn (1998)</a>, p. 15.

</li>

<li id="cite_note-FOOTNOTEWynn199817-34"><a href="#cite_ref-FOOTNOTEWynn199817_34-0">^</a> <a href="#CITEREFWynn1998">Wynn (1998)</a>, p. 17.

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<li id="cite_note-FOOTNOTEWynn199819-35"><a href="#cite_ref-FOOTNOTEWynn199819_35-0">^</a> <a href="#CITEREFWynn1998">Wynn (1998)</a>, p. 19.

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<li id="cite_note-36"><a href="#cite_ref-36">^</a> <cite id="CITEREFRanderson2008" class="citation news cs1">Randerson, James (21 August 2008). <a rel="nofollow" class="external text" href="https://www.theguardian.com/science/2008/aug/21/elephants.arithmetic">"Elephants have a head for figures"</a>. The Guardian. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150402103526/http://www.theguardian.com/science/2008/aug/21/elephants.arithmetic">Archived</a> from the original on 2 April 2015. Retrieved 29 March 2015.</cite>

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<li id="cite_note-FOOTNOTESmith2002130-37"><a href="#cite_ref-FOOTNOTESmith2002130_37-0">^</a> <a href="#CITEREFSmith2002">Smith (2002)</a>, p. 130.

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<li id="cite_note-38"><a href="#cite_ref-38">^</a> <cite id="CITEREFCarpenterFennema,_ElizabethFranke,_Megan_LoefLevi,_Linda1999" class="citation book cs1">Carpenter, Thomas; <a href="Elizabeth_Fennema" title="Elizabeth Fennema">Fennema, Elizabeth</a>; Franke, Megan Loef; Levi, Linda; <a href="Susan_Empson" title="Susan Empson">Empson, Susan</a> (1999). <a rel="nofollow" class="external text" href="https://archive.org/details/childrensmathema0000unse_i5h7">Children's mathematics: Cognitively guided instruction</a>. Portsmouth, NH: Heinemann. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-325-00137-1</bdi>.</cite>

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<li id="cite_note-Henry-39">^ <a href="#cite_ref-Henry_39-0">a</a> <a href="#cite_ref-Henry_39-1">b</a> <cite id="CITEREFHenryBrown,_Richard_S.2008" class="citation journal cs1">Henry, Valerie J.; Brown, Richard S. (2008). <a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F30034895">"First-grade basic facts: An investigation into teaching and learning of an accelerated, high-demand memorization standard"</a>. Journal for Research in Mathematics Education. 39 (2): 153–183. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F30034895">10.2307/30034895</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/30034895">30034895</a>.</cite>

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Beckmann, S. (2014). The twenty-third ICMI study: primary mathematics study on whole numbers. International Journal of STEM Education, 1(1), 1–8.

Chicago

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<li id="cite_note-41"><a href="#cite_ref-41">^</a> Schmidt, W., Houang, R., & Cogan, L. (2002). "A coherent curriculum". American Educator, 26(2), 1–18.

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<li id="cite_note-FOOTNOTEFosnotDolk200199-42">^ <a href="#cite_ref-FOOTNOTEFosnotDolk200199_42-0">a</a> <a href="#cite_ref-FOOTNOTEFosnotDolk200199_42-1">b</a> <a href="#cite_ref-FOOTNOTEFosnotDolk200199_42-2">c</a> <a href="#cite_ref-FOOTNOTEFosnotDolk200199_42-3">d</a> <a href="#cite_ref-FOOTNOTEFosnotDolk200199_42-4">e</a> <a href="#cite_ref-FOOTNOTEFosnotDolk200199_42-5">f</a> <a href="#cite_ref-FOOTNOTEFosnotDolk200199_42-6">g</a> <a href="#CITEREFFosnotDolk2001">Fosnot & Dolk (2001)</a>, p. 99.

</li>

<li id="cite_note-43"><a href="#cite_ref-43">^</a> Some authors think that "carry" may be inappropriate for education; <a href="#CITEREFvan_de_Walle2004">van de Walle (2004)</a>, p. 211 calls it "obsolete and conceptually misleading", preferring the word "trade". However, "carry" remains the standard term.

</li>

<li id="cite_note-FOOTNOTECrossleyHenry1990-44"><a href="#cite_ref-FOOTNOTECrossleyHenry1990_44-0">^</a> <a href="#CITEREFCrossleyHenry1990">Crossley & Henry (1990)</a>.

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<li id="cite_note-FOOTNOTEWingard-Nelson2014[httpsarchiveorgdetailsdecimalsfraction0000wing_v8b6page40mode1upqdecimalviewtheater_40]-46"><a href="#cite_ref-FOOTNOTEWingard-Nelson2014[httpsarchiveorgdetailsdecimalsfraction0000wing_v8b6page40mode1upqdecimalviewtheater_40]_46-0">^</a> <a href="#CITEREFWingard-Nelson2014">Wingard-Nelson (2014)</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/decimalsfraction0000wing_v8b6/page/40/mode/1up?q=decimal&view=theater">40</a>.

