CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-disabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-not-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>View source for Talk:Surreal number - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-disabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-not-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":true,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat": "dmy","wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"8ec65ad5-df32-40ad-8705-ac908acd6e6d","wgCanonicalNamespace":"Talk","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":1,"wgPageName":"Talk:Surreal_number","wgTitle":"Surreal number","wgCurRevisionId":1259017964,"wgRevisionId":0,"wgArticleId":51433,"wgIsArticle":false,"wgIsRedirect":false,"wgAction":"edit","wgUserName":null,"wgUserGroups":["*"],"wgCategories":[],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Talk:Surreal_number","wgRelevantArticleId":51433,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0, "wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":20000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgDiscussionToolsFeaturesEnabled":{"replytool":true,"newtopictool":true,"sourcemodetoolbar":true,"topicsubscription":false,"autotopicsub":false,"visualenhancements":false,"visualenhancements_reply":false,"visualenhancements_pageframe":false},"wgDiscussionToolsFallbackEditMode":"source","wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":true,"wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled": false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","oojs-ui-core.styles":"ready","oojs-ui.styles.indicators":"ready","mediawiki.widgets.styles":"ready","oojs-ui-core.icons":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.charinsert.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.discussionTools.init.styles":"ready","ext.uls.interlanguage":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["mediawiki.action.edit.collapsibleFooter","site","mediawiki.page.ready","jquery.makeCollapsible","skins.vector.js","ext.centralNotice.geoIP","ext.charinsert","ext.gadget.ReferenceTooltips","ext.gadget.charinsert", "ext.gadget.extra-toolbar-buttons","ext.gadget.refToolbar","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.discussionTools.init","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.charinsert.styles%7Cext.discussionTools.init.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cmediawiki.widgets.styles%7Coojs-ui-core.icons%2Cstyles%7Coojs-ui.styles.indicators%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="noindex,nofollow,max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="View source for Talk:Surreal number - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Talk:Surreal_number"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Talk:Surreal_number&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Talk:Surreal_number"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="ext-discussiontools-replytool-enabled ext-discussiontools-newtopictool-enabled ext-discussiontools-sourcemodetoolbar-enabled skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-1 ns-talk mw-editable page-Talk_Surreal_number rootpage-Talk_Surreal_number skin-vector-2022 action-edit"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Talk%3ASurreal+number&returntoquery=action%3Dedit" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Talk%3ASurreal+number&returntoquery=action%3Dedit" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Talk%3ASurreal+number&returntoquery=action%3Dedit" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Talk%3ASurreal+number&returntoquery=action%3Dedit" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <h1 id="firstHeading" class="firstHeading mw-first-heading">View source for Talk:Surreal number</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang mw-portlet-lang-icon-only" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector-empty" aria-label="This article exist only in this language. Add the article for other languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-portlet-lang-heading-empty" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language mw-ui-icon-wikimedia-language"></span> <span class="vector-dropdown-label-text">Add languages</span> </label> <div class="vector-dropdown-content"> <div class="mw-portlet-empty-language-selector-body"> Page contents not supported in other languages. </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="vector-tab-noicon mw-list-item"><a href="/wiki/Surreal_number" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Surreal_number" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Surreal_number"><span>Read</span></a></li><li id="ca-edit" class="selected istalk vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Talk:Surreal_number&action=edit" title="Edit this page"><span>Edit</span></a></li><li id="ca-addsection" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Talk:Surreal_number&action=edit&section=new" title="Start a new section [+]" accesskey="+"><span>Add topic</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Talk:Surreal_number&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="vector-more-collapsible-item mw-list-item"><a href="/wiki/Talk:Surreal_number"><span>Read</span></a></li><li id="ca-more-edit" class="selected istalk vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Talk:Surreal_number&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-addsection" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Talk:Surreal_number&action=edit&section=new"><span>Add topic</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Talk:Surreal_number&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Talk:Surreal_number" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Talk:Surreal_number" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Talk:Surreal_number&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fw%2Findex.php%3Ftitle%3DTalk%3ASurreal_number%26action%3Dedit"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fw%2Findex.php%3Ftitle%3DTalk%3ASurreal_number%26action%3Dedit"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects emptyPortlet" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> </div> <div id="contentSub"><div id="mw-content-subtitle">← <a href="/wiki/Talk:Surreal_number" title="Talk:Surreal number">Talk:Surreal number</a></div></div> <div id="mw-content-text" class="mw-body-content"><p>You do not have permission to edit this page, for the following reasons: </p> <ul class="permissions-errors"><li class="mw-permissionerror-blockedtext"> <div id="mw-blocked-text" style="border: 1px solid #AAA; background-color: var(--background-color-warning-subtle, ivory); color: inherit; padding: 1.5em; width: 100%; box-sizing: border-box;"> <div style="text-align: center;"><span style="font-size: 26px;"><span typeof="mw:File"><a href="/wiki/File:Stop_hand_nuvola.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/f/f1/Stop_hand_nuvola.svg/50px-Stop_hand_nuvola.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f1/Stop_hand_nuvola.svg/75px-Stop_hand_nuvola.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f1/Stop_hand_nuvola.svg/100px-Stop_hand_nuvola.svg.png 2x" data-file-width="240" data-file-height="240" /></a></span><b> This IP address has been <a href="/wiki/Wikipedia:Blocking_policy" title="Wikipedia:Blocking policy">blocked</a> from <i>editing</i> Wikipedia.</b></span><br /><span style="font-size: 18px;">This does not affect your ability to <i>read</i> Wikipedia pages.