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class="mw-body-content"><script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><section class="mf-section-0" id="mf-section-0"><style data-mw-deduplicate="TemplateStyles:r1238436933">.mw-parser-output .ombox{margin:4px 0;border-collapse:collapse;border:1px solid #a2a9b1;background-color:var(--background-color-neutral-subtle,#f8f9fa);box-sizing:border-box;color:var(--color-base,#202122)}.mw-parser-output .ombox.mbox-small{font-size:88%;line-height:1.25em}.mw-parser-output .ombox-speedy{border:2px solid #b32424;background-color:#fee7e6}.mw-parser-output .ombox-delete{border:2px solid #b32424}.mw-parser-output .ombox-content{border:1px solid #f28500}.mw-parser-output .ombox-style{border:1px solid #fc3}.mw-parser-output .ombox-move{border:1px solid #9932cc}.mw-parser-output .ombox-protection{border:2px solid #a2a9b1}.mw-parser-output .ombox .mbox-text{border:none;padding:0.25em 0.9em;width:100%}.mw-parser-output .ombox .mbox-image{border:none;padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .ombox .mbox-imageright{border:none;padding:2px 0.9em 2px 0;text-align:center}.mw-parser-output .ombox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ombox .mbox-invalid-type{text-align:center}@media(min-width:720px){.mw-parser-output .ombox{margin:4px 10%}.mw-parser-output .ombox.mbox-small{clear:right;float:right;margin:4px 0 4px 1em;width:238px}}body.skin--responsive .mw-parser-output table.ombox img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .ombox-speedy{background-color:#310402}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .ombox-speedy{background-color:#310402}}</style><table class="plainlinks ombox ombox-notice" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Blue_check.svg/30px-Blue_check.svg.png" decoding="async" width="30" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Blue_check.svg/45px-Blue_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Blue_check.svg/60px-Blue_check.svg.png 2x" data-file-width="600" data-file-height="600"></span></span></td><td class="mbox-text"><div style="padding-top: 6px; padding-bottom: 5px;"><b>This <a href="/wiki/Wikipedia:Policies_and_guidelines#guide" title="Wikipedia:Policies and guidelines">guideline</a> is a part of the English Wikipedia's <a href="/wiki/Wikipedia:Manual_of_Style" title="Wikipedia:Manual of Style">Manual of Style</a>.</b><br><div style="font-size: 90%;"> It is a generally accepted standard that editors should attempt to follow, though <a href="/wiki/Wikipedia:Ignore_all_rules" title="Wikipedia:Ignore all rules">occasional exceptions</a> may apply. Any <i>substantive</i> edit to this page should reflect <a href="/wiki/Wikipedia:Consensus" title="Wikipedia:Consensus">consensus</a>. When in doubt, discuss first on the <a href="/wiki/Wikipedia_talk:Manual_of_Style/Mathematics" title="Wikipedia talk:Manual of Style/Mathematics">talk page</a>.</div></div></td><td class="mbox-imageright"><style data-mw-deduplicate="TemplateStyles:r1252059228">.mw-parser-output .module-shortcutboxplain{float:right;margin:0 0 0 1em;border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-base,#fff);padding:0.3em 0.6em 0.2em 0.6em;text-align:center;font-size:85%}.mw-parser-output .module-shortcutboxleft{float:left;margin:0 1em 0 0}.mw-parser-output .module-shortcutlist{display:inline-block;border-bottom:1px solid var(--border-color-base,#a2a9b1);margin-bottom:0.2em}.mw-parser-output .module-shortcutboxplain ul{font-weight:bold}.mw-parser-output .module-shortcutanchordiv{position:relative;top:-3em}.mw-parser-output li .module-shortcutanchordiv{float:right}.mw-parser-output .mbox-imageright .module-shortcutboxplain{padding:0.4em 1em 0.4em 1em;line-height:1.3;margin:0}</style><div class="module-shortcutanchordiv"><span id="WP:MSM"></span><span id="MOS:MATH"></span><span id="MOS:MATHS"></span></div><div class="module-shortcutboxplain noprint" role="note"><div class="module-shortcutlist"><a href="/wiki/Wikipedia:Shortcut" title="Wikipedia:Shortcut">Shortcuts</a></div><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist"><ul><li><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Wikipedia:MSM&redirect=no">WP:MSM</a></span></li><li><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=MOS:MATH&redirect=no">MOS:MATH</a></span></li><li><span class="plainlinks"><a class="external text" 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks hlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Wikipedia:Manual_of_Style" title="Wikipedia:Manual of Style">Manual of Style (MoS)</a></th></tr><tr><td class="sidebar-image"><div class="mw-inputbox-centered" style=""><form name="searchbox" class="searchbox mw-inputbox-form" action="/wiki/Special:Search" id="style-searchbox"><div class="cdx-text-input"><input class="mw-inputbox-input mw-searchInput searchboxInput cdx-text-input__input" name="search" placeholder="" size="25" dir="ltr"></div><input type="hidden" value="Wikipedia:Manual of Style" name="prefix"><br><input type="submit" name="fulltext" value="Search the MoS" class="cdx-button"><input type="hidden" value="Search" name="fulltext"></form></div></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background-color: #def; color:black; padding-left: 2px;;color: var(--color-base)"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(content)" title="Category:Wikipedia Manual of Style (content)">Content</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Wikipedia:Manual_of_Style/Accessibility" title="Wikipedia:Manual of Style/Accessibility">Accessibility</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Biography" title="Wikipedia:Manual of Style/Biography">Biography</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Disambiguation_pages" title="Wikipedia:Manual of Style/Disambiguation pages">Disambiguation pages</a> <ul><li><a href="/wiki/Wikipedia:Manual_of_Style/Organizing_disambiguation_pages_by_subject_area" title="Wikipedia:Manual of Style/Organizing disambiguation pages by subject area">Organizing by subject</a></li></ul></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Gender_identity" title="Wikipedia:Manual of Style/Gender identity">Gender identity</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Hidden_text" title="Wikipedia:Manual of Style/Hidden text">Hidden text</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Infoboxes" title="Wikipedia:Manual of Style/Infoboxes">Infoboxes</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Linking" title="Wikipedia:Manual of Style/Linking">Linking</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Self-references_to_avoid" title="Wikipedia:Manual of Style/Self-references to avoid">Self-references</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch" title="Wikipedia:Manual of Style/Words to watch">Words to watch</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background-color: #def; color:black; padding-left: 2px;;color: var(--color-base)"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(formatting)" title="Category:Wikipedia Manual of Style (formatting)">Formatting</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Wikipedia:Manual_of_Style/Abbreviations" title="Wikipedia:Manual of Style/Abbreviations">Abbreviations</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Capital_letters" title="Wikipedia:Manual of Style/Capital letters">Capitalization</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Dates_and_numbers" title="Wikipedia:Manual of Style/Dates and numbers">Dates and numbers</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Pronunciation" title="Wikipedia:Manual of Style/Pronunciation">Pronunciation</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Spelling" title="Wikipedia:Manual of Style/Spelling">Spelling</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Text_formatting" title="Wikipedia:Manual of Style/Text formatting">Text formatting</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Titles_of_works" title="Wikipedia:Manual of Style/Titles of works">Titles of works</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background-color: #def; color:black; padding-left: 2px;;color: var(--color-base)"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(images)" title="Category:Wikipedia Manual of Style (images)">Images</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Wikipedia:Manual_of_Style/Captions" title="Wikipedia:Manual of Style/Captions">Captions</a></li> <li><a href="/wiki/Wikipedia:Image_use_policy#Adding_images_to_articles" title="Wikipedia:Image use policy">Image placement</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Icons" title="Wikipedia:Manual of Style/Icons">Icons</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Images" title="Wikipedia:Manual of Style/Images">Images</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background-color: #def; color:black; padding-left: 2px;;color: var(--color-base)"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(layout)" title="Category:Wikipedia Manual of Style (layout)">Layout</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Wikipedia:Manual_of_Style/Layout" title="Wikipedia:Manual of Style/Layout">Layout</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Lead_section" title="Wikipedia:Manual of Style/Lead section">Lead section</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Tables" title="Wikipedia:Manual of Style/Tables">Tables</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Trivia_sections" title="Wikipedia:Manual of Style/Trivia sections">Trivia sections</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background-color: #def; color:black; padding-left: 2px;;color: var(--color-base)"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(lists)" title="Category:Wikipedia Manual of Style (lists)">Lists</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Wikipedia:Manual_of_Style/Lists" title="Wikipedia:Manual of Style/Lists">Lists</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Lists_of_works" title="Wikipedia:Manual of Style/Lists of works">Lists of works</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Road_junction_lists" title="Wikipedia:Manual of Style/Road junction lists">Road junctions</a></li> <li><a href="/wiki/Wikipedia:Stand-alone_lists" title="Wikipedia:Stand-alone lists">Stand-alone lists</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background-color: #def; color:black; padding-left: 2px;;color: var(--color-base)">By topic area</div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar sidebar-collapse nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background-color: #def; color:black; padding-left: 2px;;color: var(--color-base)"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(legal)" title="Category:Wikipedia Manual of Style (legal)">Legal</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Wikipedia:Manual_of_Style/Legal" title="Wikipedia:Manual of Style/Legal">Legal</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Trademarks" title="Wikipedia:Manual of Style/Trademarks">Trademarks</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background-color: #def; color:black; padding-left: 2px;;color: var(--color-base)"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(arts)" title="Category:Wikipedia Manual of Style (arts)">Arts</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Wikipedia:Manual_of_Style/Anime_and_manga" title="Wikipedia:Manual of Style/Anime and manga">Anime and manga</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Comics" title="Wikipedia:Manual of Style/Comics">Comics</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Film" title="Wikipedia:Manual of Style/Film">Film</a></li> <li><a href="/wiki/Wikipedia:Lyrics_and_poetry" title="Wikipedia:Lyrics and poetry">Lyrics and poetry</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Novels" title="Wikipedia:Manual of Style/Novels">Novels</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Television" title="Wikipedia:Manual of Style/Television">Television</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Video_games" title="Wikipedia:Manual of Style/Video games">Video games</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Visual_arts" title="Wikipedia:Manual of Style/Visual arts">Visual arts</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Writing_about_fiction" title="Wikipedia:Manual of Style/Writing about fiction">Writing about fiction</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Category:Wikipedia_Manual_of_Style_(music)" title="Category:Wikipedia Manual of Style (music)">Music</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Wikipedia:Manual_of_Style/Music" title="Wikipedia:Manual of Style/Music">Music</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Music_samples" title="Wikipedia:Manual of Style/Music samples">Music samples</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Record_charts" title="Wikipedia:Manual of Style/Record charts">Record charts</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Stringed_instrument_tunings" title="Wikipedia:Manual of Style/Stringed instrument tunings">Stringed instruments</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background-color: #def; color:black; padding-left: 2px;;color: var(--color-base)"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(history)" title="Category:Wikipedia Manual of Style (history)">History</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Wikipedia:Manual_of_Style/Blazon" title="Wikipedia:Manual of Style/Blazon">Blazons</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Military_history" title="Wikipedia:Manual of Style/Military history">Military history</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background-color: #def; color:black; padding-left: 2px;;color: var(--color-base)"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(regional)" title="Category:Wikipedia Manual of Style (regional)">Regional</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Wikipedia:Naming_conventions_(geographic_names)#Specific_topics" title="Wikipedia:Naming conventions (geographic names)">Specific naming conventions</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Canada-related_articles" title="Wikipedia:Manual of Style/Canada-related articles">Canada</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/China-_and_Chinese-related_articles" title="Wikipedia:Manual of Style/China- and Chinese-related articles">China (and Chinese)</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Egypt-related_articles" title="Wikipedia:Manual of Style/Egypt-related articles">Egypt</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/France-_and_French-related_articles" title="Wikipedia:Manual of Style/France- and French-related articles">France (and French)</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Hawaii-related_articles" title="Wikipedia:Manual of Style/Hawaii-related articles">Hawaii</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/India-related_articles" title="Wikipedia:Manual of Style/India-related articles">India</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Indonesia-related_articles" title="Wikipedia:Manual of Style/Indonesia-related articles">Indonesia</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Ireland-related_articles" title="Wikipedia:Manual of Style/Ireland-related articles">Ireland</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Japan-related_articles" title="Wikipedia:Manual of Style/Japan-related articles">Japan</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Korea-related_articles" title="Wikipedia:Manual of Style/Korea-related articles">Korea</a></li> <li><a href="/wiki/Wikipedia:Naming_conventions_(Macedonia)" title="Wikipedia:Naming conventions (Macedonia)">Macedonia</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Malaysia-related_articles" title="Wikipedia:Manual of Style/Malaysia-related articles">Malaysia</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Pakistan-related_articles" title="Wikipedia:Manual of Style/Pakistan-related articles">Pakistan</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Philippines-related_articles" title="Wikipedia:Manual of Style/Philippines-related articles">Philippines</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Poland-related_articles" title="Wikipedia:Manual of Style/Poland-related articles">Poland</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Singapore-related_articles" title="Wikipedia:Manual of Style/Singapore-related articles">Singapore</a></li> <li><a href="/wiki/Wikipedia:WikiProject_Trinidad_and_Tobago/Style_guide" title="Wikipedia:WikiProject Trinidad and Tobago/Style guide">Trinidad and Tobago</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background-color: #def; color:black; padding-left: 2px;;color: var(--color-base)"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(religion)" title="Category:Wikipedia Manual of Style (religion)">Religion</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Wikipedia:Manual_of_Style/Islam-related_articles" title="Wikipedia:Manual of Style/Islam-related articles">Islam</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Latter_Day_Saints" title="Wikipedia:Manual of Style/Latter Day Saints">Latter Day Saints</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background-color: #def; color:black; padding-left: 2px;;color: var(--color-base)"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(science)" title="Category:Wikipedia Manual of Style (science)">Science</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Wikipedia:Manual_of_Style/Chemistry" title="Wikipedia:Manual of Style/Chemistry">Chemistry</a> <ul><li><a href="/wiki/Wikipedia:Manual_of_Style/Chemistry/Compound_classes" title="Wikipedia:Manual of Style/Chemistry/Compound classes">Compound classes</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Chemistry/Chemicals" title="Wikipedia:Manual of Style/Chemistry/Chemicals">Chemicals</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Chemistry/References_and_external_links" title="Wikipedia:Manual of Style/Chemistry/References and external links">References and external links</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Chemistry/Safety" title="Wikipedia:Manual of Style/Chemistry/Safety">Safety</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Chemistry/Structure_drawing" title="Wikipedia:Manual of Style/Chemistry/Structure drawing">Structure drawing</a></li></ul></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Computer_science" title="Wikipedia:Manual of Style/Computer science">Computer science</a></li> <li><a class="mw-selflink selflink">Mathematics</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Medicine-related_articles" title="Wikipedia:Manual of Style/Medicine-related articles">Medicine</a></li> <li><a href="/wiki/Template:Taxobox/doc" title="Template:Taxobox/doc">Taxonomy</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background-color: #def; color:black; padding-left: 2px;;color: var(--color-base)"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(sports)" title="Category:Wikipedia Manual of Style (sports)">Sports</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Wikipedia:Manual_of_Style/Cue_sports" title="Wikipedia:Manual of Style/Cue sports">Cue sports</a> (<a href="/wiki/Wikipedia:Manual_of_Style/Snooker" title="Wikipedia:Manual of Style/Snooker">Snooker</a>)</li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Figure_skating" title="Wikipedia:Manual of Style/Figure skating">Figure skating</a> (<a href="/wiki/Wikipedia:Manual_of_Style/Figure_skating_terminology" title="Wikipedia:Manual of Style/Figure skating terminology">Terminology</a>)</li></ul></div></div></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background-color: #def; color:black; padding-left: 2px;;color: var(--color-base)"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(related_guidelines)" title="Category:Wikipedia Manual of Style (related guidelines)">Related guidelines</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Wikipedia:Article_size" title="Wikipedia:Article size">Article size</a></li> <li><a href="/wiki/Wikipedia:Article_titles" title="Wikipedia:Article titles">Article titles</a></li> <li><a href="/wiki/Wikipedia:Categories,_lists,_and_navigation_templates" title="Wikipedia:Categories, lists, and navigation templates">Categories, lists, and navigation templates</a></li> <li><a href="/wiki/Wikipedia:Categorization" title="Wikipedia:Categorization">Categorization</a></li> <li><a href="/wiki/Wikipedia:Hatnote" title="Wikipedia:Hatnote">Hatnotes</a></li> <li><a href="/wiki/Wikipedia:Subpages" title="Wikipedia:Subpages">Subpages</a></li> <li><a href="/wiki/Wikipedia:Make_technical_articles_understandable" title="Wikipedia:Make technical articles understandable">Understandability</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below"> <ul><li><a href="/wiki/Wikipedia:Simplified_Manual_of_Style" title="Wikipedia:Simplified Manual of Style">Overview</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Contents" title="Wikipedia:Manual of Style/Contents">Contents</a></li> <li><a href="/wiki/Wikipedia:Styletips" title="Wikipedia:Styletips">Tips</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Style" title="Template:Style"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Style" title="Template talk:Style"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Style" title="Special:EditPage/Template:Style"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>This subpage of the <a href="/wiki/Wikipedia:Manual_of_Style" title="Wikipedia:Manual of Style">Manual of Style</a> contains guidelines for writing and editing clear, encyclopedic, attractive, and interesting articles on <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and for the use of mathematical notation in Wikipedia articles on other subjects. For matters of style not treated on this subpage, follow the main Manual of Style and its other subpages to achieve consistency of style throughout Wikipedia. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Structure"><span class="tocnumber">1</span> <span class="toctext">Structure</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Article_introduction"><span class="tocnumber">1.1</span> <span class="toctext">Article introduction</span></a></li> <li class="toclevel-2 tocsection-3"><a href="#Article_body"><span class="tocnumber">1.2</span> <span class="toctext">Article body</span></a></li> <li class="toclevel-2 tocsection-4"><a href="#Concluding_matters"><span class="tocnumber">1.3</span> <span class="toctext">Concluding matters</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-5"><a href="#Writing_style_in_mathematics"><span class="tocnumber">2</span> <span class="toctext">Writing style in mathematics</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#Mathematical_conventions"><span class="tocnumber">3</span> <span class="toctext">Mathematical conventions</span></a> <ul> <li class="toclevel-2 tocsection-7"><a href="#Terminology_conventions"><span class="tocnumber">3.1</span> <span class="toctext">Terminology conventions</span></a> <ul> <li class="toclevel-3 tocsection-8"><a href="#Natural_numbers"><span class="tocnumber">3.1.1</span> <span class="toctext">Natural numbers</span></a></li> <li class="toclevel-3 tocsection-9"><a href="#Algebra"><span class="tocnumber">3.1.2</span> <span class="toctext">Algebra</span></a></li> <li class="toclevel-3 tocsection-10"><a href="#Algebraic_geometry"><span class="tocnumber">3.1.3</span> <span class="toctext">Algebraic geometry</span></a></li> <li class="toclevel-3 tocsection-11"><a href="#Topology"><span class="tocnumber">3.1.4</span> <span class="toctext">Topology</span></a></li> <li class="toclevel-3 tocsection-12"><a href="#Miscellaneous"><span class="tocnumber">3.1.5</span> <span class="toctext">Miscellaneous</span></a></li> </ul> </li> <li class="toclevel-2 tocsection-13"><a href="#Notational_conventions"><span class="tocnumber">3.2</span> <span class="toctext">Notational conventions</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-14"><a href="#Proofs"><span class="tocnumber">4</span> <span class="toctext">Proofs</span></a></li> <li class="toclevel-1 tocsection-15"><a href="#Algorithms"><span class="tocnumber">5</span> <span class="toctext">Algorithms</span></a></li> <li class="toclevel-1 tocsection-16"><a href="#Including_citations_and_literature_references"><span class="tocnumber">6</span> <span class="toctext">Including citations and literature references</span></a></li> <li class="toclevel-1 tocsection-17"><a href="#Typesetting_of_mathematical_formulae"><span class="tocnumber">7</span> <span class="toctext">Typesetting of mathematical formulae</span></a> <ul> <li class="toclevel-2 tocsection-18"><a href="#Using_LaTeX_markup"><span class="tocnumber">7.1</span> <span class="toctext">Using LaTeX markup</span></a> <ul> <li class="toclevel-3 tocsection-19"><a href="#Line_wrapping"><span class="tocnumber">7.1.1</span> <span class="toctext">Line wrapping</span></a></li> <li class="toclevel-3 tocsection-20"><a href="#Deprecated_formatting"><span class="tocnumber">7.1.2</span> <span class="toctext">Deprecated formatting</span></a></li> <li class="toclevel-3 tocsection-21"><a href="#Alt_text"><span class="tocnumber">7.1.3</span> <span class="toctext">Alt text</span></a></li> </ul> </li> <li class="toclevel-2 tocsection-22"><a href="#Using_HTML"><span class="tocnumber">7.2</span> <span class="toctext">Using HTML</span></a> <ul> <li class="toclevel-3 tocsection-23"><a href="#Font_formatting"><span class="tocnumber">7.2.1</span> <span class="toctext">Font formatting</span></a> <ul> <li class="toclevel-4 tocsection-24"><a href="#Variables"><span class="tocnumber">7.2.1.1</span> <span class="toctext">Variables</span></a></li> <li class="toclevel-4 tocsection-25"><a href="#Functions"><span class="tocnumber">7.2.1.2</span> <span class="toctext">Functions</span></a></li> <li class="toclevel-4 tocsection-26"><a href="#Sets"><span class="tocnumber">7.2.1.3</span> <span class="toctext">Sets</span></a></li> <li class="toclevel-4 tocsection-27"><a href="#Greek_letters"><span class="tocnumber">7.2.1.4</span> <span class="toctext">Greek letters</span></a></li> <li class="toclevel-4 tocsection-28"><a href="#Common_sets_of_numbers"><span class="tocnumber">7.2.1.5</span> <span class="toctext">Common sets of numbers</span></a></li> <li class="toclevel-4 tocsection-29"><a href="#Superscripts_and_subscripts"><span class="tocnumber">7.2.1.6</span> <span class="toctext">Superscripts and subscripts</span></a></li> </ul> </li> <li class="toclevel-3 tocsection-30"><a href="#Special_symbols"><span class="tocnumber">7.2.2</span> <span class="toctext">Special symbols</span></a></li> <li class="toclevel-3 tocsection-31"><a href="#Less-than_sign"><span class="tocnumber">7.2.3</span> <span class="toctext">Less-than sign</span></a></li> <li class="toclevel-3 tocsection-32"><a href="#Multiplication_sign"><span class="tocnumber">7.2.4</span> <span class="toctext">Multiplication sign</span></a></li> <li class="toclevel-3 tocsection-33"><a href="#Minus_sign"><span class="tocnumber">7.2.5</span> <span class="toctext">Minus sign</span></a></li> <li class="toclevel-3 tocsection-34"><a href="#Square_brackets"><span class="tocnumber">7.2.6</span> <span class="toctext">Square brackets</span></a></li> <li class="toclevel-3 tocsection-35"><a href="#Function_symbol"><span class="tocnumber">7.2.7</span> <span class="toctext">Function symbol</span></a></li> <li class="toclevel-3 tocsection-36"><a href="#Radical_symbol"><span class="tocnumber">7.2.8</span> <span class="toctext">Radical symbol</span></a></li> </ul> </li> <li class="toclevel-2 tocsection-37"><a href="#Explanation_of_symbols_in_formulae"><span class="tocnumber">7.3</span> <span class="toctext">Explanation of symbols in formulae</span></a></li> <li class="toclevel-2 tocsection-38"><a href="#Punctuation_after_formulae"><span class="tocnumber">7.4</span> <span class="toctext">Punctuation after formulae</span></a></li> <li class="toclevel-2 tocsection-39"><a href="#Font_usage"><span class="tocnumber">7.5</span> <span class="toctext">Font usage</span></a> <ul> <li class="toclevel-3 tocsection-40"><a href="#Multi-letter_names"><span class="tocnumber">7.5.1</span> <span class="toctext">Multi-letter names</span></a></li> <li class="toclevel-3 tocsection-41"><a href="#Roman_versus_italic"><span class="tocnumber">7.5.2</span> <span class="toctext">Roman versus italic</span></a></li> <li class="toclevel-3 tocsection-42"><a href="#Blackboard_bold"><span class="tocnumber">7.5.3</span> <span class="toctext">Blackboard bold</span></a></li> </ul> </li> <li class="toclevel-2 tocsection-43"><a href="#Fractions"><span class="tocnumber">7.6</span> <span class="toctext">Fractions</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-44"><a href="#Graphs_and_diagrams"><span class="tocnumber">8</span> <span class="toctext">Graphs and diagrams</span></a></li> <li class="toclevel-1 tocsection-45"><a href="#See_also"><span class="tocnumber">9</span> <span class="toctext">See also</span></a> <ul> <li class="toclevel-2 tocsection-46"><a href="#Help_for_those_writing_a_formula"><span class="tocnumber">9.1</span> <span class="toctext">Help for those writing a formula</span></a></li> <li class="toclevel-2 tocsection-47"><a href="#General_information"><span class="tocnumber">9.2</span> <span class="toctext">General information</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-48"><a href="#Notes"><span class="tocnumber">10</span> <span class="toctext">Notes</span></a></li> <li class="toclevel-1 tocsection-49"><a href="#Further_reading"><span class="tocnumber">11</span> <span class="toctext">Further reading</span></a></li> </ul> </div> </section><div class="mw-heading mw-heading2 ext-discussiontools-init-section section-heading" onclick="mfTempOpenSection(1)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Structure" data-mw-thread-id="h-Structure"><span data-mw-comment-start="" id="h-Structure"></span>Structure<span data-mw-comment-end="h-Structure"></span></h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=1" title="Edit section: Structure" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> <p>Probably the hardest part of writing a Wikipedia article on a mathematical topic, and generally any Wikipedia article, is addressing a reader's level of knowledge. For example, when writing about a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> in the context of <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, is it best to assume that a reader is already familiar with <a href="/wiki/Group_theory" title="Group theory">group theory</a>? A general approach to writing an article is to start simple and then move towards more abstract and technical subjects later on in the article. </p> <div class="mw-heading mw-heading3"><h3 id="Article_introduction" data-mw-thread-id="h-Article_introduction-Structure"><span data-mw-comment-start="" id="h-Article_introduction-Structure"></span>Article introduction<span data-mw-comment-end="h-Article_introduction-Structure"></span></h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=2" title="Edit section: Article introduction" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main page: <a href="/wiki/Wikipedia:Manual_of_Style/Lead_section" title="Wikipedia:Manual of Style/Lead section">Wikipedia:Manual of Style/Lead section</a></div> <p>Articles should start with a short introductory section, called the "lead". The purpose of the lead is to </p> <ul><li>describe and define the subject,</li> <li>provide context regarding the subject,</li> <li>and summarize the article's most important points.</li></ul> <p>The lead should, as much as possible, be accessible to a general reader, so specialized terminology and symbols should be avoided. Formulas should appear in the first paragraph only if necessary, since they will not be displayed in the preview that pops up when hovering over a link. For having formulae displayed when hovering, they must be written in raw html (without templates <span class="nowrap">{{</span><a href="/wiki/Template:Var" title="Template:Var">var</a><span class="nowrap">}}</span> or <span class="nowrap">{{</span><a href="/wiki/Template:Math" title="Template:Math">math</a><span class="nowrap">}}</span>), or in LaTeX (inside <math>...</math>). In the latter case the LaTeX source is displayed without the tags <math> and </math>. </p><p>In general, the lead sentence should include the article title, or some variation thereof, in bold along with any alternate names, also in bold. The lead sentence should state that the article is about a topic in mathematics, unless the title already does so. It is safe to assume that a reader is familiar with the subjects of arithmetic, algebra, geometry, and that they may have heard of calculus, but are likely unfamiliar with it. For articles that are on these subjects, or on simpler subjects, it can be assumed that the reader is not familiar with the aforementioned subjects. A reader can be assumed to be ignorant of any topics outside of that scope or more advanced topics. </p><p>The lead sentence should informally define or describe the subject. For example: </p> <blockquote><p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>topology</b> (from the <a href="/wiki/Greek_language" title="Greek language">Greek</a> <span title="Greek-language text"><span lang="el">τόπος</span></span>, 'place', and <span title="Greek-language text"><span lang="el">λόγος</span></span>, 'study') is concerned with the properties of a <a href="/wiki/Mathematical_object" title="Mathematical object">geometric object</a> that are preserved under <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> <a href="/wiki/Deformation_theory" class="mw-redirect" title="Deformation theory">deformations</a>, such as <a href="/wiki/Stretch_factor" title="Stretch factor">stretching</a>, <a href="/wiki/Twist_(mathematics)" class="mw-redirect mw-disambig" title="Twist (mathematics)">twisting</a>, crumpling and bending, but not tearing or <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">gluing</a>.</p></blockquote> <blockquote><p>In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean plane geometry</a>, <b>Apollonius's problem</b> is to construct circles that are <a href="/wiki/Tangent" title="Tangent">tangent</a> to three given circles in a plane.</p></blockquote> <p>The lead section should include, when appropriate: </p> <ul><li><i>Historical motivation</i>, including names and dates, especially if the article does not have a "History" section. The origin of the subject's name should be explained if it is not self-evident.</li> <li>An <i>informal introduction</i> to the topic, without rigor, suitable for a general audience. The appropriate audience for the overview will vary by article, but it should be as basic as reasonable. The informal introduction should clearly state that it is informal, and that it is only stated to introduce the formal approach. Include a physical or geometric analogy or diagram if it can help introduce the topic.</li> <li><i>Motivation</i> or <i>applications</i>, which can illuminate the use of the topic and its connections to other areas of mathematics or other non-mathematical subjects.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Article_body" data-mw-thread-id="h-Article_body-Structure"><span data-mw-comment-start="" id="h-Article_body-Structure"></span>Article body<span data-mw-comment-end="h-Article_body-Structure"></span></h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=3" title="Edit section: Article body" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Readers have differing levels of experience and knowledge. When in doubt, articles should define the notation they use. For example, some readers will immediately recognize that Δ(<i>K</i>) means the <a href="/wiki/Discriminant_of_an_algebraic_number_field" title="Discriminant of an algebraic number field">discriminant of a number field</a>, but others will never have encountered the notation. The latter group will be helped by an aside like "...where Δ(<i>K</i>) is the <a href="/wiki/Discriminant_of_an_algebraic_number_field" title="Discriminant of an algebraic number field">discriminant</a> of the <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <i>K</i>". </p><p>Use standard notation when possible. If an article requires non-standard or uncommon notation, they should be defined. For example, an article that uses <i>x</i>^<i>n</i> or <i>x</i>**<i>n</i> to denote exponentiation (instead of <i>x<sup>n</sup></i>) should define the notations. If an article requires extensive notation, consider introducing the notation as a bulleted list or separating it into a section titled "Notation". </p><p>An article about a <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical object</a> should provide an exact definition of the object, perhaps in a "Definition" section after section(s) of motivation. For example: </p> <blockquote> <p>Let <i>S</i> and <i>T</i> be <a href="/wiki/Topological_space" title="Topological space">topological spaces</a>, and let <i>f</i> be a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> from <i>S</i> to <i>T</i>. Then <i>f</i> is called <i>continuous</i> if, for every <a href="/wiki/Open_set" title="Open set">open set</a> <i>O</i> in <i>T</i>, the <a href="/wiki/Preimage" class="mw-redirect" title="Preimage">preimage</a> <span style="padding-right:0.15em;"><i>f</i></span><sup>−1</sup>(<i>O</i>) is an open set in <i>S</i>. </p> </blockquote> <p>The phrase "formal definition" may help to flag the actual definition of a concept for readers unfamiliar with academic terminology, in which "definition" means formal definition, and a "proof" is always a formal proof. </p><p>When the topic is a theorem, the article should provide a precise statement of the theorem. Sometimes this statement will be in the lead, for example: </p> <blockquote> <p><b>Lagrange's theorem</b>, in the <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> of <a href="/wiki/Group_theory" title="Group theory">group theory</a>, states that for any <a href="/wiki/Finite_group" title="Finite group">finite group</a> <i>G</i>, the <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> (number of elements) of every <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> <i>H</i> of <i>G</i> divides the order of <i>G</i>. </p> </blockquote> <p>Other times, it may be better to separate the statement into its own section, as for long theorems like the <a href="/wiki/Poincar%C3%A9%E2%80%93Birkhoff%E2%80%93Witt_theorem" title="Poincaré–Birkhoff–Witt theorem">Poincaré–Birkhoff–Witt theorem</a>, or to present multiple equivalent formulations, as for <a href="/wiki/Nakayama%27s_lemma" title="Nakayama's lemma">Nakayama's lemma</a>. </p><p>Representative examples and applications help to illustrate definitions and theorems and to provide context for why they might be interesting. Shorter examples may fit into the main exposition of the article, such as the discussion at <a href="/wiki/Algebraic_number_theory#Failure_of_unique_factorization" title="Algebraic number theory">Algebraic number theory § Failure of unique factorization</a>, while others may deserve their own section, as in <a href="/wiki/Chain_rule#First_example" title="Chain rule">Chain rule § First example</a>. Multiple related examples may also be given together, as in <a href="/wiki/Adjunction_formula#Applications_to_curves" title="Adjunction formula">Adjunction formula § Applications to curves</a>. Occasionally, it is appropriate to give a large number of computationally-flavored examples, as in <a href="/wiki/Lambert_W_function#Applications" title="Lambert W function">Lambert W function § Applications</a>. It may also be edifying to list non-examples, which almost-but-not-quite satisfy the definition. In keeping with the purpose and <a href="/wiki/Wikipedia:TONE" class="mw-redirect" title="Wikipedia:TONE">tone</a> of an encyclopedia, examples should be <i>informative</i> rather than <i>instructional</i> (see <a href="/wiki/Wikipedia:NOTTEXTBOOK" class="mw-redirect" title="Wikipedia:NOTTEXTBOOK">WP:NOTTEXTBOOK</a> for details). </p><p>A picture can really bring home a point, and can often precede the mathematical discussion of a concept. <a href="/wiki/Wikipedia:How_to_create_graphs_for_Wikipedia_articles" class="mw-redirect" title="Wikipedia:How to create graphs for Wikipedia articles">How to create graphs for Wikipedia articles</a> contains some details on how to create graphs and other pictures as well as how to include them in articles. </p><p>Formulas tend to repel less mathematical readers, and mathematics articles should take pains to explain (or even replace) them by words if possible. In particular, the English words "for all", "exists", and "in" should be preferred to the corresponding symbols ∀, ∃, and ∈. Similarly, definitions should be highlighted with words such as "is defined by" in the text. </p><p>If not included in the introduction, a history section can provide additional context and details on the topic's motivation and connections. </p> <div class="mw-heading mw-heading3"><h3 id="Concluding_matters" data-mw-thread-id="h-Concluding_matters-Structure"><span data-mw-comment-start="" id="h-Concluding_matters-Structure"></span>Concluding matters<span data-mw-comment-end="h-Concluding_matters-Structure"></span></h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=4" title="Edit section: Concluding matters" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Most mathematical ideas are capable of some form of generalization. If appropriate, such material can be put under a "Generalizations" section. As an example, multiplication of the rational numbers can be generalized to other <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a>. </p><p>It is also generally good to have a "See also" section in an article. The section should link to related subjects, or to pages which could provide more insight into the contents of the article. More details on "See also" sections can be found at <a href="/wiki/Wikipedia:Manual_of_Style/Layout#%22See_also%22_section" title="Wikipedia:Manual of Style/Layout">Wikipedia:Manual of Style/Layout § "See also" section</a>. Lastly, a well-written and complete article should have a "References" section. This topic is discussed in detail in the section <a href="#Including_literature_and_references">§ Including literature and references</a>. </p> </section><div class="mw-heading mw-heading2 ext-discussiontools-init-section section-heading" onclick="mfTempOpenSection(2)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Writing_style_in_mathematics" data-mw-thread-id="h-Writing_style_in_mathematics"><span data-mw-comment-start="" id="h-Writing_style_in_mathematics"></span><span class="anchor" id="TONE"></span><span class="anchor" id="NOWE"></span>Writing style in mathematics<span data-mw-comment-end="h-Writing_style_in_mathematics"></span></h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=5" title="Edit section: Writing style in mathematics" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1252059228"><div class="module-shortcutanchordiv"><span id="MOS:MATH#TONE"></span><span id="MOS:MATH#NOWE"></span></div><div class="module-shortcutboxplain noprint" role="note"><div class="module-shortcutlist"><a href="/wiki/Wikipedia:Shortcut" title="Wikipedia:Shortcut">Shortcuts</a></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><li><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=MOS:MATH&redirect=no#TONE">MOS:MATH#TONE</a></span></li><li><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=MOS:MATH&redirect=no#NOWE">MOS:MATH#NOWE</a></span></li></ul></div></div> <p>There are several issues of writing style that are particularly relevant in mathematical writing. </p><p>In the interest of clarity, sentences should not begin with a symbol. Do not write: </p> <ul><li><span class="example deprecated-content example-bad" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-removed, #8B0000);">Suppose that <i>G</i> is a group. <i>G</i> can be decomposed into cosets, as follows.</span></li> <li><span class="example deprecated-content example-bad" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-removed, #8B0000);">Let <i>H</i> be the corresponding subgroup of <i>G</i>. <i>H</i> is then finite.</span></li></ul> <p>Instead, write something like: </p> <ul><li><span class="example" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-added, #006400);">A group <i>G</i> may be decomposed into cosets, as follows.</span></li> <li><span class="example" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-added, #006400);">Let <i>H</i> be the corresponding subgroup of <i>G</i>. Then <i>H</i> is finite.</span></li></ul><p> Mathematics articles are often written in a conversational style similar to a whiteboard lecture. However, a narrative pedagogical style runs counter to Wikipedia's recommended encyclopedic tone. While opinions vary on the most edifying style, authors should generally strike a balance between bare lists of facts and formulae, and relying too much on addressing the reader directly and referring to "we". Also avoid contentless clichés as <span class="example deprecated-content example-bad" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-removed, #8B0000);">Note that</span>, <span class="example deprecated-content example-bad" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-removed, #8B0000);">It should be noted that</span>, <span class="example deprecated-content example-bad" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-removed, #8B0000);">It must be mentioned that</span>, <span class="example deprecated-content example-bad" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-removed, #8B0000);">It must be emphasized that</span>, <span class="example deprecated-content example-bad" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-removed, #8B0000);">Consider that</span>, and <span class="example deprecated-content example-bad" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-removed, #8B0000);">We see that</span>. There is no use in imploring the reader to take note of each thing being pointed out. Rather than drawing the reader's attention to crucial information buried in the text, try to reorganize and rephrase to put the crucial part first. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"></p><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/MOS:NOTED" class="mw-redirect" title="MOS:NOTED">MOS:NOTED</a></div> <p>Articles should be as accessible as possible to readers not already familiar with the subject matter. Notations not entirely standard should be properly introduced and explained. Whenever a variable or other symbol is defined by a formula, make sure to say this is a definition introducing a notation, not an equation involving a previously known object. Also identify the nature of the entity being defined. Don't write: </p> <ul><li><span class="example deprecated-content example-bad" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-removed, #8B0000);">Multiplying <i>M</i> by <i>u</i> = <i>v</i> − <i>v</i><sub>0</sub>, ...</span></li></ul> <p>Instead, write: </p> <ul><li><span class="example" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-added, #006400);">Multiplying <i>M</i> by the vector <i>u</i> defined by <i>u</i> = <i>v</i> − <i>v</i><sub>0</sub>, ...</span></li></ul> <p><span class="anchor" id="DEFSYMBOL"></span> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1252059228"><div class="module-shortcutanchordiv"><span id="MOS:MATH#DEFSYMBOL"></span></div><div class="module-shortcutboxplain noprint" role="note"><div class="module-shortcutlist"><a href="/wiki/Wikipedia:Shortcut" title="Wikipedia:Shortcut">Shortcut</a></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><li><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=MOS:MATH&redirect=no#DEFSYMBOL">MOS:MATH#DEFSYMBOL</a></span></li></ul></div></div> <p>In definitions, the symbol "=" is preferred over "≡" or "<b>:</b>=". </p><p>When defining a term, do not use the phrase "if and only if". For example, instead of </p> <ul><li><span class="example deprecated-content example-bad" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-removed, #8B0000);">A function <i>f</i> is <i>even</i> if and only if <i>f</i>(−<i>x</i>) = <i>f</i>(<i>x</i>) for all <i>x</i></span></li></ul> <p>write </p> <ul><li><span class="example" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-added, #006400);">A function <i>f</i> is <i>even</i> if <i>f</i>(−<i>x</i>) = <i>f</i>(<i>x</i>) for all <i>x</i>.</span></li></ul> <p>If it is reasonable to do so, rephrase the sentence to avoid the use of the word "if" entirely. For example, </p> <ul><li><span class="example" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-added, #006400);">An <i>even function</i> is a function <i>f</i> such that <i>f</i>(−<i>x</i>) = <i>f</i>(<i>x</i>) for all <i>x</i>.</span></li></ul> <p><span class="anchor" id="OBVIOUS"></span> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1252059228"><div class="module-shortcutanchordiv"><span id="MOS:MATH#OBVIOUS"></span></div><div class="module-shortcutboxplain noprint" role="note"><div class="module-shortcutlist"><a href="/wiki/Wikipedia:Shortcut" title="Wikipedia:Shortcut">Shortcut</a></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><li><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=MOS:MATH&redirect=no#OBVIOUS">MOS:MATH#OBVIOUS</a></span></li></ul></div></div> <p>Avoid, as far as possible, useless phrases such as: </p> <ul><li><span class="example deprecated-content example-bad" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-removed, #8B0000);">It is easily seen that ...</span></li> <li><span class="example deprecated-content example-bad" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-removed, #8B0000);">Clearly ...</span></li> <li><span class="example deprecated-content example-bad" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-removed, #8B0000);">Obviously ...</span></li></ul> <p>The reader might not find what you write obvious. Instead, try to hint <i>why</i> something must hold, such as: </p> <ul><li><span class="example" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-added, #006400);">It follows directly from this definition that ...</span></li> <li><span class="example" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-added, #006400);">By a straightforward, if lengthy, algebraic calculation, ...