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<li id="cite_note-47"><a href="#cite_ref-47">^</a> <cite id="CITEREFCassidyHoltonRutherford2002" class="citation book cs1">Cassidy, David; Holton, Gerald; Rutherford, James (2002). "Reviewing Units, Mathematics, and Scientific Notation". Understanding Physics. New York: Springer. p. 11. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F0-387-21660-X_3">10.1007/0-387-21660-X_3</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-98755-2</bdi>.</cite>

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<li id="cite_note-48"><a href="#cite_ref-48">^</a> Dale R. Patrick, Stephen W. Fardo, Vigyan Chandra (2008) Electronic Digital System Fundamentals The Fairmont Press, Inc. p. 155

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<li id="cite_note-49"><a href="#cite_ref-49">^</a> P.E. Bates Bothman (1837) The common school arithmetic. Henry Benton. p. 31

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<li id="cite_note-FOOTNOTETruittRogers19601,_44–49,_2,_77–78-50"><a href="#cite_ref-FOOTNOTETruittRogers19601,_44–49,_2,_77–78_50-0">^</a> <a href="#CITEREFTruittRogers1960">Truitt & Rogers (1960)</a>, pp. 1, 44–49, 2, 77–78.

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<li id="cite_note-FOOTNOTEGschwindMcCluskey1975[httpbooksgooglecombooksidVLmrCAAAQBAJpgPA233_233]-51"><a href="#cite_ref-FOOTNOTEGschwindMcCluskey1975[httpbooksgooglecombooksidVLmrCAAAQBAJpgPA233_233]_51-0">^</a> <a href="#CITEREFGschwindMcCluskey1975">Gschwind & McCluskey (1975)</a>, p. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=VLmrCAAAQBAJ&pg=PA233">233</a>.

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<li id="cite_note-52"><a href="#cite_ref-52">^</a> <cite id="CITEREFIfrah2001" class="citation book cs1">Ifrah, Georges (2001). <a rel="nofollow" class="external text" href="https://archive.org/details/unset0000unse_w3q2">The Universal History of Computing: From the Abacus to the Quantum Computer</a>. New York: Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-39671-0</bdi>.</cite> p. 11

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<li id="cite_note-53"><a href="#cite_ref-53">^</a> <a href="#CITEREFMarguin1994">Marguin (1994)</a>, p. 48. Quoting <a href="#CITEREFTaton1963">Taton (1963)</a>.

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<li id="cite_note-54"><a href="#cite_ref-54">^</a> <cite id="CITEREFKistermann1998" class="citation journal cs1">Kistermann, F. W. (1998). "Blaise Pascal's adding machine: new findings and conclusions". IEEE Annals of the History of Computing. 20 (1): 69–76. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F85.646211">10.1109/85.646211</a>.</cite>

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<li id="cite_note-55"><a href="#cite_ref-55">^</a> <cite id="CITEREFCampanile2024" class="citation book cs1 cs1-prop-foreign-lang-source">Campanile, Benedetta (2024). "La girandola di Poleni: un progetto destinato a scomparire". In Di Mauro, Marco; Romano, Luigi; Zanini, Valeria (eds.). Atti del XLIII Convegno annuale SISFA (in Italian). pp. 151–158. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.6093%2F978-88-6887-278-6">10.6093/978-88-6887-278-6</a>.</cite>

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<li id="cite_note-FOOTNOTEFlynnOberman20012,_8-56"><a href="#cite_ref-FOOTNOTEFlynnOberman20012,_8_56-0">^</a> <a href="#CITEREFFlynnOberman2001">Flynn & Oberman (2001)</a>, pp. 2, 8.

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<li id="cite_note-FOOTNOTEFlynnOberman20011–9LiuTanSongChen2010194-57"><a href="#cite_ref-FOOTNOTEFlynnOberman20011–9LiuTanSongChen2010194_57-0">^</a> <a href="#CITEREFFlynnOberman2001">Flynn & Oberman (2001)</a>, pp. 1–9; <a href="#CITEREFLiuTanSongChen2010">Liu et al. (2010)</a>, p. 194.

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<li id="cite_note-58"><a href="#cite_ref-58">^</a> <cite class="citation book cs1"><a rel="nofollow" class="external text" href="http://bitsavers.org/pdf/rca/301/93-17-000_RCA_301_Programmers_Reference_Manual_Jan62.pdf">301 - Programmer's Reference Manual</a> (PDF). January 1962. 93-17-000. Retrieved July 9, 2025.</cite>

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<li id="cite_note-ModOne.1620-59"><a href="#cite_ref-ModOne.1620_59-0">^</a> <cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20171009212302/http://www.bitsavers.org/pdf/ibm/1620/A26-5706-3_IBM_1620_CPU_Model_1_Jul65.pdf">IBM 1620 Central Processing Unit, Model 1</a> (PDF). Archived from <a rel="nofollow" class="external text" href="https://www.bitsavers.org/pdf/ibm/1620/A26-5706-3_IBM_1620_CPU_Model_1_Jul65.pdf">the original</a> (PDF) on 2017-10-09. Retrieved 2017-12-18.</cite>

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<li id="cite_note-60"><a href="#cite_ref-60">^</a> Joshua Bloch, <a rel="nofollow" class="external text" href="http://googleresearch.blogspot.com/2006/06/extra-extra-read-all-about-it-nearly.html">"Extra, Extra – Read All About It: Nearly All Binary Searches and Mergesorts are Broken"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160401140544/http://googleresearch.blogspot.com/2006/06/extra-extra-read-all-about-it-nearly.html">Archived</a> 2016-04-01 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Official Google Research Blog, June 2, 2006.