</span></div><div class="paragraphbreak" style="margin-top:0.5em"></div><b>Most people who see this message have done nothing wrong.</b> Some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism, and can sometimes affect other users who are unrelated to that abuse. Review the information below for assistance if you do not believe that you have done anything wrong.<div class="paragraphbreak" style="margin-top:0.5em"></div> <p>The IP address or range 8.222.128.0/17 has been <a href="/wiki/Wikipedia:Blocking_policy" title="Wikipedia:Blocking policy">blocked</a> by <a href="/wiki/User:L235" title="User:L235">‪L235‬</a> for the following reason(s): </p> <div style="padding:10px; background:var(--background-color-base, white); color:inherit; border:1px #666 solid;"> <div class="user-block colocation-webhost" style="margin-bottom: 0.5em; background-color: #ffefd5; border: 1px solid #AAA; padding: 0.7em;"> <figure class="mw-halign-left" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Server-multiple.svg/40px-Server-multiple.svg.png" decoding="async" width="40" height="57" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Server-multiple.svg/60px-Server-multiple.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Server-multiple.svg/80px-Server-multiple.svg.png 2x" data-file-width="744" data-file-height="1052" /></span><figcaption></figcaption></figure><b>The <a href="/wiki/IP_address" title="IP address">IP address</a> that you are currently using has been blocked because it is believed to be a <a href="/wiki/Web_hosting_service" title="Web hosting service">web host provider</a> or <a href="/wiki/Colocation_centre" title="Colocation centre">colocation provider</a>.</b> To prevent abuse, <a href="/wiki/Wikipedia:Open_proxies" title="Wikipedia:Open proxies">web hosts and colocation providers may be blocked</a> from editing Wikipedia. <div style="border-top: 1px solid #AAA; clear: both">You will not be able to edit Wikipedia using a web host or colocation provider because it hides your IP address, much like a <a href="/wiki/Wikipedia:Open_proxies" title="Wikipedia:Open proxies">proxy</a> or <a href="/wiki/Virtual_private_network" title="Virtual private network">VPN</a>. <p><b>We recommend that you attempt to use another connection to edit.</b> For example, if you use a proxy or VPN to connect to the internet, turn it off when editing Wikipedia. If you edit using a mobile connection, try using a Wi-Fi connection, and vice versa. If you are using a corporate internet connection, switch to a different Wi-Fi network. If you have a Wikipedia account, please log in. </p><p>If you do not have any other way to edit Wikipedia, you will need to <a href="/wiki/Wikipedia:IP_block_exemption#Requesting_and_granting_exemption" title="Wikipedia:IP block exemption">request an IP block exemption</a>. </p> <style data-mw-deduplicate="TemplateStyles:r1214851843">.mw-parser-output .hidden-begin{box-sizing:border-box;width:100%;padding:5px;border:none;font-size:95%}.mw-parser-output .hidden-title{font-weight:bold;line-height:1.6;text-align:left}.mw-parser-output .hidden-content{text-align:left}@media all and (max-width:500px){.mw-parser-output .hidden-begin{width:auto!important;clear:none!important;float:none!important}}</style><div class="hidden-begin mw-collapsible mw-collapsed" style=""><div class="hidden-title skin-nightmode-reset-color" style="text-align:center;">How to appeal if you are confident that your connection does not use a colocation provider's IP address:</div><div class="hidden-content mw-collapsible-content" style=""> If you are confident that you are not using a web host, you may <a href="/wiki/Wikipedia:Appealing_a_block" title="Wikipedia:Appealing a block">appeal this block</a> by adding the following text on your <a href="/wiki/Help:Talk_pages" title="Help:Talk pages">talk page</a>: <code>{{<a href="/wiki/Template:Unblock" title="Template:Unblock">unblock</a>|reason=Caught by a colocation web host block but this host or IP is not a web host. My IP address is _______. <i>Place any further information here.</i> ~~~~}}</code>. <b>You must fill in the blank with your IP address for this block to be investigated.</b> Your IP address can be determined <span class="plainlinks"><b><a class="external text" href="https://en.wikipedia.org/wiki/Wikipedia:Get_my_IP_address?withJS=MediaWiki:Get-my-ip.js">here</a></b></span>. Alternatively, if you wish to keep your IP address private you can use the <a href="/wiki/Wikipedia:Unblock_Ticket_Request_System" title="Wikipedia:Unblock Ticket Request System">unblock ticket request system</a>. There are several reasons you might be editing using the IP address of a web host or colocation provider (such as if you are using VPN software or a business network); please use this method of appeal only if you think your IP address is in fact not a web host or colocation provider.</div></div> <p><span class="sysop-show" style="font-size: 85%;"><span style="border:#707070 solid 1px;background-color:#ffe0e0;padding:2px"><b>Administrators:</b></span> The <a href="/wiki/Wikipedia:IP_block_exemption" title="Wikipedia:IP block exemption">IP block exemption</a> user right should only be applied to allow users to edit using web host in exceptional circumstances, and requests should usually be directed to the functionaries team via email. If you intend to give the IPBE user right, a <a href="/wiki/Wikipedia:CheckUser" title="Wikipedia:CheckUser">CheckUser</a> needs to take a look at the account. This can be requested most easily at <a href="/wiki/Wikipedia:SPI#Quick_CheckUser_requests" class="mw-redirect" title="Wikipedia:SPI">SPI Quick Checkuser Requests</a>. <b>Unblocking</b> an IP or IP range with this template <b>is highly discouraged</b> without at least contacting the blocking administrator.</span> </p> </div></div> </div> <p>This block will expire on 18:23, 24 August 2026. Your current IP address is 8.222.208.146. </p> <div class="paragraphbreak" style="margin-top:0.5em"></div><div style="font-size: 16px;"> <p>Even when blocked, you will <i>usually</i> still be able to edit your <a href="/wiki/Special:MyTalk" title="Special:MyTalk">user talk page</a>, as well as <a href="/wiki/Wikipedia:Emailing_users" title="Wikipedia:Emailing users">email</a> administrators and other editors. </p> </div> <div class="paragraphbreak" style="margin-top:0.5em"></div><div style="font-size: 16px;"> <p>For information on how to proceed, please read the <b><a href="/wiki/Wikipedia:Appealing_a_block#Common_questions" title="Wikipedia:Appealing a block">FAQ for blocked users</a></b> and the <a href="/wiki/Wikipedia:Appealing_a_block" title="Wikipedia:Appealing a block">guideline on block appeals</a>. The <a href="/wiki/Wikipedia:Guide_to_appealing_blocks" title="Wikipedia:Guide to appealing blocks">guide to appealing blocks</a> may also be helpful. </p> </div> <p>Other useful links: <a href="/wiki/Wikipedia:Blocking_policy" title="Wikipedia:Blocking policy">Blocking policy</a> · <a href="/wiki/Help:I_have_been_blocked" title="Help:I have been blocked">Help:I have been blocked</a> </p> </div></li><li class="mw-permissionerror-globalblocking-blockedtext-range"> <div id="mw-blocked-text" style="border: 1px solid #AAA; background-color: var(--background-color-warning-subtle, ivory); color: inherit; padding: 1.5em; width: 100%; box-sizing: border-box;"> <div style="text-align: center;"><span style="font-size: 26px;"><span typeof="mw:File"><a href="/wiki/File:Stop_hand_nuvola.