MathJax basic tutorial and quick reference

Question

(Deutsch: MathJax: LaTeX Basic Tutorial und Referenz)

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1. For inline formulas, enclose the formula in $...$. For displayed formulas, use $$...$$.

   • These render differently. For example, type the following to show inline mode:
     $\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$
     $\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$
   • or type the following for display mode:
     $$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$
     $$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$

2. For Greek letters, use \alpha, \beta, ..., \omega: $\alpha$, $\beta$, ..., $\omega$.

   • For uppercase letters, use \Gamma, \Delta, ..., \Omega: $\Gamma$, $\Delta$, ..., $\Omega$.
   • Other Greek capital letters are the same as the Latin ones: A,B,E,Z and so on: $A, B, E, Z$....
   • Some Greek letters have variant forms: \epsilon \varepsilon $\epsilon$, $\varepsilon$, \phi \varphi $\phi$, $\varphi$, and others.

3. For superscripts and subscripts, use ^ and _. For example, x_i^2: $x_i^2$, \log_2 x: $\log_2 x$. For the prime symbol, use an apostrophe x' x'' x''': $x'\ x''\ x'''$.

4. Groups. Superscripts, subscripts, and other operations apply only to the next "group". A "group" is either a single symbol, or any formula surrounded by curly braces {...}.

   • If you do 10^10, you will get a surprise: 10ドル^10$. But 10^{10} gives what you probably wanted: 10ドル^{10}$.
   • Use curly braces to delimit a formula to which a superscript or subscript applies: x^y^z is an error; {x^y}^z is ${x^y}^z$, and x^{y^z} is $x^{y^z}$. Observe the differences between x_i^2 $x_i^2$, x_{i^2} $x_{i^2}$ and {x_i}^2 ${x_i}^2$.

5. Parentheses Ordinary symbols ()[] make parentheses and brackets $(2+3)[4+4]$. Use \{ and \} for curly braces $\{\}$.

   • These do not scale with the formula in between, so if you write (\frac{\sqrt x}{y^3}) the parentheses will be too small: $(\frac{\sqrt x}{y^3})$. Using \left(...\right) will make the sizes adjust automatically to the formula they enclose: \left(\frac{\sqrt x}{y^3}\right) is $\left(\frac{\sqrt x}{y^3}\right)$.

   • \left and\right apply to all the following sorts of parentheses: ( and ) $(x)$, [ and ] $[x]$, \{ and \} $\{ x \}$, | $|x|$, \vert $\vert x \vert$, \Vert $\Vert x \Vert$, \langle and \rangle $\langle x \rangle$, \lceil and \rceil $\lceil x \rceil$, and \lfloor and \rfloor $\lfloor x \rfloor$. \middle can be used to add additional dividers. There are also invisible parentheses, denoted by .: use \left.x^2\right\rvert_3^5 = 5^2-3^2 to get $$\left.x^2\right\rvert_3^5 = 5^2-3^2$$

6. Sums and integrals \sum and \int; the subscript is the lower limit and the superscript is the upper limit, so for example \sum_1^n $\sum_1^n$. Don't forget {...} if the limits are more than a single symbol. For example, \sum_{i=0}^\infty i^2 is $\sum_{i=0}^\infty i^2$.

   • Similarly, \prod $\prod$, \int $\int$, \bigcup $\bigcup$, \bigcap $\bigcap$, \iint $\iint$, \iiint $\iiint$, \idotsint $\idotsint$.

7. Fractions There are three ways to make fractions. \frac ab applies to the next two groups, and produces $\frac ab$; for more complicated numerators and denominators use {...}: \frac{a+1}{b+1} is $\frac{a+1}{b+1}$.

   • If the numerator and denominator are complicated, you may prefer \over, which splits up the group that it is in: {a+1\over b+1} is ${a+1\over b+1}$.
   • For continued fractions, use \cfrac instead of \frac.

8. Fonts

• Use \mathbb or \Bbb for "blackboard bold": $\mathbb{CHNQRZ}$.
• Use \mathbf for boldface: $\mathbf{CHNQRZ}$ $\mathbf{chnqrz}$.
  • For expression based characters, use \boldsymbol instead: $\boldsymbol{\alpha}$
• Use \mathit for italics: $\mathit{CHNQRZ}$ $\mathit{chnqrz}$.
• Use \pmb for boldfaced italics: $\pmb{CHNQRZ}$ $\pmb{chnqrz}$.
• Use \mathtt for "typewriter" font: $\mathtt{CHNQRZ}$ $\mathtt{chnqrz}$.
• Use \mathrm for roman font: $\mathrm{CHNQRZ}$ $\mathrm{chnqrz}$.
• Use \mathsf for sans-serif font: $\mathsf{CHNQRZ}$ $\mathsf{chnqrz}$.
• Use \mathcal for "calligraphic" letters: $\mathcal{CHNQRZ}$ (Uppercase only.)
• Use \mathscr for script letters: $\mathscr{CHNQRZ}$ $\mathscr{chnqrz}$
• Use \mathfrak for "Fraktur" (old German style) letters: $\mathfrak{CHNQRZ}$ $\mathfrak{chnqrz}$.

9. Radical signs / roots Use sqrt, which adjusts to the size of its argument: \sqrt{x^3} $\sqrt{x^3}$; \sqrt[3]{\frac xy} $\sqrt[3]{\frac xy}$. For complicated expressions, consider using {...}^{1/2} instead.

10. Some special functions such as "lim", "sin", "max", "ln", and so on are normally set in roman font instead of italic font. Use \lim, \sin, etc. to make these: \sin x $\sin x$, not sin x $sin x$. Use subscripts to attach a notation to \lim: \lim_{x\to 0} $$\lim_{x\to 0}$$ Nonstandard function names can be set with \operatorname{foo}(x) $\operatorname{foo}(x)$.

11. There are a very large number of special symbols and notations, too many to list here; see the short listing $\LaTeX$ and $\mathcal{A}_{\Large\mathcal{M}}\mathcal{S}$-$\LaTeX$ Symbols prepared by Dr. Emre Sermutlu, or the exhaustive listing The Comprehensive $\LaTeX$ Symbol List by Scott Pakin. Some of the most common include:

• \lt \gt \le \ge \neq $\lt$, $\gt$, $\le$, $\ge$,$\neq$. You can use \not to put a slash through almost anything: \not\lt $\not\lt$ but it often looks bad.

• \times \div \pm \mp $\times$, $\div$, $\pm$, $\mp$. \cdot is a centered dot: $x\cdot y$

• \cup \cap \setminus \subset \subseteq \subsetneq \supset \in \notin \emptyset \varnothing $\cup$, $\cap$, $\setminus$, $\subset$, $\subseteq$, $\subsetneq$, $\supset$, $\in$, $\notin$, $\emptyset$, $\varnothing$

• {n+1 \choose 2k} or \binom{n+1}{2k} ${n+1 \choose 2k}$

• \to \gets \rightarrow \leftarrow \Rightarrow \Leftarrow \mapsto \implies \iff $\to$, $\gets$, $\rightarrow$, $\leftarrow$, $\Rightarrow$, $\Leftarrow$, $\mapsto$, $\implies$, $\iff$

• \land \lor \lnot \forall \exists \top \bot \vdash \vDash $\land$, $\lor$, $\lnot$, $\forall$, $\exists$, $\top$, $\bot$, $\vdash$, $\vDash$

• \star \ast \oplus \circ \bullet $\star$, $\ast$, $\oplus$, $\circ$, $\bullet$

• \approx \sim \simeq \cong \equiv \prec \lhd $\approx$, $\sim $, $\simeq$, $\cong$, $\equiv$, $\prec$, $\lhd$

• \infty \aleph_0 $\infty,円 \aleph_0$ \nabla \partial $\nabla$, $\partial$ \Im \Re $\Im$, $\Re$

• For modular equivalence, use \pmod like this: a\equiv b\pmod n $a\equiv b\pmod n$. For the binary mod operator, use \bmod like this: a\bmod 17 $a\bmod 17$.

• Use \dots for the triple dots in $a_1, a_2, \dots, a_n$ and $a_1+a_2+\dots+a_n$

• Script lowercase l is \ell $\ell$.

  Detexify lets you draw a symbol on a web page and then lists the $\TeX$ symbols that seem to resemble it. These are not guaranteed to work in MathJax, but it's a good place to start. To check that a command is supported, note that MathJax.org maintains a list of currently supported $\LaTeX$ commands, and one can also check Dr. Carol JVF Burns's page of $\TeX$ Commands Available in MathJax.

12. Spaces MathJax usually decides for itself how to space formulas, using a complex set of rules. Putting extra literal spaces into formulas will not change the amount of space MathJax puts in: a␣b and a␣␣␣␣b are both $a b$. To add more space, use ,円 for a thin space $a,円b$; \; for a wider space $a\;b$. \quad and \qquad are large spaces: $a\quad b$, $a\qquad b$.

    To set plain text, use \text{...}: $\{x\in s\mid x\text{ is extra large}\}$. You can nest $...$ inside of \text{...}, for example to access spaces.

13. Accents and diacritical marks Use \hat for a single symbol $\hat x$, \widehat for a larger formula $\widehat{xy}$. If you make it too wide, it will look silly. Similarly, there are \bar $\bar x$ and \overline $\overline{xyz}$, and \vec $\vec x$ and \overrightarrow $\overrightarrow{xy}$ and \overleftrightarrow $\overleftrightarrow{xy}$. For dots, as in $\frac d{dx}x\dot x = \dot x^2 + x\ddot x$, use \dot and \ddot.

14. Special characters used for MathJax interpreting can be escaped using the \ character: \\$ $\$$, \{ $\{$, \} $\}$, \_ $\_$, \# $\#$, \& $\&$. If you want \ itself, you should use \backslash (symbol) or \setminus (binary operation) for $\backslash$, because \\ is for a new line.

(Tutorial ends here.)

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It is important that this note be reasonably short and not suffer from too much bloat. To include more topics, please create short addenda and post them as answers instead of inserting them into this post.

Contents

Alphabetical list of links to MathJax topics, by title:

• Absolute values and norms • Additional symbolic decorations • Aligning Equations
• Alternative Ways of Writing in LaTeX • Annotations of reasoning • Arbitrary operators
• Arrays • Big braces • Colors
• Commutative diagrams • Continued fractions • Crossing things out
• Definitions by cases (piecewise functions) • Degree symbol • Display style
• Equation numbering • Fussy spacing issues • Highlighting expressions
• Left and right arrows • Limits • Linear programming
• Long division • Matrices • Markov Chains
• Mixing code and MathJax formatting on lines • The \newcommand function
• Numbering Equations • Overlaying Symbols • Packs of cards
• Symbols • System of equations • Tables
• Tags and references • Tensor indices • Units
• Vertical bars • Vertical spacing