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<li id="cite_note-FOOTNOTENeumann1987-61"><a href="#cite_ref-FOOTNOTENeumann1987_61-0">^</a> <a href="#CITEREFNeumann1987">Neumann (1987)</a>.

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<li id="cite_note-goldberg-62">^ <a href="#cite_ref-goldberg_62-0">a</a> <a href="#cite_ref-goldberg_62-1">b</a> <cite id="CITEREFGoldberg1991" class="citation journal cs1">Goldberg, David (March 1991). <a rel="nofollow" class="external text" href="https://scholar.archive.org/work/xkhddnsu4bd4nnn7zdpykybiea">"What every computer scientist should know about floating-point arithmetic"</a>. ACM Computing Surveys. 23 (1). Association for Computing Machinery (ACM): 5–48. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F103162.103163">10.1145/103162.103163</a>.</cite>

</li>

<li id="cite_note-63"><a href="#cite_ref-63">^</a> <a href="Herbert_Enderton" title="Herbert Enderton">Enderton</a> chapters 4 and 5, for example, follow this development.

</li>

<li id="cite_note-FOOTNOTESchmidtHouangCogan20024-64"><a href="#cite_ref-FOOTNOTESchmidtHouangCogan20024_64-0">^</a> <a href="#CITEREFSchmidtHouangCogan2002">Schmidt, Houang & Cogan (2002)</a>, p. 4.

</li>

<li id="cite_note-66"><a href="#cite_ref-66">^</a> <a href="#CITEREFBaezDolan2001">Baez & Dolan (2001)</a>, p. 37 explains the historical development, in "stark contrast" with the set theory presentation: "Apparently, half an apple is easier to understand than a negative apple!"

</li>

<li id="cite_note-FOOTNOTEBegle197549Johnson1975120DevineOlsonOlson199175-67"><a href="#cite_ref-FOOTNOTEBegle197549Johnson1975120DevineOlsonOlson199175_67-0">^</a> <a href="#CITEREFBegle1975">Begle (1975)</a>, p. 49; <a href="#CITEREFJohnson1975">Johnson (1975)</a>, p. 120; <a href="#CITEREFDevineOlsonOlson1991">Devine, Olson & Olson (1991)</a>, p. 75.

</li>

<li id="cite_note-FOOTNOTEEnderton1977[httpbooksgooglecombooksidJlR-Ehk35XkCpgPA79_79]-68"><a href="#cite_ref-FOOTNOTEEnderton1977[httpbooksgooglecombooksidJlR-Ehk35XkCpgPA79_79]_68-0">^</a> <a href="#CITEREFEnderton1977">Enderton (1977)</a>, p. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA79">79</a>.

</li>

<li id="cite_note-69"><a href="#cite_ref-69">^</a> For a version that applies to any poset with the <a href="Descending_chain_condition" class="mw-redirect" title="Descending chain condition">descending chain condition</a>, see <a href="#CITEREFBergman2005">Bergman (2005)</a>, p. 100

</li>

<li id="cite_note-70"><a href="#cite_ref-70">^</a> <a href="#CITEREFEnderton1977">Enderton (1977)</a>, p. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA79">79</a> observes, "But we want one binary operation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./fe6ef363cd19902d1a7a71fb1c8b21e8ede52406.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" loading="lazy">, not all these little one-place functions."

</li>

<li id="cite_note-FOOTNOTEFerreirós1999223-71"><a href="#cite_ref-FOOTNOTEFerreirós1999223_71-0">^</a> <a href="#CITEREFFerreirós1999">Ferreirós (1999)</a>, p. 223.

</li>

<li id="cite_note-FOOTNOTESmith1980234SparksRees197966-72"><a href="#cite_ref-FOOTNOTESmith1980234SparksRees197966_72-0">^</a> <a href="#CITEREFSmith1980">Smith (1980)</a>, p. 234; <a href="#CITEREFSparksRees1979">Sparks & Rees (1979)</a>, p. 66.

</li>

<li id="cite_note-FOOTNOTECampbell1970[httpsarchiveorgdetailsstructureofarith00camppage83_83]-73"><a href="#cite_ref-FOOTNOTECampbell1970[httpsarchiveorgdetailsstructureofarith00camppage83_83]_73-0">^</a> <a href="#CITEREFCampbell1970">Campbell (1970)</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/structureofarith00camp/page/83">83</a>.

