[{"data":1,"prerenderedAt":26},["ShallowReactive",2],{"clay-page:\u002Fwriting\u002Fcomponents\u002Fmath":3},{"html":4,"toc":5,"description":22,"hero":23,"list":24,"socialImage":25},"\u003Ch1 id=\"math\">Math\u003C\u002Fh1>\n\u003Cp>Write formulas in LaTeX between dollar signs. Clay typesets them with \u003Ca href=\"https:\u002F\u002Fkatex.org\" target=\"_blank\" rel=\"noopener noreferrer\">\u003Ci class=\"fa-solid fa-arrow-up-right-from-square clay-external-icon\" aria-hidden=\"true\">\u003C\u002Fi>KaTeX\u003C\u002Fa> when it builds the site, so pages show them right away, without JavaScript.\u003C\u002Fp>\n\u003Ch2 id=\"inline-and-on-their-own-line\">\u003Ca class=\"heading-anchor\" href=\"#inline-and-on-their-own-line\" aria-label=\"Link to this section\" data-route=\"#inline-and-on-their-own-line\">#\u003C\u002Fa>Inline and on their own line\u003C\u002Fh2>\n\u003Cp>One dollar sign on each side puts a formula into the text:\u003C\u002Fp>\n\u003Cdiv class=\"code-block\">\u003Cpre class=\"shiki shiki-themes light-plus dark-plus\" style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4;--shiki-light-bg:#FFFFFF;--shiki-dark-bg:#1E1E1E\" tabindex=\"0\">\u003Ccode>\u003Cspan class=\"line\">\u003Cspan style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4\">The area of a circle is $A = \\pi r^2$.\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fcode>\u003C\u002Fpre>\u003Cbutton type=\"button\" class=\"copy-code btn btn-sm btn-square btn-ghost\" title=\"Copy code\" aria-label=\"Copy code\">\u003Ci class=\"fa-solid fa-copy\">\u003C\u002Fi>\u003C\u002Fbutton>\u003C\u002Fdiv>\n\u003Cp>The area of a circle is \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Cmrow>\u003Cmi>A\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>π\u003C\u002Fmi>\u003Cmsup>\u003Cmi>r\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsup>\u003C\u002Fmrow>A = \\pi r^2\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8141em\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em\">π\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em\">r\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em\">\u003Cspan class=\"pstrut\" style=\"height:2.7em\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>.\u003C\u002Fp>\n\u003Cp>Two dollar signs on lines of their own set a formula apart, centered:\u003C\u002Fp>\n\u003Cdiv class=\"code-block\">\u003Cpre class=\"shiki shiki-themes light-plus dark-plus\" style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4;--shiki-light-bg:#FFFFFF;--shiki-dark-bg:#1E1E1E\" tabindex=\"0\">\u003Ccode>\u003Cspan class=\"line\">\u003Cspan style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4\">$$\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4\">\\sum_{i=1}^{n} i = \\frac{n(n+1)}{2}\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4\">$$\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fcode>\u003C\u002Fpre>\u003Cbutton type=\"button\" class=\"copy-code btn btn-sm btn-square btn-ghost\" title=\"Copy code\" aria-label=\"Copy code\">\u003Ci class=\"fa-solid fa-copy\">\u003C\u002Fi>\u003C\u002Fbutton>\u003C\u002Fdiv>\n\u003Cdiv class=\"clay-math\">\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Cmrow>\u003Cmunderover>\u003Cmo>∑\u003C\u002Fmo>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmunderover>\u003Cmi>i\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmfrac>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmfrac>\u003C\u002Fmrow>\\sum_{i=1}^{n} i = \\frac{n(n+1)}{2}\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:2.9291em;vertical-align:-1.2777em\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.6514em\">\u003Cspan style=\"top:-1.8723em;margin-left:0em\">\u003Cspan class=\"pstrut\" style=\"height:3.05em\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">=\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.05em\">\u003Cspan class=\"pstrut\" style=\"height:3.05em\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop op-symbol large-op\">∑\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-4.3em;margin-left:0em\">\u003Cspan class=\"pstrut\" style=\"height:3.05em\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.2777em\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">i\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:2.113em;vertical-align:-0.686em\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.427em\">\u003Cspan style=\"top:-2.314em\">\u003Cspan