• robjohn: Some capital Greek letters are the same as the Roman equivalents, so they are not separated in $\LaTeX$. For a capital beta, one must use something like \mathrm{B}: $\mathrm{B}$
• Martin Sleziak: Two related questions: How do I insert a table when asking a question? and How to show the integral symbol on this site?
• Charles Staats: A quick addition to point 11: If you want to use a $\sin$-like symbol that is not already defined, the command is \operatorname: e.g., \operatorname{Spec} A gives $\operatorname{Spec} A$.
• axblount: It might be useful to mention hanging subscripts for things like _5C_3 $_5C_3$. You could also mention \frac vs \dfrac.
• MJD: My basic idea is that if a beginner can express a formula clearly, then someone else can come in and clean up the typesetting afterwards. I am considering getting rid of the section about \big, \left, and \right for this reason, and trimming the section on spacing.
• Andrew Stacey: Most of the references to TeX or LaTeX in this and the answers ought to be to MathJaX (the exception that I can see being the output of Detexify). I know this is a bit pedantic, but would it be alright to correct this?
• MJD: @AndrewStacey Thanks for pointing this out. Let's by all means be as correct as possible, particularly when there's no extra cost.
• Andrew Stacey: @MJD Okay, I've had a go (also the answer about arrays). I wonder also whether or not it is worth a sentence at the end pointing out that whilst MathJaX does its best to emulate TeX, it isn't TeX and so while knowing how something is done in TeX gives you a starting point, it isn't a guarantee that the same thing works in MathJaX. (As a case in point, questions about MathJaX are generally off-topic over on TeX-SX.)
• MJD: @AndrewStacey I wouldn't. They are close enough that it seems to me to be a needless refinement. I might even argue that MathJax is $\TeX,ドル although an alternative implementation. We're willing to accept that other programming languages (JavaScript, for example) that have slightly incompatible implementations are nevertheless the same language; why not in this case as well?
• Andrew Stacey: @MJD Except that this is meant as a tutorial for those who aren't familiar with the distinction (and there really is a distinction: "slightly incompatible implementations" doesn't really fit the bill here). One thing tutorials often include is a "Where to find out more" section. This doesn't. Someone who doesn't know the distinction might be tempted to search for help on TeX or LaTeX instead and wonder why it doesn't work.
• axblount: @AndrewStacey All the tips given here would work in any $\TeX$/$\LaTeX$ environment with the proper packages. MathJax is just the service used to render it. You wouldn't say "Miktex tutorial" or "texlive tutorial".
• Andrew Stacey: @axblount But that's precisely the wrong way around to think about it! The likelihood is that someone will look at this tutorial to figure out how to write something on the Maths-SX site: i.e., to use MathJaX. If they can't find help here, where do they go? If they have the idea that MathJaX is "just a javascript implementation of TeX" then they might think to look for help with TeX, but that is quite possibly not going to be helpful.
• Andrew Stacey: @axblount For a start, you've changed the goalposts: "LaTeX math expressions". LaTeX is so much more than just a way of typesetting maths! Second, I don't really know but it wouldn't take me long to cook one up. I don't use MathJaX so I haven't explored it. But I know, for example, that it can't handle catcode changes. Which means that I can't make ( and ) automatically resizeable. I can in LaTeX.
• user1527227: I wish I saw this post when I first joined. This post should be a main link on the home page. There should be a button under each box: NEW TO LATEX, CLICK HERE FOR EXAMPLES. This is extremely useful, concise.
• robjohn: @MJD: I use \mathrm in many places; e.g. $\mathrm{d}x$ in integrals and derivatives and for operator names that don't need the full force of \operatorname. \mathrm was intended for roman symbols in math mode; \text was intended for text because of the way it spaces things. See this TEX thread. Since I don't believe we can use preambles in MathJax, we can't use \DeclareMathOperator, though we can use \newcommand, but that is orthogonal to the use of \mathrm vs \text for math symbols.

Answer 1

Matrices

1. Use $$\begin{matrix}...\end{matrix}$$ In between the \begin and \end, put the matrix elements. End each matrix row with \\, and separate matrix elements with &. For example,

   $$
   \begin{matrix}
   1 & x & x^2 \\
   1 & y & y^2 \\
   1 & z & z^2 \\
   \end{matrix}
   $$
   

   produces:

$$ \begin{matrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \\ \end{matrix} $$

MathJax will adjust the sizes of the rows and columns so that everything fits.

2. To add brackets, either use \left...\right as in section 6 of the tutorial, or replace matrix with pmatrix $\begin{pmatrix}1&2\3円&4\\ \end{pmatrix}$, bmatrix $\begin{bmatrix}1&2\3円&4\\ \end{bmatrix}$, Bmatrix $\begin{Bmatrix}1&2\3円&4\\ \end{Bmatrix}$, vmatrix $\begin{vmatrix}1&2\3円&4\\ \end{vmatrix}$, Vmatrix $\begin{Vmatrix}1&2\3円&4\\ \end{Vmatrix}$.

3. Use \cdots $\cdots$ \ddots $\ddots$ \vdots $\vdots$ when you want to omit some of the entries:

   $$\begin{pmatrix} 1 & a_1 & a_1^2 & \cdots & a_1^n \\ 1 & a_2 & a_2^2 & \cdots & a_2^n \\ \vdots & \vdots& \vdots & \ddots & \vdots \\ 1 & a_m & a_m^2 & \cdots & a_m^n \end{pmatrix}$$

4. For horizontally "augmented" matrices, put parentheses or brackets around a suitably-formatted table; see arrays below for details. Here is an example:

$$ \left[\begin{array}{cc|c} 1&2&3\\ 4&5&6 \end{array}\right] $$

is produced by:

 $$ \left[
\begin{array}{cc|c}
 1&2&3\\
 4&5&6
\end{array}
\right] $$

The cc|c is the crucial part here; it says that there are three centered columns with a vertical bar between the second and third.

5. For vertically "augmented" matrices, use \hline. For example

$$ \begin{pmatrix} a & b \\ c & d\\ \hline 1 & 0\\ 0 & 1 \end{pmatrix} $$ is produced by

$$
 \begin{pmatrix}
 a & b\\
 c & d\\
 \hline
 1 & 0\\
 0 & 1
 \end{pmatrix}
$$

6. For small inline matrices use \bigl(\begin{smallmatrix} ... \end{smallmatrix}\bigr), e.g. $\bigl( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \bigr)$ is produced by:

    $\bigl( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \bigr)$
   

• Michael Hardy: This says "End each matrix row with \\". But there is no reason to end the LAST row of the matrix that way. The double backslash means: now go on to the next row. But there isn't any next row after the last one.
• trichoplax is on Codidact now: I can't edit, but that could be phrased "Separate matrix rows with \\".
• Tom Hale: as in section 6 of the tutorial... Which tutorial? Is there a link to this tutorial section?
• MJD: @tom The tutorial is at the top of this page. It has numbered sections.
• Reb.Cabin: @MichaelHardy but a \\ on every line is harmless, and it makes the editing of matrices easier because swapping with the last line can be done with one quick keystroke in many editors.
• linear_combinatori_probabi: Is it possible to get smallpmatrix or something?
• j.c.: There's something strange about the second matrix (right after "produces:"); it seems to render properly as a matrix on this page, but on the revision permalink math.meta.stackexchange.com/revisions/5023/7 it shows up as raw LaTeX in a code block.
• alhelal: vmatrix does not show the vertical bar.
• alhelal: vmatrix does not show the vertical bar. (\begin{vmatrix}0 & 3 & 2x+7\\ 2 & 7x & 9+5x\\ 0 & 0 & 2x+5\\ \end{vmatrix}=0) হলে x এর মান-
• alhelal: I am using <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> <script type="text/javascript" id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"> </script>
• Hosein Rahnama: Is there a way to combine the horizontal and vertical lines in a single matrix to show the partitions of a matrix?
• Manfred Weis: for a reflected version of \ddots that run from SW to NE this answer on mathematics meta works: \newcommand\iddots{\mathinner{ \kern1mu\raise1pt{.} \kern2mu\raise4pt{.} \kern2mu\raise7pt{\Rule{0pt}{7pt}{0pt}.} \kern1mu }}

Answer 2

Symbols

In general, you have to search in long tables about a specific symbol you're looking for, things like $\Psi,ドル $\delta,ドル $\zeta,ドル $\ge,ドル $\subseteq$ ... And it turns out that this operation can be frustrating and time consuming, which can cause the buddy to abandon writing the complete $\LaTeX$ sentence in his answer, or in some cases, the complete answer itself.

That's why the tool that I will present you in this post was conceived. Basically, it is a $\LaTeX$ handwritten symbol recognition. Example in image:

enter image description here

Here is the website: Detexify2 No more frustration.

• Gottfried Helms: How to implement usepackage ? I'd like to have \iddots from package mathdots available.
• Floris: @GottfriedHelms see this question - I think the answer is "you can't do that"...
• Gottfried Helms: @Floris: thanks, that is indeed the informative answer!
• Manfred Weis: It recognized my horrible drawing with a finger on my notebook's mousepad! Unfortunately the symbol it recognized (mapsfrom) isn't part of MathJax

Answer 3

Aligned equations

Often people want a series of equations where the equals signs are aligned. To get this, use \begin{align}...\end{align}. Each line should end with \\, and should contain an ampersand at the point to align at, typically immediately before the equals sign.

For example,

\begin{align} \sqrt{37} & = \sqrt{\frac{73^2-1}{12^2}} \\ & = \sqrt{\frac{73^2}{12^2}\cdot\frac{73^2-1}{73^2}} \\ & = \sqrt{\frac{73^2}{12^2}}\sqrt{\frac{73^2-1}{73^2}} \\ & = \frac{73}{12}\sqrt{1 - \frac{1}{73^2}} \\ & \approx \frac{73}{12}\left(1 - \frac{1}{2\cdot73^2}\right) \end{align}

is produced by

\begin{align}
\sqrt{37} & = \sqrt{\frac{73^2-1}{12^2}} \\
 & = \sqrt{\frac{73^2}{12^2}\cdot\frac{73^2-1}{73^2}} \\ 
 & = \sqrt{\frac{73^2}{12^2}}\sqrt{\frac{73^2-1}{73^2}} \\
 & = \frac{73}{12}\sqrt{1 - \frac{1}{73^2}} \\ 
 & \approx \frac{73}{12}\left(1 - \frac{1}{2\cdot73^2}\right)
\end{align}

The usual $$ marks that delimit the display may be omitted here.

• user856: The AMS's Short Math Guide recommends the align environment over eqnarray in LaTeX. In MathJax the spacing seems to be the same , but align requires one less ampersand per line.
• MJD: Thanks. I was not sure whether to discuss that. A detailed argument against eqnarray is in this article.
• user856: Would you mind if I changed your example to use align then?
• MJD: @Rahul: Please go ahead and change anything that seems good to change. This is all CW.
• MJD: Also, if you think you have a better example, please use it; I used the first one I found.
• TravisJ: Correct me if I'm wrong, but I don't believe the \$\$ is necessary before and after the \begin{align}. I've certainly never used it. From experience, the \begin{align} puts you into math-display mode by itself.
• David K: I sometimes find that one line of this environment is too close to another, making them uncomfortable to read. The interline spacing can be adjusted by using input such as \\[1ex] instead of \\ . (And of course the 1 can be changed to another value such as 1.5 or .7 in order to get enough space but not too much.)
• baxx: Note - in Jekyll I had to add an additional `\` to break lines.
• Alexander Day: @MJD i put the equation: \begin{align} f(x)&=\left(x^3\right)+\left(x^3+x^2+x^1\right)+\left(x^3+x^2\right)\\ f'(x)&=\left(\left(3x^2+2x+1\right)+\left(3x^2+2x\right)\\ f''(x)&=\left(6x+2\right)\\ \end{align} but the third & is a problem: according to Mathjax, it does not go there. then when i remove it, it says that "a missing close brace or unclosed brace" is present. what am i doing wrong?
• user379641: @AlexanderDay How did you used that boxed quote? What are commands or formats for it?
• Dan Henderson: @AlexanderDay I notice that right after your second &=, you have two instances of \left( in a row, and one of them is not closed. Removing one of them, I get: $\begin{align} f(x)&=\left(x^3\right)+\left(x^3+x^2+x^1\right)+\left(x^3+x^‌​2\right)\\ f'(x)&=\left(3x^2+2x+1\right)+\left(3x^2+2x\right)\\ f''(x)&=\left(6x+2\right)\\ \end{align}$ Is that what you expected?
• Dan Henderson: @AlwaysConfused the box delimits a MathJax formula with a syntax error.
• jdhao: There is a difference between aligned and align environment. Equation above are just one equation with different representations, rather than multiple aligned equations. I think the correct environment is aligned.
• Reb.Cabin: The align* variation does two things differently: left-aligns the equations and omits equation numbers; the regular align environment centers the equations and puts an equation number on each line, at least in my version of Jupyter notebooks.
• TKH: pandoc (or the underlying tex2pdf) actually complains if the \begin{align} is preceded by $$

Answer 4

Definitions by cases (piecewise functions)

Warning: If you make certain kinds of errors while entering code using this environment, you can easily screw-up live update, and your only recourse is to abandon your edit and refresh the page. Clearing out the code and re-entering it will not fix things - you will have to refresh the page. If you are learning how to use this feature it is recommended that you cut-and-paste a working example from here, and modify it bit-by-bit to the text you want.

Use \begin{cases}...\end{cases}. End each case with a \\, and use & before parts that should be aligned.

For example, you get this:

$$f(n) = \begin{cases} n/2, & \text{if $n$ is even} \\ 3n+1, & \text{if $n$ is odd} \end{cases}$$

by writing this:

 f(n) =
\begin{cases}
n/2, & \text{if $n$ is even} \\
3n+1, & \text{if $n$ is odd}
\end{cases}

The brace can be moved to the right: $$ \left. \begin{array}{l} \text{if $n$ is even:}&n/2\\ \text{if $n$ is odd:}&3n+1 \end{array} \right\} =f(n) $$ by writing this:

\left.
\begin{array}{l}
\text{if $n$ is even:}&n/2\\
\text{if $n$ is odd:}&3n+1
\end{array}
\right\}
=f(n)

To get a larger vertical space between cases we can use \\[2ex] instead of \\. For example, you get this:

$$f(n) = \begin{cases} \frac{n}{2}, & \text{if $n$ is even} \\[2ex] 3n+1, & \text{if $n$ is odd} \end{cases}$$

by writing this:

f(n) =
\begin{cases}
\frac{n}{2}, & \text{if $n$ is even} \\[2ex]
3n+1, & \text{if $n$ is odd}
\end{cases}

(An ‘ex’ is a length equal to the height of the letter x; 2ex here means the space should be two exes high.)