</li>

<li id="cite_note-FOOTNOTECampbell1970[httpsarchiveorgdetailsstructureofarith00camppage84_84]-74"><a href="#cite_ref-FOOTNOTECampbell1970[httpsarchiveorgdetailsstructureofarith00camppage84_84]_74-0">^</a> <a href="#CITEREFCampbell1970">Campbell (1970)</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/structureofarith00camp/page/84">84</a>.

</li>

<li id="cite_note-FOOTNOTEEnderton1977[httpbooksgooglecombooksidJlR-Ehk35XkCpgPA92_92]-75"><a href="#cite_ref-FOOTNOTEEnderton1977[httpbooksgooglecombooksidJlR-Ehk35XkCpgPA92_92]_75-0">^</a> <a href="#CITEREFEnderton1977">Enderton (1977)</a>, p. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA92">92</a>.

</li>

<li id="cite_note-FOOTNOTECameronCraig201329-76">^ <a href="#cite_ref-FOOTNOTECameronCraig201329_76-0">a</a> <a href="#cite_ref-FOOTNOTECameronCraig201329_76-1">b</a> <a href="#CITEREFCameronCraig2013">Cameron & Craig (2013)</a>, p. 29.

</li>

<li id="cite_note-77"><a href="#cite_ref-77">^</a> The verifications are carried out in <a href="#CITEREFEnderton1977">Enderton (1977)</a>, p. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA104">104</a> and sketched for a general field of fractions over a commutative ring in <a href="#CITEREFDummitFoote1999">Dummit & Foote (1999)</a>, p. 263.

</li>

<li id="cite_note-FOOTNOTEEnderton1977[httpbooksgooglecombooksidJlR-Ehk35XkCpgPA114_114]-78"><a href="#cite_ref-FOOTNOTEEnderton1977[httpbooksgooglecombooksidJlR-Ehk35XkCpgPA114_114]_78-0">^</a> <a href="#CITEREFEnderton1977">Enderton (1977)</a>, p. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA114">114</a>.

</li>

<li id="cite_note-79"><a href="#cite_ref-79">^</a> <a href="#CITEREFFerreirós1999">Ferreirós (1999)</a>, p. 135; see section 6 of <a rel="nofollow" class="external text" href="http://www.ru.nl/w-en-s/gmfw/bronnen/dedekind2.html">Stetigkeit und irrationale Zahlen</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20051031071536/http://www.ru.nl/w-en-s/gmfw/bronnen/dedekind2.html">Archived</a> 2005-10-31 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.

</li>

<li id="cite_note-80"><a href="#cite_ref-80">^</a> The intuitive approach, inverting every element of a cut and taking its complement, works only for irrational numbers; see <a href="#CITEREFEnderton1977">Enderton (1977)</a>, p. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA117">117</a> for details.

</li>

<li id="cite_note-81"><a href="#cite_ref-81">^</a> Schubert, E. Thomas, Phillip J. Windley, and James Alves-Foss. "Higher Order Logic Theorem Proving and Its Applications: Proceedings of the 8th International Workshop, volume 971 of." Lecture Notes in Computer Science (1995).

</li>

<li id="cite_note-82"><a href="#cite_ref-82">^</a> Textbook constructions are usually not so cavalier with the "lim" symbol; see <a href="#CITEREFBurrill1967">Burrill (1967)</a>, p. 138 for a more careful, drawn-out development of addition with Cauchy sequences.

</li>

<li id="cite_note-FOOTNOTEFerreirós1999128-83"><a href="#cite_ref-FOOTNOTEFerreirós1999128_83-0">^</a> <a href="#CITEREFFerreirós1999">Ferreirós (1999)</a>, p. 128.

</li>

<li id="cite_note-FOOTNOTEBurrill1967140-84"><a href="#cite_ref-FOOTNOTEBurrill1967140_84-0">^</a> <a href="#CITEREFBurrill1967">Burrill (1967)</a>, p. 140.

</li>

<li id="cite_note-85"><a href="#cite_ref-85">^</a> <cite id="CITEREFConway1986" class="citation cs2">Conway, John B. (1986), Functions of One Complex Variable I, Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90328-6</bdi></cite>

</li>

<li id="cite_note-86"><a href="#cite_ref-86">^</a> <cite id="CITEREFJoshi1989" class="citation cs2">Joshi, Kapil D (1989), Foundations of Discrete Mathematics, New York: Wiley, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-470-21152-6</bdi></cite>

</li>

<li id="cite_note-FOOTNOTEÖzhan2022[httpbooksgooglecombooksidIS51EAAAQBAJpgPA10_10]-87"><a href="#cite_ref-FOOTNOTEÖzhan2022[httpbooksgooglecombooksidIS51EAAAQBAJpgPA10_10]_87-0">^</a> <a href="#CITEREFÖzhan2022">Özhan (2022)</a>, p. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=IS51EAAAQBAJ&pg=PA10">10</a>.

