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<title>Mathematical notation</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Mathematical notation</span></span>

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<p><b>Mathematical notation</b> consists of using <a href="Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">symbols</a> for representing <a href="Operation_(mathematics)" title="Operation (mathematics)">operations</a>, unspecified <a href="Number" title="Number">numbers</a>, <a href="Relation_(mathematics)" title="Relation (mathematics)">relations</a>, and any other <a href="Mathematical_object" title="Mathematical object">mathematical objects</a> and assembling them into <a href="Expression_(mathematics)" title="Expression (mathematics)">expressions</a> and <a href="Formula" title="Formula">formulas</a>. Mathematical notation is widely used in <a href="Mathematics" title="Mathematics">mathematics</a>, <a href="Science" title="Science">science</a>, and <a href="Engineering" title="Engineering">engineering</a> for representing complex <a href="Concept" title="Concept">concepts</a> and <a href="Property_(philosophy)" title="Property (philosophy)">properties</a> in a concise, unambiguous, and accurate way.

</p><p>For example, the physicist <a href="Albert_Einstein" title="Albert Einstein">Albert Einstein</a>'s formula <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=mc^{2}}">

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

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</math></span><img src="./9f73dbd37a0cac34406ee89057fa1b36a1e6a18e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.976ex; height:2.676ex;" alt="{\displaystyle E=mc^{2}}" loading="lazy"></span> is the quantitative representation in mathematical notation of <a href="Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">mass–energy equivalence</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>Mathematical notation was first introduced by <a href="Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">François Viète</a> at the end of the 16th century and largely expanded during the 17th and 18th centuries by <a href="Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a>, <a href="Isaac_Newton" title="Isaac Newton">Isaac Newton</a>, <a href="Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a>, and overall <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>.

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<div class="mw-heading mw-heading2"><h2 id="Symbols_and_typeface">Symbols and typeface</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary of mathematical symbols</a></div>

<p>The use of many symbols is the basis of mathematical notation. They play a similar role as words in <a href="Natural_language" title="Natural language">natural languages</a>. They may play different roles in mathematical notation similarly as verbs, adjective and nouns play different roles in a sentence.

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<div class="mw-heading mw-heading3"><h3 id="Letters_as_symbols">Letters as symbols</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="List_of_letters_used_in_mathematics%2C_science%2C_and_engineering" title="List of letters used in mathematics, science, and engineering">List of letters used in mathematics, science, and engineering</a></div>

<p>Letters are typically used for naming—in <a href="List_of_mathematical_jargon" class="mw-redirect" title="List of mathematical jargon">mathematical jargon</a>, one says <i>representing</i>—<a href="Mathematical_object" title="Mathematical object">mathematical objects</a>. The <a href="Latin_alphabet" title="Latin alphabet">Latin</a> and <a href="Greek_alphabet" title="Greek alphabet">Greek</a> alphabets are used extensively, but a few letters of other alphabets are also used sporadically, such as the <a href="Hebrew_alphabet" title="Hebrew alphabet">Hebrew</a> <span class="nowrap">⁠<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph }">

</mstyle>

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

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</math></span><img src="./306c55e6bc96d94db729ff5821c8f45a34c72bce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \aleph }" loading="lazy"></span>⁠</span>, <a href="Cyrillic_script" title="Cyrillic script">Cyrillic</a> <span class="texhtml">Ш</span>, and <a href="Hiragana" title="Hiragana">Hiragana</a> <span class="texhtml">よ</span>. Uppercase and lowercase letters are considered as different symbols. For Latin alphabet, different typefaces also provide different symbols. For example, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r,R,\mathbb {R} ,{\mathcal {R}},{\mathfrak {r}},}">

</mrow>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle r,R,\mathbb {R} ,{\mathcal {R}},{\mathfrak {r}},}</annotation>

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</math></span><img src="./6628084b4eca01231c8b54ea09de51eb22591411.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.149ex; height:2.509ex;" alt="{\displaystyle r,R,\mathbb {R} ,{\mathcal {R}},{\mathfrak {r}},}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {R}}}">

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

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</math></span><img src="./b5d31f64c0e02f0dc73d5aeb636a1caf2011dce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {R}}}" loading="lazy"></span> could theoretically appear in the same mathematical text with six different meanings. Normally, roman upright typeface is not used for symbols, except for symbols representing a standard function, such as the symbol "<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin }">