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/f/f1/Stop_hand_nuvola.svg/50px-Stop_hand_nuvola.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f1/Stop_hand_nuvola.svg/75px-Stop_hand_nuvola.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f1/Stop_hand_nuvola.svg/100px-Stop_hand_nuvola.svg.png 2x" data-file-width="240" data-file-height="240" /></a></span><b> This IP address range has been <a href="https://meta.wikimedia.org/wiki/Global_blocks" class="extiw" title="m:Global blocks">globally blocked</a>.</b></span><br /><span style="font-size: 18px;">This does not affect your ability to <i>read</i> Wikipedia pages.</span></div><div class="paragraphbreak" style="margin-top:0.5em"></div><b>Most people who see this message have done nothing wrong.</b> Some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism, and can sometimes affect other users who are unrelated to that abuse. Review the information below for assistance if you do not believe that you have done anything wrong.<div class="paragraphbreak" style="margin-top:0.5em"></div><div class="paragraphbreak" style="margin-top:0.5em"></div> <p>This block affects editing on all Wikimedia wikis. </p><p>The IP address or range 8.222.128.0/17 has been globally <a href="/wiki/Wikipedia:Blocking_policy" title="Wikipedia:Blocking policy">blocked</a> by <a href="/wiki/User:Jon_Kolbert" title="User:Jon Kolbert">‪Jon Kolbert‬</a> for the following reason(s): </p> <div style="padding:10px; background:var(--background-color-base, white); color:inherit; border:1px #666 solid;"> <p><a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/NOP" class="extiw" title="m:Special:MyLanguage/NOP">Open proxy/Webhost</a>: See the <a href="https://meta.wikimedia.org/wiki/WM:OP/H" class="extiw" title="m:WM:OP/H">help page</a> if you are affected </p> </div> <p>This block will expire on 15:12, 27 August 2028. Your current IP address is 8.222.208.146. </p> <div class="paragraphbreak" style="margin-top:0.5em"></div><div style="font-size: 16px;"> <p>Even while globally blocked, you will <i>usually</i> still be able to edit pages on <a href="https://meta.wikimedia.org/wiki/" class="extiw" title="m:">Meta-Wiki</a>. </p> </div> <div class="paragraphbreak" style="margin-top:0.5em"></div><div style="font-size: 16px;"> <p>If you believe you were blocked by mistake, you can find additional information and instructions in the <a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/No_open_proxies" class="extiw" title="m:Special:MyLanguage/No open proxies">No open proxies</a> global policy. Otherwise, to discuss the block please <a href="https://meta.wikimedia.org/wiki/Steward_requests/Global" class="extiw" title="m:Steward requests/Global">post a request for review on Meta-Wiki</a>. You could also send an email to the <a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/Stewards" class="extiw" title="m:Special:MyLanguage/Stewards">stewards</a> <a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/VRT" class="extiw" title="m:Special:MyLanguage/VRT">VRT</a> queue at <kbd>stewards@wikimedia.org</kbd> including all above details. </p> </div> <p>Other useful links: <a href="https://meta.wikimedia.org/wiki/Global_blocks" class="extiw" title="m:Global blocks">Global blocks</a> · <a href="/wiki/Help:I_have_been_blocked" title="Help:I have been blocked">Help:I have been blocked</a> </p> </div></li></ul><hr /> <div id="viewsourcetext">You can view and copy the source of this page:</div><textarea readonly="" accesskey="," id="wpTextbox1" cols="80" rows="25" style="" class="mw-editfont-monospace" lang="en" dir="ltr" name="wpTextbox1">{{talk header}} {{WikiProject banner shell|class=B| {{WikiProject Mathematics|priority=Low}} }} == Norman Larrabee Alling, his WP entry == I thought it would be a good idea to link to his WP page from the introduction, but it turns out it doesn't exist. I woiuld think he should be notable enough... There's a biography online at http://www-history.mcs.st-andrews.ac.uk/Biographies/Alling.html and one of his colleagues published his Collected Papers in 1998: P Ribenboim (ed.), Collected papers of Norman Alling (Queen's University, Kingston, ON, 1998) -- MathSciNet review 19 13 629 912 according to an online reference -- which even has a "Vita Norman Alling" starting on page vii -- but there is no online version or structured reference that I can find, so there's about 4 lines of WP-suitable material to be gained from the online biography. It's a bit thin, all in all -- can anyone help? One last google gave me a hit on Philip Ehrlich's work page: Collected Papers of Norman Alling, edited by Paulo Ribenboim, Queen’s Papers in Pure and Applied Mathematics, Volume 107, 1998, Kingston, Ontario, Canada. Should be good enough, though an ISBN would be nice, and I'd like a peek at that Vita. <small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Lhmathies|Lhmathies]] ([[User talk:Lhmathies#top|talk]] • [[Special:Contributions/Lhmathies|contribs]]) 23:01, 18 February 2018 (UTC)</small>  :Now I'm confused, the MacTutor page above says NLA "was" a mathematician which would imply he was no longer with us, but I cannot find a death date anywhere, not even on the reference at https://mathematics.library.cornell.edu/works/A which seems up to date in this regard, judging by other entries. Since the rules for biographies of living persons are stricter, it would be nice to have this confirmed... [[User:Lhmathies|Lhmathies]] ([[User talk:Lhmathies|talk]]) 04:23, 23 February 2018 (UTC) == Controversy == There should be a section here, to preserve neutrality, noting that surreal numbers are non-existent, in the finitist form of mathematics. :That doesn't make a lot of sense. The fraction of math articles that could have such a disclaimer is huge. Nothing special about the surreals there. (I wouldn't call it a "controversy" either. Some formulations of mathematics just don't include it.) [[User:Sniffnoy|Sniffnoy]] ([[User talk:Sniffnoy|talk]]) 18:53, 10 May 2018 (UTC) :It is clear that the surreals are intended to be developped in an infinitary set theory so there is no need to specify that. Note that although they form an infinite class, they can still be defined in finitary mathematics.[[User:Vincent Bagayoko|Vincent Bagayoko]] ([[User talk:Vincent Bagayoko|talk]]) 11:13, 13 May 2018 (UTC) == Definition of y^R and y^L == It seems that there is no definition of the expressions y^R and y^L present in the equation for division :My reading is that they're the same as y_R and y_L in the previous paragraph. But I'm not confident enough to make the change. Help?[[User:DavidHobby|DavidHobby]] ([[User talk:DavidHobby|talk]]) 02:45, 7 May 2019 (UTC) ::True, this is inconsistent notation and I changed it.[[User:Lhmathies|Lhmathies]] ([[User talk:Lhmathies|talk]]) 13:40, 8 May 2019 (UTC) == When is the birthday number / older-than relation needed? == This is mostly idle musings, a.k.a. original research -- I would have put them under [[Talk:Surreal number/Archive_2#Is_the_Birthday_property_superfluous_in_the_construction?|this previous Talk page entry]], but it has been archived. I think the answer to that question is that you can construct the ordered class of surreal numbers without <i>using</i> the older-than relation, but in the process you will necessarily <i>define</i> it. On the other hand, to get from forms to numbers and thus to prove existence and consistency of arithmetic operations and define a field structure, the existence of the relation is crucial. Sketch: Given a set ''S'' of totally ordered sets, totally ordered by inclusion (order homomorphies), you can create a 'completion' by taking the union ''U'' and adding special forms {{nowrap | { ''L'' {{!