</li>

<li id="cite_note-FOOTNOTEGbur20111-88"><a href="#cite_ref-FOOTNOTEGbur20111_88-0">^</a> <a href="#CITEREFGbur2011">Gbur (2011)</a>, p. 1.

</li>

<li id="cite_note-89"><a href="#cite_ref-89">^</a> Lipschutz, S., & Lipson, M. (2001). Schaum's outline of theory and problems of linear algebra. Erlangga.

</li>

<li id="cite_note-90"><a href="#cite_ref-90">^</a> <cite id="CITEREFRileyHobsonBence2010" class="citation book cs1">Riley, K.F.; Hobson, M.P.; Bence, S.J. (2010). <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalmeth00rile">Mathematical methods for physics and engineering</a>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-86153-3</bdi>.</cite>

</li>

<li id="cite_note-FOOTNOTEOmondi2020[httpbooksgooglecombooksid_m7NDwAAQBAJpgPA142_142]-91"><a href="#cite_ref-FOOTNOTEOmondi2020[httpbooksgooglecombooksid_m7NDwAAQBAJpgPA142_142]_91-0">^</a> <a href="#CITEREFOmondi2020">Omondi (2020)</a>, p. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=_m7NDwAAQBAJ&pg=PA142">142</a>.

</li>

<li id="cite_note-FOOTNOTEPrinceton_University_Press2008[httpbooksgooglecombooksidZOfUsvemJDMCpgPA938_938]-92"><a href="#cite_ref-FOOTNOTEPrinceton_University_Press2008[httpbooksgooglecombooksidZOfUsvemJDMCpgPA938_938]_92-0">^</a> <a href="#CITEREFPrinceton_University_Press2008">Princeton University Press (2008)</a>, p. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=ZOfUsvemJDMC&pg=PA938">938</a>.

</li>

<li id="cite_note-FOOTNOTEPratt2017[httpbooksgooglecombooksidGbM-DgAAQBAJpgPA314_314]-93"><a href="#cite_ref-FOOTNOTEPratt2017[httpbooksgooglecombooksidGbM-DgAAQBAJpgPA314_314]_93-0">^</a> <a href="#CITEREFPratt2017">Pratt (2017)</a>, p. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=GbM-DgAAQBAJ&pg=PA314">314</a>.

</li>

<li id="cite_note-94"><a href="#cite_ref-94">^</a> <cite id="CITEREFFenn2012" class="citation book cs1">Fenn, Roger (2012). Geometry. Springer Undergraduate Mathematics Series. Springer Science & Business Media. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=b1HlBwAAQBAJ&pg=PA42">42</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781447103257</bdi>.</cite>

</li>

<li id="cite_note-FOOTNOTENicholson2012[httpbooksgooglecombooksidKNufK-P-6WoCpgPA70_70]BhattacharyaJainNagpaul1994[httpsbooksgooglecombooksidhiQ8e0b48swCpgPA159_159]-95"><a href="#cite_ref-FOOTNOTENicholson2012[httpbooksgooglecombooksidKNufK-P-6WoCpgPA70_70]BhattacharyaJainNagpaul1994[httpsbooksgooglecombooksidhiQ8e0b48swCpgPA159_159]_95-0">^</a> <a href="#CITEREFNicholson2012">Nicholson (2012)</a>, p. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=KNufK-P-6WoC&pg=PA70">70</a>; <a href="#CITEREFBhattacharyaJainNagpaul1994">Bhattacharya, Jain & Nagpaul (1994)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hiQ8e0b48swC&pg=PA159">159</a>.

</li>

<li id="cite_note-FOOTNOTERieffelPolak201116-96"><a href="#cite_ref-FOOTNOTERieffelPolak201116_96-0">^</a> <a href="#CITEREFRieffelPolak2011">Rieffel & Polak (2011)</a>, p. 16.

</li>

<li id="cite_note-FOOTNOTECheng2017124–132-97"><a href="#cite_ref-FOOTNOTECheng2017124–132_97-0">^</a> <a href="#CITEREFCheng2017">Cheng (2017)</a>, pp. 124–132.

</li>

<li id="cite_note-FOOTNOTESchindler201434-98"><a href="#cite_ref-FOOTNOTESchindler201434_98-0">^</a> <a href="#CITEREFSchindler2014">Schindler (2014)</a>, p. 34.

</li>

<li id="cite_note-FOOTNOTERiehl2016100-99"><a href="#cite_ref-FOOTNOTERiehl2016100_99-0">^</a> <a href="#CITEREFRiehl2016">Riehl (2016)</a>, p. 100.

</li>

<li id="cite_note-FOOTNOTEBhattacharyaJainNagpaul1994[httpsbooksgooglecombooksidhiQ8e0b48swCpgPA196_196]-100"><a href="#cite_ref-FOOTNOTEBhattacharyaJainNagpaul1994[httpsbooksgooglecombooksidhiQ8e0b48swCpgPA196_196]_100-0">^</a> <a href="#CITEREFBhattacharyaJainNagpaul1994">Bhattacharya, Jain & Nagpaul (1994)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hiQ8e0b48swC&pg=PA196">196</a>.

