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< script defer src = /congo/lib/katex/auto-render.min.640fd13eb028f3c0e5c119e7a0d56a28836740d716111cd12b4dcf0da038188262559bc61485906dd582b1479b3197e45a88677e782c0e35f353a3e890ee231b.js integrity = "sha512-ZA/RPrAo88DlwRnnoNVqKINnQNcWERzRK03PDaA4GIJiVZvGFIWQbdWCsUebMZfkWohnfngsDjXzU6PokO4jGw==" onload = renderMathInElement(document.body) > < / script > < / head > < body class = "flex flex-col h-screen px-6 m-auto text-lg leading-7 bg-neutral text-neutral-900 sm:px-14 md:px-24 lg:px-32 dark:bg-neutral-800 dark:text-neutral max-w-7xl" > < div id = the-top class = "absolute flex self-center" > < a class = "px-3 py-1 text-sm -translate-y-8 rounded-b-lg bg-primary-200 dark:bg-neutral-600 focus:translate-y-0" href = #main-content > < span class = "font-bold ltr:pr-2 rtl:pl-2 text-primary-600 dark:text-primary-400" > ↓ < / span > Skip to main content< / a > < / div > < header class = "flex justify-between py-6 font-semibold sm:items-center sm:py-10 text-neutral-900 dark:text-neutral print:hidden" > < div > < a class = "decoration-primary-500 hover:underline hover:decoration-2 hover:underline-offset-2" rel = me href = /congo/ > Congo< / a > < / div > < nav > < ul class = "flex flex-col list-none sm:flex-row" > < li class = "mb-1 ltr:text-right rtl:text-left sm:mb-0 ltr:sm:mr-7 ltr:sm:last:mr-0 rtl:sm:ml-7 rtl:sm:last:ml-0" > < a class = "decoration-primary-500 hover:underline hover:decoration-2 hover:underline-offset-2" href = /congo/docs/ title = Documentation > Docs< / a > < / li > < li class = "mb-1 ltr:text-right rtl:text-left sm:mb-0 ltr:sm:mr-7 ltr:sm:last:mr-0 rtl:sm:ml-7 rtl:sm:last:ml-0" > < a class = "decoration-primary-500 hover:underline hover:decoration-2 hover:underline-offset-2" href = /congo/samples/ title = "Content Samples" > Samples< / a > < / li > < li class = "mb-1 ltr:text-right rtl:text-left sm:mb-0 ltr:sm:mr-7 ltr:sm:last:mr-0 rtl:sm:ml-7 rtl:sm:last:ml-0" > < a class = "decoration-primary-500 hover:underline hover:decoration-2 hover:underline-offset-2" href = /congo/users/ title = Users > Users< / a > < / li > < li class = "mb-1 ltr:text-right rtl:text-left sm:mb-0 ltr:sm:mr-7 ltr:sm:last:mr-0 rtl:sm:ml-7 rtl:sm:last:ml-0" > < a class = "decoration-primary-500 hover:underline hover:decoration-2 hover:underline-offset-2" href = https://github.com/jpanther/congo title > GitHub< / a > < / li > < li class = "ltr:text-right rtl:text-left ltr:sm:mr-7 ltr:sm:last:mr-0 rtl:sm:ml-7 rtl:sm:last:ml-0" > < button id = search-button class = "text-base hover:text-primary-600 dark:hover:text-primary-400" title = "Search (/)" >
< span class = "relative inline-block align-text-bottom icon" > < svg aria-hidden = "true" focusable = "false" data-prefix = "fas" data-icon = "search" class = "svg-inline--fa fa-search fa-w-16" role = "img" xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 512 512" > < path fill = "currentcolor" d = "M505 442.7 405.3 343c-4.5-4.5-10.6-7-17-7H372c27.6-35.3 44-79.7 44-128C416 93.1 322.9.0 208 0S0 93.1.0 208s93.1 208 208 208c48.3.0 92.7-16.4 128-44v16.3c0 6.4 2.5 12.5 7 17l99.7 99.7c9.4 9.4 24.6 9.4 33.9.0l28.3-28.3c9.4-9.4 9.4-24.6.1-34zM208 336c-70.7.0-128-57.2-128-128 0-70.7 57.2-128 128-128 70.7.0 128 57.2 128 128 0 70.7-57.2 128-128 128z" / > < / svg > < / span > < / button > < / li > < / ul > < / nav > < / header > < main id = main-content class = "relative grow" > < article > < header class = max-w-prose > < ol class = "text-sm text-neutral-500 dark:text-neutral-400 print:hidden" > < li class = "inline hidden" > < a class = "hover:underline decoration-neutral-300 dark:underline-neutral-600" href = /congo/ > Welcome to Congo! :tada:< / a > < span class = "px-1 text-primary-500" > /< / span > < / li > < li class = inline > < a class = "hover:underline decoration-neutral-300 dark:underline-neutral-600" href = /congo/samples/ > Content Samples< / a > < span class = "px-1 text-primary-500" > /< / span > < / li > < li class = "inline hidden" > < a class = "hover:underline decoration-neutral-300 dark:underline-neutral-600" href = /congo/samples/mathematical-notation/ > Mathematical notation< / a > < span class = "px-1 text-primary-500" > /< / span > < / li > < / ol > < h1 class = "mt-0 text-4xl font-extrabold text-neutral-900 dark:text-neutral" > Mathematical notation< / h1 > < div class = "mt-8 mb-12 text-base text-neutral-500 dark:text-neutral-400 print:hidden" > < div class = "flex flex-row flex-wrap items-center" > < time datetime = "2019-03-08 00:00:00 +0000 UTC" > 8 March 2019< / time > < span class = "px-2 text-primary-500" > · < / span > < span title = "Reading time" > 1 min< / span > < / div > < / div > < / header > < section class = "flex flex-col max-w-full mt-0 prose lg:flex-row dark:prose-invert" > < div class = "order-first px-0 lg:max-w-xs ltr:lg:pl-8 rtl:lg:pr-8 lg:order-last" > < div class = "ltr:pl-5 rtl:pr-5 toc lg:sticky lg:top-10 print:hidden" > < details open class = "mt-0 overflow-hidden rounded-lg rtl:pr-5 ltr:pl-5 ltr:-ml-5 rtl:-mr-5 lg:mt-3" > < summary class = "block py-1 text-lg font-semibold cursor-pointer rtl:pr-5 ltr:pl-5 ltr:-ml-5 rtl:-mr-5 text-neutral-800 dark:text-neutral-100 lg:hidden bg-neutral-100 dark:bg-neutral-700" > Table of Contents< / summary > < div class = "py-2 border-dotted ltr:border-l rtl:border-r rtl:pr-5 ltr:pl-5 ltr:-ml-5 rtl:-mr-5 border-neutral-300 dark:border-neutral-600" > < nav id = TableOfContents > < ul > < li > < a href = #inline-notation > Inline notation< / a > < / li > < li > < a href = #block-notation > Block notation< / a > < / li > < / ul > < / nav > < / div > < / details > < / div > < / div > < div class = "min-w-0 min-h-0 max-w-prose" > < p > KaTeX can be used to render mathematical notation within articles.< / p > < p > Congo will only bundle the KaTeX assets into your project if you make use of mathematical notation. In order for this to work, simply include the < a href = https://jpanther.github.io/congo/docs/shortcodes/#katex > < code > katex< / code > shortcode< / a > within the article. Any KaTeX syntax on that page will then be automatically rendered.< / p > < p > Use the online reference of < a href = https://katex.org/docs/supported.html > supported TeX functions< / a > for the available syntax.< / p > < h2 id = inline-notation class = "relative group" > Inline notation < span class = "absolute top-0 w-6 transition-opacity opacity-0 ltr:-left-6 rtl:-right-6 not-prose group-hover:opacity-100" > < a class = "group-hover:text-primary-300 dark:group-hover:text-neutral-700" style = text-decoration-line:none!important href = #inline-notation aria-label = Anchor > #< / a > < / span > < / h2 > < p > Inline notation can be generated by wrapping the expression in < code > \\(< / code > and < code > \\)< / code > delimiters.< / p > < p > < strong > Example:< / strong > < / p > < div class = highlight > < pre tabindex = 0 class = chroma > < code class = language-tex data-lang = tex > < span class = line > < span class = cl > < span class = c > % KaTeX inline notation