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

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</math></span><img src="./ee55beec18afd710e7ab767964b915b020c65093.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.856ex; height:2.176ex;" alt="{\displaystyle \sin }" loading="lazy"></span>" of the <a href="Sine_function" class="mw-redirect" title="Sine function">sine function</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>In order to have more symbols, and for allowing related mathematical objects to be represented by related symbols, <a href="Diacritic" title="Diacritic">diacritics</a>, <a href="Subscript" class="mw-redirect" title="Subscript">subscripts</a> and <a href="Superscript" class="mw-redirect" title="Superscript">superscripts</a> are often used. For example, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f'_{1}}}}">

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

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</math></span><img src="./ca08b9d46d5a855ce0a075bc9372a4d13ac992cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.193ex; height:3.509ex;" alt="{\displaystyle {\hat {f'_{1}}}}" loading="lazy"></span> may denote the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of the <a href="Derivative" title="Derivative">derivative</a> of a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> called <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}.}">

<msub>

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</mstyle>

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

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</math></span><img src="./15589063e81f3e5fa2699321756b83cab936bb91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.84ex; height:2.509ex;" alt="{\displaystyle f_{1}.}" loading="lazy"></span>

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<div class="mw-heading mw-heading3"><h3 id="Other_symbols">Other symbols</h3></div>

<p>Symbols are not only used for naming mathematical objects. They can be used for <a href="Operation_(mathematics)" title="Operation (mathematics)">operations</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (+,-,/,\oplus ,\ldots ),}">

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</mstyle>

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

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</math></span><img src="./1838a740b7cd4f8586f00a1520699dcc093b5d0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.902ex; height:2.843ex;" alt="{\displaystyle (+,-,/,\oplus ,\ldots ),}" loading="lazy"></span> for <a href="Relation_(mathematics)" title="Relation (mathematics)">relations</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (=,<,\leq ,\sim ,\equiv ,\ldots ),}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (=,<,\leq ,\sim ,\equiv ,\ldots ),}</annotation>

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</math></span><img src="./7301a985a49e8c74b68584afefe74722a8f87d1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.39ex; height:2.843ex;" alt="{\displaystyle (=,<,\leq ,\sim ,\equiv ,\ldots ),}" loading="lazy"></span> for <a href="Logical_connective" title="Logical connective">logical connectives</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\implies ,\land ,\lor ,\ldots ),}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (\implies ,\land ,\lor ,\ldots ),}</annotation>

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</math></span><img src="./997ff31b3b9371a8becedb25fb044f7bc6120838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.478ex; height:2.843ex;" alt="{\displaystyle (\implies ,\land ,\lor ,\ldots ),}" loading="lazy"></span> for <a href="Quantifier_(logic)" title="Quantifier (logic)">quantifiers</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\forall ,\exists ),}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./0a07b94cb636c83b580262f3b01af09b31eed3a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.075ex; height:2.843ex;" alt="{\displaystyle (\forall ,\exists ),}" loading="lazy"></span> and for other purposes.

</p><p>Some symbols are similar to Latin or Greek letters, some are obtained by deforming letters, some are traditional <a href="Typographic_symbol" class="mw-redirect" title="Typographic symbol">typographic symbols</a>, but many have been specially designed for mathematics.

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<div class="mw-heading mw-heading3"><h3 id="International_standard_mathematical_notation">International standard mathematical notation</h3></div>

<p>The <a href="International_Organization_for_Standardization" title="International Organization for Standardization">International Organization for Standardization</a> (ISO) is an <a href="International_standard" title="International standard">international standard</a> development organization composed of representatives from the national <a href="Standards_organization" title="Standards organization">standards organizations</a> of member countries. The international standard <a href="ISO_80000-2" class="mw-redirect" title="ISO 80000-2">ISO 80000-2</a> (previously, <a href="ISO_31-11" title="ISO 31-11">ISO 31-11</a>) specifies symbols for use in mathematical equations. The standard requires use of italic fonts for variables (e.g., <span class="nowrap"><i>E</i> = <i>mc</i><sup>2</sup></span>) and roman (upright) fonts for mathematical constants (e.g., e or π).

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<div class="mw-heading mw-heading2"><h2 id="Expressions_and_formulas">Expressions and formulas</h2></div>

<p>An <a href="Expression_(mathematics)" title="Expression (mathematics)">expression </a> is a written arrangement of <a href="Symbol_(mathematics)" class="mw-redirect" title="Symbol (mathematics)">symbols</a> following the context-dependent, <a href="Syntax_(logic)" title="Syntax (logic)">syntactic</a> conventions of mathematical notation. Symbols can denote <a href="Numbers" class="mw-redirect" title="Numbers">numbers</a>, <a href="Variable_(mathematics)" title="Variable (mathematics)">variables</a>, <a href="Operation_(mathematics)" title="Operation (mathematics)">operations</a>, and <a href="Function_(mathematics)" title="Function (mathematics)">functions</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Other symbols include <a href="Punctuation" title="Punctuation">punctuation</a> marks and <a href="Bracket_(mathematics)" title="Bracket (mathematics)">brackets</a>, used for <a href="Symbols_of_grouping" title="Symbols of grouping">grouping</a> where there is not a well-defined <a href="Order_of_operations" title="Order of operations">order of operations</a>.