</li>

<li id="cite_note-FOOTNOTEKay2021[httpsbooksgooglecombooksidaw81EAAAQBAJpgPA44_44]-101"><a href="#cite_ref-FOOTNOTEKay2021[httpsbooksgooglecombooksidaw81EAAAQBAJpgPA44_44]_101-0">^</a> <a href="#CITEREFKay2021">Kay (2021)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA44">44</a>.

</li>

<li id="cite_note-102"><a href="#cite_ref-102">^</a> The set still must be nonempty. <a href="#CITEREFDummitFoote1999">Dummit & Foote (1999)</a>, p. 48 discuss this criterion written multiplicatively.

</li>

<li id="cite_note-FOOTNOTEMusserPetersonBurger2013[httpsbooksgooglecombooksid8jh7DwAAQBAJpgPA101_101]-103"><a href="#cite_ref-FOOTNOTEMusserPetersonBurger2013[httpsbooksgooglecombooksid8jh7DwAAQBAJpgPA101_101]_103-0">^</a> <a href="#CITEREFMusserPetersonBurger2013">Musser, Peterson & Burger (2013)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA101">101</a>.

</li>

<li id="cite_note-FOOTNOTEIsodaOlfosNoine2021[httpbooksgooglecombooksidowcLEAAAQBAJpgPA163_163&ndash;164]-104"><a href="#cite_ref-FOOTNOTEIsodaOlfosNoine2021[httpbooksgooglecombooksidowcLEAAAQBAJpgPA163_163&ndash;164]_104-0">^</a> <a href="#CITEREFIsodaOlfosNoine2021">Isoda, Olfos & Noine (2021)</a>, p. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=owcLEAAAQBAJ&pg=PA163">163–164</a>.

</li>

<li id="cite_note-FOOTNOTERudin1976178-105"><a href="#cite_ref-FOOTNOTERudin1976178_105-0">^</a> <a href="#CITEREFRudin1976">Rudin (1976)</a>, p. 178.

</li>

<li id="cite_note-FOOTNOTELee2003526Proposition_20.9-106"><a href="#cite_ref-FOOTNOTELee2003526Proposition_20.9_106-0">^</a> <a href="#CITEREFLee2003">Lee (2003)</a>, p. 526, Proposition 20.9.

</li>

<li id="cite_note-107"><a href="#cite_ref-107">^</a> <a href="#CITEREFLinderholm1971">Linderholm (1971)</a>, p. 49 observes, "By multiplication, properly speaking, a mathematician may mean practically anything. By addition he may mean a great variety of things, but not so great a variety as he will mean by 'multiplication'."

</li>

<li id="cite_note-108"><a href="#cite_ref-108">^</a> <a href="#CITEREFDummitFoote1999">Dummit & Foote (1999)</a>, p. 224. For this argument to work, one must assume that addition is a group operation and that multiplication has an identity.

</li>

<li id="cite_note-109"><a href="#cite_ref-109">^</a> For an example of left and right distributivity, see <a href="#CITEREFLoday2002">Loday (2002)</a>, p. 15.

</li>

<li id="cite_note-110"><a href="#cite_ref-110">^</a> <cite id="CITEREFShorttTrueblood1969" class="citation book cs1">Shortt, Roy F.; Trueblood, Cecil R. (June 1969). <a rel="nofollow" class="external text" href="https://files.eric.ed.gov/fulltext/ED076041.pdf">Teacher's Handbook; Elementary School Mathematics. Parts I and II</a> (PDF). Pennsylvania State University Computer-Assisted Instruction Lab. pp. 52, 59.</cite>

</li>

<li id="cite_note-111"><a href="#cite_ref-111">^</a> Compare <a href="#CITEREFViro2001">Viro (2001)</a>, p. 2, Figure 1.

</li>

<li id="cite_note-FOOTNOTEEnderton1977[httpbooksgooglecombooksidJlR-Ehk35XkCpgPA164_164]-113"><a href="#cite_ref-FOOTNOTEEnderton1977[httpbooksgooglecombooksidJlR-Ehk35XkCpgPA164_164]_113-0">^</a> <a href="#CITEREFEnderton1977">Enderton (1977)</a>, p. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA164">164</a>.

</li>

<li id="cite_note-FOOTNOTEMikhalkin20061-114"><a href="#cite_ref-FOOTNOTEMikhalkin20061_114-0">^</a> <a href="#CITEREFMikhalkin2006">Mikhalkin (2006)</a>, p. 1.

</li>

<li id="cite_note-FOOTNOTEAkianBapatGaubert20054-115"><a href="#cite_ref-FOOTNOTEAkianBapatGaubert20054_115-0">^</a> <a href="#CITEREFAkianBapatGaubert2005">Akian, Bapat & Gaubert (2005)</a>, p. 4.