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< / span > < / span > < / span > < span class = line > < span class = cl > < span class = c > < / span > Inline notation: < span class = k > \\< / span > (< span class = k > \varphi< / span > = < span class = k > \dfrac< / span > < span class = nb > {< / span > 1+< span class = k > \sqrt< / span > 5< span class = nb > }{< / span > 2< span class = nb > }< / span > = 1.6180339887…< span class = k > \\< / span > )
< / span > < / span > < / code > < / pre > < / div > < p > Inline notation: \(\varphi = \dfrac{1+\sqrt5}{2}= 1.6180339887…\)< / p > < h2 id = block-notation class = "relative group" > Block notation < span class = "absolute top-0 w-6 transition-opacity opacity-0 ltr:-left-6 rtl:-right-6 not-prose group-hover:opacity-100" > < a class = "group-hover:text-primary-300 dark:group-hover:text-neutral-700" style = text-decoration-line:none!important href = #block-notation aria-label = Anchor > #< / a > < / span > < / h2 > < p > Alternatively, block notation can be generated using < code > $$< / code > delimiters. This will output the expression in its own HTML block.< / p > < p > < strong > Example:< / strong > < / p > < div class = highlight > < pre tabindex = 0 class = chroma > < code class = language-tex data-lang = tex > < span class = line > < span class = cl > < span class = c > % KaTeX block notation
< / span > < / span > < / span > < span class = line > < span class = cl > < span class = c > < / span > < span class = sb > $$< / span > < span class = nb >
< / span > < / span > < / span > < span class = line > < span class = cl > < span class = nb > < / span > < span class = nv > \varphi< / span > < span class = nb > < / span > < span class = o > =< / span > < span class = nb > < / span > < span class = m > 1< / span > < span class = o > +< / span > < span class = nv > \frac< / span > < span class = nb > {< / span > < span class = m > 1< / span > < span class = nb > } {< / span > < span class = m > 1< / span > < span class = o > +< / span > < span class = nv > \frac< / span > < span class = nb > {< / span > < span class = m > 1< / span > < span class = nb > } {< / span > < span class = m > 1< / span > < span class = o > +< / span > < span class = nv > \frac< / span > < span class = nb > {< / span > < span class = m > 1< / span > < span class = nb > } {< / span > < span class = m > 1< / span > < span class = o > +< / span > < span class = nv > \cdots< / span > < span class = nb > } } }
< / span > < / span > < / span > < span class = line > < span class = cl > < span class = nb > < / span > < span class = s > $$< / span >
< / span > < / span > < / code > < / pre > < / div > < p > $$
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\varphi = 1+\frac{1} {1+\frac{1} {1+\frac{1} {1+\cdots} } }