</p><p>Expressions are commonly distinguished from <i><a href="Mathematical_formula" class="mw-redirect" title="Mathematical formula">formulas</a></i>: expressions are a kind of <a href="Mathematical_object" title="Mathematical object">mathematical object</a>, whereas formulas are statements <i>about</i> mathematical objects.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This is analogous to <a href="Natural_language" title="Natural language">natural language</a>, where a <a href="Noun_phrase" title="Noun phrase">noun phrase</a> refers to an object, and a whole <a href="Sentence_(linguistics)" title="Sentence (linguistics)">sentence</a> refers to a <a href="Fact" title="Fact">fact</a>. For example, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8x-5}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 8x-5}</annotation>

</semantics>

</math></span><img src="./f8612a66aaa50a47781f9c936cdc985dbd979e06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.495ex; height:2.343ex;" alt="{\displaystyle 8x-5}" loading="lazy"></span> is an expression, while the <a href="Inequality_(mathematics)" title="Inequality (mathematics)">inequality</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8x-5\geq 3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 8x-5\geq 3}</annotation>

</semantics>

</math></span><img src="./c798bbe752dacb43f38bc6c5d12da206ce867b93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.756ex; height:2.343ex;" alt="{\displaystyle 8x-5\geq 3}" loading="lazy"></span> is a formula.

</p><p>To <i>evaluate</i> an expression means to find a numerical <a href="Value_(mathematics)" title="Value (mathematics)">value</a> equivalent to the expression.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Expressions can be <i>evaluated</i> or <i>simplified</i> by replacing <a href="Operation_(mathematics)" title="Operation (mathematics)">operations</a> that appear in them with their result. For example, the expression <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8\times 2-5}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 8\times 2-5}</annotation>

</semantics>

</math></span><img src="./d68fc09bbe5a30bbfee1a138ffd33103ca09f0e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.168ex; height:2.343ex;" alt="{\displaystyle 8\times 2-5}" loading="lazy"></span> simplifies to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16-5}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 16-5}</annotation>

</semantics>

</math></span><img src="./7303a3de69a07585ead57108a953a6207c91c849.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.328ex; height:2.343ex;" alt="{\displaystyle 16-5}" loading="lazy"></span>, and evaluates to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 11.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 11.}</annotation>

</semantics>

</math></span><img src="./dc9b5e5fac088741f78d27719ffc4774c9815c59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.972ex; height:2.176ex;" alt="{\displaystyle 11.}" loading="lazy"></span>

</p>

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

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="History_of_mathematical_notation" title="History of mathematical notation">History of mathematical notation</a></div>

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

<p>It is believed that a notation to represent <a href="Number" title="Number">numbers</a> was first developed at least 50,000 years ago.<sup id="cite_ref-Eves_1990_7-0" class="reference"><a href="#cite_note-Eves_1990-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Early mathematical ideas such as <a href="Finger_counting" class="mw-redirect" title="Finger counting">finger counting</a><sup id="cite_ref-Ifrah_2000_8-0" class="reference"><a href="#cite_note-Ifrah_2000-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> have also been represented by collections of rocks, sticks, bone, clay, stone, wood carvings, and knotted ropes. The <a href="Tally_stick" title="Tally stick">tally stick</a> is a way of counting dating back to the <a href="Upper_Paleolithic" title="Upper Paleolithic">Upper Paleolithic</a>. Perhaps the oldest known mathematical texts are those of ancient <a href="Sumer" title="Sumer">Sumer</a>. The <a href="Census_quipu" class="mw-redirect" title="Census quipu">Census Quipu</a> of the Andes and the <a href="Ishango_Bone" class="mw-redirect" title="Ishango Bone">Ishango Bone</a> from Africa both used the <a href="Tally_mark" class="mw-redirect" title="Tally mark">tally mark</a> method of accounting for numerical concepts.