</li>

<li id="cite_note-FOOTNOTEMikhalkin20062-116"><a href="#cite_ref-FOOTNOTEMikhalkin20062_116-0">^</a> <a href="#CITEREFMikhalkin2006">Mikhalkin (2006)</a>, p. 2.

</li>

<li id="cite_note-FOOTNOTELitvinovMaslovSobolevskii19993-117"><a href="#cite_ref-FOOTNOTELitvinovMaslovSobolevskii19993_117-0">^</a> <a href="#CITEREFLitvinovMaslovSobolevskii1999">Litvinov, Maslov & Sobolevskii (1999)</a>, p. 3.

</li>

<li id="cite_note-FOOTNOTEViro20014-118"><a href="#cite_ref-FOOTNOTEViro20014_118-0">^</a> <a href="#CITEREFViro2001">Viro (2001)</a>, p. 4.

</li>

<li id="cite_note-FOOTNOTEGbur2011300-119"><a href="#cite_ref-FOOTNOTEGbur2011300_119-0">^</a> <a href="#CITEREFGbur2011">Gbur (2011)</a>, p. 300.

</li>

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

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

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</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em">

<ul><li><cite id="CITEREFAkianBapatGaubert2005" class="citation journal cs1">Akian, Marianne; Bapat, Ravindra; Gaubert, Stephane (2005). "Min-plus methods in eigenvalue perturbation theory and generalised Lidskii-Vishik-Ljusternik theorem". INRIA Reports. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.SP/0402090">math.SP/0402090</a>. <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/2004math......2090A">2004math......2090A</a>.</cite></li>

<li><cite id="CITEREFApostol1967" class="citation book cs1">Apostol, Tom M. (1967). Calculus. Vol. 1 (2nd ed.). Wiley.</cite></li>

<li><cite id="CITEREFBaezDolan2001" class="citation conference cs1"><a href="John_C._Baez" title="John C. Baez">Baez, J.</a>; Dolan, J. (2001). Mathematics Unlimited – 2001 and Beyond. From Finite Sets to Feynman Diagrams. p. 29. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.QA/0004133">math.QA/0004133</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-66913-2</bdi>.</cite></li>

<li><cite id="CITEREFBegle1975" class="citation book cs1">Begle, Edward (1975). <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicsofele0000begl">The Mathematics of the Elementary School</a>. <a href="McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-004325-1</bdi>.</cite></li>

<li><cite id="CITEREFBerg1967" class="citation book cs1">Berg, Lothar (1967). Lauwerier, H. A.; Koiter, W. T. (eds.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aGXFCUaFCW0C">Introduction To The Operational Calculus</a>. Vol. 2. North-Holland Publishing. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-323-16245-6</bdi>.</cite></li>

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<li><cite id="CITEREFBehrJungst1971" class="citation book cs1">Behr, Merlyn J.; Jungst, Dale G. (1971). Fundamentals of Elementary Mathematics. Academic Press.</cite></li>

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<li><cite id="CITEREFMartin2003" class="citation book cs1">Martin, John (2003). Introduction to Languages and the Theory of Computation (3rd ed.). McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-232200-2</bdi>.</cite></li>

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<li><cite id="CITEREFSchwartzman1994" class="citation book cs1">Schwartzman, Steven (1994). <a rel="nofollow" class="external text" href="https://archive.org/details/wordsofmathemati0000schw">The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English</a>. <a href="Mathematical_Association_of_America" title="Mathematical Association of America">MAA</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-88385-511-9</bdi>.</cite></li>

<li><cite id="CITEREFShmerkoYanushkevichLyshevski2009" class="citation book cs1">Shmerko, V. P.; <a href="Svetlana_Yanushkevich" title="Svetlana Yanushkevich">Yanushkevich, Svetlana N.</a>; Lyshevski, S. E. (2009). Computer arithmetics for nanoelectronics. CRC Press.</cite></li>

<li><cite id="CITEREFSmith2002" class="citation book cs1">Smith, Frank (2002). <a rel="nofollow" class="external text" href="https://archive.org/details/glasswallwhymath0000smit">The Glass Wall: Why Mathematics Can Seem Difficult</a>. Teachers College Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8077-4242-6</bdi>.</cite></li>

<li><cite id="CITEREFSmith1980" class="citation book cs1">Smith, Karl (1980). The Nature of Modern Mathematics (3rd ed.). Wadsworth. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8185-0352-8</bdi>.</cite></li>

<li><cite id="CITEREFSparksRees1979" class="citation book cs1">Sparks, F.; Rees, C. (1979). A Survey of Basic Mathematics. McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-059902-4</bdi>.</cite></li>

<li><cite id="CITEREFStewart1999" class="citation book cs1">Stewart, James (1999). <a rel="nofollow" class="external text" href="https://archive.org/details/calculusearlytra00stew">Calculus: Early Transcendentals</a> (4th ed.). Brooks/Cole. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-534-36298-0</bdi>.</cite></li>