• jibe: @MJD Do we have to use the additional instruction \displaystyle when the formulas displayed are more complex ?
• MJD: @jibs \displaystyle is enabled automatically in displays, for example between $$...$$. You should not ever have to use it.
• yo': @jibe In general, the separate cases in this notation should be in text style unless they are very very complex (and then, the { notation is just wrong anyways).
• wrufesh: can this be written with ascii math instead of latex @MJD
• Robert Frost: What an absurd function to use as an example. Nobody would ever consider such a function.

Answer 5

Arrays

It is often easier to read tables formatted in MathJax rather than plain text or a fixed width font. Arrays and tables are created with the array environment. Just after \begin{array} the format of each column should be listed, use c for a center aligned column, r for right aligned, l for left aligned and a | for a vertical line. Just as with matrices, cells are separated with & and rows are broken using \\. A horizontal line spanning the array can be placed before the current line with \hline.

For example, $$\begin{array}{c|lcr} n & \text{Left} & \text{Center} & \text{Right} \\ \hline 1 & 0.24 & 1 & 125 \\ 2 & -1 & 189 & -8 \\ 3 & -20 & 2000 & 1+10i \end{array} $$

$$
\begin{array}{c|lcr}
n & \text{Left} & \text{Center} & \text{Right} \\
\hline
1 & 0.24 & 1 & 125 \\
2 & -1 & 189 & -8 \\
3 & -20 & 2000 & 1+10i
\end{array}
$$

Arrays can be nested to make an array of tables.

For example, $$ % outer vertical array of arrays \begin{array}{c} % inner horizontal array of arrays \begin{array}{cc} % inner array of minimum values \begin{array}{c|cccc} \text{min} & 0 & 1 & 2 & 3\\ \hline 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 1 & 1 & 1\\ 2 & 0 & 1 & 2 & 2\\ 3 & 0 & 1 & 2 & 3 \end{array} & % inner array of maximum values \begin{array}{c|cccc} \text{max}&0&1&2&3\\ \hline 0 & 0 & 1 & 2 & 3\\ 1 & 1 & 1 & 2 & 3\\ 2 & 2 & 2 & 2 & 3\\ 3 & 3 & 3 & 3 & 3 \end{array} \end{array} \\ % inner array of delta values \begin{array}{c|cccc} \Delta&0&1&2&3\\ \hline 0 & 0 & 1 & 2 & 3\\ 1 & 1 & 0 & 1 & 2\\ 2 & 2 & 1 & 0 & 1\\ 3 & 3 & 2 & 1 & 0 \end{array} \end{array} $$

As the source for the preceding array is long, please right-click on one of the tables and choose $\mathsf{Show\ Math\ As\ }\blacktriangleright\mathsf{\ TeX\ Commands}$.

• user856: You'll have to wrap the contents of each cell in \text if you don't want $all italics, weird-looking spacing, an' odd apostrophes$.
• axblount: @RahulNarain: True. I used words just for illustration, but I guess the example was slightly misleading. If you'd like to modify it please go ahead.
• MJD: Thanks! I like your numeric example better, since the widths of the entries are different enough that the alignment differences are visually clear.
• ABC: @robjohn how do you use | while typing , i don't find it in my keyboard......
• robjohn: @exploringnet: on my keyboard, it is the shifted backslash. It may be in different places (or absent) depending on your keyboard. On my mobile device (iPhone), it is in the shifted numerics, to the right of the backslash. In mathmode, \vert gives $\vert$ and \mid gives $\mid,ドル but neither works in the column spec for an array. If you cannot type it on your keyboard, you can alwaays copy and paste it from another document.
• Gottfried Helms: It should perhaps be mentioned, that in nested arrays there seems to be no option to synchronize column-widths and/or row-heights over the top-level. I didn't find a solution such that if two arrays are stacked vertically one could make their column-widths matching/fit.
• nullgeppetto: This could also be convenient for some people, althought it destroys the joy of writing tables in $\LaTeX$ by hand!
• Incnis Mrsi: @Rahul: why did regulars not press developers to enhance HTML formatting instead of doing inconvenient and resource-devouring detours through MathJax? When a table contains (mostly) formulæ, the use of a formula-formatting engine looks determined. But when one wants just a table, why should it run software with completely different purpose? I once tried to speak about it at meta.SE, but was gagged.
• user147263: @IncnisMrsi What kind of pressure could we apply: bribery, threats, kidnapping? A feature request was made, supported by SE communities, and declined by SE (on technical grounds, as they say). At least we have the MathJax workaround, with all of its flaws: SO and others have nothing.
• GNUSupporter 8964民主女神 地下教會: Center Aligned Table Captions with Left Aligned Contents \begin{array}{ll} \hfill\mathrm{Bad}\hfill & \hfill\mathrm{Better}\hfill \\ \hline \\ e^{i\frac{\pi}2} \quad e^{\frac{i\pi}2}& e^{i\pi/2} \\ \int_{-\frac\pi2}^\frac\pi2 \sin x,円dx & \int_{-\pi/2}^{\pi/2}\sin x,円dx \\ \end{array}
• GNUSupporter 8964民主女神 地下教會: P.S. Table copied from MJD's example below so as to make an example with cells much wider than the caption.
• Тyma Gaidash: @MJD Is there any command for a vertical line, like \hline ? The {cc|c} part in the array code is confusing.
• Тyma Gaidash: There is the \rm command instead of the entire code in \text

Answer 6

Fussy spacing issues

These are issues that won't affect the correctness of formulas, but might make them look significantly better or worse. Beginners should feel free to ignore this advice; someone else will correct it for them, or more likely nobody will care.

Don't use \frac in exponents or limits of integrals; it looks bad and can be confusing, which is why it is rarely done in professional mathematical typesetting. Write the fraction horizontally, with a slash:

$$\begin{array}{cc} \mathrm{Bad} & \mathrm{Better} \\ \hline \\ e^{i\frac{\pi}2} \quad e^{\frac{i\pi}2}& e^{i\pi/2} \\ \int_{-\frac\pi2}^\frac\pi2 \sin x,円dx & \int_{-\pi/2}^{\pi/2}\sin x,円dx \\ \end{array}$$

The | symbol has the wrong spacing when it is used as a divider, for example in set comprehensions. Use \mid instead:

$$\begin{array}{cc} \mathrm{Bad} & \mathrm{Better} \\ \hline \\ \{x|x^2\in\Bbb Z\} & \{x\mid x^2\in\Bbb Z\} \\ \end{array}$$

When using stretchable delimiters (i.e. with \left and \right), it may be preferable to use ,円\middle|,円. This produces a stretchable vertical bar with a little bit of space around it. Another alternative is to use a colon instead.

$$\begin{array}{cc} \mathrm{Bad} & \mathrm{Better} \\ \hline \\ \left\{\dfrac{m}{n} \mid m,n\in\Bbb Z\right\} & \left\{\dfrac{m}{n} ,円\middle|,円 m,n\in\Bbb Z\right\} \\ \end{array}$$

For double and triple integrals, don't use \int\int or \int\int\int. Instead use the special forms \iint and \iiint: $$\begin{array}{cc} \mathrm{Bad} & \mathrm{Better} \\ \hline \\ \int\int_S f(x),円dy,円dx & \iint_S f(x),円dy,円dx \\ \int\int\int_V f(x),円dz,円dy,円dx & \iiint_V f(x),円dz,円dy,円dx \end{array}$$

Use ,円 to insert a thin space before differentials; without this $\TeX$ will mash them together:

$$\begin{array}{cc} \mathrm{Bad} & \mathrm{Better} \\ \hline \\ \iiint_V f(x)dz dy dx & \iiint_V f(x),円dz,円dy,円dx \end{array}$$

When using a function (e.g. $f$, $\sin$, etc) followed by arguments with larger parentheses, insert negative space before the parentheses using \!:

$$\begin{array}{cc} \mathrm{Bad} & \mathrm{Better} \\ \hline \\ f\left( \dfrac{1}{x} \right) & f\!\left(\dfrac{1}{x}\right) \end{array}$$

When using absolute value, use \lvert ... \rvert instead of a pair of pipes |...|.

$$\begin{array}{cc} \mathrm{Bad} & \mathrm{Better} \\ \hline \\ |\sin x| & \lvert\sin x\rvert \end{array}$$

• hmakholm left over Monica: I think the first adjusted fraction looks better than the original, but I don't like the second. In any case, this minor spacing imbalance is too peripheral to belong in a basic MathJax tutorial IMO. Too likely to scare people away rather than make them feel helped.
• MJD: @Henning Do you mean that the fraction example is too unimportant even to appear in an addendum on fussy spacing, or that the fussy spacing article is too unimportant to appear as an addendum to the tutorial?
• hmakholm left over Monica: I was talking specifically about the fraction example. Mostly I'm concerned that somebody will come away thinking, Eeek! Do I have to worry about THAT to use the site? But it's also arguable that the disclaimer at the top of the answer ought to take care of that.
• TheHopefulActuary: @MJD I like the less space, but what if we want to list the bounds for multiple integrals? Like if we have say 3 integrals and we have 3 separate bounds for each how would we list each one? Or do we have to do \int_bound1^bound2\int_bound3^bound4\int_bound5^bound6??
• MJD: @Kyle I think that's exactly what you do in that case.
• asmeurer: Worth nothing you can use \middle with | to get it to work with \left and \right, like \left\{x\middle | \frac{x^2}{2} \in \mathbb{z}\right\}: $\left\{x\middle | \frac{x^2}{2} \in \mathbb{z}\right\}$
• MJD: Thanks very much! I wanted to do that, but didn't know how.
• user76284: @asmeurer Don't forget the spacing around the bar.
• asmeurer: It seems \middle \mid doesn't work. What is the correct way to get the right spacing with automatic vertical resizing?
• mechanicious: In the case of base $e$ powers I would recommend using $\exp(i\pi/2)$ which is, in my opinion, even better than what's suggested in this post.
• Christoph: @asmeurer I always use \left\{,円 ... ,円\middle|,円 ... ,円\right\} like in $\left\{,円x\in\Bbb R,円\middle|,円 \frac{x^2}{2}\in\Bbb Z,円\right\}$.
• sreysus: I've been using \; and \: along with ,円, and I think they have different spacing, so it's pretty versatile but can get you stuck on formatting an answer for ages because you were trying to get the spaces to match on the scale of pixels :\

Answer 7

Crossing things out

Use \require{cancel} in the first formula in your post that requires cancelling; you need it only once per page. Then use:

$$\require{cancel}\begin{array}{rl} \verb|y+\cancel{x}| & y+\cancel{x}\\ \verb|\cancel{y+x}| & \cancel{y+x}\\ \verb|y+\bcancel{x}| & y+\bcancel{x}\\ \verb|y+\xcancel{x}| & y+\xcancel{x}\\ \verb|y+\cancelto{0}{x}| & y+\cancelto{0}{x}\\ \verb+\frac{1\cancel9}{\cancel95} = \frac15+& \frac{1\cancel9}{\cancel95} = \frac15 \\ \end{array} $$

Use \require{enclose} for the following:

$$\require{enclose}\begin{array}{rl} \verb|\enclose{horizontalstrike}{x+y}| & \enclose{horizontalstrike}{x+y}\\ \verb|\enclose{verticalstrike}{\frac xy}| & \enclose{verticalstrike}{\frac xy}\\ \verb|\enclose{updiagonalstrike}{x+y}| & \enclose{updiagonalstrike}{x+y}\\ \verb|\enclose{downdiagonalstrike}{x+y}| & \enclose{downdiagonalstrike}{x+y}\\ \verb|\enclose{horizontalstrike,updiagonalstrike}{x+y}| & \enclose{horizontalstrike,updiagonalstrike}{x+y}\\ \end{array} $$

\enclose can also produce enclosing boxes, circles, and other notations; see MathML menclose documentation for a complete list.

It is worth noting that MathJax should not be used for formatting non-mathematical text. The preferred way for striking out text is to use the HTML strikethrough tag, <s>[text to be striken]</s>, which renders as (削除) [text to be striken] (削除ここまで).