</p><p>The concept of <a href="Zero" class="mw-redirect" title="Zero">zero</a> and the introduction of a notation for it are important developments in early mathematics, which predates for centuries the concept of zero as a number. It was used as a placeholder by the <a href="Babylonian_numerals" class="mw-redirect" title="Babylonian numerals">Babylonians</a> and <a href="Greek_numerals" title="Greek numerals">Greek Egyptians</a>, and then as an <a href="Integer" title="Integer">integer</a> by the <a href="Maya_numerals" title="Maya numerals">Mayans</a>, <a href="Indian_numerals" class="mw-redirect" title="Indian numerals">Indians</a> and <a href="Arabic_numerals" title="Arabic numerals">Arabs</a> (see <a href="History_of_zero" class="mw-redirect" title="History of zero">the history of zero</a>).

</p>

<div class="mw-heading mw-heading3"><h3 id="Modern_notation">Modern notation</h3></div>

<p>Until the 16th century, mathematics was essentially <a href="Rhetorical_algebra" class="mw-redirect" title="Rhetorical algebra">rhetorical</a>, in the sense that everything but explicit numbers was expressed in words. However, some authors such as <a href="Diophantus" title="Diophantus">Diophantus</a> used some symbols as abbreviations.

</p><p>The first systematic use of formulas, and, in particular the use of symbols (<a href="Variable_(mathematics)" title="Variable (mathematics)">variables</a>) for unspecified numbers is generally attributed to <a href="Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">François Viète</a> (16th century). However, he used different symbols than those that are now standard.

</p><p>Later, <a href="Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> (17th century) introduced the modern notation for variables and <a href="Equation" title="Equation">equations</a>; in particular, the use of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./bbeca34b28f569a407ef74a955d041df9f360268.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}" loading="lazy"></span> for <a href="Unknown_(mathematics)" class="mw-redirect" title="Unknown (mathematics)">unknown</a> quantities and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f13f068df656c1b1911ae9f81628c49a6181194d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}" loading="lazy"></span> for known ones (<a href="Constant_(mathematics)" title="Constant (mathematics)">constants</a>). He introduced also the notation <span class="texhtml mvar" style="font-style:italic;">i</span> and the term "imaginary" for the <a href="Imaginary_unit" title="Imaginary unit">imaginary unit</a>.

</p><p>The 18th and 19th centuries saw the standardization of mathematical notation as used today. <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> was responsible for many of the notations currently in use: the <a href="Functional_notation" class="mw-redirect" title="Functional notation">functional notation</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x),}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./10535d1a7a971ffeeb216605cb846099fab2e653.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.064ex; height:2.843ex;" alt="{\displaystyle f(x),}" loading="lazy"></span> <span class="texhtml"><i>e</i></span> for the base of the <a href="Natural_logarithm" title="Natural logarithm">natural logarithm</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum }">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> for <a href="Summation" title="Summation">summation</a>, etc.<sup id="cite_ref-Boyer-Merzbach_1991_9-0" class="reference"><a href="#cite_note-Boyer-Merzbach_1991-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> He also popularized the use of <span class="texhtml mvar" style="font-style:italic;">π</span> for the <a href="Archimedes_constant" class="mw-redirect" title="Archimedes constant">Archimedes constant</a> (proposed by <a href="William_Jones_(mathematician)" title="William Jones (mathematician)">William Jones</a>, based on an earlier notation of <a href="William_Oughtred" title="William Oughtred">William Oughtred</a>).<sup id="cite_ref-Arndt-Haenel_2006_10-0" class="reference"><a href="#cite_note-Arndt-Haenel_2006-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

</p><p>Since then many new notations have been introduced, often specific to a particular area of mathematics. Some notations are named after their inventors, such as <a href="Leibniz's_notation" title="Leibniz's notation">Leibniz's notation</a>, <a href="Legendre_symbol" title="Legendre symbol">Legendre symbol</a>, the <a href="Einstein_summation_convention" class="mw-redirect" title="Einstein summation convention">Einstein summation convention</a>, etc.

</p>

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

<p>General <a href="Typesetting_system" class="mw-redirect" title="Typesetting system">typesetting systems</a> are generally not well suited for mathematical notation. One of the reasons is that, in mathematical notation, the symbols are often arranged in two-dimensional figures, such as in:

</p>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\frac {{\begin{bmatrix}a&b\\c&d\end{bmatrix}}^{n}}{n!}}.}</annotation>

</semantics>

</math></span><img src="./00f05c2431187190062e84cae7cc75ac50b68ee5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.418ex; height:10.176ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\frac {{\begin{bmatrix}a&b\\c&d\end{bmatrix}}^{n}}{n!}}.}" loading="lazy"></span></dd></dl>

<p><a href="TeX" title="TeX">TeX</a> is a mathematically oriented typesetting system that was created in 1978 by <a href="Donald_Knuth" title="Donald Knuth">Donald Knuth</a>. It is widely used in mathematics, through its extension called <a href="LaTeX" title="LaTeX">LaTeX</a>, and is a <i>de facto</i> standard. (The above expression is written in LaTeX.)