</span></li></ul> <p>Articles should avoid common blackboard abbreviations such as <i>wrt</i> (with respect to), <i>wlog</i> (without loss of generality), and <i>iff</i> (if and only if), as well as quantifier symbols ∀ and ∃ instead of <i>for all</i> and <i>there exists</i>. In addition to compromising the encyclopedic tone, these abbreviations are a form of jargon that may confuse the reader. </p><p><span class="anchor" id="ANY"></span> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1252059228"><div class="module-shortcutanchordiv"><span id="MOS:MATH#ANY"></span></div><div class="module-shortcutboxplain noprint" role="note"><div class="module-shortcutlist"><a href="/wiki/Wikipedia:Shortcut" title="Wikipedia:Shortcut">Shortcut</a></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><li><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=MOS:MATH&redirect=no#ANY">MOS:MATH#ANY</a></span></li></ul></div></div> <p>Avoid <i>any</i> when verbalizing quantifiers since it is ambiguous. Instead of <span class="example deprecated-content example-bad" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-removed, #8B0000);">if any <i>x</i> satisfies <i>F</i>(<i>x</i>) = 0</span>, write <span class="example" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-added, #006400);">if every <i>x</i> satisfies <i>F</i>(<i>x</i>) = 0</span>, or <span class="example" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-added, #006400);">if some <i>x</i> satisfies <i>F</i>(<i>x</i>) = 0</span>, depending on what you wish to express. </p><p><span class="anchor" id="PLURAL"></span> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1252059228"><div class="module-shortcutanchordiv"><span id="MOS:MATH#PLURAL"></span></div><div class="module-shortcutboxplain noprint" role="note"><div class="module-shortcutlist"><a href="/wiki/Wikipedia:Shortcut" title="Wikipedia:Shortcut">Shortcut</a></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><li><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=MOS:MATH&redirect=no#PLURAL">MOS:MATH#PLURAL</a></span></li></ul></div></div><p>The plural of <i>formula</i> is either <i>formulae</i> or <i>formulas</i>. Both are acceptable, but an article should be internally consistent. In an already consistent article, editors should refrain from changing one style to another. </p></section><div class="mw-heading mw-heading2 ext-discussiontools-init-section section-heading" onclick="mfTempOpenSection(3)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Mathematical_conventions" data-mw-thread-id="h-Mathematical_conventions"><span data-mw-comment-start="" id="h-Mathematical_conventions"></span>Mathematical conventions<span data-mw-comment-end="h-Mathematical_conventions"></span></h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=6" title="Edit section: Mathematical conventions" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Wikipedia:WikiProject_Logic/Standards_for_notation" title="Wikipedia:WikiProject Logic/Standards for notation">Wikipedia:WikiProject Logic/Standards for notation</a></div> <p>A number of conventions have been developed to make Wikipedia's mathematics articles more consistent with each other. These conventions cover choices of terminology, such as the definitions of <i>compact</i> and <i>ring</i>, as well as notation, such as the correct symbols to use for a subset. </p><p>These conventions are suggested in order to bring some uniformity between different articles, to aid a reader who moves from one article to another. However, each article may establish its own conventions. For example, an article on a specialized subject might be more clear if written using the conventions common in that area. Thus the act of changing an article from one set of conventions to another should not be undertaken lightly. </p><p>Each article should explain its own terminology as if there are no conventions, in order to minimize the chance of confusion. Not only do different articles use different conventions, but Wikipedia's readers come to articles with widely different conventions in mind. These readers will often not be familiar with our conventions, which may differ greatly from the conventions they see outside Wikipedia. Moreover, when our articles are presented in print or on other websites, there may be no simple way for readers to check what conventions have been employed. </p> <div class="mw-heading mw-heading3"><h3 id="Terminology_conventions" data-mw-thread-id="h-Terminology_conventions-Mathematical_conventions"><span data-mw-comment-start="" id="h-Terminology_conventions-Mathematical_conventions"></span>Terminology conventions<span data-mw-comment-end="h-Terminology_conventions-Mathematical_conventions"></span></h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=7" title="Edit section: Terminology conventions" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <div class="mw-heading mw-heading4"><h4 id="Natural_numbers" data-mw-thread-id="h-Natural_numbers-Terminology_conventions"><span data-mw-comment-start="" id="h-Natural_numbers-Terminology_conventions"></span>Natural numbers<span data-mw-comment-end="h-Natural_numbers-Terminology_conventions"></span></h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=8" title="Edit section: Natural numbers" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>"The <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of <a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a>" has two common meanings: <span class="nowrap">{0, 1, 2, 3, ...},</span> which may also be called <i>non-negative integers</i>, and <span class="nowrap">{1, 2, 3, ...},</span> which may also be called <i>positive integers</i>. Use the sense appropriate to the field to which the subject of the article belongs if the field has a preferred convention. If the sense is unclear, and if it is important whether or not zero is included, consider using one of the alternative phrases rather than <i>natural numbers</i> if the context permits. </p> <div class="mw-heading mw-heading4"><h4 id="Algebra" data-mw-thread-id="h-Algebra-Terminology_conventions"><span data-mw-comment-start="" id="h-Algebra-Terminology_conventions"></span>Algebra<span data-mw-comment-end="h-Algebra-Terminology_conventions"></span></h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=9" title="Edit section: Algebra" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <ul><li>A <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> is assumed to be <a href="/wiki/Associative" class="mw-redirect" title="Associative">associative</a> and <a href="/wiki/Unital_algebra" class="mw-redirect" title="Unital algebra">unital</a>. A structure satisfying all the ring axioms except the existence of a multiplicative identity is called a <a href="/wiki/Rng_(algebra)" title="Rng (algebra)">rng</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> There is an exception for rings of operators, such as <a href="/wiki/*_algebra" class="mw-redirect" title="* algebra">* algebras</a>, <a href="/wiki/B*_algebra" class="mw-redirect" title="B* algebra">B* algebras</a>, <a href="/wiki/C*_algebra" class="mw-redirect" title="C* algebra">C* algebras</a>, which we do not assume to be unital.</li> <li>The ring with one element is called the <a href="/wiki/Zero_ring" title="Zero ring">zero ring</a>.</li> <li>A <a href="/wiki/Local_ring" title="Local ring">local ring</a> is not assumed <a href="/wiki/Noetherian_ring" title="Noetherian ring">noetherian</a> (<i>contra</i> <a href="/wiki/Zariski" class="mw-redirect" title="Zariski">Zariski</a>).</li> <li>For <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebras</a> use <i>v</i><sup>2</sup> = +<i>Q</i>(<i>v</i>).</li></ul> <div class="mw-heading mw-heading4"><h4 id="Algebraic_geometry" data-mw-thread-id="h-Algebraic_geometry-Terminology_conventions"><span data-mw-comment-start="" id="h-Algebraic_geometry-Terminology_conventions"></span>Algebraic geometry<span data-mw-comment-end="h-Algebraic_geometry-Terminology_conventions"></span></h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=10" title="Edit section: Algebraic geometry" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <ul><li>An <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic variety</a> is assumed to be an <i>irreducible</i> <a href="/wiki/Algebraic_set" class="mw-redirect" title="Algebraic set">algebraic set</a>.</li> <li>A <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a> is not assumed to be separated. The term "prescheme" is not used.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Topology" data-mw-thread-id="h-Topology-Terminology_conventions"><span data-mw-comment-start="" id="h-Topology-Terminology_conventions"></span>Topology<span data-mw-comment-end="h-Topology-Terminology_conventions"></span></h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=11" title="Edit section: Topology" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <ul><li>A <a href="/wiki/Compact_space" title="Compact space">compact space</a> is not assumed to be <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> (<i>contra</i> <a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki</a>, who uses <a href="/wiki/Quasi-compact" class="mw-redirect" title="Quasi-compact">quasi-compact</a> for our notion of <a href="/wiki/Compact_space" title="Compact space">compactness</a>).</li> <li>Separation axioms for topological spaces are as described on the <a href="/wiki/Separation_axiom" title="Separation axiom">separation axiom</a> page.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Miscellaneous" data-mw-thread-id="h-Miscellaneous-Terminology_conventions"><span data-mw-comment-start="" id="h-Miscellaneous-Terminology_conventions"></span>Miscellaneous<span data-mw-comment-end="h-Miscellaneous-Terminology_conventions"></span></h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=12" title="Edit section: Miscellaneous" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <ul><li><a href="/wiki/Directed_set" title="Directed set">Directed sets</a> are <a href="/wiki/Preordered_set" class="mw-redirect" title="Preordered set">preordered sets</a> with finite joins, not <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partial orders</a> as in, e.g., <a href="/wiki/John_L._Kelley" title="John L. Kelley">Kelley</a> (<i>General Topology</i>; <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90125-6" title="Special:BookSources/0-387-90125-6">0-387-90125-6</a>).</li> <li>A <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattice</a> need not be bounded. In a bounded lattice, 0 and 1 are allowed to be equal.</li> <li><a href="/wiki/Elliptic_function" title="Elliptic function">Elliptic functions</a> are written in <i>ω</i> = <a href="/w/index.php?title=Half-period&action=edit&redlink=1" class="new" title="Half-period (page does not exist)">half-period</a> style.</li> <li>A weight <i>k</i> <a href="/wiki/Modular_form" title="Modular form">modular form</a> follows the Serre convention that <i>f</i>(−1/<i>τ</i>) = <i>τ</i><sup><i>k</i></sup><i>f</i>(<i>τ</i>), and <i>q</i> = <i>e</i><sup>2<i>πiτ</i></sup>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Notational_conventions" data-mw-thread-id="h-Notational_conventions-Mathematical_conventions"><span data-mw-comment-start="" id="h-Notational_conventions-Mathematical_conventions"></span>Notational conventions<span data-mw-comment-end="h-Notational_conventions-Mathematical_conventions"></span></h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=13" title="Edit section: Notational conventions" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <ul><li>The abstract <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> of <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> <i>n</i>, when written additively, has notation Z<sub><i>n</i></sub>, or in contexts where there may be confusion with <a href="/wiki/P-adic_integer" class="mw-redirect" title="P-adic integer"><i>p</i>-adic integers</a>, <b>Z</b>/<i>n</i><b>Z</b>; when written multiplicatively, e.g. as <a href="/wiki/Roots_of_unity" class="mw-redirect" title="Roots of unity">roots of unity</a>, C<sub><i>n</i></sub> is used (this does not affect the notation of <a href="/wiki/Isometry_group" title="Isometry group">isometry groups</a> called C<sub><i>n</i></sub>).</li> <li>The standard notation for the abstract <a href="/wiki/Dihedral_group" title="Dihedral group">dihedral group</a> of order 2<i>n</i> is D<sub><i>n</i></sub> in geometry and D<sub>2<i>n</i></sub> in finite group theory. There is no good way to reconcile these two conventions, so articles using them should make clear which they are using.</li> <li><a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli numbers</a> are denoted by B<sub><i>n</i></sub>, and are zero for <i>n</i> odd and greater than 1.</li> <li>In <a href="/wiki/Category_theory" title="Category theory">category theory</a>, write <a href="/wiki/Hom-set" class="mw-redirect" title="Hom-set">Hom-sets</a>, or <a href="/wiki/Morphism" title="Morphism">morphisms</a> from <i>A</i> to <i>B</i>, as Hom(<i>A</i>,<i>B</i>) rather than Mor(<i>A</i>,<i>B</i>) (and with the implied convention that the category is not a <a href="/wiki/Small_category" class="mw-redirect" title="Small category">small category</a> unless that is said).</li> <li>The <a href="/wiki/Semidirect_product" title="Semidirect product">semidirect product</a> of groups <i>K</i> and <i>Q</i> should be written <span class="nowrap"><i>K</i> ×<sub><i>φ</i></sub> <i>Q</i></span> or <span class="nowrap"><i>Q</i> ×<sub><i>φ</i></sub> <i>K</i></span> where <i>K</i> is the normal subgroup and <span class="nowrap"><i>φ</i> : <i>Q</i> → Aut(<i>K</i>)</span> is the <a href="/wiki/Homomorphism" title="Homomorphism">homomorphism</a> defining the product. The semidirect product may also be written <span class="nowrap"><i>K</i> ⋊ <i>Q</i></span> or <span class="nowrap"><i>Q</i> ⋉ <i>K</i></span> (with the bar on the side of the non-normal subgroup) with or without the <i>φ</i>. <ul><li>The context should clearly state that this is a semidirect product and should state which group is normal.</li> <li>The bar notation is discouraged because it is not supported by all browsers.</li> <li>If the bar notation is used it should be entered as <code>{{unicode|&#x22C9;}}</code> (⋉) or <code>{{unicode|&#x22CA;}}</code> (⋊) for maximum portability.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Dates_and_numbers#Chronological_items" title="Wikipedia:Manual of Style/Dates and numbers"><span title="Template:Unicode is deprecated (July 2023)">needs update</span></a></i>]</sup></li></ul></li> <li><a href="/wiki/Subset" title="Subset">Subset</a> is denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subseteq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊆</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subseteq }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a924f8dcb2847bb8871edfdbf4c6b5cca0669228" 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 \subseteq }"></noscript><span class="lazy-image-placeholder" style="width: 1.808ex;height: 2.176ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a924f8dcb2847bb8871edfdbf4c6b5cca0669228" data-alt="{\displaystyle \subseteq }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, proper subset by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subsetneq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊊</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subsetneq }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ccda11a9c5eb10088602771f7c0f05ffea7f41c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \subsetneq }"></noscript><span class="lazy-image-placeholder" style="width: 1.808ex;height: 2.509ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ccda11a9c5eb10088602771f7c0f05ffea7f41c" data-alt="{\displaystyle \subsetneq }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. The symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊂</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subset }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f51f0eeff0c2a9dcb9c856f87ca0359e701ef01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \subset }"></noscript><span class="lazy-image-placeholder" style="width: 1.808ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f51f0eeff0c2a9dcb9c856f87ca0359e701ef01" data-alt="{\displaystyle \subset }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> may be used if the meaning is clear from context, or if it is not important whether it is interpreted as subset or as proper subset (for example, <span class="mwe-math-element"><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\subset B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊂</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subset B}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010e98bb4c817357e3ef7e8fa7fbe2385b2aec6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\subset B}"></noscript><span class="lazy-image-placeholder" style="width: 6.606ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010e98bb4c817357e3ef7e8fa7fbe2385b2aec6e" data-alt="{\displaystyle A\subset B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> might be given as the hypothesis of a theorem whose conclusion is obviously true in the case that <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=B}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/045cafe35b1e9c9ac889481fd7178d6f59a77fdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A=B}"></noscript><span class="lazy-image-placeholder" style="width: 6.606ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/045cafe35b1e9c9ac889481fd7178d6f59a77fdb" data-alt="{\displaystyle A=B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>). All other uses of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊂</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subset }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f51f0eeff0c2a9dcb9c856f87ca0359e701ef01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \subset }"></noscript><span class="lazy-image-placeholder" style="width: 1.808ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f51f0eeff0c2a9dcb9c856f87ca0359e701ef01" data-alt="{\displaystyle \subset }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> symbol should be explicitly explained in the text.</li> <li>For a matrix <a href="/wiki/Transpose" title="Transpose">transpose</a>, use superscript non-italic capital letter T: <i>X</i><sup>T</sup>, <span class="mwe-math-element"><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^{\mathrm {T} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{\mathrm {T} }}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4492698e0bacc68f8b1db02074e5f6372892b190" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.416ex; height:2.676ex;" alt="{\displaystyle X^{\mathrm {T} }}"></noscript><span class="lazy-image-placeholder" style="width: 3.416ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4492698e0bacc68f8b1db02074e5f6372892b190" data-alt="{\displaystyle X^{\mathrm {T} }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> or <span class="mwe-math-element"><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^{\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{\mathsf {T}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10c92ba91b82568321949fe3842b338a0800d30b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.348ex; height:2.676ex;" alt="{\displaystyle X^{\mathsf {T}}}"></noscript><span class="lazy-image-placeholder" style="width: 3.348ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10c92ba91b82568321949fe3842b338a0800d30b" data-alt="{\displaystyle X^{\mathsf {T}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, and not <i>X</i><sup><i>T</i></sup>, <span class="mwe-math-element"><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^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{T}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fafdb86d262cb17cb5b920d159c35b5bf10c4ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.386ex; height:2.676ex;" alt="{\displaystyle X^{T}}"></noscript><span class="lazy-image-placeholder" style="width: 3.386ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fafdb86d262cb17cb5b920d159c35b5bf10c4ff" data-alt="{\displaystyle X^{T}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, or <span class="mwe-math-element"><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^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{\top }}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8503bdc53e70271a35a72feb4741033e1ee028d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.508ex; height:2.676ex;" alt="{\displaystyle X^{\top }}"></noscript><span class="lazy-image-placeholder" style="width: 3.508ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8503bdc53e70271a35a72feb4741033e1ee028d" data-alt="{\displaystyle X^{\top }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</li> <li>In a <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattice</a>, infima are written as <span class="nowrap"><i>a</i> ∧ <i>b</i></span> or as a product <i>ab</i>, suprema as <span class="nowrap"><i>a</i> ∨ <i>b</i></span> or as a sum <span class="nowrap"><i>a</i> + <i>b</i></span>. In a pure lattice theoretical context the first notation is used, usually without any precedence rules. In a pure engineering or "ideals in a ring" context the second notation is used and multiplication has higher precedence than addition. In any other context the confusion of readers of all backgrounds should be minimized. In an abstract bounded lattice, the smallest and greatest elements are denoted by 0 and 1.</li> <li>The scalar or <a href="/wiki/Dot_product" title="Dot product">dot product</a> of vectors should be denoted with a centre-dot <span class="nowrap"><b>a</b> ⋅ <b>b</b></span>, as an <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a> <span class="nowrap">⟨<b>a</b>,<b>b</b>⟩</span> or (<b>a</b>,<b>b</b>), or as a <a href="/wiki/Matrix_product" class="mw-redirect" title="Matrix product">matrix product</a> <b>a</b><sup>T</sup><b>b</b>, never with juxtaposition <b>ab</b>.</li></ul> </section><div class="mw-heading mw-heading2 ext-discussiontools-init-section section-heading" onclick="mfTempOpenSection(4)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Proofs" data-mw-thread-id="h-Proofs"><span data-mw-comment-start="" id="h-Proofs"></span>Proofs<span data-mw-comment-end="h-Proofs"></span></h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=14" title="Edit section: Proofs" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> <p>This is an encyclopedia and not a collection of mathematical texts, but we often want to include proofs to explain a theorem or definition. A downside of including proofs is that they may interrupt the flow of the article, whose goal is usually expository. Use your judgment; as a rule of thumb, include proofs when they expose or illuminate the concept or idea; don't include them when they serve only to establish the correctness of a result. </p><p>Since many readers will want to skip proofs, it is a good idea to set them apart in some way, for instance by giving them a separate section. Additional discussion and guidelines can be found at <a href="/wiki/Wikipedia:WikiProject_Mathematics/Proofs" title="Wikipedia:WikiProject Mathematics/Proofs">Wikipedia:WikiProject Mathematics/Proofs</a>. </p> </section><div class="mw-heading mw-heading2 ext-discussiontools-init-section section-heading" onclick="mfTempOpenSection(5)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Algorithms" data-mw-thread-id="h-Algorithms"><span data-mw-comment-start="" id="h-Algorithms"></span>Algorithms<span data-mw-comment-end="h-Algorithms"></span></h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=15" title="Edit section: Algorithms" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Wikipedia:WikiProject_Computer_science/Manual_of_style_(computer_science)#Algorithms" class="mw-redirect" title="Wikipedia:WikiProject Computer science/Manual of style (computer science)">Wikipedia:WikiProject Computer science/Manual of style (computer science) § Algorithms</a></div> <p>An article about an <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> may include <a href="/wiki/Pseudocode" title="Pseudocode">pseudocode</a> or in some cases <a href="/wiki/Source_code" title="Source code">source code</a> in some <a href="/wiki/Programming_language" title="Programming language">programming language</a>. Wikipedia does not have a standard programming language or languages, and not all readers will understand any particular language even if the language is well-known and easy to read, so consider whether the algorithm could be expressed in some other way. If source code is used always choose a programming language that expresses the algorithm as clearly as possible. </p><p>Articles should not include multiple implementations of the same algorithm in different programming languages unless there is encyclopedic interest in each implementation. </p><p>Source code should always use <a href="https://www.mediawiki.org/wiki/Extension:SyntaxHighlight_GeSHi" class="extiw" title="mw:Extension:SyntaxHighlight GeSHi">syntax highlighting</a>. For example this markup:<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> <pre><syntaxhighlight lang="Haskell"> primes = sieve [2..] sieve (p : xs) = p : sieve [x | x <- xs, x `mod` p > 0] </syntaxhighlight> </pre> <p>generates the following: </p> <div class="mw-highlight mw-highlight-lang-haskell mw-content-ltr" dir="ltr"><pre><span></span><span class="w"> </span><span class="n">primes</span><span class="w"> </span><span class="ow">=</span><span class="w"> </span><span class="n">sieve</span><span class="w"> </span><span class="p">[</span><span class="mi">2</span><span class="o">..