• Akiva Weinberger: Can I use \enclose{counterstrike}? :P
• Darth Geek: That sneaky 19ドル/95 = 1/5$. Nice one!
• alan2here: I see you can further resolve existing resolutions, $\cancelto{\cancelto{\cancelto{x^{2+x}}{\cancelto{x^2}{x}+4}}4}0$
• Tim Thayer: Is enclose a $\LaTeX$ package, or only a MathML option?
• Martin Sleziak: Here is a related post on meta: Striking out equations.
• user803596: Nice command to make an arrow: $\cancelto{}{}$ \cancelto{}{}
• The Amplitwist: In fact, strikethrough markup should be avoided even in text as far as possible, in the interest of accessibility: they are not picked up by screen readers (see veroniiiica.com/2020/05/29/…). CC @XanderHenderson

Answer 8

Commutative diagrams

(For more examples, see this meta question.)

AMScd diagrams must start with a "require":

$\require{AMScd}$
\begin{CD}
A @>a>> B\\
@V b V V= @VV c V\\
C @>>d> D
\end{CD}

to get this diagram: $\require{AMScd}$ \begin{CD} A @>a>> B\\ @V b V V= @VV c V\\ C @>>d> D \end{CD}

@>>> is used for arrow right

@<<< is used for arrow left

@VVV is used for arrow down

@AAA is used for arrow up

@= is used for horizontal double line

@| is used for vertical double line

@. is used for no arrow

Another example:

 \begin{CD}
 A @>>> B @>{\text{very long label}}>> C \\
 @. @AAA @| \\
 D @= E @<<< F
\end{CD}

\begin{CD} A @>>> B @>{\text{very long label}}>> C \\ @. @AAA @| \\ D @= E @<<< F \end{CD}

Long labels increase the length of the arrow and in this version also automatically increase corresponding arrows.

$\require{AMScd}$
\begin{CD}
 RCOHR'SO_3Na @>{\text{Hydrolysis,$\Delta, Dil.HCl$}}>> (RCOR')+NaCl+SO_2+ H_2O 
\end{CD}

$\require{AMScd}$ \begin{CD} \text{RCOHR'SO$_3$Na} @>{\text{Hydrolysis, $\Delta,$ Dil. HCl}}>> \text{(RCOR')+NaCl+SO$_2$+ H$_2$O} \end{CD}

• Quark: $\require{AMScd}$\begin{CD} RCOHR'SO_3Na @>{\text{Hydrolysis,\Delta, Dil.HCl}>> (RCOR')+NaCl+SO_2+ H_2O \end{CD} Why does this code not give the correct output?
• Lehs: @Quark: The main error was a missing bracket after HCl.
• Quark: @Lehs Thanks. That was a silly mistake :| What if I wanted to write something below the arrow? Also, could you suggest some online website to learn MathJax?
• Lehs: @Quark: then you move a > sign: @>>{\text{very long label}}> I learn MathJax from the examples i.e. in this tread.
• Najib Idrissi: @Lehs Why did you rollback my edit...? You removed the formatting from the list, broke again (in Chrome) the example diagram, and reinserted your duplicate example. Why?
• Lehs: @NajibIdrissi: because your edit appeared as a mess in IE. The diagram wasn't even written out. Maybe there is something wrong in your or in my web-program. Now it looks good in IE.
• Lehs: @NajibIdrissi: Now it also looks good in Chrome for Windows and for Android, plus Safari for Androids. I don't know what the problem is with the current version.
• A. Thomas Yerger: I realize this thread is quite old, but what about diagonal arrows?
• Lehs: @AlfredYerger: there are no such possibilities in AMScd.
• Martin Sleziak: @AlfredYerger Maybe presheaf can help there? See also answer and suggestions about this here: How to draw a commutative diagram?
• Uncool: how to draw a double arrow pointing both directions to the left as well as to the right between two points in commutative diagram?
• Тyma Gaidash: How does one draw a curved arrow like in quiver?

Answer 9

Additional decorations

$\def\demo#1#2{#1{#2}\ #1{#2#2}\ #1{#2#2#2}}$

\overline: $\demo\overline A$

\underline: $\demo\underline B$

\widetilde: $\demo\widetilde C$

\widehat: $\demo\widehat D$

\fbox: $\demo\fbox {$E$}$

\underleftarrow: $\demo\underleftarrow{F}\qquad$ variant: \xleftarrow{}: $\xleftarrow{abc}$

\underrightarrow: $\demo\underrightarrow{G}\qquad$ variant: \xrightarrow{}: $\xrightarrow{abc}$

\underleftrightarrow: $\demo\underleftrightarrow{H}$

\overrightarrow $\demo\overrightarrow{AB}$

\overbrace: $\overbrace{(n - 2) + (n - 1) + (n + 0) + (n + 1) + (n + 2)}$

\underbrace: $\underbrace{(n - 2) + (n - 1) + (n + 0) + (n + 1) + (n + 2)}$

\underbrace: underbraces can be nested, like this: $\underbrace{(n - 2) + \underbrace{(n - 1) + \underbrace{(n + 0)} + (n + 1)} + (n + 2)}$

\overbrace and \underbrace accept a superscript or a subscript, respectively, to annotate the brace. For example, \underbrace{a\cdot a\cdots a}_{b\text{ times}} is $$\underbrace{a\cdot a\cdots a}_{b\text{ times}}$$

Note: \varliminf: $\varliminf$ and \varlimsup:$\varlimsup$ have special symbol of their own.

Single character accents

\check: $\check{I}$

\acute: $\acute{J}$

\grave: $\grave{K}$

\vec: $\vec u\ \vec{AB}$ (c.f. \overrightarrow above)

\bar: $\bar z$

\hat: $\hat x$

\tilde: $\tilde x$

\dot \ddot \dddot: $\dot x,\ddot x,\dddot x$

\mathring: $\mathring A$

General stacking

If you cannot find your symbol remember that you can stack various symbols using

\overset{above}{level}: $\overset{@}{ABC}\ \overset{x^2}{\longmapsto}\ \overset{\bullet\circ\circ\bullet}{T}$

\underset{below}{level}: $\underset{@}{ABC}\ \underset{x^2}{\longmapsto}\ \underset{\bullet\circ\circ\bullet}{T}$

You can use these together too. You can type $X \overset{a}{\underset{b}{\to}} Y$ with X\overset{a}{\underset{b}{\to}}Y.

Arc over points

\overset{ \huge\frown}{PQ}: $\overset{ \huge\frown}{PQ}$ denotes the arc over points $P$ and $Q$ (As per comment of @Calvin Khor to @Paul Sinclair's question)

• Mr Pie: Instead of using $\fbox$, you could also use $\boxed{...}$
• zwim: added arrows with text variants, some new single char accents and general stacking section.
• Reb.Cabin: stackrel also seems to work well, as in \stackrel{\text{def}}{=} $\longrightarrow\stackrel{\text{def}}{=}$
• Paul Sinclair: Is there a way to do arcs over points, such as to indicate the arc of a curve between two points $P$ and $Q$? \widearc {PQ} doesn't seem to work.
• user645636: $\frown{PQ}$ kind of.
• Paul Sinclair: The best I've been able to come up with is \overset{\frown}{PQ} : $\overset{\frown}{PQ}$. But since \frown doesn't adjust in size, it doesn't look right. Does anyone know how get a properly sized arc?
• Calvin Khor: @PaulSinclair I offer the following \overset{\frown}{AB}\overset{ \large\frown}{CD}\overset{\Large\frown}{EF}\overset{ \huge\frown}{GH}\overset{\Huge\frown}{ABC} $\overset{\frown}{AB}\overset{ \large\frown}{CD}\overset{\Large\frown}{EF}\overset{ \huge\frown}{GH}\overset{\Huge\frown}{ABC}$

Answer 10

System of equations

• Use \begin{array}...\end{array} and \left\{...\right.. For example, you get this:

$$ \left\{ \begin{array}{c} a_1x+b_1y+c_1z=d_1 \\ a_2x+b_2y+c_2z=d_2 \\ a_3x+b_3y+c_3z=d_3 \end{array} \right. $$

by writing this:

$$
\left\{ 
\begin{array}{c}
a_1x+b_1y+c_1z=d_1 \\ 
a_2x+b_2y+c_2z=d_2 \\ 
a_3x+b_3y+c_3z=d_3
\end{array}
\right. 
$$

• Alternatively we can use \begin{cases}...\end{cases}. The same system

$$ \begin{cases} a_1x+b_1y+c_1z=d_1 \\ a_2x+b_2y+c_2z=d_2 \\ a_3x+b_3y+c_3z=d_3 \end{cases} $$

is produced by the following code

$$\begin{cases}
a_1x+b_1y+c_1z=d_1 \\ 
a_2x+b_2y+c_2z=d_2 \\ 
a_3x+b_3y+c_3z=d_3
\end{cases}
$$

• To align the = signs use \begin{aligned}...\end{aligned} and \left\{...\right. (see asmeurer's comment) $$\left\{\begin{aligned} a_1x+b_1y+c_1z&=d_1+e_1 \\ a_2x+b_2y&=d_2 \\ a_3x+b_3y+c_3z&=d_3 \end{aligned} \right. $$

whose code is

$$
\left\{
\begin{aligned} 
a_1x+b_1y+c_1z &=d_1+e_1 \\ 
a_2x+b_2y&=d_2 \\ 
a_3x+b_3y+c_3z &=d_3 
\end{aligned} 
\right. 
$$

• To align the = signs and the terms as in $$\left\{\begin{array}{ll}a_1x+b_1y+c_1z &=d_1+e_1 \\ a_2x+b_2y &=d_2 \\ a_3x+b_3y+c_3z &=d_3 \end{array} \right.$$

use array with l (for "align left"; there are also c and r) parameters

$$
\left\{
\begin{array}{ll}
a_1x+b_1y+c_1z &=d_1+e_1 \\ 
a_2x+b_2y &=d_2 \\ 
a_3x+b_3y+c_3z &=d_3 
\end{array} 
\right.
$$

• Vertical space between equations. As explained in Definition by cases to get a larger vertical space between equations we can use \\[2ex] instead of \\. The system

$$\begin{cases} a_1x+b_1y+c_1z=\frac{p_1}{q_1} \\[2ex] a_2x+b_2y+c_2z=\frac{p_2}{q_2} \\[2ex] a_3x+b_3y+c_3z=\frac{p_3}{q_3} \end{cases} $$

is generated by the following code

$$\begin{cases} a_1x+b_1y+c_1z=d_1 \\[2ex] a_2x+b_2y+c_2z=d_2 \\[2ex] a_3x+b_3y+c_3z=d_3 \end{cases} $$

in comparison with

$$\begin{cases} a_1x+b_1y+c_1z=\frac{p_1}{q_1} \\ a_2x+b_2y+c_2z=\frac{p_2}{q_2} \\ a_3x+b_3y+c_3z=\frac{p_3}{q_3} \end{cases} $$

whose code is

$$\begin{cases} a_1x+b_1y+c_1z=\frac{p_1}{q_1} \\ a_2x+b_2y+c_2z=\frac{p_2}{q_2} \\ a_3x+b_3y+c_3z=\frac{p_3}{q_3} \end{cases} $$

• In response to elect's comment. The following code

  $$ \left\{ \begin{array}{l} 0 = c_x-a_{x0}-d_{x0}\dfrac{(c_x-a_{x0})\cdot d_{x0}}{\|d_{x0}\|^2} + c_x-a_{x1}-d_{x1}\dfrac{(c_x-a_{x1})\cdot d_{x1}}{\|d_{x1}\|^2} \\[2ex] 0 = c_y-a_{y0}-d_{y0}\dfrac{(c_y-a_{y0})\cdot d_{y0}}{\|d_{y0}\|^2} + c_y-a_{y1}-d_{y1}\dfrac{(c_y-a_{y1})\cdot d_{y1}}{\|d_{y1}\|^2} \end{array} \right. $$

produces

$$ \left\{ \begin{array}{l} 0 = c_x-a_{x0}-d_{x0}\dfrac{(c_x-a_{x0})\cdot d_{x0}}{\|d_{x0}\|^2} + c_x-a_{x1}-d_{x1}\dfrac{(c_x-a_{x1})\cdot d_{x1}}{\|d_{x1}\|^2} \\[2ex] 0 = c_y-a_{y0}-d_{y0}\dfrac{(c_y-a_{y0})\cdot d_{y0}}{\|d_{y0}\|^2} + c_y-a_{y1}-d_{y1}\dfrac{(c_y-a_{y1})\cdot d_{y1}}{\|d_{y1}\|^2} \end{array} \right. $$

• Steeven: Is it possible to rotate text? To have a vertical word written in front of the large curly bracket that spans over all the equations?
• Mr Pie: @Steeven Go here $\longrightarrow$ math.meta.stackexchange.com/questions/27798/…
• Steeven: Thank you, @user477343. This would be a useful feature on this list.