</p><p>More recently, another approach for mathematical typesetting is provided by <a href="MathML" title="MathML">MathML</a>. However, it is not well supported in web browsers, which is its primary target.

</p>

<div class="mw-heading mw-heading2"><h2 id="Non-Latin-based_mathematical_notation">Non-Latin-based mathematical notation</h2></div>

<p><a href="Modern_Arabic_mathematical_notation" title="Modern Arabic mathematical notation">Modern Arabic mathematical notation</a> is based mostly on the <a href="Arabic_alphabet" title="Arabic alphabet">Arabic alphabet</a> and is used widely in the <a href="Arab_world" title="Arab world">Arab world</a>, especially in pre-<a href="Tertiary_education" title="Tertiary education">tertiary education</a>. (Western notation uses <a href="Arabic_numerals" title="Arabic numerals">Arabic numerals</a>, but the Arabic notation also replaces Latin letters and related symbols with Arabic script.)

</p><p>In addition to Arabic notation, mathematics also makes use of <a href="Greek_alphabet" title="Greek alphabet">Greek letters</a> to denote a wide variety of mathematical objects and variables. On some occasions, certain <a href="Hebrew_alphabet" title="Hebrew alphabet">Hebrew letters</a> are also used (such as in the context of <a href="Infinite_cardinal" class="mw-redirect" title="Infinite cardinal">infinite cardinals</a>).

</p><p>Some mathematical notations are mostly diagrammatic, and so are almost entirely script independent. Examples are <a href="Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a> and <a href="Coxeter%E2%80%93Dynkin_diagram" title="Coxeter–Dynkin diagram">Coxeter–Dynkin diagrams</a>.

</p><p>Braille-based mathematical notations used by blind people include <a href="Nemeth_Braille" title="Nemeth Braille">Nemeth Braille</a> and <a href="GS8_Braille" class="mw-redirect" title="GS8 Braille">GS8 Braille</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Meaning_and_interpretation">Meaning and interpretation</h2></div>

<p>The <a href="Syntax" title="Syntax">syntax</a> of notation defines how symbols can be combined to make <a href="Expression_(mathematics)#Well-defined_expressions" title="Expression (mathematics)">well-formed expressions</a>, without any given meaning or interpretation. The <a href="Semantics" title="Semantics">semantics</a> of notation interprets what the symbols represent and assigns a meaning to the expressions and formulas. The reverse process of taking a statement and writing it in logical or mathematical notation is called <a href="Logic_translation" title="Logic translation">translation</a>.

</p>

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

<p>Given a <a href="Formal_language" title="Formal language">formal language</a>, an <a href="Interpretation_(logic)" title="Interpretation (logic)">interpretation</a> assigns a <a href="Domain_of_discourse" title="Domain of discourse">domain of discourse</a> to the language. Specifically, it assigns each of the constant symbols to objects of the domain, function letters to functions within the domain, predicate letters to statments, and vairiables are assumed to range over the domain.

</p>

<div class="mw-heading mw-heading3"><h3 id="Map–territory_relation">Map–territory relation</h3></div>

<p>The <a href="Map%E2%80%93territory_relation" title="Map–territory relation">map–territory relation</a> describes the relationship between an object and the representation of that object, such as the <a href="Earth" title="Earth">Earth</a> and a <a href="Map" title="Map">map</a> of it. In mathematics, this is how the number 4 relates to its representation "4". The quotation marks are the formally correct usage, distinguishing the number from its name. However, it is fairly common practice in math to commit this fallacy saying "Let x denote...", rather than "Let "x" denote..." which is generally harmless.