</span><span class="p">]</span> <span class="w"> </span><span class="n">sieve</span><span class="w"> </span><span class="p">(</span><span class="n">p</span><span class="w"> </span><span class="kt">:</span><span class="w"> </span><span class="n">xs</span><span class="p">)</span><span class="w"> </span><span class="ow">=</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="kt">:</span><span class="w"> </span><span class="n">sieve</span><span class="w"> </span><span class="p">[</span><span class="n">x</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="ow"><-</span><span class="w"> </span><span class="n">xs</span><span class="p">,</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="p">`</span><span class="n">mod</span><span class="p">`</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="mi">0</span><span class="p">]</span> </pre></div> </section><div class="mw-heading mw-heading2 ext-discussiontools-init-section section-heading" onclick="mfTempOpenSection(6)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Including_citations_and_literature_references" data-mw-thread-id="h-Including_citations_and_literature_references"><span data-mw-comment-start="" id="h-Including_citations_and_literature_references"></span>Including citations and literature references<span data-mw-comment-end="h-Including_citations_and_literature_references"></span></h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=16" title="Edit section: Including citations and literature references" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> <p><span class="anchor" id="MATHSCITATIONS"></span><span class="anchor" id="Mathscitations"></span><span class="anchor" id="Including_citations_and_literature_references"></span> Per the Wikipedia policy, <a href="/wiki/Wikipedia:VERIFY" class="mw-redirect" title="Wikipedia:VERIFY">WP:VERIFY</a>, it is essential for article content to have inline citations, and thus to have a well-chosen list of references and pointers to the literature. Some reasons for this are the following: </p> <ul><li>Wikipedia articles cannot be a substitute for a textbook (that is what <a href="/wiki/Wikibooks" title="Wikibooks">Wikibooks</a> is for). Also, often one might want to find out more details (like the proof of a theorem stated in the article).</li> <li>Some notions are defined differently depending on context or author. Articles should contain some references that support the given usage.</li> <li>Important theorems should cite historical papers as an additional information (not necessarily for looking them up).</li> <li>Today many research papers or even books are freely available online and thus virtually just one click away from Wikipedia. Newcomers would greatly profit from having an immediate connection to further discussions of a topic.</li> <li>Providing further reading enables other editors to verify and to extend the given information, as well as to discuss the quality of a particular source.</li></ul> <p>The <a href="/wiki/Wikipedia:Cite_sources" class="mw-redirect" title="Wikipedia:Cite sources">Wikipedia:Cite sources</a> article has more information on this and also several examples for how the cited literature should look. </p> </section><div class="mw-heading mw-heading2 ext-discussiontools-init-section section-heading" onclick="mfTempOpenSection(7)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Typesetting_of_mathematical_formulae" data-mw-thread-id="h-Typesetting_of_mathematical_formulae"><span data-mw-comment-start="" id="h-Typesetting_of_mathematical_formulae"></span>Typesetting of mathematical formulae<span data-mw-comment-end="h-Typesetting_of_mathematical_formulae"></span></h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=17" title="Edit section: Typesetting of mathematical formulae" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-7 collapsible-block" id="mf-section-7"> <p><span class="anchor" id="Formula"></span><span class="anchor" id="Formulae"></span><span class="anchor" id="FORMULA"></span><span class="anchor" id="FORMULAE"></span><span class="anchor" id="Typesetting_of_mathematical_formulas"></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1252059228"></p><div class="module-shortcutanchordiv"><span id="MOS:FORMULA"></span><span id="MOS:FORMULAE"></span></div><div class="module-shortcutboxplain noprint" role="note"><div class="module-shortcutlist"><a href="/wiki/Wikipedia:Shortcut" title="Wikipedia:Shortcut">Shortcuts</a></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><li><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=MOS:FORMULA&redirect=no">MOS:FORMULA</a></span></li><li><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=MOS:FORMULAE&redirect=no">MOS:FORMULAE</a></span></li></ul></div></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Help:Displaying_a_formula" title="Help:Displaying a formula">Help:Displaying a formula</a></div> <p>One may set formulae using LaTeX (the <code class="nowrap" style=""><math></code> tag, described in the next subsection) or, in certain cases, using other means of formatting that render in HTML; both are acceptable and widely used, except for <i>section headings,</i> which should use HTML only, as LaTeX markup might cause uneven spacing in the table of contents, as well as the appearance of illegible <a href="/wiki/Wikipedia:ANCHOR" class="mw-redirect" title="Wikipedia:ANCHOR">anchor links</a> to sections. Some of the issues presented by using LaTeX or HTML are discussed below. </p><p>Large-scale formatting changes to an article or group of articles are likely to be controversial. One should not change formatting <a href="/wiki/Wikipedia:Be_bold" title="Wikipedia:Be bold">boldly</a> from LaTeX to HTML, nor from non-LaTeX to LaTeX without a clear improvement. Proposed changes should generally be discussed on the <a href="/wiki/Wikipedia:Talk_pages" class="mw-redirect" title="Wikipedia:Talk pages">talk page of the article</a> before implementation. If there is no positive response, or if planned changes affect more than one article, consider notifying an appropriate Wikiproject, such as <a href="/wiki/Wikipedia:WikiProject_Mathematics" title="Wikipedia:WikiProject Mathematics">WikiProject Mathematics</a> for mathematical articles. </p><p>For inline formulae, such as <span class="texhtml"><i>a</i><sup>2</sup> − <i>b</i><sup>2</sup></span>, the community of mathematical editors of English Wikipedia currently has no consensus about preferred formatting; see <a href="/wiki/Wikipedia:Rendering_math" title="Wikipedia:Rendering math">WP:Rendering math</a> for details. </p><p>For a formula <i>on its own line</i> the preferred formatting is the LaTeX markup, with a possible exception for simple <a href="/wiki/String_(computer_science)" title="String (computer science)">strings</a> of Latin letters, digits, <a href="/wiki/Template:Punctuation_marks" title="Template:Punctuation marks">common punctuation marks</a>, and arithmetical operators. Even for simple formulae the LaTeX markup might be preferred if required for uniformity within an article. For readability, it is also strongly preferred not to mix HTML and LaTeX markup in the same expression. </p> <div class="mw-heading mw-heading3"><h3 id="Using_LaTeX_markup" data-mw-thread-id="h-Using_LaTeX_markup-Typesetting_of_mathematical_formulae"><span data-mw-comment-start="" id="h-Using_LaTeX_markup-Typesetting_of_mathematical_formulae"></span>Using LaTeX markup<span data-mw-comment-end="h-Using_LaTeX_markup-Typesetting_of_mathematical_formulae"></span></h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=18" title="Edit section: Using LaTeX markup" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Wikipedia allows editors to typeset mathematical formulae in (a subset of) <a href="/wiki/LaTeX" title="LaTeX">LaTeX</a> markup (see also <a href="/wiki/TeX" title="TeX">TeX</a>); the formulae are, for a default reader, translated into <a href="/wiki/Portable_Network_Graphics" class="mw-redirect" title="Portable Network Graphics">PNG</a> images. They may also be rendered as <a href="/wiki/MathML" title="MathML">MathML</a> or <a href="/wiki/HTML" title="HTML">HTML</a> (using <a href="/wiki/MathJax" title="MathJax">MathJax</a>), depending on user preferences. For more details on this, see <a href="/wiki/Help:Displaying_a_formula" title="Help:Displaying a formula">Help:Displaying a formula</a>. </p><p>The LaTeX formulae can be displayed inline (like this: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} \in \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \in \mathbb {R} ^{2}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e455f6dedb0742db1c32842ed601c9b1aa978a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.984ex; height:2.676ex;" alt="{\displaystyle \mathbf {x} \in \mathbb {R} ^{2}}"></noscript><span class="lazy-image-placeholder" style="width: 6.984ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e455f6dedb0742db1c32842ed601c9b1aa978a" data-alt="{\displaystyle \mathbf {x} \in \mathbb {R} ^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>), as well as on their own line: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="{\displaystyle \int _{0}^{\pi }\sin x\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\pi }\sin x\,dx.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2420b791435d32ab80e338b3b61c21c338e81312" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.177ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\pi }\sin x\,dx.}"></noscript><span class="lazy-image-placeholder" style="width: 12.177ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2420b791435d32ab80e338b3b61c21c338e81312" data-alt="{\displaystyle \int _{0}^{\pi }\sin x\,dx.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span> </p><p>A frequent method for displaying formulas on their own line has been to indent the line with one or more colons (:). Although this produces the intended visual appearance, it produces invalid html (see <a href="/wiki/Wikipedia:Manual_of_Style/Accessibility#Indentation" title="Wikipedia:Manual of Style/Accessibility">Wikipedia:Manual of Style/Accessibility § Indentation</a>). Instead, formulas may be placed on their own line using <code class="mw-highlight mw-highlight-lang-html mw-content-ltr" dir="ltr"><span class="p"><</span><span class="nt">math</span> <span class="na">display</span><span class="o">=</span><span class="s">block</span><span class="p">></span></code>. For instance, the formula above was typeset using <code><math display=block><code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><span class="k">\int</span><span class="nb">_</span>0<span class="nb">^</span><span class="k">\pi</span> <span class="k">\sin</span> x<span class="k">\,</span>dx.</code></math></code>. </p><p>If you find an article which indents lines with spaces in order to achieve some formula layout effect, you should convert the formula to LaTeX markup. </p><p>Having LaTeX-based formulae inline has the following drawbacks: </p> <ul><li>The font size can be slightly larger than that of the surrounding text on some browsers, making text containing inline formulae harder to read.</li> <li>The download speed of a page is negatively affected if it contains many formulas.</li> <li>Until bug <a href="https://phabricator.wikimedia.org/T263572" class="extiw" title="phabricator:T263572">T263572</a> is fixed, it will not work in image captions when readers click through to see full-size images.</li></ul> <p>If an inline formula needs to be typeset in LaTeX, keep the height down by using text-style or horizontal fractions: <code><math><code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><span class="k">\tfrac</span>12 x</code></math></code> produces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}x}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2ae7f723e152f65ed1ae241fd20b79eed3d68b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.988ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}x}"></noscript><span class="lazy-image-placeholder" style="width: 2.988ex;height: 3.509ex;vertical-align: -1.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2ae7f723e152f65ed1ae241fd20b79eed3d68b5" data-alt="{\displaystyle {\tfrac {1}{2}}x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> and <code><math><code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr">x / 2</code></math></code> produces <span class="mwe-math-element"><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/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x/2}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c68203be699cce32f38b291115d5bc2c6220dc9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle x/2}"></noscript><span class="lazy-image-placeholder" style="width: 3.655ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c68203be699cce32f38b291115d5bc2c6220dc9d" data-alt="{\displaystyle x/2}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, but <code><math><code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><span class="k">\frac</span><span class="nb">{</span>x<span class="nb">}{</span>2<span class="nb">}</span></code></math></code> is too tall to fit inline. </p><p>Often better formatting can be achieved with <code class="mw-highlight mw-highlight-lang-html mw-content-ltr" dir="ltr"><span class="p"><</span><span class="nt">math</span> <span class="na">display</span><span class="o">=</span><span class="s">inline</span><span class="p">></span></code> tag, which translates to the <code>\textstyle</code> LaTeX command. By default, LaTeX code is rendered as if it were a displayed equation (not inline), and this can frequently be too big. For example, the formula <code><math><code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><span class="k">\sum</span><span class="nb">_{</span>n=1<span class="nb">}^</span><span class="k">\infty</span> <span class="k">\frac</span><span class="nb">{</span>1<span class="nb">}{</span>n<span class="nb">^</span>2<span class="nb">}</span> = <span class="k">\frac</span><span class="nb">{</span><span class="k">\pi</span><span class="nb">^</span>2<span class="nb">}{</span>6<span class="nb">}</span></code></math></code> is too large to be used inline, rendering as </p> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent" style="padding-left: 1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>6</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/664617be29449b83fe4c47625907f1713ec4b523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.997ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}.}"></noscript><span class="lazy-image-placeholder" style="width: 13.997ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/664617be29449b83fe4c47625907f1713ec4b523" data-alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></div> <p>Adding <code>display=inline</code> generates a smaller summation symbol and moves the limits to its right side. The rewritten formula <code><math display=inline><code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><span class="k">\sum</span><span class="nb">_{</span>n=1<span class="nb">}^</span><span class="k">\infty</span> 1/n<span class="nb">^</span>2 = <span class="k">\pi</span><span class="nb">^</span>2/6</code></math></code> renders as <span class="nowrap"><span class="mwe-math-element"><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 _{n=1}^{\infty }1/n^{2}=\pi ^{2}/6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{n=1}^{\infty }1/n^{2}=\pi ^{2}/6}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87707e25b93ef1216297e968e4376fe20b109ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.746ex; height:3.176ex;" alt="{\textstyle \sum _{n=1}^{\infty }1/n^{2}=\pi ^{2}/6}"></noscript><span class="lazy-image-placeholder" style="width: 18.746ex;height: 3.176ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87707e25b93ef1216297e968e4376fe20b109ac" data-alt="{\textstyle \sum _{n=1}^{\infty }1/n^{2}=\pi ^{2}/6}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>,</span> which fits much better inline. Adding <code class="mw-highlight mw-highlight-lang-html mw-content-ltr" dir="ltr">display=inline</code> renders exponents lower, especially under square roots, often resulting in a smaller square root which fits better in inline text: compare <code><math><code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><span class="k">\sqrt</span><span class="nb">{</span>x<span class="nb">^</span>2+y<span class="nb">^</span>2<span class="nb">}</span></code></math></code> to <code><math display=inline><code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><span class="k">\sqrt</span><span class="nb">{</span>x<span class="nb">^</span>2+y<span class="nb">^</span>2<span class="nb">}</span></code></math></code> which render as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {x^{2}+y^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x^{2}+y^{2}}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bde512f387a0677a158fd76d3f558ed8b55b3ba1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:9.763ex; height:4.843ex;" alt="{\displaystyle {\sqrt {x^{2}+y^{2}}}}"></noscript><span class="lazy-image-placeholder" style="width: 9.763ex;height: 4.843ex;vertical-align: -1.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bde512f387a0677a158fd76d3f558ed8b55b3ba1" data-alt="{\displaystyle {\sqrt {x^{2}+y^{2}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> and <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {x^{2}+y^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {x^{2}+y^{2}}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00f4319b4270574ab98a1379e89a4848ac308417" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.763ex; height:3.509ex;" alt="{\textstyle {\sqrt {x^{2}+y^{2}}}}"></noscript><span class="lazy-image-placeholder" style="width: 9.763ex;height: 3.509ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00f4319b4270574ab98a1379e89a4848ac308417" data-alt="{\textstyle {\sqrt {x^{2}+y^{2}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></span>, respectively. </p><p>HTML-generating formatting, as <a href="#Using_HTML">described below</a>, is adequate for articles that use only simple inline formulae and better for text-only browsers. </p> <div class="mw-heading mw-heading4"><h4 id="Line_wrapping" data-mw-thread-id="h-Line_wrapping-Using_LaTeX_markup"><span data-mw-comment-start="" id="h-Line_wrapping-Using_LaTeX_markup"></span>Line wrapping<span data-mw-comment-end="h-Line_wrapping-Using_LaTeX_markup"></span></h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=19" title="Edit section: Line wrapping" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Directly using <code><math>...</math></code> tags results in line wrapping points that allow wrapping between it and adjacent text (typically punctuation). Including such text within the tags to avoid this wrapping results in its font being discordant with other text. This can be remedied by wrapping the LaTeX markup in a suitable template (optionally excluding the adjacent punctuation), e.g. <code>({{nobr|<math>...</math>}})</code>, or by replacing the tags with a template, e.g. <code>({{tmath|1=...}})</code>. Take care of some necessary substitutions documented for <span class="nowrap">{{</span><a href="/wiki/Template:Tmath" title="Template:Tmath">tmath</a><span class="nowrap">}}</span>. </p> <div class="mw-heading mw-heading4"><h4 id="Deprecated_formatting" data-mw-thread-id="h-Deprecated_formatting-Using_LaTeX_markup"><span data-mw-comment-start="" id="h-Deprecated_formatting-Using_LaTeX_markup"></span>Deprecated formatting<span data-mw-comment-end="h-Deprecated_formatting-Using_LaTeX_markup"></span></h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=20" title="Edit section: Deprecated formatting" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p><span class="anchor" id="Very_simple_formulae"></span><span class="anchor" id="Forcing_output_to_be_an_image"></span> </p><p>Older versions of the MediaWiki software supported displaying simple LaTeX formulae as HTML rather than as an image. Although this is no longer an option, some formulae have formatting in them intended to force them to display as an image, such as an invisible quarter space (<code>\,</code>) added at the end of the formula, or <code>\displaystyle</code> at the beginning. Such formatting can be removed if a formula is edited and need not be added to new formulae. </p> <div class="mw-heading mw-heading4"><h4 id="Alt_text" data-mw-thread-id="h-Alt_text-Using_LaTeX_markup"><span data-mw-comment-start="" id="h-Alt_text-Using_LaTeX_markup"></span>Alt text<span data-mw-comment-end="h-Alt_text-Using_LaTeX_markup"></span></h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=21" title="Edit section: Alt text" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Images generated from LaTeX markup have <a href="/wiki/Wikipedia:ALT" class="mw-redirect" title="Wikipedia:ALT">alt text</a>, which is displayed to visually impaired readers and other readers who cannot see the images. The default alt text is the LaTeX markup that produced the image. You can override this by explicitly specifying an <code>alt</code> attribute for the <code>math</code> element. For example, <code><math alt="Square root of pi">\sqrt{\pi}</math></code> generates an image <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\pi }}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae18ec124928c74818b516e6350ca9610966c6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.268ex; height:3.009ex;" alt="{\displaystyle {\sqrt {\pi }}}"></noscript><span class="lazy-image-placeholder" style="width: 3.268ex;height: 3.009ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae18ec124928c74818b516e6350ca9610966c6e" data-alt="{\displaystyle {\sqrt {\pi }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> whose alt text is "Square root of pi". Small and easily explained formulas used in less technical articles can benefit from explicitly specified alt text. More complicated formulas, or formulas used in more technical articles, are often better off with the default alt text. </p> <div class="mw-heading mw-heading3"><h3 id="Using_HTML" data-mw-thread-id="h-Using_HTML-Typesetting_of_mathematical_formulae"><span data-mw-comment-start="" id="h-Using_HTML-Typesetting_of_mathematical_formulae"></span>Using HTML<span data-mw-comment-end="h-Using_HTML-Typesetting_of_mathematical_formulae"></span></h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=22" title="Edit section: Using HTML" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>The following sections cover the way of presenting simple inline formulae in HTML, instead of using LaTeX. </p><p>Templates supporting HTML formatting are listed in <a href="/wiki/Category:Mathematical_formatting_templates" title="Category:Mathematical formatting templates">Category:Mathematical formatting templates</a>. Not all templates are recommended for use; in particular, use of the <span class="nowrap">{{</span><a href="/wiki/Template:Frac" class="mw-redirect" title="Template:Frac">frac</a><span class="nowrap">}}</span> template to format fractions is discouraged in mathematics articles. </p> <div class="mw-heading mw-heading4"><h4 id="Font_formatting" data-mw-thread-id="h-Font_formatting-Using_HTML"><span data-mw-comment-start="" id="h-Font_formatting-Using_HTML"></span>Font formatting<span data-mw-comment-end="h-Font_formatting-Using_HTML"></span></h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=23" title="Edit section: Font formatting" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>By default, regular text is rendered in a <a href="/wiki/Sans_serif" class="mw-redirect" title="Sans serif">sans serif</a> font. The TeX markup of <code class="nowrap" style=""><math>...</math></code> uses a <a href="/wiki/Serif" title="Serif">serif</a> font to display a formula (whether as SVG or HTML). HTML math expressions should use the <code>{{<a href="/wiki/Template:Math" title="Template:Math">math</a>}}</code> template so they display in a serif font as well. Doing so will also ensure that the text within a formula will not line-wrap, and that the font size will closely match the surrounding text in any <a href="/wiki/Wikipedia:Skins" class="mw-redirect" title="Wikipedia:Skins">skin</a>. Using <span class="nowrap">{{</span><a href="/wiki/Template:Math" title="Template:Math">math</a><span class="nowrap">}}</span> and <span class="nowrap">{{</span><a href="/wiki/Template:Mvar" title="Template:Mvar">mvar</a><span class="nowrap">}}</span> also helps some spell checkers and screen readers treat math markup properly. Note that certain special characters (equal signs, absolute value bars) require <a href="/wiki/Template:Math#Use_of_equals-sign_and_absolute_value_bars" title="Template:Math">special attention</a>. </p> <dl><dd>The relationship is defined as <code class="mw-highlight mw-highlight-lang-wikitext mw-content-ltr" dir="ltr"><span class="p">{{</span><span class="nt">math</span><span class="p">|</span><span class="ge">''x''</span> <span class="p">{{</span><span class="nf">=</span><span class="p">}}</span> −(<span class="ge">''y''</span><span class="p"><</span><span class="nt">sup</span><span class="p">></span>2<span class="p"></</span><span class="nt">sup</span><span class="p">></span> + 2)<span class="p">}}</span></code>.</dd></dl> <p>will result in: </p> <dl><dd>The relationship is defined as <span class="texhtml"><i>x</i> = −(<i>y</i><sup>2</sup> + 2)</span>.