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$$< / p > < / div > < / section > < footer class = "pt-8 max-w-prose print:hidden" > < div class = flex > < img class = "w-24 h-24 !mt-0 !mb-0 ltr:mr-4 rtl:ml-4 rounded-full" width = 96 height = 96 alt = Author src = /congo/img/author_hu052d24d60828c1cca42a6aefe45b8b44_54195_192x192_fill_q75_box_smart1.jpg > < div class = place-self-center > < div class = "text-[0.6rem] leading-3 text-neutral-500 dark:text-neutral-400 uppercase" > Author< / div > < div class = "font-semibold leading-6 text-neutral-800 dark:text-neutral-300" > Congo< / div > < div class = "text-sm text-neutral-700 dark:text-neutral-400" > This is an example author bio, and although there’ s a stock photo of a dog here, this article was actually created by a human. 🐶< / div > < div class = "text-2xl sm:text-lg" > < div class = "flex flex-wrap text-neutral-400 dark:text-neutral-500" > < a class = "px-1 hover:text-primary-700 dark:hover:text-primary-400" href = https://twitter.com/ target = _blank aria-label = Twitter rel = "me noopener noreferrer" > < span class = "relative inline-block align-text-bottom icon" > < svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 512 512" > < path fill = "currentcolor" d = "M459.37 151.716c.325 4.548.325 9.097.325 13.645.0 138.72-105.583 298.558-298.558 298.558-59.452.0-114.68-17.219-161.137-47.106 8.447.974 16.568 1.299 25.34 1.299 49.055.0 94.213-16.568 130.274-44.832-46.132-.975-84.792-31.188-98.112-72.772 6.498.974 12.995 1.624 19.818 1.624 9.421.0 18.843-1.3 27.614-3.573-48.081-9.747-84.143-51.98-84.143-102.985v-1.299c13.969 7.797 30.214 12.67 47.431 13.319-28.264-18.843-46.781-51.005-46.781-87.391.0-19.492 5.197-37.36 14.294-52.954 51.655 63.675 129.3 105.258 216.365 109.807-1.624-7.797-2.599-15.918-2.599-24.04.0-57.828 46.782-104.934 104.934-104.934 30.213.0 57.502 12.67 76.67 33.137 23.715-4.548 46.456-13.32 66.599-25.34-7.798 24.366-24.366 44.833-46.132 57.827 21.117-2.273 41.584-8.122 60.426-16.243-14.292 20.791-32.161 39.308-52.628 54.253z" / > < / svg > < / span > < / a > < a class = "px-1 hover:text-primary-700 dark:hover:text-primary-400" href = https://facebook.com/ target = _blank aria-label = Facebook rel = "me noopener noreferrer" > < span class = "relative inline-block align-text-bottom icon" > < svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 512 512" > < path fill = "currentcolor" d = "M504 256C504 119 393 8 256 8S8 119 8 256c0 123.78 90.69 226.38 209.25 245V327.69h-63V256h63v-54.64c0-62.15 37-96.48 93.67-96.48 27.14.0 55.52 4.84 55.52 4.84v61h-31.28c-30.8.0-40.41 19.12-40.41 38.73V256h68.78l-11 71.69h-57.78V501C413.31 482.38 504 379.78 504 256z" / > < / svg > < / span > < / a > < a class = "px-1 hover:text-primary-700 dark:hover:text-primary-400" href = https://linkedin.com/ target = _blank aria-label = Linkedin rel = "me noopener noreferrer" > < span class = "relative inline-block align-text-bottom icon" > < svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 448 512" > < path fill = "currentcolor" d = "M416 32H31.9C14.3 32 0 46.5.0 64.3v383.4C0 465.5 14.3 480 31.9 480H416c17.6.0 32-14.5 32-32.3V64.3c0-17.8-14.4-32.3-32-32.3zM135.4 416H69V202.2h66.5V416zm-33.2-243c-21.3.0-38.5-17.3-38.5-38.5S80.9 96 102.2 96c21.2.0 38.5 17.3 38.5 38.5.0 21.3-17.2 38.5-38.5 38.5zm282.1 243h-66.4V312c0-24.8-.5-56.7-34.5-56.7-34.6.0-39.9 27-39.9 54.9V416h-66.4V202.2h63.7v29.2h.9c8.9-16.8 30.6-34.5 62.9-34.5 67.2.0 79.7 44.3 79.7 101.9V416z" / > < / svg > < / span > < / a > < a class = "px-1 hover:text-primary-700 dark:hover:text-primary-400" href = https://youtube.com/ target = _blank aria-label = Youtube rel = "me noopener noreferrer" > < span class = "relative inline-block align-text-bottom icon" > < svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 576 512" > < path fill = "currentcolor" d = "M549.655 124.083c-6.281-23.65-24.787-42.276-48.284-48.597C458.781 64 288 64 288 64S117.22 64 74.629 75.486c-23.497 6.322-42.003 24.947-48.284 48.597-11.412 42.867-11.412 132.305-11.412 132.305s0 89.438 11.412 132.305c6.281 23.65 24.787 41.5 48.284 47.821C117.22 448 288 448 288 448s170.78.0 213.371-11.486c23.497-6.321 42.003-24.171 48.284-47.821 11.412-42.867 11.412-132.305 11.412-132.305s0-89.438-11.412-132.305zm-317.51 213.508V175.185l142.739 81.205-142.739 81.201z" / > < / svg > < / span > < / a > < / div > < / div > < / div > < / div > < div class = pt-8 > < hr class =