</p>

<div class="mw-heading mw-heading2"><h2 id="Software_for_mathematical_typesetting">Software for mathematical typesetting</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Typesetting" title="Typesetting">Typesetting</a>, <a href="Comparison_of_TeX_editors" title="Comparison of TeX editors">Comparison of TeX editors</a>, and <a href="Mathematical_markup_language" title="Mathematical markup language">Mathematical markup language</a></div>

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</style><div>

<ul><li><a href="AUCTeX" title="AUCTeX">AUCTeX</a></li>

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

<li><a href="Apache_OpenOffice#Components" title="Apache OpenOffice">Apache OpenOffice Math</a></li>

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

<li><a href="Calligra_Words#Formula_editor" title="Calligra Words">Calligra Words - Formula editor</a></li>

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

<li><a href="GNOME_LaTeX" title="GNOME LaTeX">GNOME LaTeX</a></li>

<li><a href="GNU_TeXmacs" title="GNU TeXmacs">GNU TeXmacs</a></li>

<li><a href="Gummi_(software)" title="Gummi (software)">Gummi</a></li>

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

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

<li><a href="LibreOffice_Math" class="mw-redirect" title="LibreOffice Math">LibreOffice Math</a></li>

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

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

<li><a href="MathML" title="MathML">MathML</a></li></ul>

</td>

<ul><li><a href="MathType" title="MathType">MathType</a></li>

<li><a href="Notepad%2B%2B" title="Notepad++">Notepad++</a></li>

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

<li><a href="Scientific_WorkPlace" title="Scientific WorkPlace">Scientific WorkPlace</a></li>

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

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

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

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

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

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

<li><a href="Visual_Studio_Code" title="Visual Studio Code">Visual Studio Code</a> - <a rel="nofollow" class="external text" href="https://github.com/James-Yu/latex-workshop/wiki">LaTeX Workshop</a></li>

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

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

<li><a href="WinShell" title="WinShell">WinShell</a></li></ul>

<p>

</p>

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

<ul><li><a href="Abuse_of_notation" title="Abuse of notation">Abuse of notation</a></li>

<li>Chemistry notation</li>

<li><a href="Denotation" title="Denotation">Denotation</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="Language_of_mathematics" title="Language of mathematics">Language of mathematics</a></li>

<li><a href="List_of_open-source_software_for_mathematics" title="List of open-source software for mathematics">List of open-source software for mathematics</a></li>

<li><a href="Mathematical_Alphanumeric_Symbols" title="Mathematical Alphanumeric Symbols">Mathematical Alphanumeric Symbols</a></li>

<li><a href="Modern_Arabic_mathematical_notation" title="Modern Arabic mathematical notation">Modern Arabic mathematical notation</a></li>

<li><a href="Notation_in_probability_and_statistics" title="Notation in probability and statistics">Notation in probability and statistics</a></li>

<li><a href="Principle_of_compositionality" title="Principle of compositionality">Principle of compositionality</a></li>

<li><a href="Scientific_notation" title="Scientific notation">Scientific notation</a></li>

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

<li><a href="Syntactic_sugar" title="Syntactic sugar">Syntactic sugar</a></li>

<li><a href="Vector_notation" title="Vector notation">Vector notation</a></li></ul>

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

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</style><cite id="CITEREFEinstein1905" class="citation journal cs1 cs1-prop-foreign-lang-source">Einstein, Albert (1905). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/10.1002/andp.19053231314">"Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?"</a>. <i>Annalen der Physik</i> (in German). <b>323</b> (13): <span class="nowrap">639–</span>641. <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/1905AnP...323..639E">1905AnP...323..639E</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19053231314">10.1002/andp.19053231314</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-3804">0003-3804</a>.</cite></span>

</li>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a>, s.v. “<a href="https://doi.org/10.1093/OED/4555505636" class="extiw external" title="doi:10.1093/OED/4555505636">Expression (n.), sense II.7</a>,” "<i>A group of symbols which together represent a numeric, algebraic, or other mathematical quantity or function.</i>"</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFStoll1963" class="citation book cs1">Stoll, Robert R. (1963). <i>Set Theory and Logic</i>. San Francisco, CA: Dover Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-63829-4</bdi>.</cite> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{cite book}}</code>: </span><span class="cs1-hidden-error citation-comment">ISBN / Date incompatibility (help)</span></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a>, s.v. "<a href="https://doi.org/10.1093/OED/3423541985" class="extiw external" title="doi:10.1093/OED/3423541985">Evaluate (v.), sense a</a>", "<i>Mathematics. To work out the ‘value’ of (a quantitative expression); to find a numerical expression for (any quantitative fact or relation).</i>"</span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a>, s.v. “<a href="https://doi.org/10.1093/OED/1018661347" class="extiw external" title="doi:10.1093/OED/1018661347">Simplify (v.), sense 4.a</a>”, "<i>To express (an equation or other mathematical expression) in a form that is easier to understand, analyse, or work with, e.g. by collecting like terms or substituting variables.</i>"</span>