Answer 11

Colors

Named colors are browser-dependent; if a browser doesn't know a particular color name, it may render the text as black. The following colors are standard in HTML4 and CSS2 and should be interpreted the same by most browsers: $$\begin{array}{|rc|} \hline \verb+\color{black}{text}+ & \color{black}{text} \\ \verb+\color{gray}{text}+ & \color{gray}{text} \\ \verb+\color{silver}{text}+ & \color{silver}{text} \\ \verb+\color{white}{text}+ & \color{white}{text} \\ \hline \verb+\color{maroon}{text}+ & \color{maroon}{text} \\ \verb+\color{red}{text}+ & \color{red}{text} \\ \verb+\color{yellow}{text}+ & \color{yellow}{text} \\ \verb+\color{lime}{text}+ & \color{lime}{text} \\ \verb+\color{olive}{text}+ & \color{olive}{text} \\ \verb+\color{green}{text}+ & \color{green}{text} \\ \verb+\color{teal}{text}+ & \color{teal}{text} \\ \verb+\color{aqua}{text}+ & \color{aqua}{text} \\ \verb+\color{blue}{text}+ & \color{blue}{text} \\ \verb+\color{navy}{text}+ & \color{navy}{text} \\ \verb+\color{purple}{text}+ & \color{purple}{text} \\ \verb+\color{fuchsia}{text}+ & \color{magenta}{text} \\ \hline \end{array}$$

HTML5 and CSS 3 define an additional 124 color names that will be supported on many browsers.

Math Stack Exchange's default style uses a light-colored page background, so avoid using light colors for text. Stick to darker colors like maroon, green, blue, and purple, and remember also that 7–10% of men are color-blind and have difficulty distinguishing red and green. (Some people have difficulty distinguishing other colors too, so don't rely on colors saying "the blue part" over and over again.)

The color may also have the form #rgb where $r, g, b$ are in the range or 0–9, a–f and represent the intensity of red, green, and blue on a scale of 0ドル–15$, with a=10, b=11, ... f=15. For example:

$$\begin{array}{|rrrrrrrr|}\hline \verb+#000+ & \color{#000}{text} & & & \verb+#00F+ & \color{#00F}{text} & & \\ & & \verb+#0F0+ & \color{#0F0}{text} & & & \verb+#0FF+ & \color{#0FF}{text}\\ \verb+#F00+ & \color{#F00}{text} & & & \verb+#F0F+ & \color{#F0F}{text} & & \\ & & \verb+#FF0+ & \color{#FF0}{text} & & & \verb+#FFF+ & \color{#FFF}{text}\\ \hline \end{array} $$

$$\begin{array}{|rrrrrrrr|} \hline \verb+#000+ & \color{#000}{text} & \verb+#005+ & \color{#005}{text} & \verb+#00A+ & \color{#00A}{text} & \verb+#00F+ & \color{#00F}{text} \\ \verb+#500+ & \color{#500}{text} & \verb+#505+ & \color{#505}{text} & \verb+#50A+ & \color{#50A}{text} & \verb+#50F+ & \color{#50F}{text} \\ \verb+#A00+ & \color{#A00}{text} & \verb+#A05+ & \color{#A05}{text} & \verb+#A0A+ & \color{#A0A}{text} & \verb+#A0F+ & \color{#A0F}{text} \\ \verb+#F00+ & \color{#F00}{text} & \verb+#F05+ & \color{#F05}{text} & \verb+#F0A+ & \color{#F0A}{text} & \verb+#F0F+ & \color{#F0F}{text} \\ \hline \verb+#080+ & \color{#080}{text} & \verb+#085+ & \color{#085}{text} & \verb+#08A+ & \color{#08A}{text} & \verb+#08F+ & \color{#08F}{text} \\ \verb+#580+ & \color{#580}{text} & \verb+#585+ & \color{#585}{text} & \verb+#58A+ & \color{#58A}{text} & \verb+#58F+ & \color{#58F}{text} \\ \verb+#A80+ & \color{#A80}{text} & \verb+#A85+ & \color{#A85}{text} & \verb+#A8A+ & \color{#A8A}{text} & \verb+#A8F+ & \color{#A8F}{text} \\ \verb+#F80+ & \color{#F80}{text} & \verb+#F85+ & \color{#F85}{text} & \verb+#F8A+ & \color{#F8A}{text} & \verb+#F8F+ & \color{#F8F}{text} \\ \hline \verb+#0F0+ & \color{#0F0}{text} & \verb+#0F5+ & \color{#0F5}{text} & \verb+#0FA+ & \color{#0FA}{text} & \verb+#0FF+ & \color{#0FF}{text} \\ \verb+#5F0+ & \color{#5F0}{text} & \verb+#5F5+ & \color{#5F5}{text} & \verb+#5FA+ & \color{#5FA}{text} & \verb+#5FF+ & \color{#5FF}{text} \\ \verb+#AF0+ & \color{#AF0}{text} & \verb+#AF5+ & \color{#AF5}{text} & \verb+#AFA+ & \color{#AFA}{text} & \verb+#AFF+ & \color{#AFF}{text} \\ \verb+#FF0+ & \color{#FF0}{text} & \verb+#FF5+ & \color{#FF5}{text} & \verb+#FFA+ & \color{#FFA}{text} & \verb+#FFF+ & \color{#FFF}{text} \\ \hline \end{array}$$

You can have a look here for quick reference on colors in HTML.

• Rory Daulton: We should add that colors can be used on items other than text, such as variables and operators. The '\color' command applies to the next item: surround anything longer with braces.
• Harald Hanche-Olsen: One can in fact use any CSS-compatible colour specification here, including rgb, rgba, hsl, and hsla colours. (I'd edit the answer, but have no time now. Maybe later, if nobody else beats me to it.)
• ThankYouForFlyingRyanair: Can colors be used on striking out text and equations. If so, how?

Answer 12

Tags & References

For longer calculations (or referring to other post's results) it is convenient to use the tagging/labelling/referencing system. To tag an equation use \tag{yourtag}, and if you want to refer to that tag later on, add \label{somelabel} right after the \tag. It is not necessary that yourtag and somelabel are the same, but it usually is more convenient to do so:

$$ a := x^2-y^3 \tag{*}\label{*} $$

$$ a := x^2-y^3 \tag{*}\label{*} $$

In order to refer to an equation, just use \eqref{somelabel}

$$ a+y^3 \stackrel{\eqref{*}}= x^2 $$

$$ a+y^3 \stackrel{\eqref{*}}= x^2 $$

or \ref{somelabel}

Equations are usually referred to as $\eqref{*},ドル but you can also use $\ref{*}$.

Equations are usually referred to as $\eqref{*},ドル but you can also use $\ref{*}$.

As you can see, references are even turned into hyperlinks, which you can use externally as well, e.g. like this. Note that you can also reference labels in other posts as long as they appear on the same site, which is especially useful when referring to a question with multiple equations, or when commenting on a post.

---

(削除) Due to a bug blocks containing a \label will break in preview, as a workaround you can put $\def\label#1{}$ in your post while editing and remove that on submission - unfortunately this means you won't spot misspelled references before submitting... Just don't forget to remove that \def again (削除ここまで)

• Tobias Kienzler: Also works in comments: \eqref{*} yields a clickable $\eqref{*}$
• Gerald Senarclens de Grancy: To enable automatically tagging your queations with incremental numbers, add <script type="text/x-mathjax-config"> MathJax.Hub.Config({TeX: { equationNumbers: {autoNumber: "all"} }}); </script> to your header.
• Tobias Kienzler: @GeraldSenarclensdeGrancy That would however yield a global numbering on all answers to one question, not per-answer. And it would break the current expectation of by default not having tags despite using unstarred \begin{align} etc.... Though personally I'd agree with this
• Crescendo: I'm just curious, is there a way to have the tags on the left side of the equation? Something like$$(1)\qquad\qquad\sum\limits_{j}k\tag*{}$$But the $(1)$ tag is all the way to the left.
• Mr Pie: How do we write a tag without brackets (because I want to tag a little square as a box of accomplishment)?
• Tobias Kienzler: @user477343 No idea, you could ask at tex.stackexchange.com or see if the manual of amsmath has something...
• Tobias Kienzler: @Crescendo You could ask on tex.stackexchange.com for a solution. A workaround could be \begin{array}{lc} or similar.
• Mr Pie: Hey, I figured how to tag without brackets. You simply put what is inside the braces: $\{$\tag*{$\ldots$}$\}$ which I learnt from here $\longrightarrow$ math.meta.stackexchange.com/questions/27731/…
• Dmitrii Demenev: How do I add several tags analogously to empheq? e.g. \begin{cases} 2x_1 + 3x_2 = 8 \qquad (1a) \\ 7x_1 + 9x_2 = -13 \qquad (1b) \end{cases}
• HelloGoodbye: I can't make \tag{*}\label{*} work—when I use it, the equation is never rendered into an image and in the code that is displayed in place of the image this show up as "\tag{}\label{}". \tag{1}\label{1} on the other hand does work.

Answer 13

Continued fractions

To make a continued fraction, use \cfrac, which works just like \frac but typesets the results differently:

$$ x = a_0 + \cfrac{1^2}{a_1 + \cfrac{2^2}{a_2 + \cfrac{3^2}{a_3 + \cfrac{4^4}{a_4 + \cdots}}}}$$

Don't use regular \frac or \over, or it will look awful:

$$ x = a_0 + \frac{1^2}{a_1 + \frac{2^2}{a_2 + \frac{3^2}{a_3 + \frac{4^4}{a_4 + \cdots}}}}$$

You can of course use \frac for the compact notation:

$$ x = a_0 + \frac{1^2}{a_1+} \frac{2^2}{a_2+} \frac{3^2}{a_3 +} \frac{4^4}{a_4 +} \cdots$$

Continued fractions are too big to put inline. Display them with $$...$$ or use a notation like $[a_0; a_1, a_2, a_3, \ldots]$.

• Américo Tavares: The RHS of the following continued fraction $$ \cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cfrac{a_{3}}{b_{3}+\ddots }}}= {\genfrac{}{}{}{}{a_1}{b_1}} {\genfrac{}{}{0pt}{}{}{+}} {\genfrac{}{}{}{}{a_2}{b_2}} {\genfrac{}{}{0pt}{}{}{+}} {\genfrac{}{}{}{}{a_3}{b_3}} {\genfrac{}{}{0pt}{}{}{+\dots}} $$ can be typeset with the \genfrac command '{\genfrac{}{}{}{}{a_1}{b_1}} {\genfrac{}{}{0pt}{}{}{+}} {\genfrac{}{}{}{}{a_2}{b_2}} {\genfrac{}{}{0pt}{}{}{+}} {\genfrac{}{}{}{}{a_3}{b_3}} {\genfrac{}{}{0pt}{}{}{+\dots}}'
• MJD: I wonder if something like $\frac12{\vphantom{1}\atop+}\frac34$ would be good enough? It is much simpler. (\frac12{\vphantom{1}\atop+}\frac34)
• Américo Tavares: Yes, it is. I didn't mention it because in User’s Guide for the amsmath Package it is written the following: "Note. For technical reasons, using the primitive fraction commands \over, \atop, \above in a LATEX document is not recommended (see, e.g., amsmath.faq)."
• MJD: Happily, we are not writing $\LaTeX$ documents here.
• Américo Tavares: Or write \underset{j=1}{\overset{\infty}{\LARGE\mathrm K}}\frac{a_j}{b_j}=\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+\ddots}}} to get $$\underset{j=1}{\overset{\infty}{\LARGE\mathrm K}}\frac{a_j}{b_j}=\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+\ddots}}}.$$
• AlexR: @AméricoTavares Or use \mathop instead of \overset and \underset: \mathop{\LARGE\mathrm K}_{i=1}^\infty \frac{a_i}{b_i} $$\mathop{\LARGE\mathrm K}_{i=1}^\infty \frac{a_i}{b_i}$$
• user249332: @AméricoTavares, Why don't you edit the answer and put this extremely helpful command into there, I think that would be more helpful.