}} ''R'' }, ''L'' < ''R'', ''L'' ∪ ''R'' {{=}} ''U'' }} -- there is an obvious extension of the ordering and you can extend ''S'' with the completion to get a new totally ordered set of totally ordered sets. Now start with the empty set of sets and apply transfinite induction to get the surreals as the union of all the sets in the union of the sets of sets (where the sets of numbers are the same as the generations in the Conway construction). The step of taking the union is only strictly necessary for the steps corresponding to limit ordinals, including 0; in other cases ''S'' has a maximal element. Note that the induction step only needs a total order on the set of sets. The well-ordering is a result of using (transfinite) induction. And instead of the 'special forms' you can just take the set of upwards (or downwards) closed sets -- each 'new' number corresponds to exactly one closed set, there are no equivalence classes involved and no double or parallel induction happening. [[User:Lhmathies|Lhmathies]] ([[User talk:Lhmathies|talk]]) 13:24, 10 January 2020 (UTC) :Actually this came out a bit misleading -- you do need a total order relation on ''S'' to make the induction step work and that is in effect an older-than relation, but the order type of ''S'' is not important so you might not be able to express it as a number. <small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Lhmathies|Lhmathies]] ([[User talk:Lhmathies#top|talk]] • [[Special:Contributions/Lhmathies|contribs]]) 14:29, 13 January 2020 (UTC)</small>  == Mistake in article image == Where the value of a third is shown in the diagram is in fact where two thirds should be. The tree branches right, left, right, left, r, l, r, l etc. for 2/3rds and right, left, '''left''', right, left, right left, r, l, r, l etc. for 1/3rd. --[[Special:Contributions/82.30.170.253|82.30.170.253]] ([[User talk:82.30.170.253|talk]]) 02:51, 20 April 2020 (UTC) : Yup, that's an error alright. I'd suggest commenting on the page for the image, or perhaps directly messaging the person who made it. (You could also try editing it yourself, obviously, but that may be difficult; I tried it myself and didn't get very far...) [[User:Sniffnoy|Sniffnoy]] ([[User talk:Sniffnoy|talk]]) 17:13, 20 April 2020 (UTC) I agree. It is an error. You beat me to it. I tried to comment on the page for the image, but could not find it. The answer is 2/3 as per a video from John Conway himself. <small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/2001:16B8:170A:4700:E942:4EC6:B9E3:1C31|2001:16B8:170A:4700:E942:4EC6:B9E3:1C31]] ([[User talk:2001:16B8:170A:4700:E942:4EC6:B9E3:1C31#top|talk]]) 21:35, 1 June 2020 (UTC)</small>  == Does Conway need to be defended? == One of the most recent edits has "remove conflicting naming claims" as the edit summary -- but the old text did say that the name was first used by Conway. It also said that equivalent treatments had been developed earlier, and that is gone too... so what was the real reason to change it? (The previous formulation was mine from a few years ago, full disclosure). It looks like Conway has to be the first to ever come up with anything that looks like surreal numbers -- confer the emphasis in "History of the concept" on Alling's prior treatment not being as useful, which may be true but possibly not worth so many words. Is this a cult? [[User:Lhmathies|Lhmathies]] ([[User talk:Lhmathies|talk]]) 08:59, 1 June 2020 (UTC) == Error in epsilon definition? == Am I misunderstanding or should :{{nowrap begin}}{ 0 | 1, {{sfrac|1|2}}, {{sfrac|1|4}}, {{sfrac|1|8}}, ... }{{nowrap end}} = &epsilon; actually be :{{nowrap begin}}{ 1, {{sfrac|1|2}}, {{sfrac|1|4}}, {{sfrac|1|8}}, ... | 0 }{{nowrap end}} = &epsilon; ? In the former version, as it currently is in the article, 0 | 1 would be 1/2 and the subsequent infinite series of fractions seems misplaced. [[User:Kufat|Kufat]] ([[User talk:Kufat|talk]]) 02:03, 13 August 2022 (UTC) :Order does not matter in sets. The order used in the article is more convenient in that it describes an infinite sequence of simpler numbers that are greater than ε. The latter is not a surreal number because 0 is not less than 1. –[[User:LaundryPizza03|<b style="color:#77b">Laundry</b><b style="color:#fb0">Pizza</b><b style="color:#b00">03</b>]] ([[User talk:LaundryPizza03|<span style="color:#0d0">d</span>]][[Special:Contribs/LaundryPizza03|<span style="color:#0bf">c̄</span>]]) 03:16, 13 August 2022 (UTC) == Exponentiation == Should the article contain a paragraph on [[exponentiation]] of surreal numbers, and perhaps even a hint of [[tetration]], [[pentation]], and other [[hyperoperation]]s? —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 15:45, 7 November 2022 (UTC) :Did you see [[Surreal number#Exponentiation]] (a subsection under the treatment of the [[Surreal number#Exponential function]])? This is a general definition for any two surreal numbers -- but it seems to me that hyperoperations are only defined for integer arguments (because you need to count down to zero) and I'm at a loss as to any possible generalizations. (Of course you could allow infinite numbers as the first argument, but I'm not sure if anything more interesting happens than with integers). :Exponentiation with an exponent that is a dyadic fraction (of which the integers are a subset) doesn't even need the ''log'' and ''exp'' functions, but that is not treated separately. :[[User:Lhmathies|Lhmathies]] ([[User talk:Lhmathies|talk]]) 13:41, 21 February 2023 (UTC) == Surcomplex numbers == One passage of the article reads as follows: "''A surcomplex number is a number of the form a+bi, where a and b are surreal numbers and i is the square root of −1. The surcomplex numbers form an algebraically closed field (except for being a proper class), isomorphic to the algebraic closure of the field generated by extending the rational numbers by a proper class of algebraically independent transcendental elements.''" I find a great similarity between a) "(except for being a proper class)" and b) "Other than that, Mrs. Lincoln, what did you think of the play?". My point is that there is a tremendous, indescribably large distinction between : a) an algebraically closed field that is a set and : b) one that is not a set, that cannot be a set. [[User:2601:200:c082:2ea0:48d7:7772:5ebd:f634]] 00:04, 29 March 2023‎ :Sure. But to my naive perception, this wanders off into [[model theory]] and requires posing questions like "is there a model of the surreals in the [[Grothendieck universe]] or maybe in NBG?" or maybe this is a question about "how does the Yoneda lemma work out in this case?" or maybe the question is "how does the [[transfer principle]] work out in this case?" -- and I have no clue about this. But since the transfer principle is mostly about transferring over statements written in first-order logic, maybe something goes wonky higher logics, ascending up the sigma-pi hierarchy? Who knows? On the other hand, it is my understanding that there is one and only one model of the complex numbers. Since the paragraph closes with the statement: ''Up to field isomorphism, this fact characterizes the field of surcomplex numbers within any fixed set theory.