</dd></dl> <div class="mw-heading mw-heading5"><h5 id="Variables" data-mw-thread-id="h-Variables-Font_formatting"><span data-mw-comment-start="" id="h-Variables-Font_formatting"></span>Variables<span data-mw-comment-end="h-Variables-Font_formatting"></span></h5><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=24" title="Edit section: Variables" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Use italic text for variables, but never for numbers or symbols. To ensure an italic serif typeface, use the <span class="nowrap">{{</span><a href="/wiki/Template:Mvar" title="Template:Mvar">mvar</a><span class="nowrap">}}</span> template to enclose a simple mention of a variable by name. This helps distinguish certain characters such as <span class="texhtml mvar" style="font-style:italic;">I</span> and <span class="texhtml mvar" style="font-style:italic;">l</span>. Within <span class="nowrap">{{</span><a href="/wiki/Template:Math" title="Template:Math">math</a><span class="nowrap">}}</span> templates (which will set a serif font but not italics), use the wikitext markup of double apostrophes to make variables italic. For example: </p> <dl><dd><code class="mw-highlight mw-highlight-lang-wikitext mw-content-ltr" dir="ltr"><span class="p">{{</span><span class="nt">mvar</span><span class="p">|</span>x<span class="p">}}</span> is a value on the horizontal axis</code></dd></dl> <p>displays as: </p> <dl><dd><span class="texhtml mvar" style="font-style:italic;">x</span> is a value on the horizontal axis</dd></dl> <p>and </p> <dl><dd><code class="mw-highlight mw-highlight-lang-wikitext mw-content-ltr" dir="ltr"><span class="p">{{</span><span class="nt">math</span><span class="p">|</span><span class="ge">''x''</span> <span class="p">{{</span><span class="nf">=</span><span class="p">}}</span> −(<span class="ge">''y''</span><span class="p"><</span><span class="nt">sup</span><span class="p">></span>2<span class="p"></</span><span class="nt">sup</span><span class="p">></span> + 2)<span class="p">}}</span></code></dd></dl> <p>results in: </p> <dl><dd><span class="texhtml"><i>x</i> = −(<i>y</i><sup>2</sup> + 2)</span></dd></dl> <p>While italicizing variables, things like parentheses, digits, equal and plus signs should be kept outside of the double-apostrophed sections. Using double apostrophes for math content instead of <span class="nowrap">{{</span><a href="/wiki/Template:Mvar" title="Template:Mvar">mvar</a><span class="nowrap">}}</span> or without <span class="nowrap">{{</span><a href="/wiki/Template:Math" title="Template:Math">math</a><span class="nowrap">}}</span> is undesirable for readers because it will render in a sans serif font; this is especially confusing when other articles or sections render the same variables or equations in a serif font. Descriptive subscripts should not be in italics, because they are not variables. For example, <span class="texhtml"><i>m</i><sub>foo</sub></span> is the mass of a foo. <a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI units</a> are never italicized: <span class="texhtml"><i>x</i> = 5 cm</span>. </p> <div class="mw-heading mw-heading5"><h5 id="Functions" data-mw-thread-id="h-Functions-Font_formatting"><span data-mw-comment-start="" id="h-Functions-Font_formatting"></span>Functions<span data-mw-comment-end="h-Functions-Font_formatting"></span></h5><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=25" title="Edit section: Functions" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Names for <a href="/wiki/List_of_mathematical_functions" title="List of mathematical functions">standard functions</a>, such as <span class="texhtml">sin</span> and <span class="texhtml">cos</span>, are not in italic font, but we use italic names such as <span class="texhtml mvar" style="font-style:italic;"> f </span> for functions in other cases; for example when we define the function as in <span class="nowrap"> <span class="texhtml"> <i>f</i>(<i>x</i>) = sin(<i>x</i>) cos(<i>x</i>) </span> .</span> </p> <div class="mw-heading mw-heading5"><h5 id="Sets" data-mw-thread-id="h-Sets-Font_formatting"><span data-mw-comment-start="" id="h-Sets-Font_formatting"></span>Sets<span data-mw-comment-end="h-Sets-Font_formatting"></span></h5><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=26" title="Edit section: Sets" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Sets are usually written in upper case italics; for example: </p> <dl><dd><span class="texhtml"><i>A</i> = {<i>x</i> : <i>x</i> > 0</span>}</dd></dl> <p>would be written: </p> <dl><dd><code class="mw-highlight mw-highlight-lang-wikitext mw-content-ltr" dir="ltr"><span class="p">{{</span><span class="nt">math</span><span class="p">|</span><span class="ge">''A''</span> <span class="p">{{</span><span class="nf">=</span><span class="p">}}</span> {<span class="ge">''x''</span> : <span class="ge">''x''</span> > 0<span class="p">}}</span>}</code> .</dd></dl> <div class="mw-heading mw-heading5"><h5 id="Greek_letters" data-mw-thread-id="h-Greek_letters-Font_formatting"><span data-mw-comment-start="" id="h-Greek_letters-Font_formatting"></span>Greek letters<span data-mw-comment-end="h-Greek_letters-Font_formatting"></span></h5><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=27" title="Edit section: Greek letters" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Italicize lower-case Greek letters when they are variables or constants (in line with the general advice to italicize variables): the example expression <span class="texhtml"><i>λ</i> + <i>y</i> = <i>πr</i><sup>2</sup></span> would then be typeset by: </p><p><code class="mw-highlight mw-highlight-lang-wikitext mw-content-ltr" dir="ltr"><span class="p">{{</span><span class="nt">math</span><span class="p">|</span><span class="nl">''&lambda;'' + ''y'' </span><span class="o">=</span> <span class="ge">''</span><span class="ni">&pi;</span><span class="ge">r''</span><span class="p"><</span><span class="nt">sup</span><span class="p">></span>2<span class="p"></</span><span class="nt">sup</span><span class="p">>}}</span></code> </p><p>(It is also possible to enter <a href="/wiki/Greek_alphabet#Greek_in_Unicode" title="Greek alphabet">Greek letters</a> directly.) </p><p>For consistency with the LaTeX style, do not italicize <i>capital</i> Greek letters; e.g. <span class="nowrap"> <span class="texhtml"> <i>n</i>! = <a href="/wiki/Gamma_function" title="Gamma function">Γ</a>(<i>n</i> + 1) </span>.</span> </p><p>According to the Unicode Consortium,<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> the characters <span class="nowrap"><style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style><span class="monospaced">U+00B5</span> </span><span style="font-size:125%;line-height:1em">µ</span> <span style="font-variant: small-caps; text-transform: lowercase;">MICRO SIGN</span> and <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2126</span> </span><span style="font-size:125%;line-height:1em">Ω</span> <span style="font-variant: small-caps; text-transform: lowercase;">OHM SIGN</span> are intended for compatibility with legacy character sets, and Unicode-capable environments (like Wikipedia) should use the Greek letters instead (<span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+03BC</span> </span><span style="font-size:125%;line-height:1em">μ</span> <span style="font-variant: small-caps; text-transform: lowercase;">GREEK SMALL LETTER MU</span> and <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+03A9</span> </span><span style="font-size:125%;line-height:1em">Ω</span> <span style="font-variant: small-caps; text-transform: lowercase;">GREEK CAPITAL LETTER OMEGA</span>). This is also required for "micro" by <a href="/wiki/MOS:UNITSYMBOLS" class="mw-redirect" title="MOS:UNITSYMBOLS">MOS:UNITSYMBOLS</a>. </p> <div class="mw-heading mw-heading5"><h5 id="Common_sets_of_numbers" data-mw-thread-id="h-Common_sets_of_numbers-Font_formatting"><span data-mw-comment-start="" id="h-Common_sets_of_numbers-Font_formatting"></span>Common sets of numbers<span data-mw-comment-end="h-Common_sets_of_numbers-Font_formatting"></span></h5><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=28" title="Edit section: Common sets of numbers" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Commonly used sets of numbers are typeset in boldface, as in the set of real numbers <span class="texhtml"><b>R</b></span>. Again, typically we use wiki markup: three apostrophes (<code class="mw-highlight mw-highlight-lang-wikitext mw-content-ltr" dir="ltr"><span class="gs">'''</span></code>) rather than the HTML <code class="mw-highlight mw-highlight-lang-wikitext mw-content-ltr" dir="ltr"><span class="p"><</span><span class="nt">b</span><span class="p">></span></code> tag for making text bold. On chalkboards and increasingly in printed publications, mathematicians alternately use blackboard bold for sets such as the real numbers, which may be encoded in LaTeX as <code class="mw-highlight mw-highlight-lang-wikitext mw-content-ltr" dir="ltr"><span class="p"><</span><span class="nt">math</span><span class="p">></span><span class="k">\mathbb</span><span class="nb">{</span>R<span class="nb">}</span><span class="p"></</span><span class="nt">math</span><span class="p">></span></code> (preferred shortcut: <span class="nowrap"><code class="mw-highlight mw-highlight-lang-wikitext mw-content-ltr" dir="ltr"><span class="p"><</span><span class="nt">math</span><span class="p">></span><span class="k">\R</span><span class="p"></</span><span class="nt">math</span><span class="p">></span></code>),</span> which renders as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} .}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9de9049e03e5e5a0cab57076dbe4a369c1e3a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} .}"></noscript><span class="lazy-image-placeholder" style="width: 2.325ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9de9049e03e5e5a0cab57076dbe4a369c1e3a7" data-alt="{\displaystyle \mathbb {R} .}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> However, the special <a href="/wiki/Unicode" title="Unicode">Unicode</a> characters, such as <span class="nowrap">U+211D</span> (plain text ℝ or math font <span class="texhtml">ℝ</span>) and its adjacent characters should be avoided at present, since these characters are not yet universally supported and may have an inconsistent appearance. Either bold or blackboard bold is acceptable in Wikipedia articles, but should be consistent within each article. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="#Blackboard_bold">§ Blackboard bold</a></div> <div class="mw-heading mw-heading5"><h5 id="Superscripts_and_subscripts" data-mw-thread-id="h-Superscripts_and_subscripts-Font_formatting"><span data-mw-comment-start="" id="h-Superscripts_and_subscripts-Font_formatting"></span>Superscripts and subscripts<span data-mw-comment-end="h-Superscripts_and_subscripts-Font_formatting"></span></h5><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=29" title="Edit section: Superscripts and subscripts" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main page: <a href="/wiki/Wikipedia:Manual_of_Style_(superscripts_and_subscripts)" class="mw-redirect" title="Wikipedia:Manual of Style (superscripts and subscripts)">Wikipedia:Manual of Style (superscripts and subscripts)</a></div> <p>Superscripts and subscripts should be wrapped in <code><sup></code> and <code><sub></code> tags, respectively, with no other formatting info. Font sizes and such should be entrusted to be handled with stylesheets. For example, to write <span class="texhtml"><i>c</i><sub>3+5</sub></span>, use </p> <dl><dd><code class="mw-highlight mw-highlight-lang-wikitext mw-content-ltr" dir="ltr"><span class="p">{{</span><span class="nt">math</span><span class="p">|</span><span class="ge">''c''</span><span class="p"><</span><span class="nt">sub</span><span class="p">></span>3+5<span class="p"></</span><span class="nt">sub</span><span class="p">>}}</span></code>.</dd></dl> <p>Do not use special characters like <code><a href="/wiki/Unicode_subscripts_and_superscripts" title="Unicode subscripts and superscripts">²</a></code> (<code>&sup2;</code>) for squares. This does not combine well with other powers, as the following comparison shows: </p> <dl><dd><span class="texhtml">1 + <i>x</i> + <i>x</i>² + <i>x</i><sup>3</sup> + <i>x</i><sup>4</sup></span> (with <code>&sup2;</code>) versus</dd> <dd><span class="texhtml">1 + <i>x</i> + <i>x</i><sup>2</sup> + <i>x</i><sup>3</sup> + <i>x</i><sup>4</sup></span> (with <code><sup>2</sup></code>).</dd></dl> <p>Moreover, the TeX engine used on Wikipedia may format simple superscripts using <code class="mw-highlight mw-highlight-lang-wikitext mw-content-ltr" dir="ltr"><span class="p"><</span><span class="nt">sup</span><span class="p">></span>...<span class="p"></</span><span class="nt">sup</span><span class="p">></span></code> depending on user preferences. Thus, for some users <span class="mwe-math-element"><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^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}"></noscript><span class="lazy-image-placeholder" style="width: 2.384ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" data-alt="{\displaystyle x^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> will be an image, and for others it will be HTML like <span class="texhtml"><i>x</i><sup>2</sup></span>. Formulae formatted without using TeX should use the same syntax to maintain the same appearance. </p> <div class="mw-heading mw-heading4"><h4 id="Special_symbols" data-mw-thread-id="h-Special_symbols-Using_HTML"><span data-mw-comment-start="" id="h-Special_symbols-Using_HTML"></span>Special symbols<span data-mw-comment-end="h-Special_symbols-Using_HTML"></span></h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=30" title="Edit section: Special symbols" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1252059228"><div class="module-shortcutanchordiv"><span id="MOS:MATHSPECIAL"></span></div><div class="module-shortcutboxplain noprint" role="note"><div class="module-shortcutlist"><a href="/wiki/Wikipedia:Shortcut" title="Wikipedia:Shortcut">Shortcut</a></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><li><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=MOS:MATHSPECIAL&redirect=no">MOS:MATHSPECIAL</a></span></li></ul></div></div> <p>The <a href="/wiki/List_of_mathematical_symbols" class="mw-redirect" title="List of mathematical symbols">list of mathematical symbols</a>, <a href="/wiki/List_of_mathematical_symbols_by_subject" class="mw-redirect" title="List of mathematical symbols by subject">list of mathematical symbols by subject</a> and the list at <a href="/wiki/Wikipedia:Mathematical_symbols" class="mw-redirect" title="Wikipedia:Mathematical symbols">Wikipedia:Mathematical symbols</a> may be useful when editing mathematics articles. <a href="/wiki/Mathematical_operators_and_symbols_in_Unicode" title="Mathematical operators and symbols in Unicode">Almost all mathematical operator symbols</a> have their specific <a href="/wiki/Code_point" title="Code point">code points</a> in <a href="/wiki/Unicode" title="Unicode">Unicode</a> outside both <a href="/wiki/ASCII" title="ASCII">ASCII</a> and <a href="/wiki/General_Punctuation" title="General Punctuation">General Punctuation</a> (with notable exception of "+", "=", "|", as well as ",", ":", and three sorts of <a href="/wiki/Bracket" title="Bracket">brackets</a>). The <a href="/wiki/List_of_mathematical_symbols_by_subject" class="mw-redirect" title="List of mathematical symbols by subject">list of mathematical symbols by subject</a> includes markup for LaTeX and HTML, and Unicode code points. </p><p>Keep in mind: </p> <ol><li>Not all of the symbols in these lists are displayed correctly on all browsers (see <a href="/wiki/Help:Special_characters" title="Help:Special characters">Help:Special characters</a>). Although the symbols that correspond to <a href="/wiki/List_of_XML_and_HTML_character_entity_references" title="List of XML and HTML character entity references">named entities</a> are very likely to be displayed correctly, a significant number of viewers will have problems seeing all the characters listed at <a href="/wiki/Mathematical_operators_and_symbols_in_Unicode" title="Mathematical operators and symbols in Unicode">Mathematical operators and symbols in Unicode</a>. One way to guarantee that an uncommon symbol is rendered correctly for all readers is to <a href="#Forcing_output_to_be_an_image">force the symbol to display as an image</a>, using the <code class="nowrap" style=""><math>...</math></code> with LaTeX markup.</li> <li>Not all readers will be familiar with mathematical notation. Thus, to maximize the size of the audience who can read an article, it is better to be conservative in using symbols. For example, writing "<i>a</i> divides <i>b</i>" rather than "<span class="nowrap"><i>a</i> | <i>b</i></span>" in an elementary article may make it more accessible.</li></ol> <p><br> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1252059228"><div class="module-shortcutanchordiv"><span id="MOS:ROMANNUM"></span></div><div class="module-shortcutboxplain noprint" role="note"><div class="module-shortcutlist"><a href="/wiki/Wikipedia:Shortcut" title="Wikipedia:Shortcut">Shortcut</a></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><li><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=MOS:ROMANNUM&redirect=no">MOS:ROMANNUM</a></span></li></ul></div></div> <ul><li>For <a href="/wiki/Roman_numerals" title="Roman numerals">Roman numerals</a>, <a href="/wiki/Basic_Latin_(Unicode_block)" title="Basic Latin (Unicode block)">Basic Latin</a> (ASCII) letters should be used instead of the equivalent <a href="/wiki/Numerals_in_Unicode#Roman_numerals" title="Numerals in Unicode">Unicode characters in the U+21XX range</a>. For example, <span class="example" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-added, #006400);">L</span> and <span class="example" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-added, #006400);">VI</span>, not <span class="example deprecated-content example-bad" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-removed, #8B0000);">Ⅼ</span>, and not precomposed characters like <span class="example deprecated-content example-bad" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-removed, #8B0000);">Ⅵ</span>. (The only exception is when discussing the Unicode characters themselves.) <dl><dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For when to use or not use Roman numerals, see <a href="/wiki/Wikipedia:Manual_of_Style/Dates_and_numbers" title="Wikipedia:Manual of Style/Dates and numbers">Wikipedia:Manual of Style/Dates and numbers</a>.</div></dd></dl></li> <li>Use <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2032</span> </span><span style="font-size:125%;line-height:1em">′</span> <span style="font-variant: small-caps; text-transform: lowercase;">PRIME</span> or <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2033</span> </span><span style="font-size:125%;line-height:1em">″</span> <span style="font-variant: small-caps; text-transform: lowercase;">DOUBLE PRIME</span> where the <a href="/wiki/Prime_(symbol)" title="Prime (symbol)">prime symbol</a> is appropriate; do not use the ASCII apostrophe (') or double quote (") in these cases. <span class="nowrap">{{</span><a href="/wiki/Template:Prime" title="Template:Prime">prime</a><span class="nowrap">}}</span> and <span class="nowrap">{{</span><a href="/wiki/Template:Pprime" title="Template:Pprime">pprime</a><span class="nowrap">}}</span> can be useful to prevent overlap with italicized characters.</li> <li>Use <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+002A</span> </span><span style="font-size:125%;line-height:1em">*</span> <span style="font-variant: small-caps; text-transform: lowercase;">ASTERISK</span> when the character should render like a <a href="/wiki/Superscript" class="mw-redirect" title="Superscript">superscript</a>, and is typical when used as a postfix. This character also appears on keyboards, and is thus easier to type and search for. Example: <span class="example" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-added, #006400);">C*-algebra</span>. Use <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2217</span> </span><span style="font-size:125%;line-height:1em">∗</span> <span style="font-variant: small-caps; text-transform: lowercase;">ASTERISK OPERATOR</span> (&lowast;) for subscripts and when the bottom of the character should roughly align with the <a href="/wiki/Typeface_anatomy" title="Typeface anatomy">baseline</a> of neighboring characters, which is typical when used as a prefix or infix operator, or a standalone character. Usage should be consistent across articles covering the same subfield of mathematics; see <a href="/wiki/Asterisk#Mathematics" title="Asterisk">Asterisk#Mathematics</a> for a canonical list.</li> <li>Use <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+002F</span> </span><span style="font-size:125%;line-height:1em">/</span> <span style="font-variant: small-caps; text-transform: lowercase;">SOLIDUS</span> instead of <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2215</span> </span><span style="font-size:125%;line-height:1em">∕</span> <span style="font-variant: small-caps; text-transform: lowercase;">DIVISION SLASH</span></li> <li>Use <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+003A</span> </span><span style="font-size:125%;line-height:1em">:</span> <span style="font-variant: small-caps; text-transform: lowercase;">COLON</span> instead of <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2236</span> </span><span style="font-size:125%;line-height:1em">∶</span> <span style="font-variant: small-caps; text-transform: lowercase;">RATIO</span></li> <li>Use <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+007E</span> </span><span style="font-size:125%;line-height:1em">~</span> <span style="font-variant: small-caps; text-transform: lowercase;">TILDE</span> instead of <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+223C</span> </span><span style="font-size:125%;line-height:1em">∼</span> <span style="font-variant: small-caps; text-transform: lowercase;">TILDE OPERATOR</span>. Use <a href="/wiki/Approximation#Typography" title="Approximation">other symbols</a> to mean "approximately" (and <span class="nounderlines" style="border: 1px solid var(--border-color-muted,#ddd); color: var(--color-base); background-color: var( --background-color-neutral-subtle, #fdfdfd); padding: 1px 1px;">¬</span> for <a href="/wiki/Negation" title="Negation">negation</a>) in mathematical expressions, because <a href="/wiki/Tilde#Mathematics" title="Tilde">tilde has other mathematical meanings</a>.</li> <li>Use <code><math>... \setminus ...</math></code> or <code><math>... \smallsetminus ...</math></code> instead of <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2216</span> </span><span style="font-size:125%;line-height:1em">∖</span> <span style="font-variant: small-caps; text-transform: lowercase;">SET MINUS</span> or <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+005C</span> </span><span style="font-size:125%;line-height:1em">\</span> <span style="font-variant: small-caps; text-transform: lowercase;">REVERSE SOLIDUS</span> for set substraction. (Either Unicode character can be used where <code class="nowrap" style=""><math>...</math></code> markup cannot be used for technical reasons.)</li> <li>Use <code><math>... \circ ...</math></code> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }"></noscript><span class="lazy-image-placeholder" style="width: 1.162ex;height: 1.509ex;vertical-align: 0.125ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" data-alt="{\displaystyle \circ }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>) instead of <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2218</span> </span><span style="font-size:125%;line-height:1em">∘</span> <span style="font-variant: small-caps; text-transform: lowercase;">RING OPERATOR</span> (which on some systems is too small and can be confused with <a href="/wiki/Interpunct" title="Interpunct">interpunct</a>) or <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+25CB</span> </span><span style="font-size:125%;line-height:1em">○</span> <span style="font-variant: small-caps; text-transform: lowercase;">WHITE CIRCLE</span> (which on some systems is too large).</li> <li>Use <code><math>... \ngtr ...</math></code> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ngtr }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ngtr }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0159b396d12788e88b9be3dfce2152d90ad09726" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \ngtr }"></noscript><span class="lazy-image-placeholder" style="width: 1.808ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0159b396d12788e88b9be3dfce2152d90ad09726" data-alt="{\displaystyle \ngtr }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>) instead of <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+226F</span> </span><span style="font-size:125%;line-height:1em">≯</span> <span style="font-variant: small-caps; text-transform: lowercase;">NOT GREATER-THAN</span>, which on some systems renders as two separate characters.