Answer 14

Using \newcommand

I would like to remark that it is possible to define LaTeX commands as you do in your TeX files. I felt so happy when I first discovered it! It's enough to insert something like

$ \newcommand{\SES}[3]{ 0 \to #1 \to #2 \to #3 \to 0 } $

$ \newcommand{\SES}[3]{ 0 \to #1 \to #2 \to #3 \to 0 }$ at the top of your post (remember the dollars!). Then you can just use your commands as you are used to do: in my example typing $$ \SES{A}{B}{C} $$ will produce the following:

$$ \SES{A}{B}{C} $$

It's also possible to use plain \def:

\def\ses#1#2#3{0 \to #1 \to #2 \to #3 \to 0}

and then $\ses{A}{B}{C}$ will produce the same output.

• AlexR: Be aware that this affects the entire post, possibly even the frontpage, so it should be used with great care.
• ericw31415: $\SES{1}{2}{3}$
• iBug: @AlexR It's been fixed since.
• joseville: This unfortunately doesn't work in stackedit.io which I sometimes use to compose and edit longer stackexchange posts. Anyone aware of possible workarounds or alternatives?
• Тyma Gaidash: Why does this not work with \def?

Answer 15

\implies ($\implies$) is a marginally preferable alternative to \Rightarrow ($\Rightarrow$) for implication.

There's also \iff $\iff$ and \impliedby $\impliedby$.

\to ($\to$) is preferable to \rightarrow or \longrightarrow for things like $f\colon A \to B$. The reverse is \gets ($\gets$).

• MJD: Why is it preferable?
• John Salvatierrez: implies looks nicer as the arrow is longer and \to is quicker to right (and it's also what you say in your head while typing it). at least that's what I think.
• yo': Remember the difference between \to and \mapsto as in $T:\mathbb R\to \mathbb R,\; x\mapsto x+1$ produced by T:\mathbb R\to \mathbb R,\; x\mapsto x+1
• Mario Carneiro: I prefer using \to when it appears as part of a larger propositional formula, rather than at the top level, i.e. $p\land((q\lor r)\to s),ドル because the spacing is similar to that of other binary operators. \implies is better for sentence- or clause-level implications, or in displays, i.e. $$x+2=4-x\implies x=1.$$
• Mr Pie: I have always used \Longleftarrow for \impliedby. It generates the same thing anyway, for which the former generates $\Longleftarrow$ and the latter generates $\impliedby$ with \Leftarrow $\Leftarrow$ as an alternate for reverse implication.
• Mr Pie: @yo' instead of \mathbb you could also use \Bbb as a matter of fact :)
• Alexandros: Is there a way to add some text above impllies? Such as "by (1)" to refer to another equation that is used for substitution and similar cases.
• user645636: @Alexandros yes \overset{3.1415}{\underset{26535}{\implies}} produces $$\overset{3.1415}{\underset{26535}{\implies}}$$

Answer 16

Big braces

Use \left and \right to make braces - (round), [square] and {curly} - scale up to be the size of their arguments. Thus

$$
f\left(
 \left[ 
 \frac{
 1+\left\{x,y\right\}
 }{
 \left(
 \frac{x}{y}+\frac{y}{x}
 \right)
 \left(u+1\right)
 }+a
 \right]^{3/2}
\right)
$$

renders as $$ f\left(\left[ \frac{1+\left\{x,y\right\}}{\left(\frac{x}{y}+\frac{y}{x}\right)\left(u+1\right)}+a\right]^{3/2}\right). $$

Note that curly braces need to be escaped as \{ \}.

If you start a big brace with \left and then need to match that to a \right brace that's on a different line, use the forms \right. and \left. to make "shadow" braces. Thus,

$$
\begin{aligned}
a=&\left(1+2+3+ \cdots \right. \\
& \cdots+ \left. \infty-2+\infty-1+\infty\right)
\end{aligned}
$$

renders as $$ \begin{aligned} a=&\left(1+2+3+ \cdots \right. \\ & \cdots+ \left. \infty-2+\infty-1+\infty\right). \end{aligned} $$

There is also a \middle construct which is useful when one has a mid-expression brace which must also scale up:

$$
\left\langle 
 q
\middle\|
 \frac{\frac{x}{y}}{\frac{u}{v}}
\middle| 
 p 
\right\rangle
$$

renders as $$ \left\langle q\middle\|\frac{\frac{x}{y}}{\frac{u}{v}} \middle| p \right\rangle. $$

Note that constructs like \left\langle, \left| and \left\| are also possible.

Alternatively there also exists the \big hierarchy whose pairing is not mandatory, you can type \big(\frac 1x\big) $\big(\frac 1x\big)$

The advantage of left/right is that it dimensions automatically, but has the inconvenient of not producing consistent results depending of the vertical extension of its inner content, instead the big hierarchy has fixed size:

\Bigg(\bigg(\Big(\big((x)\big)\Big)\bigg)\Bigg) $\Bigg(\bigg(\Big(\big((x)\big)\Big)\bigg)\Bigg)$

• Nick: Note: \Big( ... \Big) produces $\Big(\dots\Big)$ but this bracket size is fixed in all situations unlike \left( ... \right) which varies in size with its contents. \Big can be useful in various situations.
• zwim: Added a paragraph about the big hierarchy.

Answer 17

Limits

To make a limit (like $\lim \limits_{x \to 1} \frac{x^2-1}{x-1}$), use this syntax:

First, start off with $\lim. This renders as $\lim$. The backslash is there to prevent things like $lim,ドル where the letters are slanted.

Second, add \limits_{x \to 1} inside. The code now looks like $\lim \limits_{x \to 1}$, and renders as $\lim \limits_{x \to 1}$. The \to inside makes the right arrow, rendered as $\to$. The _ makes the $x \to 1$ go underneath the $\lim$. Finally, the pair of curly braces { } makes sure that $x \to 1$ is treated as a whole object, and not two separate things.

Lastly, add the function you want to apply the limit to. To make the limit mentioned above, $\lim \limits_{x \to 1} \frac{x^2-1}{x-1},ドル simply use $\lim\limits_{x \to 1} \frac{x^2-1}{x-1}$.

And that is how you make a limit using MathJax.

• MJD: Why not just \lim_{x\to 1} $$\lim_{x\to 1}?$$ As I understand it \limits is only needed for operations that don't already understand limits, for example if you want to use + and get $$\mathop{+}\limits_{i=1}^k\text{ instead of }+_{i=1}^k$$ When used inline, your suggestion will produce $\lim\limits_{x\to 1}$ instead of the more compact form $\lim_{x\to 1}$ that mathjax normally chooses. Are you sure this is good advice?
• Anonymous Computer: @MJD $\lim_{x\to 1} renders to $\lim_{x\to 1},ドル and $\lim\limits_{x\to 1 renders as $lim\limits_{x\to 1}$. Note how the $x\to 1$ is separated from the first limit, and not directly underneath. We do not write limits like that in real life, so we use \limits.
• Anonymous Computer: I meant that the second limit renders to $\lim \limits_{x \to 1}$
• MJD: Limits are usually written that way in typeset materials like papers and books when the limit is inline, rather than a displayed formula, and that's why MathJax typesets it that way.
• user1729: The issue with this answer is that it is trying to "force" display mode on inline code. Doing so makes the text look less pretty. For example, see how the spacing between the lines change when I force display mode using \lim\limits_{x\mapsto 1}\dfrac1x: $\lim\limits_{x\mapsto 1}\dfrac1x$. On the other hand, when I let $\TeX$ do what it wants to do, using \lim_{x\mapsto 1}\frac1x, the spacing between the lines stays the same, which is much neater: $\lim_{x\mapsto 1}\frac1x$. This is much easier on the eyes. If you want to make your math mode more prominent then take a new line using $$-$$
• user1729: The moral is: $\TeX$ was written by a jolly clever chap. Let it do what it wants, because it does it for a reason!
• David K: Part 11 of the "question" shows how to write limits in the way they were meant to be written in LaTeX and MathJax.

Answer 18

Highlighting equation

To highlight an equation, \bbox can be used. E.g,

$$ \bbox[yellow]
{
e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n
\qquad (1)
}
$$

produces

$$ \bbox[yellow] { e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n \qquad (1) } $$

By default, the bounding box is "tight", so it doesn't extend beyond the characters used in the formula. You can add a little space around the equation by adding a measurement after the color. E.g.,

$$ \bbox[yellow,5px]
{
e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n
\qquad (1)
}
$$

produces

$$ \bbox[yellow,5px] { e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n \qquad (1) } $$

To add a border, use

$$ \bbox[5px,border:2px solid red]
{
e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n
\qquad (2) 
}
$$

produces

$$ \bbox[5px,border:2px solid red] { e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n \qquad (2) } $$

You can do both border and background, as well:

$$ \bbox[yellow,5px,border:2px solid red]
{
e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n
\qquad (1)
}
$$

produces

$$ \bbox[yellow,5px,border:2px solid red] { e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n \qquad (1) } $$

• Lord_Farin: When using constructs like this, please heed the points raised in this discussion on usage of colour.
• user379641: This would be a very helpful feature.

Answer 19

Arbitrary operators

If an operator is not available as a built-in command, use \operatorname{...}. So for things like $$\operatorname{arsinh}(x)$$ write \operatorname{arsinh}(x) since \arsinh(x) will give an error and arsinh(x) has wrong font and spacing: $arsinh(x)$.

This was already mentioned in a comment by Charles Staats. You might consider this an addition to the FAQ section on \lim, \sin and so on.

For operators which need limits above and below the operator, use \operatorname*{...}, as in $$ \operatorname*{Res}_{z=1}\left(\frac1{z^2-z}\right)=1 $$

New operators may also be defined using the \DeclareMathOperator syntax: \DeclareMathOperator{newOperatorCommand}{newOperator}$\DeclareMathOperator{newOperatorCommand}{newOperator}$ defines a new operator. On the page where this code occurs, \newOperatorCommand will be rendered as $\newOperatorCommand$.

• Felix Marin: We can also use $\verb*{\rm ...}*$. For example, $\verb*{\rm arsinh}*$ yields ${\rm arsinh}$.
• MvG: @Felix: \rm will change the font but not the spacing. \operatorname{arsinh}x renders as "$\operatorname{arsinh}x$" while {\rm arsinh}x renders as "${\rm arsinh}x$". Notice the added space between operator and operand in the first example, which is missing in the second. On the whole, I'd say that operatorname is a lot more in the spirit of semantic markup, declaring what you want to write instead of how you want to write it, so I'd strongly suggest using this.
• Felix Marin: Thanks. I didn't know there was a difference between them. I always avoided ${\tt operatorname}$ because it was too long.
• MJD: Thanks for this. I thought carefully about whether to put \operatorname in the main post, and decided to leave it out. The reason is simple: If a beginner omits \operatorname, the resulting formula will still be perfectly clear, and a more experienced user will have no trouble inserting the \operatorname where it is needed. So including it in the main post would not be a good use of space.
• JP McCarthy: ... I always use "\text{operator }". Hmmm, $\text{arsinh }x$ vs $\operatorname{arsinh}x$.
• MickG: If you use the same operator many times, I think you can do \DeclareMathOperator{\arsinh}{arsinh} at the post's top. Never tried it though...
• Laxmi Narayan Bhandari: What is the code for the last one?
• MvG: @Laxmi you can right-click on MathJax formulas and select "Show Math As / TeX Commands" to see the code for any formula. You can also click on the date of the edits to see edit history, and in that history use "Side-by-side Markdown" rendering to see the source of the whole post.

Answer 20

Absolute values and norms

The absolute value of some expression can be denoted as \lvert x\rvert or, more generally, as \left\lvert ... \right\rvert. It renders as $\lvert x\rvert$.

The norm of a vector (or similar) can be denoted as \lVert v\rVert or, more generally, as \left\lVert ... \right\rVert. It renders as $\lVert v\rVert$. (You may also write \left\|...\right\| instead.)

In both cases, the rendering is better than what you'd get from |x| or ||v||, which render with bars that don't descend low enough and sub-optimal spacing. At least on some browsers, so here is a screenshot how it looks for me, using Firefox 31 on OS X:

Screenshot

And here is the same formula rendered by your browser:

$$|x|, ||v|| \quad\longrightarrow\quad \lvert x\rvert, \lVert v\rVert$$

It was typeset as

$$|x|, ||v|| \quad\longrightarrow\quad \lvert x\rvert, \lVert v\rVert$$

• Martin Sleziak: You can use \|x\| instead of \lVert x \rVert; $\|x\|$ and $\lVert x \rVert$. (I don't think that there is a difference between them. I've tried [asking on SE](tex.stackexchange.com/questions/77767/whats-the-correct-way-to-write-norm).)
• MJD: On my browser |x| and \lvert x\rvert ($|x|$ and $\lvert x\rvert$) look identical, contrary to your claim. Perhaps you need to show an example more complicated than just 'x'?
• MvG: @MJD: What's your browser? I included a screenshot to support my claim.
• MJD: Usually various versions of Firefox on either Linux or Windows. I happen to have Windows 8 booted now, so here's a screenshot from there: a.pomf.se/jrujkq.PNG The bar height looks good on both pairs of symbols; the spacing is a little off for the || version. On Linux they looked the same.
• MJD: Here's a screenshot with FF 31.0 under Linux: a.pomf.se/fhwmjo.png
• Davide Cervone: The difference in output that you are seeing has to do with whether you have the STIX fonts installed locally on your computer or not. The | in STIX doesn't descend below the baseline, while in the MathJax TeX fonts it does.
• Don Hatch: "\lvert x\rvert or, more generally, as \left\lvert ... \right\rvert" What do you mean by "or more generally"? Why would one use one vs the other?