[6]: Th.27'' and so that feels very cut-n-dried and unambiguous. "Any fixed set theory" would seem to include ZFC and [[New Foundations]] and etc. I tried to rephrase your statement as a menu of plausible questions. Rephrasing statements as questions allows them to get answered. I am not sure what you might have been thinking of. [[Special:Contributions/67.198.37.16|67.198.37.16]] ([[User talk:67.198.37.16|talk]]) 20:51, 30 April 2023 (UTC) == "Esoteric Pun" Parenthetical. == In "Gaps and Continuity" there's a part describing gap "On" and its relation to "No". Then, in parenthesis it is described as an "Esoteric Pun", followed by a colon (indicating a link between ideas), followed by some more text. I have a few issues with this section that I will do my best to loosely describe, and I'm hoping that either my sentiments will be reflected and something will be done, or that justification will be given and I will understand the choices that have been made in the writing of the section. In no particular order: # The word "Esoteric" links to "Western Esotericism". Forgive my ignorance, but I see no way in which that is actually relevant here. I believe it is a different use of the word "Esoteric", possibly an informal (or even wrong) one. # Even if it is somehow relevant, the portion after the colon following "Esoteric Pun" fails to explain how it is related to esotericism at all, or just how it is "esoteric". # It also fails to elaborate on the pun. There is the obvious observation that "No" and "On" are English inverses of one another, but it isn't mentioned or clued into at all, in favor of simply elaborating on the mathematical relationship between the two. If this does actually serve to explain the pun a bit more, I'm just not seeing how. # Even without the issues of the first sentence, I'm not sure what function the rest serves as a parenthetical. It's fairly long and, at least to me, just feels poorly written. I think that if there's something important that needs saying in there, it should be given it's own paragraph, rather than being stuffed into parentheses. [Conclusive paragraph which sums up my thoughts] [[User:TheJonyMyster|TheJonyMyster]] ([[User talk:TheJonyMyster|talk]]) 03:09, 19 May 2023 (UTC) :{{ping|TheJonyMyster}} Fixed it. Esoteric is one of the types of [[mathematical joke]] as explained in the lead of that article. So the first link was just a red herring and I've removed it. Though of course since the ''book'' includes its own creation myth, it's possible that there is some esoteric content, should a third-party source report on it... [[User:Skyerise|Skyerise]] ([[User talk:Skyerise|talk]]) 12:38, 19 May 2023 (UTC) == sup and inf == Do the surreals have the [[least-upper-bound property]]? That is, does every non-empty set of surreals with an upper bound have a least upper bound ({{math|sup}}) in the surreals? Whether yes or no, mention of this in the article would be nice. I assume the answer would be the same for a greatest-lower-bound property ({{math|inf}}), and mention of that too would be nice. Thank you —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 13:13, 7 September 2023 (UTC) :This question is answered in the negative in the section "[[Surreal_number#Gaps_and_continuity|Gaps and continuity]]". –[[User:LaundryPizza03|<b style="color:#77b">Laundry</b><b style="color:#fb0">Pizza</b><b style="color:#b00">03</b>]] ([[User talk:LaundryPizza03|<span style="color:#0d0">d</span>]][[Special:Contribs/LaundryPizza03|<span style="color:#0bf">c̄</span>]]) 13:17, 16 September 2023 (UTC) == Dead Link == This link in the section Further reading is dead: http://www-cs-faculty.stanford.edu/~knuth/sn.html [[User:Hubert1965|Hubert1965]] ([[User talk:Hubert1965|talk]]) 14:39, 3 November 2023 (UTC) :{{ping|Hubert1965}} Thanks, I've fixed it. [[User:Skyerise|Skyerise]] ([[User talk:Skyerise|talk]]) 14:43, 3 November 2023 (UTC) == magnitudes and absolutes == :{{tq|... [[Infinity|infinite]] and [[infinitesimal|infinitesimal numbers]], respectively larger or smaller in [[absolute value]] than any positive real number.}} The word {{tq|positive}} was removed, and restored with "This is necessary; an infinitisimal is not smaller than -1." But it is smaller ''in absolute value'', unless I missed a memo. It is not smaller than the real number zero, though. [[User:Tamfang|—Tamfang]] ([[User talk:Tamfang|talk]]) 05:23, 3 May 2024 (UTC) :I think the sentence should be reworded. "X is smaller in absolute value than Y" is unclear and could be interpreted as "the absolute value of X is smaller than Y", which is how I initially read it. Lumping infinites and infinitesimals together in one phrase also makes the sentence a little awkward. Perhaps {{tq|... but also infinite numbers, whose absolute value is larger than any real number, and infinitesimal numbers, whose absolute value is larger than zero but smaller than any positive real number.}} It's a little wordy but is less ambiguous than the current sentence. [[User:CodeTalker|CodeTalker]] ([[User talk:CodeTalker|talk]]) 07:06, 3 May 2024 (UTC) == Games & Applications to combinatorial game theory—merge sections? == It seems like these sections repeat some information and are in general about a similar topic. Should they be merged? [[User:Theanswertolifetheuniverseandeverything|Theanswertolifetheuniverseandeverything]] ([[User talk:Theanswertolifetheuniverseandeverything|talk]]) 12:44, 17 June 2024 (UTC) == Replacing an ambiguous conclusion to a paragraph == A paragraph in [[Surreal number#Induction_rule]] closes with <big>''"These labels will also be justified by the rules for surreal addition and multiplication below"''</big>. I am concerned about an ambiguity: When I reached the aforementioned "below", I expected to see a definition of addition that establishes that the labels were correct (i.e., a derivation or proof.) Instead, I found a circular definition of addition that requires a ''a priori'' knowledge of the labels. On my first few readings, I thought that I had skipped over (or forgotten) a step. I am thinking of a replacement conclusion to the paragraph with something like <big>''"These labels will used to define addition and multiplication below in a way that permits any given surreal number to be described by a bewildering array number of different forms ( such as <math>\{2\tfrac 1 2|4\tfrac 1 2\}=\{2|\} = 3 \ne 3\tfrac 1 2.)\,</math>"''</big> While this sentence clarifies the true nature of the temporary assignment of labels, it is a heavy sentence. Ordinarily I would place some or all of it in a footnote. But Wikipedia is not too keen on footnotes. For that reason I decided not to "Be Bold" on this edit. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|talk]]) 16:32, 18 June 2024 (UTC) :It would suffice to emend that to {{tq|1="by many different forms"}}. -- [[User:Elphion|Elphion]] ([[User talk:Elphion|talk]]) 03:51, 19 June 2024 (UTC) == Is this a typo? == At the time of writing, [https://en.wikipedia.org/w/index.php?title=Surreal_number&oldid=1245486655#Transfinite_induction '''Surreal_number#Transfinite_induction'''] contains the statement: <big><big>"</big></big>"The surreal number ''ω'' − 1 is not an ordinal; the ordinal ''ω'' is not the successor of any ordinal. This is a surreal number with birthday ''ω''+1, which is labeled ''ω''− 1 on the basis that it coincides with the sum of ''ω'' = { 1, 2, 3, 4, ... } and −1 = {&nbsp;|&nbsp;0&nbsp;}<big><big>"</big></big> It seems to me that "''ω'' = { 1, 2, 3, 4, ... }" should be replaced by "''ω'' = { 1, 2, 3, 4, ... | }". Am I right??? [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|talk]]) 22:42, 16 September 2024 (UTC) :Yes. I made an edit for this and another edit for another typo in that section. —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 01:29, 17 September 2024 (UTC) == Potentially problematic images == [[File:Surreal less than or equal to.svg|thumb|Illustrating x ≤ y]][[File:Surreal 0 less than or equal to 3.svg|thumb|upright=.8|Illustrating 0 ≤ 3]]I mostly work on Wikiversity, but have also done lots of [[v:User:Guy vandegrift/Images on other wikis|images on other wikis]]. Those experiences taught me that sometimes it is necessary to consult with an article's active editors, especially with images that contain too many innovations. One innovation in these images is the left to right ranking of the sets and surreal numbers: <math display="block">X_L\leq^*x\leq^*X_R\leq^*Y_L\leq^*y\leq^*Y_R</math> By placing the x and y above the left and right sets, the reader can view it in terms of this ordering. The "problematic" aspect of this is the use of an asterisk to denote inequalities between sets of numbers. [https://www.tondering.dk/download/sur.pdf Tøndering] does something similar, using things like <math>a\not\geq A</math> (where a is a number and A is a set.) As somewhat of a novice to set theory, I find <math>\leq^*</math> much easier on the eye and brain (compared with entities such as <math>a\not\geq A</math> or <math>a\neg\geq A</math>.) For one thing, I like to see the "wide" part of < next to the larger element. Also, the asterisk warns the novice that "inequalites" between sets and numbers are tricky. It notifies readers of the complexities of comparing numbers with sets, but in a way that encourages them to ignore the details on a first reading. Also, I attempted to use a chatbot to find a convention for describing inequalities between numbers and sets of numbers. Gemini was unable to find anything. See [[v:Chatbot_math/Gemini/Ranking_sets_of_numbers]] --- [[User:Guy vandegrift|Guy vandegrift]] --([[User talk:Guy vandegrift|talk]]) 21:48, 22 November 2024 (UTC) </textarea><div class="templatesUsed"><div class="mw-templatesUsedExplanation"><p><span id="templatesused">Pages transcluded onto the current version of this page<span class="posteditwindowhelplinks"> (<a href="/wiki/Help:Transclusion" title="Help:Transclusion">help</a>)</span>:</span> </p></div><ul> <li><a href="/wiki/Surreal_number" title="Surreal number">Surreal number</a> (<a href="/w/index.php?title=Surreal_number&action=edit" title="Surreal number">edit</a>) </li><li><a href="/wiki/Template:Archive_list" title="Template:Archive list">Template:Archive list</a> (<a href="/w/index.php?title=Template:Archive_list&action=edit" title="Template:Archive list">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Comma_separated_entries" title="Template:Comma separated entries">Template:Comma separated entries</a> (<a href="/w/index.php?title=Template:Comma_separated_entries&action=edit" title="Template:Comma separated entries">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Encodefirst" title="Template:Encodefirst">Template:Encodefirst</a> (<a href="/w/index.php?title=Template:Encodefirst&action=edit" title="Template:Encodefirst">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Find_general_sources" title="Template:Find general sources">Template:Find general sources</a> (<a href="/w/index.php?title=Template:Find_general_sources&action=edit" title="Template:Find general sources">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Find_page_text" title="Template:Find page text">Template:Find page text</a> (<a href="/w/index.php?title=Template:Find_page_text&action=edit" title="Template:Find page text">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Find_sources" title="Template:Find sources">Template:Find sources</a> (<a href="/w/index.php?title=Template:Find_sources&action=edit" title="Template:Find sources">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Find_sources/proj/is_biography" title="Template:Find sources/proj/is biography">Template:Find sources/proj/is biography</a> (<a href="/w/index.php?title=Template:Find_sources/proj/is_biography&action=edit" title="Template:Find sources/proj/is biography">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Find_sources/proj/is_med" title="Template:Find sources/proj/is med">Template:Find sources/proj/is med</a> (<a href="/w/index.php?title=Template:Find_sources/proj/is_med&action=edit" title="Template:Find sources/proj/is med">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Find_sources/proj/is_video" title="Template:Find sources/proj/is video">Template:Find sources/proj/is video</a> (<a href="/w/index.php?title=Template:Find_sources/proj/is_video&action=edit" title="Template:Find sources/proj/is video">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Find_sources/top_proj" title="Template:Find sources/top proj">Template:Find sources/top proj</a> (<a href="/w/index.php?title=Template:Find_sources/top_proj&action=edit" title="Template:Find sources/top proj">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:For_loop" title="Template:For loop">Template:For loop</a> (<a href="/w/index.php?title=Template:For_loop&action=edit" title="Template:For loop">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:HasTemplate" title="Template:HasTemplate">Template:HasTemplate</a> (<a href="/w/index.php?title=Template:HasTemplate&action=edit" title="Template:HasTemplate">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Ifexist_not_redirect" title="Template:Ifexist not redirect">Template:Ifexist not redirect</a> (<a href="/w/index.php?title=Template:Ifexist_not_redirect&action=edit" title="Template:Ifexist not redirect">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:LASTYEAR" title="Template:LASTYEAR">Template:LASTYEAR</a> (<a href="/w/index.php?title=Template:LASTYEAR&action=edit" title="Template:LASTYEAR">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Main_other" title="Template:Main other">Template:Main other</a> (<a href="/w/index.php?title=Template:Main_other&action=edit" title="Template:Main other">view source</a>) (protected)</li><li><a href="/wiki/Template:Math" title="Template:Math">Template:Math</a> (<a href="/w/index.php?title=Template:Math&action=edit" title="Template:Math">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Nowrap" title="Template:Nowrap">Template:Nowrap</a> (<a href="/w/index.php?title=Template:Nowrap&action=edit" title="Template:Nowrap">view source</a>) (protected)</li><li><a href="/wiki/Template:Nowrap_begin" title="Template:Nowrap begin">Template:Nowrap begin</a> (<a href="/w/index.php?title=Template:Nowrap_begin&action=edit" title="Template:Nowrap begin">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Nowrap_end" title="Template:Nowrap end">Template:Nowrap end</a> (<a href="/w/index.php?title=Template:Nowrap_end&action=edit" title="Template:Nowrap end">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Pagetype" title="Template:Pagetype">Template:Pagetype</a> (<a href="/w/index.php?title=Template:Pagetype&action=edit" title="Template:Pagetype">view source</a>) (protected)</li><li><a href="/wiki/Template:Ping" class="mw-redirect" title="Template:Ping">Template:Ping</a> (<a href="/w/index.php?title=Template:Ping&action=edit" class="mw-redirect" title="Template:Ping">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Portal" title="Template:Portal">Template:Portal</a> (<a href="/w/index.php?title=Template:Portal&action=edit" title="Template:Portal">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Reply_to" title="Template:Reply to">Template:Reply to</a> (<a href="/w/index.php?title=Template:Reply_to&action=edit" title="Template:Reply