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< span class = "ml-3 ltr:hidden rtl:inline text-neutral-700 dark:text-neutral group-hover:text-primary-600 dark:group-hover:text-primary-400" > → < / span >
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< span class = "flex flex-col" > < span class = "mt-[0.1rem] leading-6 group-hover:underline group-hover:decoration-primary-500" > Charts< / span >
< span class = "mt-[0.1rem] text-xs text-neutral-500 dark:text-neutral-400" > < time datetime = "2019-03-06 00:00:00 +0000 UTC" > 6 March 2019< / time > < / span > < / span > < / a > < / span >
< span > < a class = "flex text-right group" href = /congo/samples/placeholder-text/ > < span class = "flex flex-col" > < span class = "mt-[0.1rem] leading-6 group-hover:underline group-hover:decoration-primary-500" > Placeholder Text< / span >
< span class = "mt-[0.1rem] text-xs text-neutral-500 dark:text-neutral-400" > < time datetime = "2019-03-09 00:00:00 +0000 UTC" > 9 March 2019< / time > < / span > < / span >
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< span class = "ml-3 ltr:inline rtl:hidden text-neutral-700 dark:text-neutral group-hover:text-primary-600 dark:group-hover:text-primary-400" > → < / span >
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< span class = "mr-3 ltr:hidden rtl:inline text-neutral-700 dark:text-neutral group-hover:text-primary-600 dark:group-hover:text-primary-400" > ← < / span > < / a > < / span > < / div > < / div > < / footer > < / article > < / main > < div id = search-wrapper class = "fixed inset-0 z-50 flex flex-col p-4 sm:p-6 md:p-[10vh] lg:p-[12vh] w-screen h-screen cursor-default bg-neutral-500/50 backdrop-blur-sm dark:bg-neutral-900/50 invisible" data-url = https://jpanther.github.io/congo/ > < div id = search-modal class = "flex flex-col w-full max-w-3xl min-h-0 mx-auto border rounded-md shadow-lg border-neutral-200 top-20 bg-neutral dark:bg-neutral-800 dark:border-neutral-700" > < header class = "relative z-10 flex items-center justify-between flex-none px-2" > < form class = "flex items-center flex-auto min-w-0" > < div class = "flex items-center justify-center w-8 h-8 text-neutral-400" > < span class = "relative inline-block align-text-bottom icon" > < svg aria-hidden = "true" focusable = "false" data-prefix = "fas" data-icon = "search" class = "svg-inline--fa fa-search fa-w-16" role = "img" xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 512 512" > < path fill = "currentcolor" d = "M505 442.7 405.3 343c-4.5-4.5-10.6-7-17-7H372c27.6-35.3 44-79.7 44-128C416 93.1 322.9.0 208 0S0 93.1.0 208s93.1 208 208 208c48.3.0 92.7-16.4 128-44v16.3c0 6.4 2.5 12.5 7 17l99.7 99.7c9.4 9.4 24.6 9.4 33.9.0l28.3-28.3c9.4-9.4 9.4-24.6.1-34zM208 336c-70.7.0-128-57.2-128-128 0-70.7 57.2-128 128-128 70.7.0 128 57.2 128 128 0 70.7-57.2 128-128 128z" / > < / svg > < / span > < / div > < input type = search id = search-query class = "flex flex-auto h-12 mx-1 bg-transparent appearance-none focus:outline-dotted focus:outline-transparent focus:outline-2" placeholder = Search tabindex = 0 > < / form > < button id = close-search-button class = "flex items-center justify-center w-8 h-8 text-neutral-700 dark:text-neutral hover:text-primary-600 dark:hover:text-primary-400" title = "Close (Esc)" >