Answer 21

Giving reasons on each line of a sequence of equations

To produce this: \begin{align} v + w & = 0 &&\text{Given} \tag 1\\ -w & = -w + 0 && \text{additive identity} \tag 2\\ -w + 0 & = -w + (v + w) && \text{equations $(1)$ and $(2)$} \end{align}

write this:

\begin{align}
 v + w & = 0 &&\text{Given} \tag 1\\
 -w & = -w + 0 && \text{additive identity} \tag 2\\
 -w + 0 & = -w + (v + w) && \text{equations $(1)$ and $(2)$}
\end{align}

• code_dredd: Using multiple \tag commands in my equations causes them to break. It only takes one tag per equation and it labels the entire thing instead of allowing tagging on a per-line basis. Any ideas?
• David K: @code_dredd The particular formatting in this answer still seems to work. Perhaps you could post your formulas in a new meta question to get help with them.
• Some Guy: Why would you use \tag, instead of just using ()?
• David K: @SomeGuy First, this is what \tag is meant for. It puts the (1) exactly where it should be, at the right margin. Second, if you have an equation like 0 = ax^2+bx+c and you just use (1) on it instead of \tag1, you end up with 0ドル = ax^2+bx+c(1)$. To fix this, instead of \tag you have to insert other commands to make enough blank space. Not a net gain, in my opinion.
• Andre Lin: @DavidK but I couldn't create a space between them
• David K: @AndreLin Space between what and what?
• Andre Lin: @DavidK Between the equation and$(1)$
• David K: @AndreLin That's strange. I sometimes don't see the $(1)$ at all if the page is loading too slowly, but when I do see it it's as far to the right as any of the math or text gets, which is as much space from the rest of the equation as you can get.
• Andre Lin: @DavidK I don’t think your situation is common
• Andre Lin: @DavidK Isn’t it a WiFi issue
• David K: @AndreLin In any case \tag is a commonly-used LaTeX/MathJax command, for example here.

Answer 22

Pack of cards

If you are asking (or answering) a combinatorics question involving packs of cards you can make it look more elegant by using \spadesuit, \heartsuit, \diamondsuit, \clubsuit in math mode: $$\spadesuit\quad\heartsuit\quad\diamondsuit\quad\clubsuit$$ Or if you're really fussy:
\color{red}{\heartsuit} and \color{red}{\diamondsuit}
$$\color{red}{\heartsuit}\quad\color{red}{\diamondsuit}$$

You can also enter the standard Unicode characters (U+2660 BLACK SPADE SUIT etc.) literally, or copy them from here:

$$♠\quad♡\quad♢\quad♣\\ ♤\quad♥\quad♦\quad♧ $$

• user379641: This is very nice! Is there other auto-shapes or stickers?
• user379641: Is it also possible to draw the spade and club in outlines and fill the heart and diamond with a colour?
• David: @AlwaysConfused None that come to mind. Google search turned up this which might help. Otherwise search for a TeX/LaTeX/MathJax symbol table.
• MJD: @AlwaysConfused Unicode has those characters, so you can enter them however you normally enter Unicode characters, or you can now use copy-paste to copy them from this answer.
• David: @MJD Not sure that your edit is a good idea, firstly because I think we would prefer questions and answers on MSE to be in MathJax as far as possible, secondly because this page is specifically a MathJax tutorial. However I'm not really bothered - if you still think it's a good idea, let me know and I'll approve the edit.
• code_dredd: Is there a way to force the heart and diamond suit symbols to be filled, like the club and spade?
• David: @code_dredd See my previous comment in reply to "Always Confused", also the comment by MJD.
• code_dredd: @David I guess nothing has changed since then... Thanks.
• Calvin Khor: To the above commenters - it is possible, for instance $$\color{yellow}♥\!\!\!\color{blue}♡ $$ achieved via the code $$\color{yellow}♥\!\!\!\color{blue}♡$$ but you will need to fiddle with the number of \!s depending on where you put it, because I don't think there is a command in mathJax to place characters on top of each other. Another example,$$\Huge \color{green}\Huge \color{green}♥\!\!\!\!\!\!\!\!\!\!,円\color{red}♡$$ gives $$\Huge \color{green}♥\!\!\!\!\!\!\!\!\!\!,円\color{red}♡ $$

Answer 23

Left and Right Implication Arrows

Another way to display the arrows for right and left implication instead of using

$\Rightarrow$, $\Leftarrow$ and $\Leftrightarrow$

which produces $\Rightarrow,ドル $\Leftarrow$ and $\Leftrightarrow$ respectively, you can use

$\implies$ for $\implies,ドル $\impliedby$ for $\impliedby$ and $\iff$ for $\iff$

The latter of which produces longer arrows which may be more desirable to some.

Answer 24

Degree symbol

Standard Mathjax does not yet support a dedicated degree symbol, so here are some of the ways to try and emulate one :

$$ \begin{array} \\ \text{45^\text{o}} & \text{renders as} & 45^\text{o} \\ \text{45^o} & \text{renders as} & 45^o \\ \text{45^\circ} & \text{renders as} & 45^\circ \\ \text{45^{\large\circ}} & \text{renders as} & 45^{\large\circ}\\ \text{45\unicode{xB0}} & \text{renders as} & 45\unicode{xB0} & \text{Actual Unicode character}\\ \text{90°} & \text{renders as} & 90° & \text{Using keyboard entry of symbol} % % Use the following line as a template for additional entries % % \text{} & \text{renders as} & \\ \end{array} $$

The degree symbol for angles is not ^\circ. Although many people use this notation, the result looks quite different from the canonical degree symbol shipped with the font, as seen above.

If your keyboard doesn't have a ° key, feel free to copy from this post here, or follow these suggestions.

Note that comments below indicate that on some configurations at least, ° renders inferior to ^\circ. And I recently had a post of mine edited just for the sake of turning ° into ^\circ, indicating that someone felt rather strongly about this. So the suggestion above does seem somewhat controversial at the moment. I maintain that from a semantic point of view, ° is superior to ^\circ, and if the rendering suffers from this, then it's a bug in MathJax. After all, LaTeX offers a proper degree symbol in the tex companion fonts, indicating that someone there, too, decided that ^\circ is not perfect. But if things are broken now, I can't fault people from pragmatically sticking with the rendering they prefer. Personally I prefer semantics, also for the sake of screen readers.

Accessibility

Aside from appearance, one consideration in choosing which notation to use is how it will get parsed by screen readers. For example, ChromeVox reads both 45^\circ and 45° as "forty-five degrees", while the other two are pronounced as "forty-five oh", which may be a reason to avoid them.

Usepackage

Commonly in Latex you can \usepackage{gensymb} to get the \degree symbol, however on Stack Exchange this is not an option. Note that even if you can do this it will typically affect the entire page, which may have side effects for other users. So don't rely on this approach.

• dustin: If mathjax loads siunitx or gensymb, there is then \degree in latex which is the degree symbol.
• MvG: @dustin: I couldn't find siunitx or gensymb mentioned anywhere in the MatJax source repository. Are they available as some kind of third-party extension? If so, where? Since MathJax is not LaTeX, packages can't be loaded unless they have been migrated. By the way, all occurrences of "degree" in the MathJax sources refer to something else, as far as I can tell, so there really doesn't seem to be a \degree macro. There should be one, imho.
• dustin: I am not a mathjax expert. I just know latex. I just gave that suggestion in case they were available. Siunitx would be a great package to have. If you aren't familiar, you will see the advantage by scanning the documentation on ctan.
• MJD: On my display, ° looks bad and ^\circ looks good: a.pomf.se/xnlfyg.png
• Joffan: Degree sign can generally be typed by holding down Alt and typing 0176 on the numeric keypad. ° (I don't know how international the actual number is). The leading zero is required.
• MvG: @Joffan: 167 is the decimal representation of the Codepoint for ° in Latin 1, Unicode and CP-1252. Without the leading zero, CP-437 gets applied instead, at least in typical English-speaking countries, so you'd use Alt+248 there. The Wikipedia article I linked to already describes those two ways of entering the symbol, and en.wikipedia.org/wiki/Alt_code has some more details.
• user379641: How to use Radian (c) , gradian (g) and Steradian (sr) ? And also, Angstrom (though a lenght unit)?
• user427802: Actually we can write degrees by 90^o (O for Orange, using lowercase o, like 'o'), and it'd render it close to degrees symbol $90ドル^o + 30 ^o + 45^o$$
• Tom Hale: @AbhasKumarSinha It looks quite slanty to me.
• MvG: @StephenG: I'm not happy with your latest edit. I feel that it is not helpful to users if we suggest even more ways to poorly format that symbol (like ^o imho), or to mention a LaTeX approach just to say it won't work. You deleted the example for 45°, but kept the sentence talking about it, including the colon. I'm reluctant to revert your edit on a CW page without a conversation, but as it stands I see the edit as a change for the worse. Can we find a combined solution?
• MvG: I just wrote a feature request for a \degree symbol, since I believe it would be technically easy and conceptually beneficial to have such a symbol defined for the whole site.
• StephenG - Help Ukraine: @MvG I have added an entry to the "renders as" table for keyboard entry (which frankly looks awful IMO) but regarding your "unhappiness" note only one line was deleted from the version preceding my first edit and I regard your belief that this justifies your claim my edit was "unhelpful" is nonsense. I fail to see how undoing my edit helps anyone but you.
• Misha Lavrov: While we're at it, I included my comment on accessibility from the feature request post, since it may be more useful here. It would be nice if other people tested other screen readers to get a sample size of higher than one.
• Calvin Khor: I recently discovered \mathring and hence there is a further variant a\mathring{}$a\mathring{}$ which is neither circ$a^\circ$ nor the actual unicode symbol $a°$

Answer 25

Long division

$$
\require{enclose}
\begin{array}{r}
 13 \\[-3pt]
4 \enclose{longdiv}{52} \\[-3pt]
 \underline{4}\phantom{2} \\[-3pt]
 12 \\[-3pt]
 \underline{12}
\end{array}
$$

$$ \require{enclose} \begin{array}{r} 13 \\[-3pt] 4 \enclose{longdiv}{52} \\[-3pt] \underline{4}\phantom{2} \\[-3pt] 12 \\[-3pt] \underline{12} \end{array} $$

One important trick shown here is the use of \phantom{2} to make a blank space that is the same size and shape as the digit 2 just above it.

This is adapted from https://stackoverflow.com/a/22871404/3466415 (which uses slightly different but not less valid formatting).

• Américo Tavares: Synthetic division. Example to find that $$x^3−6x^2+11x−6=(x−{\color{red}1})(x^2−5x+6)+{\color{blue}0}$$ \begin{array}{c|rrrr}& x^3 & x^2 & x^1 & x^0\\ & 1 & -6 & 11 & -6\\ {\color{red}1} & \downarrow & 1 & -5 & 6\\ \hline & 1 & -5 & 6 & |\phantom{-} {\color{blue}0} \end{array} \begin{array}{c|rrrr}& x^3 & x^2 & x^1 & x^0\\ & 1 & -6 & 11 & -6\\ {\color{red}1} & \downarrow & 1 & -5 & 6\\ \hline & 1 & -5 & 6 & |\phantom{-} {\color{blue}0} \end{array}
• Américo Tavares: @Maria Mazur For the same example $ \dfrac{x^3-6x^2+11x-6}{x-1}=x^2-5x+6$: $$\begin{array}{rrrr|ll} x^3 & -6x^2 & +11x & -6 & x - 1 \\ -x^3 & +x^2 & & & x^2-5x+6 \\ \hline & -5x^2 & +11x & -6\\ & \phantom{-}5x^2 & -5x & & & & \\ \hline & & +6x & -6 \\ & & -6x & +6 \\ \hline & & 0 & 0 \end{array}$$ I've used this code \begin{array}{rrrr|ll} x^3 & -6x^2 & +11x & -6 & x - 1 \\ -x^3 & +x^2 & & & x^2-5x+6 \\ \hline & -5x^2 & +11x & -6\\ & \phantom{-}5x^2 & -5x & & & & \\ \hline & & +6x & -6 \\ & & -6x & +6 \\ \hline & & 0 & 0 \end{array}

Answer 26

Displaystyle and Textstyle

Many things like fractions, sums, limits, and integrals display differently when written inline versus in a displayed formula. You can switch styles back and forth with \displaystyle and \textstyle in order to achieve the desired appearance.