to">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Search_box" title="Template:Search box">Template:Search box</a> (<a href="/w/index.php?title=Template:Search_box&action=edit" title="Template:Search box">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Sfrac" title="Template:Sfrac">Template:Sfrac</a> (<a href="/w/index.php?title=Template:Sfrac&action=edit" title="Template:Sfrac">view source</a>) (extended confirmed protected)</li><li><a href="/wiki/Template:Sfrac/styles.css" title="Template:Sfrac/styles.css">Template:Sfrac/styles.css</a> (<a href="/w/index.php?title=Template:Sfrac/styles.css&action=edit" title="Template:Sfrac/styles.css">view source</a>) (extended confirmed protected)</li><li><a href="/wiki/Template:Strong" title="Template:Strong">Template:Strong</a> (<a href="/w/index.php?title=Template:Strong&action=edit" title="Template:Strong">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Talk_header" title="Template:Talk header">Template:Talk header</a> (<a href="/w/index.php?title=Template:Talk_header&action=edit" title="Template:Talk header">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Talk_header/archivebotparse" title="Template:Talk header/archivebotparse">Template:Talk header/archivebotparse</a> (<a href="/w/index.php?title=Template:Talk_header/archivebotparse&action=edit" title="Template:Talk header/archivebotparse">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Talk_header/styles.css" title="Template:Talk header/styles.css">Template:Talk header/styles.css</a> (<a href="/w/index.php?title=Template:Talk_header/styles.css&action=edit" title="Template:Talk header/styles.css">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Talk_quote_inline" title="Template:Talk quote inline">Template:Talk quote inline</a> (<a href="/w/index.php?title=Template:Talk_quote_inline&action=edit" title="Template:Talk quote inline">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Talk_quote_inline/styles.css" title="Template:Talk quote inline/styles.css">Template:Talk quote inline/styles.css</a> (<a href="/w/index.php?title=Template:Talk_quote_inline/styles.css&action=edit" title="Template:Talk quote inline/styles.css">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Th/abp" class="mw-redirect" title="Template:Th/abp">Template:Th/abp</a> (<a href="/w/index.php?title=Template:Th/abp&action=edit" class="mw-redirect" title="Template:Th/abp">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Tq" class="mw-redirect" title="Template:Tq">Template:Tq</a> (<a href="/w/index.php?title=Template:Tq&action=edit" class="mw-redirect" title="Template:Tq">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Trim" title="Template:Trim">Template:Trim</a> (<a href="/w/index.php?title=Template:Trim&action=edit" title="Template:Trim">view source</a>) (protected)</li><li><a href="/wiki/Template:Void" title="Template:Void">Template:Void</a> (<a href="/w/index.php?title=Template:Void&action=edit" title="Template:Void">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:WikiProject_Mathematics" title="Template:WikiProject Mathematics">Template:WikiProject Mathematics</a> (<a href="/w/index.php?title=Template:WikiProject_Mathematics&action=edit" title="Template:WikiProject Mathematics">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:WikiProject_banner_shell" title="Template:WikiProject banner shell">Template:WikiProject banner shell</a> (<a href="/w/index.php?title=Template:WikiProject_banner_shell&action=edit" title="Template:WikiProject banner shell">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Yearly_archive_list" title="Template:Yearly archive list">Template:Yearly archive list</a> (<a href="/w/index.php?title=Template:Yearly_archive_list&action=edit" title="Template:Yearly archive list">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Yearly_archive_list/display" title="Template:Yearly archive list/display">Template:Yearly archive list/display</a> (<a href="/w/index.php?title=Template:Yearly_archive_list/display&action=edit" title="Template:Yearly archive list/display">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Yesno" title="Template:Yesno">Template:Yesno</a> (<a href="/w/index.php?title=Template:Yesno&action=edit" title="Template:Yesno">view source</a>) (protected)</li><li><a href="/wiki/Template:Yesno-no" title="Template:Yesno-no">Template:Yesno-no</a> (<a href="/w/index.php?title=Template:Yesno-no&action=edit" title="Template:Yesno-no">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Archive_list" title="Module:Archive list">Module:Archive list</a> (<a href="/w/index.php?title=Module:Archive_list&action=edit" title="Module:Archive list">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Arguments" title="Module:Arguments">Module:Arguments</a> (<a href="/w/index.php?title=Module:Arguments&action=edit" title="Module:Arguments">view source</a>) (protected)</li><li><a href="/wiki/Module:Banner_shell" title="Module:Banner shell">Module:Banner shell</a> (<a href="/w/index.php?title=Module:Banner_shell&action=edit" title="Module:Banner shell">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Banner_shell/config" title="Module:Banner shell/config">Module:Banner shell/config</a> (<a href="/w/index.php?title=Module:Banner_shell/config&action=edit" title="Module:Banner shell/config">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Banner_shell/styles.css" title="Module:Banner shell/styles.css">Module:Banner shell/styles.css</a> (<a href="/w/index.php?title=Module:Banner_shell/styles.css&action=edit" title="Module:Banner shell/styles.css">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Check_for_deprecated_parameters" title="Module:Check for deprecated parameters">Module:Check for deprecated parameters</a> (<a href="/w/index.php?title=Module:Check_for_deprecated_parameters&action=edit" title="Module:Check for deprecated parameters">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Check_for_unknown_parameters" title="Module:Check for unknown parameters">Module:Check for unknown parameters</a> (<a href="/w/index.php?title=Module:Check_for_unknown_parameters&action=edit" title="Module:Check for unknown parameters">view source</a>) (protected)</li><li><a href="/wiki/Module:Disambiguation/templates" title="Module:Disambiguation/templates">Module:Disambiguation/templates</a> (<a href="/w/index.php?title=Module:Disambiguation/templates&action=edit" title="Module:Disambiguation/templates">view source</a>) (protected)</li><li><a href="/wiki/Module:Find_sources" title="Module:Find sources">Module:Find sources</a> (<a href="/w/index.php?title=Module:Find_sources&action=edit" title="Module:Find sources">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Find_sources/config" title="Module:Find sources/config">Module:Find sources/config</a> (<a href="/w/index.php?title=Module:Find_sources/config&action=edit" title="Module:Find sources/config">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Find_sources/links" title="Module:Find sources/links">Module:Find sources/links</a> (<a href="/w/index.php?title=Module:Find_sources/links&action=edit" title="Module:Find sources/links">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Find_sources/templates/Find_general_sources" title="Module:Find sources/templates/Find general sources">Module:Find sources/templates/Find general sources</a> (<a href="/w/index.php?title=Module:Find_sources/templates/Find_general_sources&action=edit" title="Module:Find sources/templates/Find general