</li></ul> <p><br> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Wikipedia:Manual_of_Style/Abbreviations#Unicode_abbreviation_ligatures" title="Wikipedia:Manual of Style/Abbreviations">Wikipedia:Manual of Style/Abbreviations § Unicode abbreviation ligatures</a></div> <div class="mw-heading mw-heading4"><h4 id="Less-than_sign" data-mw-thread-id="h-Less-than_sign-Using_HTML"><span data-mw-comment-start="" id="h-Less-than_sign-Using_HTML"></span>Less-than sign<span data-mw-comment-end="h-Less-than_sign-Using_HTML"></span></h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=31" title="Edit section: Less-than sign" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Although the MediaWiki markup engine is fairly smart about differentiating between unescaped "<" characters that are used to denote the start of an embedded HTML or HTML-like tag and those that are just being used as literal less-than symbols, it is ideal to use <code>&lt;</code> when writing the less-than sign, just like in HTML and XML. For example, to write <span class="texhtml"><i>x</i> < 3</span>, use </p> <dl><dd><code class="mw-highlight mw-highlight-lang-wikitext mw-content-ltr" dir="ltr"><span class="p">{{</span><span class="nt">math</span><span class="p">|</span><span class="ge">''x''</span> <span class="ni">&lt;</span> 3<span class="p">}}</span></code>,</dd></dl> <p>not </p> <dl><dd><code class="mw-highlight mw-highlight-lang-wikitext mw-content-ltr" dir="ltr"><span class="p">{{</span><span class="nt">math</span><span class="p">|</span><span class="ge">''x''</span> < 3<span class="p">}}</span></code>.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Multiplication_sign" data-mw-thread-id="h-Multiplication_sign-Using_HTML"><span data-mw-comment-start="" id="h-Multiplication_sign-Using_HTML"></span>Multiplication sign<span data-mw-comment-end="h-Multiplication_sign-Using_HTML"></span></h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=32" title="Edit section: Multiplication sign" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1252059228"><div class="module-shortcutanchordiv"><span id="WP:⋅"></span></div><div class="module-shortcutboxplain noprint" role="note"><div class="module-shortcutlist"><a href="/wiki/Wikipedia:Shortcut" title="Wikipedia:Shortcut">Shortcut</a></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><li><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Wikipedia:%E2%8B%85&redirect=no">WP:⋅</a></span></li></ul></div></div> <p>Standard algebraic notation is best for formulae, so two variables <i>q</i> and <i>d</i> being multiplied are best written as <i>qd</i> when presented in a formula. That is, when <i>citing</i> a formula, don't use <code>×</code>. </p><p>However, when <i>explaining</i> the formula for a general audience (not just mathematicians), or giving examples of its application, it is prudent to use the <a href="/wiki/Multiplication_sign" title="Multiplication sign">multiplication sign</a>: "×", coded as <code>&times;</code> in HTML or accessed via <code>{{<a href="/wiki/Template:Times" title="Template:Times">times</a>}}</code>. Do not use the letter "x" to indicate multiplication. </p> <ul><li>When <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">dividing</a> 26 by 4, 6 is the <a href="/wiki/Quotient" title="Quotient">quotient</a> and 2 is the <a href="/wiki/Remainder" title="Remainder">remainder</a>, because <span class="nowrap">26 = 6 × 4 + 2.</span></li> <li><span class="nowrap">−42 = 9 × (−5) + 3</span></li></ul> <p>An alternative to <code>×</code> is the <a href="/wiki/Dot_operator" class="mw-redirect" title="Dot operator">dot operator</a> <code>&sdot;</code> (also encoded <code class="mw-highlight mw-highlight-lang-wikitext mw-content-ltr" dir="ltr"><span class="p"><</span><span class="nt">math</span><span class="p">></span><span class="k">\cdot</span><span class="p"></</span><span class="nt">math</span><span class="p">></span></code> and reachable in the "Math and logic" drop-down list below the edit box or via template <code>{{<a href="/wiki/Template:Sdot" title="Template:Sdot">sdot</a>}}</code>), which produces a symmetrically spaced centered dot: "<i>a</i> ⋅ <i>b</i>". </p><p>Do not use the ASCII <a href="/wiki/Asterisk" title="Asterisk">asterisk</a> (*) as a multiplication sign outside of <a href="/wiki/Source_code" title="Source code">source code</a>. It is not used for this purpose in professionally published mathematics, and most fonts render it in an inappropriate vertical position (above the midline of the text rather than centered on it). For the dot operator, do not use punctuation symbols, such as a simple interpunct <code>&middot;</code> (the choice offered in the "Wiki markup" drop-down list below the edit box), as in many fonts it does not kern properly. The use of <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2022</span> </span><span style="font-size:125%;line-height:1em">•</span> <span style="font-variant: small-caps; text-transform: lowercase;"><a href="/wiki/%E2%80%A2" class="mw-redirect" title="•">BULLET</a></span> as an operator symbol is also discouraged except in abstract contexts (e.g. to denote an unspecified operator). </p><p>Metric units often embed the notion of multiplication and division. <a href="/wiki/NIST" class="mw-redirect" title="NIST">NIST</a> endorses the half-high dot (⋅) or a bare space for this purpose. </p> <div class="mw-heading mw-heading4"><h4 id="Minus_sign" data-mw-thread-id="h-Minus_sign-Using_HTML"><span data-mw-comment-start="" id="h-Minus_sign-Using_HTML"></span>Minus sign<span data-mw-comment-end="h-Minus_sign-Using_HTML"></span></h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=33" title="Edit section: Minus sign" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>The correct encoding of the minus sign "−" is different from all varieties of <a href="/wiki/Hyphen" title="Hyphen">hyphen</a> "-‐‑",<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> as well as from en-<a href="/wiki/Dash" title="Dash">dash</a> "–". To really get a minus sign, use the "minus" character "−" (reachable via selecting "Math and logic" in the drop-down list below the edit box or using <code>{{<a href="/wiki/Template:Minus" title="Template:Minus">subst:minus</a>}}</code>) or use the "<code>&minus;</code>" entity. </p> <div class="mw-heading mw-heading4"><h4 id="Square_brackets" data-mw-thread-id="h-Square_brackets-Using_HTML"><span data-mw-comment-start="" id="h-Square_brackets-Using_HTML"></span>Square brackets<span data-mw-comment-end="h-Square_brackets-Using_HTML"></span></h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=34" title="Edit section: Square brackets" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Square brackets have two problems; they can occasionally cause problems with wiki markup, and editors sometimes 'fix' the brackets in asymmetrical <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">intervals</a> to make them symmetrical. The nowiki tag can be used as a general solution to problems like this, as in <code><nowiki>]</nowiki></code> to have the ] treated as literal text. </p><p>The use of intervals for the range or domain of a function is very common. A solution which makes the reason for the different brackets around an interval more plain is to use one of the templates <span class="nowrap">{{</span><a href="/wiki/Template:Open-closed" title="Template:Open-closed">open-closed</a><span class="nowrap">}}</span>, <span class="nowrap">{{</span><a href="/wiki/Template:Closed-open" title="Template:Closed-open">closed-open</a><span class="nowrap">}}</span>, <span class="nowrap">{{</span><a href="/wiki/Template:Open-open" title="Template:Open-open">open-open</a><span class="nowrap">}}</span>, <span class="nowrap">{{</span><a href="/wiki/Template:Closed-closed" title="Template:Closed-closed">closed-closed</a><span class="nowrap">}}</span>. For instance: </p> <dl><dd><code class="mw-highlight mw-highlight-lang-wikitext mw-content-ltr" dir="ltr"><span class="p">{{</span><span class="nt">open-closed</span><span class="p">|</span>−π, π<span class="p">}}</span></code></dd></dl> <p>produces </p> <dl><dd><span class="texhtml">(−π, π]</span>.</dd></dl> <p>These templates use the <span class="nowrap">{{</span><a href="/wiki/Template:Math" title="Template:Math">math</a><span class="nowrap">}}</span> template to avoid line breaks and use the TeX font. </p> <div class="mw-heading mw-heading4"><h4 id="Function_symbol" data-mw-thread-id="h-Function_symbol-Using_HTML"><span data-mw-comment-start="" id="h-Function_symbol-Using_HTML"></span>Function symbol<span data-mw-comment-end="h-Function_symbol-Using_HTML"></span></h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=35" title="Edit section: Function symbol" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1252059228"><div class="module-shortcutanchordiv"><span id="MOS:FNOF"></span></div><div class="module-shortcutboxplain noprint" role="note"><div class="module-shortcutlist"><a href="/wiki/Wikipedia:Shortcut" title="Wikipedia:Shortcut">Shortcut</a></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><li><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=MOS:FNOF&redirect=no">MOS:FNOF</a></span></li></ul></div></div> <p>Functions are conventionally denoted by the lowercase italic letters <span class="nowrap">⁠<span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></noscript><span class="lazy-image-placeholder" style="width: 1.279ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" data-alt="{\displaystyle f}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>⁠</span>, <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></noscript><span class="lazy-image-placeholder" style="width: 1.116ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" data-alt="{\displaystyle g}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>⁠</span>, and <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" 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}"></noscript><span class="lazy-image-placeholder" style="width: 1.339ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" data-alt="{\displaystyle h}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>⁠</span>, often with a serif font whose <i>f</i> extends below the baseline. Many readers see Wikipedia body copy in a sans-serif font whose italic <i>f</i> has a different appearance without a "tail". Circa 2010, some editors proposed using the Unicode symbol <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+0192</span> </span><span style="font-size:125%;line-height:1em">ƒ</span> <span style="font-variant: small-caps; text-transform: lowercase;">LATIN SMALL LETTER F WITH HOOK</span> (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">&fnof;</span>) as a replacement function symbol; however, this symbol should not be used, because the mathematical function symbol is nothing more or less than an ordinary italic <i>f</i>, and its appearance should be consistent with the font otherwise used for mathematical notation in the same article. Furthermore, some screen readers, such as <a href="/wiki/JAWS_(screen_reader)" title="JAWS (screen reader)">JAWS</a> and <a href="/wiki/NonVisual_Desktop_Access" title="NonVisual Desktop Access">NonVisual Desktop Access</a>, pronounce the Unicode ƒ symbol as "florin" (it can be used as a <a href="/wiki/Florin_sign" title="Florin sign">Florin currency symbol</a><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>) or a numerical code, which can be confusing. </p><p>An ordinary italicized letter should be used instead, as in <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">''f''</code>, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">{{mvar|f}}</code>, or <code><math>f</math></code>, which render as <i>f</i>, <span class="texhtml mvar" style="font-style:italic;">f</span>, or <span class="nowrap"><span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></noscript><span class="lazy-image-placeholder" style="width: 1.279ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" data-alt="{\displaystyle f}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>,</span> respectively.<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> </p> <div class="mw-heading mw-heading4"><h4 id="Radical_symbol" data-mw-thread-id="h-Radical_symbol-Using_HTML"><span data-mw-comment-start="" id="h-Radical_symbol-Using_HTML"></span>Radical symbol<span data-mw-comment-end="h-Radical_symbol-Using_HTML"></span></h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=36" title="Edit section: Radical symbol" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1252059228"><div class="module-shortcutanchordiv"><span id="MOS:RADICAL"></span></div><div class="module-shortcutboxplain noprint" role="note"><div class="module-shortcutlist"><a href="/wiki/Wikipedia:Shortcut" title="Wikipedia:Shortcut">Shortcut</a></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><li><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=MOS:RADICAL&redirect=no">MOS:RADICAL</a></span></li></ul></div></div> <p>The <a href="/wiki/Radical_symbol" title="Radical symbol">radical symbol</a> √ can be used when written on its own, but when part of a larger expression, can be problematic. <span class="nowrap">{{</span><a href="/wiki/Template:Radic" title="Template:Radic">radic</a><span class="nowrap">}}</span> (often seen as <span class="nowrap">{{</span><a href="/wiki/Template:Sqrt" class="mw-redirect" title="Template:Sqrt">sqrt</a><span class="nowrap">}}</span>) is the best way to write such expressions in HTML, but the result is unattractive due to the hole between the overline and the radical symbol in many web browsers: </p> <dl><dd><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">9</span></span>, <span class="nowrap"><sup style="margin-right: -0.5em; vertical-align: 0.8em;">3</sup>√<span style="border-top:1px solid; padding:0 0.1em;">27</span></span></dd></dl> <p>This method should be avoided whenever technically possible to do so. Instead, use <code class="nowrap" style=""><math>...</math></code> tags and \sqrt{}, even if inline. For example: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {9}},{\sqrt[{3}]{27}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>9</mn> </msqrt> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>27</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {9}},{\sqrt[{3}]{27}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86256c76319740596fb7c6cb8cd8b34fa2a55b8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.393ex; height:3.009ex;" alt="{\displaystyle {\sqrt {9}},{\sqrt[{3}]{27}}}"></noscript><span class="lazy-image-placeholder" style="width: 8.393ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86256c76319740596fb7c6cb8cd8b34fa2a55b8a" data-alt="{\displaystyle {\sqrt {9}},{\sqrt[{3}]{27}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>Because of Mediawiki bug <a href="https://phabricator.wikimedia.org/T263572" class="extiw" title="phabricator:T263572">T263572</a>, <code class="nowrap" style=""><math>...</math></code> markup is incompatible with the Media Viewer (used for full-screen image viewing on mobile devices), so until that is fixed, the <span class="nowrap">{{</span><a href="/wiki/Template:Radic" title="Template:Radic">radic</a><span class="nowrap">}}</span> method or √ with no overline should be used in image captions. </p><p>The use of √ with no overline is acceptable for simple expressions, as long as the operand is unambiguous.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Explanation_of_symbols_in_formulae" data-mw-thread-id="h-Explanation_of_symbols_in_formulae-Typesetting_of_mathematical_formulae"><span data-mw-comment-start="" id="h-Explanation_of_symbols_in_formulae-Typesetting_of_mathematical_formulae"></span>Explanation of symbols in formulae<span data-mw-comment-end="h-Explanation_of_symbols_in_formulae-Typesetting_of_mathematical_formulae"></span></h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=37" title="Edit section: Explanation of symbols in formulae" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1252059228"><div class="module-shortcutanchordiv"><span id="MOS:EXPLAINSYMBOLS"></span></div><div class="module-shortcutboxplain noprint" role="note"><div class="module-shortcutlist"><a href="/wiki/Wikipedia:Shortcut" title="Wikipedia:Shortcut">Shortcut</a></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><li><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=MOS:EXPLAINSYMBOLS&redirect=no">MOS:EXPLAINSYMBOLS</a></span></li></ul></div></div> <p>A list as in </p> <blockquote> <p>Example 1: The foocity is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =\mathbf {b} \times a\mathbf {r} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>×</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\mathbf {b} \times a\mathbf {r} ,}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c336efa08115ae298130868cb0e7baf743ffc1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.086ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} =\mathbf {b} \times a\mathbf {r} ,}"></noscript><span class="lazy-image-placeholder" style="width: 12.086ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c336efa08115ae298130868cb0e7baf743ffc1dd" data-alt="{\displaystyle \mathbf {F} =\mathbf {b} \times a\mathbf {r} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf4628a1adf07133a6009e4a78bdd990c6eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} }"></noscript><span class="lazy-image-placeholder" style="width: 1.485ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf4628a1adf07133a6009e4a78bdd990c6eb9" data-alt="{\displaystyle \mathbf {b} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the barness vector,</li> <li><span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" 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}"></noscript><span class="lazy-image-placeholder" style="width: 1.23ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" data-alt="{\displaystyle a}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the bazness coefficient,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></noscript><span class="lazy-image-placeholder" style="width: 1.102ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" data-alt="{\displaystyle \mathbf {r} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the quuxance vector.</li></ul> </blockquote> <p>should be written as prose, to avoid using more vertical space than necessary: </p> <blockquote> <p>Example 2: The foocity is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =\mathbf {b} \times a\mathbf {r} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>×</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\mathbf {b} \times a\mathbf {r} ,}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c336efa08115ae298130868cb0e7baf743ffc1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.086ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} =\mathbf {b} \times a\mathbf {r} ,}"></noscript><span class="lazy-image-placeholder" style="width: 12.086ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c336efa08115ae298130868cb0e7baf743ffc1dd" data-alt="{\displaystyle \mathbf {F} =\mathbf {b} \times a\mathbf {r} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf4628a1adf07133a6009e4a78bdd990c6eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} }"></noscript><span class="lazy-image-placeholder" style="width: 1.485ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf4628a1adf07133a6009e4a78bdd990c6eb9" data-alt="{\displaystyle \mathbf {b} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the barness vector, <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" 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}"></noscript><span class="lazy-image-placeholder" style="width: 1.23ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" data-alt="{\displaystyle a}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the bazness coefficient, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></noscript><span class="lazy-image-placeholder" style="width: 1.102ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" data-alt="{\displaystyle \mathbf {r} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the quuxance vector. </p> </blockquote> <p>An exception would be if some of the definitions are very long (as in <a href="/wiki/Heat_equation#Three-dimensional_problem" title="Heat equation">Heat equation</a>, for example). In any case, each definition should end with a comma or semicolon, and the last one should end with a period if it terminates a sentence. </p> <div class="mw-heading mw-heading3"><h3 id="Punctuation_after_formulae" data-mw-thread-id="h-Punctuation_after_formulae-Typesetting_of_mathematical_formulae"><span data-mw-comment-start="" id="h-Punctuation_after_formulae-Typesetting_of_mathematical_formulae"></span><span class="anchor" id="PUNC"></span>Punctuation after formulae<span data-mw-comment-end="h-Punctuation_after_formulae-Typesetting_of_mathematical_formulae"></span></h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=38" title="Edit section: Punctuation after formulae" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1252059228"><div class="module-shortcutanchordiv"><span id="MOS:MATH#PUNC"></span></div><div class="module-shortcutboxplain noprint" role="note"><div class="module-shortcutlist"><a href="/wiki/Wikipedia:Shortcut" title="Wikipedia:Shortcut">Shortcut</a></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><li><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=MOS:MATH&redirect=no#PUNC">MOS:MATH#PUNC</a></span></li></ul></div></div> <p>Just as in mathematics publications, a sentence which ends with a formula must have a period at the end of the formula.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> This equally applies to displayed formulae (that is, formulae that take up a line by themselves). Similarly, if the conventional punctuation rules would require a question mark, comma, semicolon, or other punctuation at that place, the formula must have that punctuation at the end. </p><p>If the formula is written in LaTeX, that is, surrounded by the <code><math></code> and <code></math></code> tags, then the punctuation should also be inside the tags, because otherwise the punctuation could wrap to a new line if the formula is at the edge of the browser window. Alternatively—since the previous result can be unaesthetic, especially for inlined formulae presented as an image whose baseline does not line up with that of the running text—the punctuation can be placed after the <code></math></code> tag and then the whole formula (including the punctuation) can be enclosed with the <span class="nowrap">{{</span><a href="/wiki/Template:Nowrap" title="Template:Nowrap">nowrap</a><span class="nowrap">}}</span> template, as in <code>This shows that {{nowrap|<math>\tfrac{1}{2} = 0.5</math>.}}</code>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Font_usage" data-mw-thread-id="h-Font_usage-Typesetting_of_mathematical_formulae"><span data-mw-comment-start="" id="h-Font_usage-Typesetting_of_mathematical_formulae"></span>Font usage<span data-mw-comment-end="h-Font_usage-Typesetting_of_mathematical_formulae"></span></h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=39" title="Edit section: Font usage" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <div class="mw-heading mw-heading4"><h4 id="Multi-letter_names" data-mw-thread-id="h-Multi-letter_names-Font_usage"><span data-mw-comment-start="" id="h-Multi-letter_names-Font_usage"></span>Multi-letter names<span data-mw-comment-end="h-Multi-letter_names-Font_usage"></span></h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=40" title="Edit section: Multi-letter names" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Functions that have multi-letter names should always be in an upright font. The most well-known functions—trigonometric functions, logarithms, etc.—can be written without parentheses for as long as the result does not become ambiguous. For example: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\sin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\sin x}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0851ace05a363d7392081f9463553be0d89a1afb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.122ex; height:2.176ex;" alt="{\displaystyle 2\sin x}"></noscript><span class="lazy-image-placeholder" style="width: 6.122ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0851ace05a363d7392081f9463553be0d89a1afb" data-alt="{\displaystyle 2\sin x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (parentheses may be omitted here, as the argument consists of a single term only; typeset from <code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><math>2<span class="k">\sin</span> x</math></code>)</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\sin(x+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\sin(x+1)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27f7500c970c894aa40992113d2915ba232d7cc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.547ex; height:2.843ex;" alt="{\displaystyle 2\sin(x+1)}"></noscript><span class="lazy-image-placeholder" style="width: 11.547ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27f7500c970c894aa40992113d2915ba232d7cc5" data-alt="{\displaystyle 2\sin(x+1)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (parentheses are required to clarify the intended argument)</dd></dl> <p>but not </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2sinx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2sinx}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b45fb5f63ce89feae3356aa9b92ffb011ebca423" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.78ex; height:2.176ex;" alt="{\displaystyle 2sinx}"></noscript><span class="lazy-image-placeholder" style="width: 5.78ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b45fb5f63ce89feae3356aa9b92ffb011ebca423" data-alt="{\displaystyle 2sinx}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (<i>incorrect</i>—typeset from <code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><math>2sin x</math></code>).