</li>

<li id="cite_note-Eves_1990-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Eves_1990_7-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFEves1990" class="citation book cs1"><a href="Howard_Eves" title="Howard Eves">Eves, Howard</a> (1990). <i>An Introduction to the History of Mathematics</i> (6 ed.). Saunders College Pub. p. 9. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-03-029558-4</bdi>.</cite></span>

</li>

<li id="cite_note-Ifrah_2000-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ifrah_2000_8-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFIfrah2000" class="citation book cs1"><a href="Georges_Ifrah" title="Georges Ifrah">Ifrah, Georges</a> (2000). <i>The Universal History of Numbers: From prehistory to the invention of the computer</i>. Translated by Bellos, David; Harding, E. F.; Wood, Sophie; Monk, Ian. <a href="John_Wiley_and_Sons" class="mw-redirect" title="John Wiley and Sons">John Wiley and Sons</a>. p. 48. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-39340-1</bdi>.</cite> (NB. Ifrah supports his thesis by quoting idiomatic phrases from languages across the entire world. He notes that humans learned to count on their hands. He shows, for example, a picture of <a href="Boethius" title="Boethius">Boethius</a> (who lived 480–524 or 525) reckoning on his fingers.)</span>

</li>

<li id="cite_note-Boyer-Merzbach_1991-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Boyer-Merzbach_1991_9-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFBoyerMerzbach1991" class="citation book cs1"><a href="Carl_Benjamin_Boyer" title="Carl Benjamin Boyer">Boyer, Carl Benjamin</a>; <a href="Uta_Merzbach" title="Uta Merzbach">Merzbach, Uta C.</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00boye/page/442"><i>A History of Mathematics</i></a>. <a href="John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. pp. <span class="nowrap">442–</span>443. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-54397-8</bdi>.</cite></span>

</li>

<li id="cite_note-Arndt-Haenel_2006-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Arndt-Haenel_2006_10-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFArndtHaenel2006" class="citation book cs1">Arndt, Jörg; Haenel, Christoph (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QwwcmweJCDQC&pg=PA166"><i>Pi Unleashed</i></a>. <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. p. 166. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-66572-4</bdi>.</cite></span>

</li>

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

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

<ul><li><a href="Florian_Cajori" title="Florian Cajori">Florian Cajori</a>, <i><a href="A_History_of_Mathematical_Notations" title="A History of Mathematical Notations">A History of Mathematical Notations</a></i> (1929), <a rel="nofollow" class="external text" href="https://archive.org/details/b29980343_0001/page/32/mode/2up">Vol. 1</a>, <a rel="nofollow" class="external text" href="https://archive.org/details/b29980343_0002/page/n3/mode/2up">Vol. 2</a>. (Dover reprint 2011, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-67766-4</bdi>)</li>

<li>Mazur, Joseph (2014), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YZLzjwEACAAJ&q=enlightening+symbols"><i>Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers</i></a>. Princeton, New Jersey: Princeton University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-15463-3</bdi></li></ul>

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

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Mathematical_notation" class="extiw external" title="commons:Category:Mathematical notation">Mathematical notation</a></span>.</div></div>

</div>

<ul><li><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/mathsym.html">Earliest Uses of Various Mathematical Symbols</a></li>

<li><a rel="nofollow" class="external text" href="http://www.apronus.com/math/mrwmath.htm">Mathematical ASCII Notation</a> how to type math notation in any text editor.</li>

<li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/language/index.shtml">Mathematics as a Language</a> at <a href="Alexander_Bogomolny#Cut-the-Knot" title="Alexander Bogomolny">Cut-the-Knot</a></li>

<li><a href="Stephen_Wolfram" title="Stephen Wolfram">Stephen Wolfram</a>: <a rel="nofollow" class="external text" href="http://www.stephenwolfram.com/publications/mathematical-notation-past-future/">Mathematical Notation: Past and Future</a>. October 2000. Transcript of a keynote address presented at <a href="MathML" title="MathML">MathML</a> and Math on the Web: MathML International Conference.</li></ul>

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</style><div id="Common_mathematical_notation,_symbols,_and_formulas167" style="font-size:114%;margin:0 4em">Common <a href="Mathematics" title="Mathematics">mathematical</a> , <a href="List_of_mathematical_symbols_by_subject" class="mw-redirect" title="List of mathematical symbols by subject">symbols</a>, and formulas</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible expanded navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Lists_of_Unicode_and_LaTeX_mathematical_symbols167" style="font-size:114%;margin:0 4em">Lists of <a href="Unicode" title="Unicode">Unicode</a> and <a href="LaTeX" title="LaTeX">LaTeX</a> mathematical symbols</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_mathematical_symbols_by_subject" class="mw-redirect" title="List of mathematical symbols by subject">List of mathematical symbols by subject</a></li>