<li><cite id="CITEREFTaton1963" class="citation book cs1 cs1-prop-foreign-lang-source">Taton, René (1963). Le Calcul Mécanique. Que Sais-Je ? n° 367 (in French). Presses universitaires de France. pp. 20–28.</cite></li>

<li><cite id="CITEREFTruittRogers1960" class="citation book cs1">Truitt, T.; Rogers, A. (1960). Basics of Analog Computers. John F. Rider. <a href="LCC_(identifier)" class="mw-redirect" title="LCC (identifier)">LCC</a> <a rel="nofollow" class="external text" href="https://catalog.loc.gov/vwebv/search?searchCode=CALL%2B&searchArg=QA76.4+T7&searchType=1&recCount=25">QA76.4 T7</a>.</cite></li>

<li><cite id="CITEREFvan_de_Walle2004" class="citation book cs1">van de Walle, John (2004). <a rel="nofollow" class="external text" href="https://archive.org/details/elementarymiddle00vand">Elementary and Middle School Mathematics: Teaching developmentally</a> (5e ed.). Pearson. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-205-38689-5</bdi>.</cite></li>

<li><cite id="CITEREFViro2001" class="citation book cs1">Viro, Oleg (2001). Cascuberta, Carles; Miró-Roig, Rosa Maria; Verdera, Joan; Xambó-Descamps, Sebastià (eds.). <a rel="nofollow" class="external text" href="http://www.math.uu.se/~oleg/dequant/dequantH1.html">European Congress of Mathematics: Barcelona, July 10–14, 2000, Volume I. Dequantization of Real Algebraic Geometry on Logarithmic Paper</a>. Progress in Mathematics. Vol. 201. Basel: Birkhäuser. pp. 135–146. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0005163">math/0005163</a>. <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/2000math......5163V">2000math......5163V</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-7643-6417-5</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1024.14026">1024.14026</a>.</cite></li>

<li><cite id="CITEREFWingard-Nelson2014" class="citation book cs1">Wingard-Nelson, Rebecca (2014). Decimals and Fractions: It's Easy. Enslow Publishers, Inc.</cite></li>

<li><cite id="CITEREFWynn1998" class="citation conference cs1">Wynn, Karen (1998). "Numerical competence in infants". The Development of Mathematical Skills. Taylor & Francis. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-86377-816-X</bdi>.</cite></li></ul>

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

<ul><li><cite id="CITEREFBaroodyTiilikainen2003" class="citation conference cs1">Baroody, Arthur; Tiilikainen, Sirpa (2003). <a rel="nofollow" class="external text" href="https://archive.org/details/developmentofari0000unse/page/75">The Development of Arithmetic Concepts and Skills. Two perspectives on addition development</a>. Routledge. p. <a rel="nofollow" class="external text" href="https://archive.org/details/developmentofari0000unse/page/75">75</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8058-3155-X</bdi>.</cite></li>

<li><cite id="CITEREFDavisonLandauMcCrackenThompson1999" class="citation book cs1">Davison, David M.; Landau, Marsha S.; McCracken, Leah; Thompson, Linda (1999). Mathematics: Explorations & Applications (TE ed.). Prentice Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-435817-8</bdi>.</cite></li>

<li><cite id="CITEREFBuntJonesBedient1976" class="citation book cs1">Bunt, Lucas N.H.; Jones, Phillip S.; Bedient, Jack D. (1976). <a rel="nofollow" class="external text" href="https://archive.org/details/historicalrootso0000bunt">The Historical roots of Elementary Mathematics</a>. Prentice-Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-389015-0</bdi>.</cite></li>

<li><cite id="CITEREFPoonen2010" class="citation journal cs1">Poonen, Bjorn (2010). <a rel="nofollow" class="external text" href="http://www.girlsangle.org/page/bulletin.php">"Addition"</a>. Girls' Angle Bulletin. 3 (3–5). <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2151-5743">2151-5743</a>.</cite></li>

<li><cite id="CITEREFWeaver1982" class="citation conference cs1">Weaver, J. Fred (1982). "Addition and Subtraction: A Cognitive Perspective". Addition and Subtraction: A Cognitive Perspective. Interpretations of Number Operations and Symbolic Representations of Addition and Subtraction. Taylor & Francis. p. 60. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-89859-171-6</bdi>.</cite></li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Elementary_arithmetic25" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Elementary_arithmetic25" style="font-size:114%;margin:0 4em"><a href="Elementary_arithmetic" title="Elementary arithmetic">Elementary arithmetic</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0;background:transparent;color:inherit;"><div style="padding:0px"><table class="navbox-columns-table" style="border-spacing: 0px; text-align:left;width:auto; margin-left:auto; margin-right:auto;"><tbody><tr style="vertical-align:top"><td style="width:5em">   </td><td class="navbox-list" style="padding:0px;text-align: center;width:25%;"><div>

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

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

<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="Multiplication" title="Multiplication">Multiplication (2)</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>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Inverse for right argument</th><td class="navbox-list-with-group navbox-list navbox-odd 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="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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