</dd></dl> <dl><dd>Note: For potential pitfalls of forms not understood consistently across the board, see <a href="/wiki/Order_of_operations" title="Order of operations">order of operations</a> and <a href="/wiki/Implied_multiplication" class="mw-redirect" title="Implied multiplication">implied multiplication</a>; if there is any risk that a term could become ambiguous for our readership, use parentheses.</dd></dl> <p>When operator (function) names do not have a pre-defined abbreviation, we may use <code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><span class="k">\operatorname</span></code>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\operatorname {csch} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>csch</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\operatorname {csch} x}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c07d8c1226a9e08b2412415ac8bf3433ec5b88e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.54ex; height:2.176ex;" alt="{\displaystyle 2\operatorname {csch} x}"></noscript><span class="lazy-image-placeholder" style="width: 7.54ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c07d8c1226a9e08b2412415ac8bf3433ec5b88e" data-alt="{\displaystyle 2\operatorname {csch} x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (typeset from <code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><math>2<span class="k">\operatorname</span><span class="nb">{</span>csch<span class="nb">}</span>x</math></code>).</dd> <dd><span class="mwe-math-element"><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\operatorname {tr} (A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\operatorname {tr} (A)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6661a3bfb9dec9af6b20fdf4c2de0247ddb63c21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.986ex; height:2.843ex;" alt="{\displaystyle a\operatorname {tr} (A)}"></noscript><span class="lazy-image-placeholder" style="width: 6.986ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6661a3bfb9dec9af6b20fdf4c2de0247ddb63c21" data-alt="{\displaystyle a\operatorname {tr} (A)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (typeset from <code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><math>a<span class="k">\operatorname</span><span class="nb">{</span>tr<span class="nb">}</span>(A)</math></code>).</dd></dl> <p><code>\operatorname</code> includes correct spacing that would not be present with other means such as <code>\rm</code>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2{\rm {sin}}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{\rm {sin}}x}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/686a52742614752815f51bed46edd9403dffaae5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.348ex; height:2.176ex;" alt="{\displaystyle 2{\rm {sin}}x}"></noscript><span class="lazy-image-placeholder" style="width: 5.348ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/686a52742614752815f51bed46edd9403dffaae5" data-alt="{\displaystyle 2{\rm {sin}}x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (<i>incorrect</i>—typeset from <code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><math>2<span class="nb">{</span><span class="k">\rm</span> sin<span class="nb">}</span> x</math></code>).</dd></dl> <p>Special care is needed with subscripted labels to distinguish the purpose of the subscript (as this is a common error): variables and constants in subscripts should be italic, while textual labels should be in normal text font (Roman, upright). For example: </p> <dl><dd><span class="mwe-math-element"><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_{\text{this one}}=y_{\text{that one}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>this one</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>that one</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{this one}}=y_{\text{that one}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ae47e6fb3d2cd11b90764196211c0ac891adcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.799ex; height:2.009ex;" alt="{\displaystyle x_{\text{this one}}=y_{\text{that one}}}"></noscript><span class="lazy-image-placeholder" style="width: 17.799ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ae47e6fb3d2cd11b90764196211c0ac891adcd" data-alt="{\displaystyle x_{\text{this one}}=y_{\text{that one}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (correct—typeset from <code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><math> x<span class="nb">_</span><span class="k">\text</span><span class="nb">{</span>this one<span class="nb">}</span> = y<span class="nb">_</span><span class="k">\text</span><span class="nb">{</span>that one<span class="nb">}</span></math></code>),</dd></dl> <p>and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}{y_{i}^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}{y_{i}^{2}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/866e4a7957052e4d87f4884dd9480591de3b6def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:5.957ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}{y_{i}^{2}}}"></noscript><span class="lazy-image-placeholder" style="width: 5.957ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/866e4a7957052e4d87f4884dd9480591de3b6def" data-alt="{\displaystyle \sum _{i=1}^{n}{y_{i}^{2}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (correct—typeset from <code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><math><span class="k">\sum</span><span class="nb">_{</span>i=1<span class="nb">}^</span>n <span class="nb">{</span> y<span class="nb">_</span>i<span class="nb">^</span>2 <span class="nb">}</span></math></code>),</dd></dl> <p>but not </p> <dl><dd><span class="mwe-math-element"><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=x_{predicted}-x_{observed}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>r</mi> <mi>e</mi> <mi>d</mi> <mi>i</mi> <mi>c</mi> <mi>t</mi> <mi>e</mi> <mi>d</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>b</mi> <mi>s</mi> <mi>e</mi> <mi>r</mi> <mi>v</mi> <mi>e</mi> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=x_{predicted}-x_{observed}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6956a7be90a487d40bacad2f3345522b5666f26a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.009ex; height:2.676ex;" alt="{\displaystyle r=x_{predicted}-x_{observed}}"></noscript><span class="lazy-image-placeholder" style="width: 23.009ex;height: 2.676ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6956a7be90a487d40bacad2f3345522b5666f26a" data-alt="{\displaystyle r=x_{predicted}-x_{observed}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (<i>incorrect</i>—typeset from <code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><math>r = x<span class="nb">_{</span>predicted<span class="nb">}</span> - x<span class="nb">_{</span>observed<span class="nb">}</span></math></code>).</dd></dl> <p>For several years this manual recommended <code>\mbox</code> as a workaround for lack of <code>\text</code>, but this is now considered undesirable. See <i><a href="/wiki/Wikipedia:Manual_of_Style/Mathematics/Why_you_should_never_use_%5Cmbox_within_Wikipedia" title="Wikipedia:Manual of Style/Mathematics/Why you should never use \mbox within Wikipedia">An opinion: Why you should never use \mbox within Wikipedia</a>.</i> </p> <div class="mw-heading mw-heading4"><h4 id="Roman_versus_italic" data-mw-thread-id="h-Roman_versus_italic-Font_usage"><span data-mw-comment-start="" id="h-Roman_versus_italic-Font_usage"></span>Roman versus italic<span data-mw-comment-end="h-Roman_versus_italic-Font_usage"></span></h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=41" title="Edit section: Roman versus italic" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>For single-letter variables, constants, and operators such as the <a href="/wiki/Derivative" title="Derivative">differential</a>, <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>, and <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">Euler's number</a>, Wikipedia articles usually use an italic font. One writes </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\pi }\sin x\,dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\pi }\sin x\,dx,}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2bf5587dd568c034872a85bada61f47fdff354" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.177ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\pi }\sin x\,dx,}"></noscript><span class="lazy-image-placeholder" style="width: 12.177ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2bf5587dd568c034872a85bada61f47fdff354" data-alt="{\displaystyle \int _{0}^{\pi }\sin x\,dx,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (typeset from <code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><math><span class="k">\int</span><span class="nb">_</span>0<span class="nb">^</span><span class="k">\pi</span> <span class="k">\sin</span> x <span class="k">\,</span> dx ,</math></code>—note the thin space (<code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><span class="k">\,</span></code>) before <code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr">dx</code>),</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}},}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a610aa8446be002e2e30d7121f6a87273d4caa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.395ex; height:5.843ex;" alt="{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}},}"></noscript><span class="lazy-image-placeholder" style="width: 15.395ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a610aa8446be002e2e30d7121f6a87273d4caa" data-alt="{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (typeset from <code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><math><span class="k">\frac</span><span class="nb">{</span>dz<span class="nb">}{</span>dx<span class="nb">}</span> = <span class="k">\frac</span><span class="nb">{</span>dz<span class="nb">}{</span>dy<span class="nb">}</span> <span class="k">\cdot</span> <span class="k">\frac</span><span class="nb">{</span>dy<span class="nb">}{</span>dx<span class="nb">}</span> ,</math></code>),</dd> <dd><span class="mwe-math-element"><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+iy,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+iy,}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8e572b73f9e0a614cefa2e32fae2917f74417c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.775ex; height:2.509ex;" alt="{\displaystyle x+iy,}"></noscript><span class="lazy-image-placeholder" style="width: 6.775ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8e572b73f9e0a614cefa2e32fae2917f74417c1" data-alt="{\displaystyle x+iy,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (typeset from <code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><math>x+iy,</math></code>), and</dd> <dd><span class="mwe-math-element"><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^{i\theta }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\theta }.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c88bf216e9c6c748e4786506a8fb503c3ef428e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.301ex; height:2.676ex;" alt="{\displaystyle e^{i\theta }.}"></noscript><span class="lazy-image-placeholder" style="width: 3.301ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c88bf216e9c6c748e4786506a8fb503c3ef428e7" data-alt="{\displaystyle e^{i\theta }.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (typeset from <code class="mw-highlight mw-highlight-lang-tex mw-content-ltr" dir="ltr"><math>e<span class="nb">^{</span>i<span class="k">\theta</span><span class="nb">}</span> .</math></code>).</dd></dl> <p>Some authors prefer to use an upright (Roman) font for operators, as in <span class="texhtml">d</span>, for the differential operator, as opposed to <span class="texhtml mvar" style="font-style:italic;">d</span> for a variable. Upright fonts are sometimes used for standard, nearly universal constants, as in <span class="texhtml">i</span>, <span class="texhtml">e</span>, and <span class="texhtml">π</span>; other authors use Roman boldface, as in <b><span class="texhtml">i</span></b>. Changes from one style to another should be done only to make an article consistent with itself. Formatting changes should <i>not</i> be made solely to make articles consistent with each other, nor to make articles conform to a particular style guide or standards body. It is inappropriate for an editor to go through articles doing mass changes from one style to another. When there is dispute over the correct style to use, follow the same principles as <a href="/wiki/MOS:STYLERET" class="mw-redirect" title="MOS:STYLERET">MOS:STYLERET</a>. </p><p>Generally, one way to determine which usage is appropriate on Wikipedia is to look at prevalence in reliable sources in addition to relevant style guides, per <a href="/wiki/Wikipedia:WEIGHT" class="mw-redirect" title="Wikipedia:WEIGHT">WP:WEIGHT</a>. For example, the ISO 80000-2 recommends that the mathematical constant <span class="texhtml"><i>e</i></span> should be typeset in an upright Roman font: <span class="texhtml">e</span>. But this guide is rarely followed in reliable mathematical sources, and it is contradicted by other style guides, like <a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a>'s <i>TeXbook</i>. This makes the more common practice to use an italic face for the constant <span class="texhtml"><i>e</i></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Blackboard_bold" data-mw-thread-id="h-Blackboard_bold-Font_usage"><span data-mw-comment-start="" id="h-Blackboard_bold-Font_usage"></span>Blackboard bold<span data-mw-comment-end="h-Blackboard_bold-Font_usage"></span></h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=42" title="Edit section: Blackboard bold" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1252059228"><div class="module-shortcutanchordiv"><span id="MOS:BBB"></span></div><div class="module-shortcutboxplain noprint" role="note"><div class="module-shortcutlist"><a href="/wiki/Wikipedia:Shortcut" title="Wikipedia:Shortcut">Shortcut</a></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><li><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=MOS:BBB&redirect=no">MOS:BBB</a></span></li></ul></div></div> <p>The <a href="/wiki/Blackboard_bold" title="Blackboard bold">blackboard bold</a> letter style originated in the 1960s to distinguish bold letters from ordinary letters on a blackboard or using a typewriter; in professionally typeset documents, bold fonts were used for the same purpose. Since then, blackboard bold has gradually gained currency, and is now commonly used in mathematical printing to denote certain common objects in a style distinct from other uses of bold letters. </p><p>Today, either blackboard bold or ordinary bold letters can be used interchangeably to represent the standard <a href="/wiki/Number_system" class="mw-redirect" title="Number system">number systems</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} ,}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b1252d3e79f5246bb7650b1beb5c0336cd48d66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {N} ,}"></noscript><span class="lazy-image-placeholder" style="width: 2.325ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b1252d3e79f5246bb7650b1beb5c0336cd48d66" data-alt="{\displaystyle \mathbb {N} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ,}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3aa4cb112cbe4f94a3ff8569f869c31dce5fce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.197ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} ,}"></noscript><span class="lazy-image-placeholder" style="width: 2.197ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3aa4cb112cbe4f94a3ff8569f869c31dce5fce4" data-alt="{\displaystyle \mathbb {Z} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ,}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91185244fbdded6ea99a5e9e6603299128b10928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} ,}"></noscript><span class="lazy-image-placeholder" style="width: 2.455ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91185244fbdded6ea99a5e9e6603299128b10928" data-alt="{\displaystyle \mathbb {Q} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ,}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0522388d36b55de7babe4bbfc49475eaf590c2bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ,}"></noscript><span class="lazy-image-placeholder" style="width: 2.325ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0522388d36b55de7babe4bbfc49475eaf590c2bd" data-alt="{\displaystyle \mathbb {R} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ,}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ff6a3dc2982018ff20f1d2c927afc74a217be6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {C} ,}"></noscript><span class="lazy-image-placeholder" style="width: 2.325ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ff6a3dc2982018ff20f1d2c927afc74a217be6" data-alt="{\displaystyle \mathbb {C} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} ,}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d178e5ac94e706fdb8d8733d567b7c087b23195" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {H} ,}"></noscript><span class="lazy-image-placeholder" style="width: 2.455ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d178e5ac94e706fdb8d8733d567b7c087b23195" data-alt="{\displaystyle \mathbb {H} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q},}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e7d145216d6fd8a79e6a869ac9e5c54c6334be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.056ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q},}"></noscript><span class="lazy-image-placeholder" style="width: 3.056ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e7d145216d6fd8a79e6a869ac9e5c54c6334be8" data-alt="{\displaystyle \mathbb {F} _{q},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p},}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7967f468b6a942f77dd96ada0815be530fcda626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.256ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p},}"></noscript><span class="lazy-image-placeholder" style="width: 3.256ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7967f468b6a942f77dd96ada0815be530fcda626" data-alt="{\displaystyle \mathbb {Z} _{p},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"></noscript><span class="lazy-image-placeholder" style="width: 2.867ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" data-alt="{\displaystyle \mathbb {Q} _{p}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>) and for certain other mathematical objects, including <a href="/wiki/Affine_space" title="Affine space">affine space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{n}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d64ccca0c6fbd18be5d4ec26e26efd89e9c0d76e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {A} ^{n}}"></noscript><span class="lazy-image-placeholder" style="width: 2.897ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d64ccca0c6fbd18be5d4ec26e26efd89e9c0d76e" data-alt="{\displaystyle \mathbb {A} ^{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, <a href="/wiki/Projective_space" title="Projective space">projective space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{n}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e712abd7ff167cfb12ffa7c8c9bf7b0029cf192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.639ex; height:2.343ex;" alt="{\displaystyle \mathbb {P} ^{n}}"></noscript><span class="lazy-image-placeholder" style="width: 2.639ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e712abd7ff167cfb12ffa7c8c9bf7b0029cf192" data-alt="{\displaystyle \mathbb {P} ^{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, <a href="/wiki/Adele_ring" title="Adele ring">adele rings</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b88e22526256c10a2a2356a4f6bc9c7ad895bd86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}"></noscript><span class="lazy-image-placeholder" style="width: 3.371ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b88e22526256c10a2a2356a4f6bc9c7ad895bd86" data-alt="{\displaystyle \mathbb {A} _{K}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, the additive and multiplicative <a href="/wiki/Group_scheme" title="Group scheme">group schemes</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{a}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a3a118277795dec18b43d8e6b839e6af5856180" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.91ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{a}}"></noscript><span class="lazy-image-placeholder" style="width: 2.91ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a3a118277795dec18b43d8e6b839e6af5856180" data-alt="{\displaystyle \mathbb {G} _{a}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{m}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/338f0e58c3bbaf1b4f67cd73a3fac1f29c40e39c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.483ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{m}}"></noscript><span class="lazy-image-placeholder" style="width: 3.483ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/338f0e58c3bbaf1b4f67cd73a3fac1f29c40e39c" data-alt="{\displaystyle \mathbb {G} _{m}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>), and <a href="/wiki/Hypercohomology" class="mw-redirect" title="Hypercohomology">hypercohomology</a> (e.g., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} ^{i}(X,\Omega _{X}^{\bullet })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} ^{i}(X,\Omega _{X}^{\bullet })}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42036bcd6789790a8e8c2d82fd2b71179fe25529" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.741ex; height:3.343ex;" alt="{\displaystyle \mathbb {H} ^{i}(X,\Omega _{X}^{\bullet })}"></noscript><span class="lazy-image-placeholder" style="width: 10.741ex;height: 3.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42036bcd6789790a8e8c2d82fd2b71179fe25529" data-alt="{\displaystyle \mathbb {H} ^{i}(X,\Omega _{X}^{\bullet })}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>). Font preferences vary from one mathematical author or publisher to another. </p><p>A particular concern for the use of blackboard bold on Wikipedia is that the Unicode symbols for blackboard bold characters are not supported by all systems, and font substitution in browsers often renders these symbols in discordant fonts. The use of Unicode characters for blackboard bold is discouraged in English Wikipedia; instead, either the LaTeX rendering (for example <code><math>\mathbb{Z}</math></code> or <code><math>\Z</math></code>) or standard bold fonts should be used. As with all such choices, each article should be consistent with itself, and editors should not change articles from one choice of typeface to another, except to maintain internal consistency. When there is dispute, follow <a href="/wiki/MOS:STYLERET" class="mw-redirect" title="MOS:STYLERET">MOS:STYLERET</a>. </p><p>Due to <a href="https://phabricator.wikimedia.org/T279805" class="extiw" title="phab:T279805">a rendering bug</a>, LaTeX blackboard bold currently does not work with numerals. Use bold instead (e.g. {{math|'''1'''}} or <math>\bold{1}</math>), which is more common anyway. If absolutely necessary (e.g. when discussing the notation itself), use the Unicode character (e.g. 𝟙). </p><p>Due to bug <a href="https://phabricator.wikimedia.org/T263572" class="extiw" title="phabricator:T263572">T263572</a>, <code class="nowrap" style=""><math>...</math></code> markup should not be used in image captions. </p> <div class="mw-heading mw-heading3"><h3 id="Fractions" data-mw-thread-id="h-Fractions-Typesetting_of_mathematical_formulae"><span data-mw-comment-start="" id="h-Fractions-Typesetting_of_mathematical_formulae"></span>Fractions<span data-mw-comment-end="h-Fractions-Typesetting_of_mathematical_formulae"></span></h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=43" title="Edit section: Fractions" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Wikipedia:Manual_of_Style/Dates_and_numbers#Fractions" title="Wikipedia:Manual of Style/Dates and numbers">Wikipedia:Manual of Style/Dates and numbers § Fractions</a></div> <p>In mathematics articles, fractions should always be written either with a horizontal fraction bar (as in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{2}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f4b147775b52f71eacf8937ff124ac92f0574d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle \textstyle {\frac {1}{2}}}"></noscript><span class="lazy-image-placeholder" style="width: 1.658ex;height: 3.509ex;vertical-align: -1.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f4b147775b52f71eacf8937ff124ac92f0574d2" data-alt="{\displaystyle \textstyle {\frac {1}{2}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>), or with a forward <a href="/wiki/Slash_(punctuation)" title="Slash (punctuation)">slash</a> and with the baseline of the numbers aligned with the baseline of the surrounding text (as in 1/2). The use of <code>{{<a href="/wiki/Template:Frac" class="mw-redirect" title="Template:Frac">frac</a>}}</code> (such as <style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>) is discouraged in mathematics articles. The use of <a href="/wiki/Unicode" title="Unicode">Unicode</a> <a href="/wiki/Precomposed_fraction" class="mw-redirect" title="Precomposed fraction">precomposed fractions</a> (such as ½) is discouraged entirely, for <a href="/wiki/Web_accessibility" title="Web accessibility">accessibility</a> reasons<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> among others.