sources">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:For_loop" title="Module:For loop">Module:For loop</a> (<a href="/w/index.php?title=Module:For_loop&action=edit" title="Module:For loop">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:GetParameters" title="Module:GetParameters">Module:GetParameters</a> (<a href="/w/index.php?title=Module:GetParameters&action=edit" title="Module:GetParameters">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Message_box" title="Module:Message box">Module:Message box</a> (<a href="/w/index.php?title=Module:Message_box&action=edit" title="Module:Message box">view source</a>) (protected)</li><li><a href="/wiki/Module:Message_box/tmbox.css" title="Module:Message box/tmbox.css">Module:Message box/tmbox.css</a> (<a href="/w/index.php?title=Module:Message_box/tmbox.css&action=edit" title="Module:Message box/tmbox.css">view source</a>) (protected)</li><li><a href="/wiki/Module:MultiReplace" title="Module:MultiReplace">Module:MultiReplace</a> (<a href="/w/index.php?title=Module:MultiReplace&action=edit" title="Module:MultiReplace">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Pagetype" title="Module:Pagetype">Module:Pagetype</a> (<a href="/w/index.php?title=Module:Pagetype&action=edit" title="Module:Pagetype">view source</a>) (protected)</li><li><a href="/wiki/Module:Pagetype/config" title="Module:Pagetype/config">Module:Pagetype/config</a> (<a href="/w/index.php?title=Module:Pagetype/config&action=edit" title="Module:Pagetype/config">view source</a>) (protected)</li><li><a href="/wiki/Module:Pagetype/disambiguation" class="mw-redirect" title="Module:Pagetype/disambiguation">Module:Pagetype/disambiguation</a> (<a href="/w/index.php?title=Module:Pagetype/disambiguation&action=edit" class="mw-redirect" title="Module:Pagetype/disambiguation">view source</a>) (protected)</li><li><a href="/wiki/Module:Pagetype/rfd" title="Module:Pagetype/rfd">Module:Pagetype/rfd</a> (<a href="/w/index.php?title=Module:Pagetype/rfd&action=edit" title="Module:Pagetype/rfd">view source</a>) (protected)</li><li><a href="/wiki/Module:Pagetype/setindex" title="Module:Pagetype/setindex">Module:Pagetype/setindex</a> (<a href="/w/index.php?title=Module:Pagetype/setindex&action=edit" title="Module:Pagetype/setindex">view source</a>) (protected)</li><li><a href="/wiki/Module:Pagetype/softredirect" title="Module:Pagetype/softredirect">Module:Pagetype/softredirect</a> (<a href="/w/index.php?title=Module:Pagetype/softredirect&action=edit" title="Module:Pagetype/softredirect">view source</a>) (protected)</li><li><a href="/wiki/Module:Portal" title="Module:Portal">Module:Portal</a> (<a href="/w/index.php?title=Module:Portal&action=edit" title="Module:Portal">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Portal/images/m" title="Module:Portal/images/m">Module:Portal/images/m</a> (<a href="/w/index.php?title=Module:Portal/images/m&action=edit" title="Module:Portal/images/m">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Portal/styles.css" title="Module:Portal/styles.css">Module:Portal/styles.css</a> (<a href="/w/index.php?title=Module:Portal/styles.css&action=edit" title="Module:Portal/styles.css">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Reply_to" title="Module:Reply to">Module:Reply to</a> (<a href="/w/index.php?title=Module:Reply_to&action=edit" title="Module:Reply to">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Separated_entries" title="Module:Separated entries">Module:Separated entries</a> (<a href="/w/index.php?title=Module:Separated_entries&action=edit" title="Module:Separated entries">view source</a>) (protected)</li><li><a href="/wiki/Module:String" title="Module:String">Module:String</a> (<a href="/w/index.php?title=Module:String&action=edit" title="Module:String">view source</a>) (protected)</li><li><a href="/wiki/Module:String2" title="Module:String2">Module:String2</a> (<a href="/w/index.php?title=Module:String2&action=edit" title="Module:String2">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:TableTools" title="Module:TableTools">Module:TableTools</a> (<a href="/w/index.php?title=Module:TableTools&action=edit" title="Module:TableTools">view source</a>) (protected)</li><li><a href="/wiki/Module:Template_parameter_value" title="Module:Template parameter value">Module:Template parameter value</a> (<a href="/w/index.php?title=Module:Template_parameter_value&action=edit" title="Module:Template parameter value">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Unsubst" title="Module:Unsubst">Module:Unsubst</a> (<a href="/w/index.php?title=Module:Unsubst&action=edit" title="Module:Unsubst">view source</a>) (protected)</li><li><a href="/wiki/Module:WikiProject_banner" title="Module:WikiProject banner">Module:WikiProject banner</a> (<a href="/w/index.php?title=Module:WikiProject_banner&action=edit" title="Module:WikiProject banner">view source</a>) (protected)</li><li><a href="/wiki/Module:WikiProject_banner/config" title="Module:WikiProject banner/config">Module:WikiProject banner/config</a> (<a href="/w/index.php?title=Module:WikiProject_banner/config&action=edit" title="Module:WikiProject banner/config">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:WikiProject_banner/styles.css" title="Module:WikiProject banner/styles.css">Module:WikiProject banner/styles.css</a> (<a href="/w/index.php?title=Module:WikiProject_banner/styles.css&action=edit" title="Module:WikiProject banner/styles.css">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Wikitext_Parsing" title="Module:Wikitext Parsing">Module:Wikitext Parsing</a> (<a href="/w/index.php?title=Module:Wikitext_Parsing&action=edit" title="Module:Wikitext Parsing">view source</a>) (protected)</li><li><a href="/wiki/Module:Yesno" title="Module:Yesno">Module:Yesno</a> (<a href="/w/index.php?title=Module:Yesno&action=edit" title="Module:Yesno">view source</a>) (protected)</li></ul></div><p id="mw-returnto">Return to <a href="/wiki/Talk:Surreal_number" title="Talk:Surreal number">Talk:Surreal number</a>.</p> <noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/wiki/Talk:Surreal_number">https://en.wikipedia.org/wiki/Talk:Surreal_number</a>"</div></div> <div id="catlinks" class="catlinks catlinks-allhidden" data-mw="interface"></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Talk:Surreal_number&action=edit&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-7dfb9d98f5-fpq4m","wgBackendResponseTime":387,"wgPageParseReport":{"limitreport":{"cputime":"0.065","walltime":"0.089","ppvisitednodes":{"value":418,"limit":1000000},"postexpandincludesize":{"value":17744,"limit":2097152},"templateargumentsize":{"value":6556,"limit":2097152},"expansiondepth":{"value":9,"limit":100},"expensivefunctioncount":{"value":0,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":469,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 63.478 1 -total"," 99.86% 63.390 2 Template:Blocked_text"," 41.27% 26.196 1 Template:Colocationwebhost"," 40.32% 25.595 2 Template:Replace"," 37.49% 23.796 1 Template:Hidden"," 16.36% 10.385 1 Template:Hidden_begin"," 15.31% 9.720 1 Template:Tlx"," 2.09% 1.325 1 Template:Hidden_end"," 2.04% 1.298 1 MediaWiki:Wikimedia-globalblocking-blockedtext-mistake"," 1.78% 1.132 1 MediaWiki:Wikimedia-globalblocking-blockedtext-mistake-email-steward"]},"scribunto":{"limitreport-timeusage":{"value":"0.013","limit":"10.000"},"limitreport-memusage":{"value":1043303,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-7dfb9d98f5-fpq4m","timestamp":"20241125202615","ttl":2592000,"transientcontent":false}}});});</script> </body> </html>