class=\"pstrut\" style=\"height:3em\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em\">\u003Cspan class=\"pstrut\" style=\"height:3em\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.677em\">\u003Cspan class=\"pstrut\" style=\"height:3em\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.686em\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fdiv>\n\u003Ch2 id=\"more-examples\">\u003Ca class=\"heading-anchor\" href=\"#more-examples\" aria-label=\"Link to this section\" data-route=\"#more-examples\">#\u003C\u002Fa>More examples\u003C\u002Fh2>\n\u003Cdiv class=\"code-block\">\u003Cpre class=\"shiki shiki-themes light-plus dark-plus\" style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4;--shiki-light-bg:#FFFFFF;--shiki-dark-bg:#1E1E1E\" tabindex=\"0\">\u003Ccode>\u003Cspan class=\"line\">\u003Cspan style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4\">$$\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4\">\\begin{pmatrix} a & b \u003C\u002Fspan>\u003Cspan style=\"--shiki-light:#ae0000;--shiki-dark:#D7BA7D\">\\\\\u003C\u002Fspan>\u003Cspan style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4\"> c & d \\end{pmatrix}\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4\">\\begin{pmatrix} x \u003C\u002Fspan>\u003Cspan style=\"--shiki-light:#ae0000;--shiki-dark:#D7BA7D\">\\\\\u003C\u002Fspan>\u003Cspan style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4\"> y \\end{pmatrix}\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4\">=\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4\">\\begin{pmatrix} ax + by \u003C\u002Fspan>\u003Cspan style=\"--shiki-light:#ae0000;--shiki-dark:#D7BA7D\">\\\\\u003C\u002Fspan>\u003Cspan style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4\"> cx + dy \\end{pmatrix}\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4\">$$\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fcode>\u003C\u002Fpre>\u003Cbutton type=\"button\" class=\"copy-code btn btn-sm btn-square btn-ghost\" title=\"Copy code\" aria-label=\"Copy code\">\u003Ci class=\"fa-solid fa-copy\">\u003C\u002Fi>\u003C\u002Fbutton>\u003C\u002Fdiv>\n\u003Cdiv class=\"clay-math\">\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Cmrow>\u003Cmrow>\u003Cmo fence=\"true\">(\u003C\u002Fmo>\u003Cmtable rowspacing=\"0.16em\" columnalign=\"center center\" columnspacing=\"1em\">\u003Cmtr>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmi>a\u003C\u002Fmi>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmi>b\u003C\u002Fmi>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003C\u002Fmtr>\u003Cmtr>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmi>c\u003C\u002Fmi>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmi>d\u003C\u002Fmi>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003C\u002Fmtr>\u003C\u002Fmtable>\u003Cmo fence=\"true\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmo fence=\"true\">(\u003C\u002Fmo>\u003Cmtable rowspacing=\"0.16em\" columnalign=\"center\" columnspacing=\"1em\">\u003Cmtr>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003C\u002Fmtr>\u003Cmtr>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmi>y\u003C\u002Fmi>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003C\u002Fmtr>\u003C\u002Fmtable>\u003Cmo fence=\"true\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmo>=\u003C\u002Fmo>\u003Cmrow>\u003Cmo fence=\"true\">(\u003C\u002Fmo>\u003Cmtable rowspacing=\"0.16em\" columnalign=\"center\" columnspacing=\"1em\">\u003Cmtr>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003Cmi>x\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>b\u003C\u002Fmi>\u003Cmi>y\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003C\u002Fmtr>\u003Cmtr>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmrow>\u003Cmi>c\u003C\u002Fmi>\u003Cmi>x\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>d\u003C\u002Fmi>\u003Cmi>y\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003C\u002Fmtr>\u003C\u002Fmtable>\u003Cmo fence=\"true\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmrow>\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}\n\\begin{pmatrix} x \\\\ y \\end{pmatrix}\n=\n\\begin{pmatrix} ax + by \\\\ cx + dy \\end{pmatrix}\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:2.4em;vertical-align:-0.95em\">\u003C\u002Fspan>\u003Cspan class=\"minner\">\u003Cspan class=\"mopen delimcenter\" style=\"top:0em\">\u003Cspan class=\"delimsizing