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Metric units are given in decimal fractions (e.g., <span class="nowrap"><span data-sort-value="6998520000000000000♠"></span>5.2 cm</span>); non-metric units can be either type of fraction, but the fraction style should be consistent throughout the article. </p><p>Only "/" is used for <a href="/wiki/Quotient_object" class="mw-redirect" title="Quotient object">quotient objects</a> in abstract algebra: <span class="texhtml"><i>R</i> / <i>A</i></span> or <span class="mwe-math-element"><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/A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R/A}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f4726ff2ae24c183855f7561b6aa992e411a0b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.67ex; height:2.843ex;" alt="{\displaystyle R/A}"></noscript><span class="lazy-image-placeholder" style="width: 4.67ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f4726ff2ae24c183855f7561b6aa992e411a0b5" data-alt="{\displaystyle R/A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> – markup: <code>{{<a href="/wiki/Template:Math" title="Template:Math">math</a>|''R'' / ''A''}}</code> or <code class="nowrap"><<a href="https://www.mediawiki.org/wiki/Extension:Math" class="extiw" title="mw:Extension:Math">math</a>>R / A</math></code> </p><p>For simple fractional subscripts or superscripts, the horizontal style is visually the least confusing: <code>{{<a href="/wiki/Template:Math" title="Template:Math">math</a>|x<sup>1/2</sup>}}</code> (<span class="texhtml">x<sup>1/2</sup></span>) or <code class="nowrap"><<a href="https://www.mediawiki.org/wiki/Extension:Math" class="extiw" title="mw:Extension:Math">math</a>>x^{1/2}</math></code> (<span class="mwe-math-element"><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^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{1/2}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32077e2f0f843fd11f5440a9818e6b2d353fb6c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.028ex; height:2.843ex;" alt="{\displaystyle x^{1/2}}"></noscript><span class="lazy-image-placeholder" style="width: 4.028ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32077e2f0f843fd11f5440a9818e6b2d353fb6c8" data-alt="{\displaystyle x^{1/2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>). </p> </section><div class="mw-heading mw-heading2 ext-discussiontools-init-section section-heading" onclick="mfTempOpenSection(8)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Graphs_and_diagrams" data-mw-thread-id="h-Graphs_and_diagrams"><span data-mw-comment-start="" id="h-Graphs_and_diagrams"></span>Graphs and diagrams<span data-mw-comment-end="h-Graphs_and_diagrams"></span></h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=44" title="Edit section: Graphs and diagrams" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-8 collapsible-block" id="mf-section-8"> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:MOS_triangle.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/MOS_triangle.svg/220px-MOS_triangle.svg.png" decoding="async" width="220" height="129" class="mw-file-element" data-file-width="196" data-file-height="115"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 129px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/MOS_triangle.svg/220px-MOS_triangle.svg.png" data-width="220" data-height="129" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/MOS_triangle.svg/330px-MOS_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/MOS_triangle.svg/440px-MOS_triangle.svg.png 2x" data-class="mw-file-element"> </span></a><figcaption>The angle <span class="texhtml mvar" style="font-style:italic;">CAB</span> is <span class="texhtml mvar" style="font-style:italic;">α</span>.<br>The length of <span class="texhtml mvar" style="font-style:italic;">CA</span> is <span class="texhtml mvar" style="font-style:italic;">b</span>.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Wikipedia:How_to_create_charts_for_Wikipedia_articles" title="Wikipedia:How to create charts for Wikipedia articles">Wikipedia:How to create charts for Wikipedia articles</a></div> <p>There is no general agreement on what fonts to use in graphs and diagrams. In geometrical diagrams points are normally labelled using upper case letters, sides with lower case and angles with lower case Greek letters. </p><p>Recent<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Dates_and_numbers#Chronological_items" title="Wikipedia:Manual of Style/Dates and numbers"><span title="The time period mentioned near this tag is ambiguous.">when?</span></a></i>]</sup> geometry books tend to use an italic serif font in diagrams as in <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" 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}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> for a point. This allows easy use in LaTeX markup. However, older books tend to use upright letters as in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {A} }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6366939c4ebbd4e8494d0dedc54c4b8dd7135a" 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 \mathrm {A} }"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6366939c4ebbd4e8494d0dedc54c4b8dd7135a" data-alt="{\displaystyle \mathrm {A} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> and many diagrams in Wikipedia use sans-serif upright A instead. Graphs in books tend to use LaTeX conventions, but yet again there are wide variations. </p><p>For ease of reference diagrams and graphs should use the same conventions as the text that refers to them. If there is a better illustration with a different convention, though, the better illustration should normally be used. </p> <div style="clear:both;" class=""></div> </section><div class="mw-heading mw-heading2 ext-discussiontools-init-section section-heading" onclick="mfTempOpenSection(9)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="See_also" data-mw-thread-id="h-See_also"><span data-mw-comment-start="" id="h-See_also"></span>See also<span data-mw-comment-end="h-See_also"></span></h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=45" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-9 collapsible-block" id="mf-section-9"> <div class="mw-heading mw-heading3"><h3 id="Help_for_those_writing_a_formula" data-mw-thread-id="h-Help_for_those_writing_a_formula-See_also"><span data-mw-comment-start="" id="h-Help_for_those_writing_a_formula-See_also"></span>Help for those writing a formula<span data-mw-comment-end="h-Help_for_those_writing_a_formula-See_also"></span></h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=46" title="Edit section: Help for those writing a formula" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <ul><li><a href="/wiki/Help:Displaying_a_formula" title="Help:Displaying a formula">Help:Displaying a formula</a></li> <li><a href="/wiki/Wikipedia:Mathematical_symbols" class="mw-redirect" title="Wikipedia:Mathematical symbols">Wikipedia:Mathematical symbols</a></li> <li><a href="/wiki/Wikipedia:Rendering_math" title="Wikipedia:Rendering math">Wikipedia:Rendering math</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="General_information" data-mw-thread-id="h-General_information-See_also"><span data-mw-comment-start="" id="h-General_information-See_also"></span>General information<span data-mw-comment-end="h-General_information-See_also"></span></h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=47" title="Edit section: General information" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <ul><li><a href="/wiki/Wikipedia:WikiProject_Mathematics" title="Wikipedia:WikiProject Mathematics">Wikipedia:WikiProject Mathematics</a></li> <li><a href="/wiki/Wikipedia:Scientific_citation_guidelines" title="Wikipedia:Scientific citation guidelines">Wikipedia:Scientific citation guidelines</a>—advice on providing references for mathematical and scientific articles</li></ul> </section><div class="mw-heading mw-heading2 ext-discussiontools-init-section section-heading" onclick="mfTempOpenSection(10)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Notes" data-mw-thread-id="h-Notes"><span data-mw-comment-start="" id="h-Notes"></span>Notes<span data-mw-comment-end="h-Notes"></span></h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=48" title="Edit section: Notes" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-10 collapsible-block" id="mf-section-10"> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Currently, <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring (mathematics)</a> and related articles attempt to cover both unital rings and non-unital rings: they do not consistently implement this interpretation. This attempt to cover multiple meanings violates <a href="/wiki/Wikipedia:DICT#Major_differences" class="mw-redirect" title="Wikipedia:DICT">WP:DICT#Major differences</a> (<a href="/wiki/Homograph" title="Homograph">homographs</a>).</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"> This example, from here <a class="external autonumber" href="https://en.wikipedia.org/w/index.php?title=Sieve_of_Eratosthenes&oldid=347438933">[1]</a>, is in <a href="/wiki/Haskell" title="Haskell">Haskell</a>, not a well-known language so generally not a good choice when showing an algorithm.</span> </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 rel="nofollow" class="external text" href="https://www.unicode.org/reports/tr25/tr25-6.html">Unicode Technical Report #25: Unicode Support for Mathematics</a>, §2.5 Duplicated Characters</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">Note that, aside of <math>, many <a href="/wiki/Help:Template" title="Help:Template">templates</a> and <a href="/wiki/Help:Parser" class="mw-redirect" title="Help:Parser">parser functions</a> accept the <a href="/wiki/Hyphen-minus" title="Hyphen-minus">hyphen-minus</a> "-" as a valid representation of the minus sign. Except situations where "-" has to represent the minus sign in a <a href="/wiki/Source_code" title="Source code">source code</a> (including wiki code), it should not be seen in a rendered page, though.</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">Latin Extended-B, <a rel="nofollow" class="external autonumber" href="http://www.unicode.org/charts/PDF/U0180.pdf">[2]</a></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="/wiki/Wikipedia_talk:WikiProject_Mathematics/Archive_68#%C6%92_or_f?" class="mw-redirect" title="Wikipedia talk:WikiProject Mathematics/Archive 68">Wikipedia talk:WikiProject Mathematics/Archive 68#ƒ or f?</a></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="/wiki/Special:PermanentLink/987584665#Radical_RFC_comments" title="Special:PermanentLink/987584665">October 2020 RfC</a>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">This style, adopted by Wikipedia, is shared by Higham (1998), Halmos (1970), the Chicago Manual of Style, and many mathematics journals.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">It is technically possible to use a <a href="/wiki/Word_joiner" title="Word joiner">word joiner</a> before the punctuation instead, but it's <a rel="nofollow" class="external text" href="http://jkorpela.fi/html/nobr.html#prevent">seldom respected by browsers</a>.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Characters in <a href="/wiki/ISO/IEC_8859-1" title="ISO/IEC 8859-1">ISO/IEC 8859-1</a> (¼, ½, and ¾) work with screen readers, but others, like ⅐, might not.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Not all fractions are available precomposed.</span> </li> </ol></div></div> </section><div class="mw-heading mw-heading2 ext-discussiontools-init-section section-heading" onclick="mfTempOpenSection(11)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Further_reading" data-mw-thread-id="h-Further_reading"><span data-mw-comment-start="" id="h-Further_reading"></span>Further reading<span data-mw-comment-end="h-Further_reading"></span></h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Wikipedia:Manual_of_Style/Mathematics&action=edit&section=49" title="Edit section: Further reading" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-11 collapsible-block" id="mf-section-11"> <p>A style guide specifically written for mathematics: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHigham1998" class="citation cs2"><a href="/wiki/Nicholas_Higham" title="Nicholas Higham">Higham, Nicholas J.</a> (1998), <a rel="nofollow" class="external text" href="http://epubs.siam.org/doi/book/10.1137/1.9780898719550"><i>Handbook of Writing for the Mathematical Sciences</i></a> (second ed.), SIAM (Society for Industrial and Applied Mathematics), <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F1.9780898719550">10.1137/1.9780898719550</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89871-420-6" title="Special:BookSources/0-89871-420-6"><bdi>0-89871-420-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Writing+for+the+Mathematical+Sciences&rft.edition=second&rft.pub=SIAM+%28Society+for+Industrial+and+Applied+Mathematics%29&rft.date=1998&rft_id=info%3Adoi%2F10.1137%2F1.9780898719550&rft.isbn=0-89871-420-6&rft.aulast=Higham&rft.aufirst=Nicholas+J.&rft_id=http%3A%2F%2Fepubs.siam.org%2Fdoi%2Fbook%2F10.1137%2F1.9780898719550&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWikipedia%3AManual+of+Style%2FMathematics" class="Z3988"></span>.</li></ul> <p>More style guidance: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalmos1970" class="citation cs2"><a href="/wiki/Paul_Halmos" title="Paul Halmos">Halmos, P.R.</a> (1970), "How to Write Mathematics", <i>Enseignements Mathématiques</i>, <b>16</b>: 123–152, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5169%2Fseals-43857">10.5169/seals-43857</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Enseignements+Math%C3%A9matiques&rft.atitle=How+to+Write+Mathematics&rft.volume=16&rft.pages=123-152&rft.date=1970&rft_id=info%3Adoi%2F10.5169%2Fseals-43857&rft.aulast=Halmos&rft.aufirst=P.R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWikipedia%3AManual+of+Style%2FMathematics" class="Z3988"></span>. Reprinted in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0821800558" title="Special:BookSources/0821800558">0821800558</a></li></ul> <p>Some finer points of typography are discussed in: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1984" class="citation cs2"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald E.</a> (1984), <i>The TeXbook</i>, Reading, Massachusetts: Addison-Wesley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-13448-9" title="Special:BookSources/0-201-13448-9"><bdi>0-201-13448-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+TeXbook&rft.place=Reading%2C+Massachusetts&rft.pub=Addison-Wesley&rft.date=1984&rft.isbn=0-201-13448-9&rft.aulast=Knuth&rft.aufirst=Donald+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWikipedia%3AManual+of+Style%2FMathematics" class="Z3988"></span>.</li></ul> <p>General style manuals often include advice on mathematics, including </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFUniversity_of_Chicago_Press_Staff2010" class="citation cs2">University of Chicago Press Staff, ed. (2010), <a rel="nofollow" class="external text" href="http://www.chicagomanualofstyle.org/home.html"><i>The Chicago Manual of Style</i></a> (16th ed.), University of Chicago Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780226104201" title="Special:BookSources/9780226104201"><bdi>9780226104201</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Chicago+Manual+of+Style&rft.edition=16th&rft.pub=University+of+Chicago+Press&rft.date=2010&rft.isbn=9780226104201&rft_id=http%3A%2F%2Fwww.chicagomanualofstyle.org%2Fhome.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWikipedia%3AManual+of+Style%2FMathematics" class="Z3988"></span></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Manual_of_Style" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="text-align: center;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Manual_of_Style" title="Template:Manual of Style"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Manual_of_Style" title="Template talk:Manual of Style"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Manual_of_Style" title="Special:EditPage/Template:Manual of Style"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Manual_of_Style" style="font-size:114%;margin:0 4em"><a href="/wiki/Wikipedia:Manual_of_Style" title="Wikipedia:Manual of Style">Manual of Style</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2" style="text-align: center;"><div> <ul><li><b><a href="/wiki/Wikipedia:Simplified_Manual_of_Style" title="Wikipedia:Simplified Manual of Style">Overview</a></b></li> <li><b><a href="/wiki/Wikipedia:Manual_of_Style/Contents" title="Wikipedia:Manual of Style/Contents">Contents</a></b></li> <li><b><a href="/wiki/Wikipedia:Styletips" title="Wikipedia:Styletips">Tips</a></b></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(content)" title="Category:Wikipedia Manual of Style (content)">Content</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="/wiki/Wikipedia:Manual_of_Style/Accessibility" title="Wikipedia:Manual of Style/Accessibility">Accessibility</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Biography" title="Wikipedia:Manual of Style/Biography">Biography</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Disambiguation_pages" title="Wikipedia:Manual of Style/Disambiguation pages">Disambiguation pages</a> <ul><li><a href="/wiki/Wikipedia:Manual_of_Style/Organizing_disambiguation_pages_by_subject_area" title="Wikipedia:Manual of Style/Organizing disambiguation pages by subject area">Organizing by subject area</a></li></ul></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Gender_identity" title="Wikipedia:Manual of Style/Gender identity">Gender identity</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Hidden_text" title="Wikipedia:Manual of Style/Hidden text">Hidden text</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Infoboxes" title="Wikipedia:Manual of Style/Infoboxes">Infoboxes</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Linking" title="Wikipedia:Manual of Style/Linking">Linking</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Self-references_to_avoid" title="Wikipedia:Manual of Style/Self-references to avoid">Self-references</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch" title="Wikipedia:Manual of Style/Words to watch">Words to watch</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(formatting)" title="Category:Wikipedia Manual of Style (formatting)">Formatting</a></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="/wiki/Wikipedia:Manual_of_Style/Abbreviations" title="Wikipedia:Manual of Style/Abbreviations">Abbreviations</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Capital_letters" title="Wikipedia:Manual of Style/Capital letters">Capitalization</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Dates_and_numbers" title="Wikipedia:Manual of Style/Dates and numbers">Dates and numbers</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Pronunciation" title="Wikipedia:Manual of Style/Pronunciation">Pronunciation</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Spelling" title="Wikipedia:Manual of Style/Spelling">Spelling</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Text_formatting" title="Wikipedia:Manual of Style/Text formatting">Text formatting</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Titles_of_works" title="Wikipedia:Manual of Style/Titles of works">Titles of works</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(images)" title="Category:Wikipedia Manual of Style (images)">Images</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="/wiki/Wikipedia:Manual_of_Style/Captions" title="Wikipedia:Manual of Style/Captions">Captions</a></li> <li><a href="/wiki/Wikipedia:Image_use_policy#Adding_images_to_articles" title="Wikipedia:Image use policy">Image placement</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Icons" title="Wikipedia:Manual of Style/Icons">Icons</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Images" title="Wikipedia:Manual of Style/Images">Images</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(layout)" title="Category:Wikipedia Manual of Style (layout)">Layout</a></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="/wiki/Wikipedia:Manual_of_Style/Layout" title="Wikipedia:Manual of Style/Layout">Layout</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Lead_section" title="Wikipedia:Manual of Style/Lead section">Lead section</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Tables" title="Wikipedia:Manual of Style/Tables">Tables</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Trivia_sections" title="Wikipedia:Manual of Style/Trivia sections">Trivia sections</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(lists)" title="Category:Wikipedia Manual of Style (lists)">Lists</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="/wiki/Wikipedia:Manual_of_Style/Lists" title="Wikipedia:Manual of Style/Lists">Lists</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Lists_of_works" title="Wikipedia:Manual of Style/Lists of works">Lists of works</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Road_junction_lists" title="Wikipedia:Manual of Style/Road junction lists">Road junctions</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Stand-alone_lists" class="mw-redirect" title="Wikipedia:Manual of Style/Stand-alone lists">Stand-alone lists</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%">By topic area</th><td class="navbox-list-with-group 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="/wiki/Category:Wikipedia_Manual_of_Style_(legal)" title="Category:Wikipedia Manual of Style (legal)">Legal</a></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="/wiki/Wikipedia:Manual_of_Style/Legal" title="Wikipedia:Manual of Style/Legal">Legal</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Trademarks" title="Wikipedia:Manual of Style/Trademarks">Trademarks</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(arts)" title="Category:Wikipedia Manual of Style (arts)">Arts</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="/wiki/Wikipedia:Manual_of_Style/Anime_and_manga" title="Wikipedia:Manual of Style/Anime and manga">Anime and manga</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Comics" title="Wikipedia:Manual of Style/Comics">Comics</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Film" title="Wikipedia:Manual of Style/Film">Film</a></li> <li><a href="/wiki/Wikipedia:Lyrics_and_poetry" title="Wikipedia:Lyrics and poetry">Lyrics and poetry</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Novels" title="Wikipedia:Manual of Style/Novels">Novels</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Television" title="Wikipedia:Manual of Style/Television">Television</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Video_games" title="Wikipedia:Manual of Style/Video games">Video games</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Visual_arts" title="Wikipedia:Manual of Style/Visual arts">Visual arts</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Writing_about_fiction" title="Wikipedia:Manual of Style/Writing about fiction">Writing about fiction</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(music)" title="Category:Wikipedia Manual of Style (music)">Music</a></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="/wiki/Wikipedia:Manual_of_Style/Music" title="Wikipedia:Manual of Style/Music">Music</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Music_samples" title="Wikipedia:Manual of Style/Music samples">Music samples</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Record_charts" title="Wikipedia:Manual of Style/Record charts">Record charts</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Stringed_instrument_tunings" title="Wikipedia:Manual of Style/Stringed instrument tunings">Stringed instruments</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(history)" title="Category:Wikipedia Manual of Style (history)">History</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="/wiki/Wikipedia:Manual_of_Style/Blazon" title="Wikipedia:Manual of Style/Blazon">Blazons</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Military_history" title="Wikipedia:Manual of Style/Military history">Military history</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Wikipedia_Manual_of_Style_(regional)" title="Category:Wikipedia Manual of Style (regional)">Regional</a></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="/wiki/Wikipedia:Naming_conventions_(geographic_names)#Specific_topics" title="Wikipedia:Naming conventions (geographic names)">Specific naming conventions</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/Canada-related_articles" title="Wikipedia:Manual of Style/Canada-related articles">Canada</a></li> <li><a href="/wiki/Wikipedia:Manual_of_Style/China-_and_Chinese-related_articles" title="Wikipedia:Manual of Style/China- and Chinese-related 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