2022-04-03 05:33:00 +00:00
< span class = "relative inline-block align-text-bottom icon" > < svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 320 512" > < path fill = "currentcolor" d = "M310.6 361.4c12.5 12.5 12.5 32.75.0 45.25C304.4 412.9 296.2 416 288 416s-16.38-3.125-22.62-9.375L160 301.3 54.63 406.6C48.38 412.9 40.19 416 32 416S15.63 412.9 9.375 406.6c-12.5-12.5-12.5-32.75.0-45.25l105.4-105.4L9.375 150.6c-12.5-12.5-12.5-32.75.0-45.25s32.75-12.5 45.25.0L160 210.8l105.4-105.4c12.5-12.5 32.75-12.5 45.25.0s12.5 32.75.0 45.25l-105.4 105.4L310.6 361.4z" / > < / svg > < / span > < / button > < / header > < section class = "flex-auto px-2 overflow-auto" > < ul id = search-results > < / ul > < / section > < / div > < / div > < footer class = "py-10 print:hidden" > < div class = "flex justify-between" > < div > < p class = "text-sm text-neutral-500 dark:text-neutral-400" > © 2022 Congo contributors< / p > < p class = "text-xs text-neutral-500 dark:text-neutral-400" > Powered by < a class = "hover:underline hover:decoration-primary-400 hover:text-primary-500" href = https://gohugo.io/ target = _blank rel = "noopener noreferrer" > Hugo< / a > & < a class = "hover:underline hover:decoration-primary-400 hover:text-primary-500" href = https://git.io/hugo-congo target = _blank rel = "noopener noreferrer" > Congo< / a > < / p > < / div > < div class = "text-sm cursor-pointer text-neutral-700 dark:text-neutral hover:text-primary-600 dark:hover:text-primary-400 ltr:mr-14 rtl:ml-14" > < button id = appearance-switcher class = "w-12 h-12" type = button title = "Switch to dark appearance" >
2022-03-13 23:47:01 +00:00
< span class = "inline dark:hidden" > < span class = "relative inline-block align-text-bottom icon" > < svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 512 512" > < path fill = "currentcolor" d = "M32 256C32 132.2 132.3 32 255.8 32c11.36.0 29.7 1.668 40.9 3.746 9.616 1.777 11.75 14.63 3.279 19.44C245 86.5 211.2 144.6 211.2 207.8c0 109.7 99.71 193 208.3 172.3 9.561-1.805 16.28 9.324 10.11 16.95C387.9 448.6 324.8 480 255.8 480 132.1 480 32 379.6 32 256z" / > < / svg > < / span > < / span > < span class = "hidden dark:inline" > < span class = "relative inline-block align-text-bottom icon" > < svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 512 512" > < path fill = "currentcolor" d = "M256 159.1c-53.02.0-95.1 42.98-95.1 95.1s41.2 96.9 95.1 96.9 95.1-42.98 95.1-95.1S309 159.1 256 159.1zM509.3 347l-63.2-91.9 63.15-91.01c6.332-9.125 1.104-21.74-9.826-23.72l-109-19.7-19.7-109c-1.975-10.93-14.59-16.16-23.72-9.824L256 65.89 164.1 2.736c-9.125-6.332-21.74-1.107-23.72 9.824L121.6 121.6 12.56 141.3C1.633 143.2-3.596 155.9 2.736 164.1L65.89 256 2.74 347.01c-6.332 9.125-1.105 21.74 9.824 23.72l109 19.7 19.7 109c1.975 10.93 14.59 16.16 23.72 9.824L256 446.1l91.01 63.15c9.127 6.334 21.75 1.107 23.72-9.822l19.7-109 109-19.7C510.4 368.8 515.6 356.1 509.3 347zM256 383.1c-70.69.0-127.1-57.31-127.1-127.1.0-70.69 57.31-127.1 127.1-127.1S383.1 186.2 383.1 256c0 70.7-56.4 127.1-127.1 127.1z" / > < / svg > < / span > < / span > < / button > < / div > < / div > < / footer > < / body > < / html >