size3\">(\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mtable\">\u003Cspan class=\"col-align-c\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.45em\">\u003Cspan style=\"top:-3.61em\">\u003Cspan class=\"pstrut\" style=\"height:3em\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.41em\">\u003Cspan class=\"pstrut\" style=\"height:3em\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.95em\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"arraycolsep\" style=\"width:0.5em\">\u003C\u002Fspan>\u003Cspan class=\"arraycolsep\" style=\"width:0.5em\">\u003C\u002Fspan>\u003Cspan class=\"col-align-c\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.45em\">\u003Cspan style=\"top:-3.61em\">\u003Cspan class=\"pstrut\" style=\"height:3em\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.41em\">\u003Cspan class=\"pstrut\" style=\"height:3em\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.95em\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose delimcenter\" style=\"top:0em\">\u003Cspan class=\"delimsizing size3\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em\">\u003C\u002Fspan>\u003Cspan class=\"minner\">\u003Cspan class=\"mopen delimcenter\" style=\"top:0em\">\u003Cspan class=\"delimsizing size3\">(\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mtable\">\u003Cspan class=\"col-align-c\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.45em\">\u003Cspan style=\"top:-3.61em\">\u003Cspan class=\"pstrut\" style=\"height:3em\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.41em\">\u003Cspan class=\"pstrut\" style=\"height:3em\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em\">y\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.95em\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose delimcenter\" style=\"top:0em\">\u003Cspan class=\"delimsizing size3\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:2.4em;vertical-align:-0.95em\">\u003C\u002Fspan>\u003Cspan class=\"minner\">\u003Cspan class=\"mopen delimcenter\" style=\"top:0em\">\u003Cspan class=\"delimsizing size3\">(\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mtable\">\u003Cspan class=\"col-align-c\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.45em\">\u003Cspan style=\"top:-3.61em\">\u003Cspan class=\"pstrut\" style=\"height:3em\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em\">y\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.41em\">\u003Cspan class=\"pstrut\" style=\"height:3em\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em\">y\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.95em\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose delimcenter\" style=\"top:0em\">\u003Cspan class=\"delimsizing size3\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fdiv>\n\u003Cdiv class=\"code-block\">\u003Cpre class=\"shiki shiki-themes light-plus dark-plus\" style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4;--shiki-light-bg:#FFFFFF;--shiki-dark-bg:#1E1E1E\" tabindex=\"0\">\u003Ccode>\u003Cspan class=\"line\">\u003Cspan style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4\">$$\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4\">f(x) = \\begin{cases} x^2 & \\text{if } x \\ge 0 \u003C\u002Fspan>\u003Cspan style=\"--shiki-light:#ae0000;--shiki-dark:#D7BA7D\">\\\\\u003C\u002Fspan>\u003Cspan style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4\"> -x & \\text{otherwise} \\end{cases}\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"--shiki-light:#000000;--shiki-dark:#D4D4D4\">$$\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fcode>\u003C\u002Fpre>\u003Cbutton type=\"button\" class=\"copy-code btn btn-sm btn-square btn-ghost\" title=\"Copy code\" aria-label=\"Copy code\">\u003Ci class=\"fa-solid fa-copy\">\u003C\u002Fi>\u003C\u002Fbutton>\u003C\u002Fdiv>\n\u003Cdiv class=\"clay-math\">\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Cmrow>\u003Cmi>f\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmrow>\u003Cmo fence=\"true\">{\u003C\u002Fmo>\u003Cmtable rowspacing=\"0.36em\" columnalign=\"left left\" columnspacing=\"1em\">\u003Cmtr>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmsup>\u003Cmi>x\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsup>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmrow>\u003Cmtext>if \u003C\u002Fmtext>\u003Cmi>x\u003C\u002Fmi>\u003Cmo>≥\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003C\u002Fmtr>\u003Cmtr>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmrow>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmtext>otherwise\u003C\u002Fmtext>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003C\u002Fmtr>\u003C\u002Fmtable>\u003C\u002Fmrow>\u003C\u002Fmrow>f(x) = \\begin{cases} x^2 & \\text{if } x \\ge 0 \\\\ -x & \\text{otherwise} \\end{cases}\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em\">f\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:3em;vertical-align:-1.25em\">\u003C\u002Fspan>\u003Cspan class=\"minner\">\u003Cspan class=\"mopen delimcenter\" style=\"top:0em\">\u003Cspan class=\"delimsizing size4\">{\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mtable\">\u003Cspan class=\"col-align-l\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.69em\">\u003Cspan style=\"top:-3.69em\">\u003Cspan class=\"pstrut\" style=\"height:3.008em\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em\">\u003Cspan class=\"pstrut\" style=\"height:2.7em\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.25em\">\u003Cspan class=\"pstrut\" style=\"height:3.008em\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">−\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.19em\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"arraycolsep\" style=\"width:1em\">\u003C\u002Fspan>\u003Cspan class=\"col-align-l\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.69em\">\u003Cspan style=\"top:-3.69em\">\u003Cspan class=\"pstrut\" style=\"height:3.008em\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord text\">\u003Cspan class=\"mord\">if \u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.25em\">\u003Cspan class=\"pstrut\" style=\"height:3.008em\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord text\">\u003Cspan class=\"mord\">otherwise\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.19em\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fdiv>\n\u003Cp>KaTeX lists every command it knows in \u003Ca href=\"https:\u002F\u002Fkatex.org\u002Fdocs\u002Fsupported.html\" target=\"_blank\" rel=\"noopener noreferrer\">\u003Ci class=\"fa-solid fa-arrow-up-right-from-square clay-external-icon\" aria-hidden=\"true\">\u003C\u002Fi>its documentation\u003C\u002Fa>.\u003C\u002Fp>\n\u003Ch2 id=\"dollar-signs-in-text\">\u003Ca class=\"heading-anchor\" href=\"#dollar-signs-in-text\" aria-label=\"Link to this section\" data-route=\"#dollar-signs-in-text\">#\u003C\u002Fa>Dollar signs in text\u003C\u002Fh2>\n\u003Cp>Prices stay text. A dollar sign only starts a formula when no space follows it, and only ends one when no space comes before it and no digit after it: “It costs $5 and $10” is text. To write a dollar sign that would start a formula, put a backslash before it: \u003Ccode>\\$\u003C\u002Fcode>.\u003C\u002Fp>\n\u003Ch2 id=\"mistakes\">\u003Ca class=\"heading-anchor\" href=\"#mistakes\" aria-label=\"Link to this section\" data-route=\"#mistakes\">#\u003C\u002Fa>Mistakes\u003C\u002Fh2>\n\u003Cp>A formula KaTeX cannot read is shown in red with its source, instead of failing the build, so you see it on the page: \u003Cspan class=\"katex-error\" title=\"ParseError: KaTeX parse error: Unexpected end of input in a macro argument, expected '}' at end of input: \\frac{1\">\\frac{1\u003C\u002Fspan>.\u003C\u002Fp>\n\u003Ch2 id=\"accessibility-and-size\">\u003Ca class=\"heading-anchor\" href=\"#accessibility-and-size\" aria-label=\"Link to this section\" data-route=\"#accessibility-and-size\">#\u003C\u002Fa>Accessibility and size\u003C\u002Fh2>\n\u003Cul>\n\u003Cli>Every formula is also written as MathML, which screen readers read out.\u003C\u002Fli>\n\u003Cli>Wide formulas scroll sideways on small screens instead of widening the page.\u003C\u002Fli>\n\u003Cli>The fonts for formulas only load on pages that have math.\u003C\u002Fli>\n\u003C\u002Ful>\n",[6,10,13,16,19],{"id":7,"text":8,"level":9},"inline-and-on-their-own-line","Inline and on their own line",2,{"id":11,"text":12,"level":9},"more-examples","More examples",{"id":14,"text":15,"level":9},"dollar-signs-in-text","Dollar signs in text",{"id":17,"text":18,"level":9},"mistakes","Mistakes",{"id":20,"text":21,"level":9},"accessibility-and-size","Accessibility and size","Formulas in LaTeX, inline or on a line of their own, typeset with KaTeX when the site is built.",null,false,"\u002F_clay\u002Fsocial\u002Fwriting\u002Fcomponents\u002Fmath.png",1792666370544]