<li><a href="Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary of mathematical symbols</a></li>

<li><a href="List_of_logic_symbols" title="List of logic symbols">List of logic symbols</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible expanded navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Lists_of_Unicode_symbols167" style="font-size:114%;margin:0 4em">Lists of <a href="Unicode" title="Unicode">Unicode</a> symbols</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_Unicode_characters" title="List of Unicode characters">List of Unicode characters</a></li>

<li><a href="Unicode_block" title="Unicode block">Unicode block</a></li></ul>

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

<ul><li><a href="Mathematical_Alphanumeric_Symbols" title="Mathematical Alphanumeric Symbols">Mathematical Alphanumeric Symbols</a></li>

<li><a href="Blackboard_bold#Usage" title="Blackboard bold">Blackboard bold</a></li>

<li><a href="Letterlike_Symbols" title="Letterlike Symbols">Letterlike Symbols</a></li>

<li><a href="Symbols_for_zero" title="Symbols for zero">Symbols for zero</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Arrow_(symbol)" title="Arrow (symbol)">Arrows</a> and <a href="Geometric_Shapes_(Unicode_block)" title="Geometric Shapes (Unicode block)">Geometric Shapes</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arrow_(symbol)" title="Arrow (symbol)">Arrows</a></li>

<li><a href="Miscellaneous_Symbols_and_Arrows" title="Miscellaneous Symbols and Arrows">Miscellaneous Symbols and Arrows</a></li>

<li><a href="Geometric_Shapes_(Unicode_block)" title="Geometric Shapes (Unicode block)">Geometric Shapes (Unicode block)</a></li></ul>

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

<ul><li><a href="Mathematical_operators_and_symbols_in_Unicode" title="Mathematical operators and symbols in Unicode">Mathematical operators and symbols</a></li>

<li><a href="Mathematical_Operators_(Unicode_block)" title="Mathematical Operators (Unicode block)">Mathematical Operators (Unicode block)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Supplemental_Mathematical_Operators" title="Supplemental Mathematical Operators">Supplemental Math Operators</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Supplemental_Mathematical_Operators" title="Supplemental Mathematical Operators">Supplemental Mathematical Operators</a></li>

<li><a href="Number_Forms" title="Number Forms">Number Forms</a></li></ul>

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

<li><a href="Miscellaneous_Technical" title="Miscellaneous Technical">Technical</a></li>

<li><a href="ISO_31-11" title="ISO 31-11">ISO 31-11</a> (Mathematical signs and symbols for use in physical sciences and technology)</li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible expanded navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Typographical_conventions_and_notations167" style="font-size:114%;margin:0 4em">Typographical conventions and notations</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Language_of_mathematics" title="Language of mathematics">Language</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="APL_syntax_and_symbols#Monadic_functions" title="APL syntax and symbols">APL syntax and symbols</a></li></ul>

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

<ul><li><a href="Diacritic" title="Diacritic">Diacritic</a></li>

<li><a href="List_of_letters_used_in_mathematics%2C_science%2C_and_engineering" title="List of letters used in mathematics, science, and engineering">Letters in STEM</a>

<ul><li><a href="Greek_letters_used_in_mathematics%2C_science%2C_and_engineering" title="Greek letters used in mathematics, science, and engineering">Greek letters in STEM</a></li>

<li><a href="Latin_letters_used_in_mathematics%2C_science%2C_and_engineering" title="Latin letters used in mathematics, science, and engineering">Latin letters in STEM</a></li></ul></li></ul>

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

<ul>

<li><a href="List_of_mathematical_abbreviations" title="List of mathematical abbreviations">Abbreviations</a></li>

<li><a href="Notation_in_probability_and_statistics" title="Notation in probability and statistics">Notation in probability and statistics</a></li>

<li><a href="List_of_common_physics_notations" title="List of common physics notations">List of common physics notations</a></li></ul>

<ul><li><a href="Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary of mathematical symbols</a></li>

<li><a href="List_of_mathematical_constants" title="List of mathematical constants">List of mathematical constants</a></li>

<li><a href="Physical_constant" title="Physical constant">Physical constants</a></li>

<li><a href="Table_of_mathematical_symbols_by_introduction_date" title="Table of mathematical symbols by introduction date">Table of mathematical symbols by introduction date</a></li>

<li><a href="List_of_typographical_symbols_and_punctuation_marks" title="List of typographical symbols and punctuation marks">List of typographical symbols and punctuation marks</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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