Here's an example switching back and forth in a displayed equation:

$$\sum_{n=1}^\infty \frac{1}{n^2} \to
 \textstyle \sum_{n=1}^\infty \frac{1}{n^2} \to
 \displaystyle \sum_{n=1}^\infty \frac{1}{n^2}$$

$$\sum_{n=1}^\infty \frac{1}{n^2} \to \textstyle \sum_{n=1}^\infty \frac{1}{n^2} \to \displaystyle \sum_{n=1}^\infty \frac{1}{n^2}$$

It is possible to switch style inline as well:

Compare $\displaystyle \lim_{t \to 0} \int_t^1 f(t),円 dt$
versus $\lim_{t \to 0} \int_t^1 f(t),円 dt$.

Compare $\displaystyle \lim_{t \to 0} \int_t^1 f(t),円 dt$ versus $\lim_{t \to 0} \int_t^1 f(t),円 dt$.

Do observe that the taller formulas gotten with \displaystyle distort the line spacing.

Filler text, more filler text and even more filler text, and an outrageous amount of filler text. It would not occur to me to use $\displaystyle \lim_{t \to 0} \int_t^1 f(x),円 dx$ here. As we see, a formula typeset in displaystyle makes it necessary to move the lines further apart. A ridiculous amount of filler text to make a point. Not pleasing to the eye at all.

In other words, there is also a reason TeX defaults to \textstyle when typesetting inline formulas.

• Simply Beautiful Art: Oh!! I was always confused on why some people had \displaystyle.
• user486983: @SimplyBeautifulArt I was always wondering on why the math expressions of some people looked nicer than mine..
• emacs drives me nuts: There is also $\scriptstyle{AbC}$ $\scriptstyle{AbC}$ and $\scriptscriptstyle{AbC}$ $\scriptscriptstyle{AbC}$.

Answer 27

Vertical Spacing

Some formulas such as $\overline a+\overline b=\overline {a\cdot b}$, $\sqrt{a}-\sqrt{b}$, do not look quite right when it comes to vertical spacing. Fortunately, there is more than one way to fix this. One can for instance employ the \mathstrut command as follows:

$\sqrt{\mathstrut a} - \sqrt{\mathstrut b}$

Which yields: $\sqrt{\mathstrut a} - \sqrt{\mathstrut b}$. Or using \vphantom (vertical phantom) command, which measures the height of its argument and places a math strut of that height into the formula.

$\sqrt{\vphantom{b} a} - \sqrt{b}$

Which renders as: $\sqrt{\vphantom{b} a} - \sqrt{b}$.

Another issue is with the spacing within lines in situations like this,

Based on the previous technique, we can simplify $\dfrac{1}{\sqrt{\vphantom{b} a} - \sqrt{b}}$, and we thus get the result of the previous limit. [this text is added to show alignment with the above smashed object]

These two lines are too far apart, but this is unnecessary since the second line is very short. We can solve this by using the \smash command, to get:

Based on the previous technique, we can simplify $\smash{\dfrac{1}{\sqrt{\vphantom{b} a} - \sqrt{b}}}$, and we thus get the result of the previous limit. [this text is added to show alignment with the above smashed object]

Beware - as above - the smashed text may overlap the next line if that line extends far enough to reach the smashed object, so this solution is not always feasible (it is esp. likely to occur in slim-width browsers, e.g. phones). Analogous overlapping may occur with any prior lines. Note that smash can be restricted to top or bottom with an argument: \smash[t]... or \smash[b]...

• on4aa: Alternatively, one can also sneak in a rule of zero width \rule{0pt}{2ex}, as explained here.
• dfeuer: On Android, at least, the results of \smash look awful. The formula overlaps the text.
• Bill Dubuque: @dfeuer Android is not the source of the problem. Rather it is the fact that the browser window width is so short that the 2nd line extends far enough to be below the radical, so smashing the radical causes it to overlap the 2nd line. The same thing occurs in any browser if you make its width small enough.
• dfeuer: @BillDubuque, okay, but it's still a problem for mobile browsers.
• Bill Dubuque: @dfeuer Again, it's not "mobile" browsers that are the source of the problem - rather it is the fact that there is something displayed below the smashed object. That is more likely to occur on phones since they are more likely to have shorter width screens. If you shrink the width of a desktop browser window to be small enough (or extend the length of the following line) then the above answer will show the problem there too (e.g. I edited the answer to show that).

Answer 28

Equation numbering

Simple equation

To give an equation a number, use the \tag{}. To refer to it later, use \label{} to label this equation. When you want to refer to it, use \eqref{}. For example,

$$e=mc^2 \tag{1}\label{eq1}$$

Equation $\eqref{eq1}$ is one of the greatest equations in mankind's history. Equation $\eqref{eq1}$ is produced using the following code,

$$e=mc^2 \tag{1}\label{eq1}$$

To refer to it, use \eqref{eq1}.

Multi-line equation

Multi-line equation is actually just one equation rather than several equations. So the correct environment is aligned instead of align.

$$\begin{equation}\begin{aligned} a &= b + c \\ &= d + e + f + g \\ &= h + i \end{aligned}\end{equation}\tag{2}\label{eq2}$$

Equation $\eqref{eq2}$ is a multi-line equation. The code to produce equation $\eqref{eq2}$ is

$$\begin{equation}\begin{aligned}
a &= b + c \\
 &= d + e + f + g \\
 &= h + i
\end{aligned}\end{equation}\tag{2}\label{eq2}$$

Multiple aligned equations

For multiple aligned equations, we use the align environment.

$$\begin{align} a &= b + c \tag{3}\label{eq3} \\ x &= yz \tag{4}\label{eq4}\\ l &= m - n \tag{5}\label{eq5} \end{align}$$

Equation $\eqref{eq3}$, $\eqref{eq4}$ and $\eqref{eq5}$ are multiple equations aligned together. The code to produce these equations is,

$$\begin{align}
a &= b + c \tag{3}\label{eq3} \\
x &= yz \tag{4}\label{eq4}\\
l &= m - n \tag{5}\label{eq5}
\end{align}$$

• Mr Pie: I don’t believe there is any difference between align and aligned, but whatever feels comfortable I suppose.
• jdhao: There is actaully a difference, read here for a detailed discussions.
• Mr Pie: thank you very much for clearing up that understanding :)
• jdhao: You are welcome. When in doubt, always google it first :).
• code_dredd: If there's an equation with multiple lines, is there a way to add tags on a per-line basis, i.e. \tag{1} for line 1, \tag{2} for line 2, etc? If I use the \tag{...} commands, I can only use one per equation and it labels the entire equation, not each line.
• jdhao: I am not aware of this kind of command. What is your use case?
• user173262: the last equation numbering can also be used with align* instead of align
• Jiri Kriz: I get all tags on the first line: "a = b + c (3)(4)(5)".
• Randy Cragun: Why do we need both $$ and \begin{equation} with aligned? I noticed that when I omitted \begin{equation} that I did not get an equation number, but that does not happen when I use only $$ without any other environments inside.
• A.J.: If equations are not being numbered, is there any (practical) difference between aligned and align? (I understand they should be used with single equations and multiple equations respectively.) I've checked the link mentioned in an earlier comment which shows several examples of differences, but none of them seem to apply when using Mathjax here on Math Stack Exchange; they use environments like tabular or list or enumerate, none of which are recognized by Mathjax. Also, is there any reason to use \begin{equation} and \end{equation} if equations are not being numbered?
• Leon Chang: I found in physics stackexchange, MathJax doesn't support \label{ } for reference. To align multiple equations, we need to escape twice like: \begin{align} a &= b + c \tag{Eq.3}\\\\ x &= yz \tag{Eq.4}\\\\ l &= m - n \tag{Eq.5} \end{align}. Also, you don't need $$, which is for a single equation and it is equivalent to \begin{equation}

Answer 29

Linear programming

Formulation

A theoretical LPP can be typeset as

\begin{array}{ll}
\text{maximize} & c^T x \\
\text{subject to}& d^T x = \alpha \\
&0 \le x \le 1.
\end{array}

\begin{array}{ll} \text{maximize} & c^T x \\ \text{subject to}& d^T x = \alpha \\ &0 \le x \le 1. \end{array}

To input a numerical LPP, use alignat instead of align to get better alignment between signs, variables and coefficients.

\begin{alignat}{5}
 \max \quad & z = & x_1 & + & 12 x_2 & & & && \\
 \mbox{s.t.} \quad & & 13 x_1 & + & x_2 & + & 12x_3 & \geq 5 && \tag{constraint 1} \\
 & & x_1 & & & + & x_3 & \leq 16 && \tag{constraint 2} \\
 & & 15 x_1 & + & 201 x_2 & & & = 14 && \tag{constraint 3} \\
 & & \rlap{x_i \ge 0, i = 1, 2, 3}
\end{alignat}

\begin{alignat}{5} \max \quad & z = & x_1 & + & 12 x_2 & & & && \\ \mbox{s.t.} \quad & & 13 x_1 & + & x_2 & + & 12x_3 & \geq 5 && \tag{constraint 1} \\ & & x_1 & & & + & x_3 & \leq 16 && \tag{constraint 2} \\ & & 15 x_1 & + & 201 x_2 & & & = 14 && \tag{constraint 3} \\ & & \rlap{x_i \ge 0, i = 1, 2, 3} \end{alignat}

We treat $\max$, $z$, each variable, $\pm$ sign and RHS as one separate column, while leaving an extra empty column on the right. Then we count the number of separators &, add one into this number then divide it by two. (e.g. (9 + 1) ÷ 2 = 5)

\rlap is used so that the last row spans over one column.

Optional: \tag is used to label the constraints.

Change MATLAB/Octave matrices to $\rm\LaTeX$ code

To get fractions, execute format rat at the beginning.

Writing manually the $\rm\LaTeX$ code for a matrix with many rows and columns in Octave is tedious. The Octave function

strcat("\\begin{bmatrix}\n",strrep(strrep(mat2str(A)," "," & "), ...
";"," \\\\\n")(2:end-1),"\n\\end{bmatrix}\n")

converts

A = [1 2 2; 2 3 4; 4 4 2]
A =
 1 2 2
 2 3 4
 4 4 2

to

\begin{bmatrix}
1 & 2 & 2 \\
2 & 3 & 4 \\
4 & 4 & 2
\end{bmatrix}

so that pasting the generated code gives

$$ \begin{bmatrix} 1 & 2 & 2 \\ 2 & 3 & 4 \\ 4 & 4 & 2 \end{bmatrix}. $$

Simplex tableaux

Since the coefficient of the objective value variable $z$ never changes, my habit is to omit the $z$-column to save ink.

Normal simplex tableau

\begin{array}{rrrrrr|r}
 & x_1 & x_2 & s_1 & s_2 & s_3 & \\ \hline
 s_1 & 0 & 1 & 1 & 0 & 0 & 8 \\
 s_2 & 1 & -1 & 0 & 1 & 0 & 4 \\
 s_3 & 1 & 1 & 0 & 0 & 1 & 12 \\ \hline
 & -1 & -1 & 0 & 0 & 0 & 0
\end{array}

\begin{array}{rrrrrr|r} & x_1 & x_2 & s_1 & s_2 & s_3 & \\ \hline s_1 & 0 & 1 & 1 & 0 & 0 & 8 \\ s_2 & 1 & -1 & 0 & 1 & 0 & 4 \\ s_3 & 1 & 1 & 0 & 0 & 1 & 12 \\ \hline & -1 & -1 & 0 & 0 & 0 & 0 \end{array}

It can be stacked up to give an illustration of the entering of variables at different stages.

\begin{array}{rrrrrrr|rr}
 & x_1 & x_2 & s_1 & s_2 & s_3 & w & & \text{ratio} \\ \hline
 s_1 & 0 & 1 & 1 & 0 & 0 & 0 & 8 & - \\
 w & 1^* & -1 & 0 & -1 & 0 & 1 & 4 & 4 \\
 s_3 & 1 & 1 & 0 & 0 & 1 & 0 & 12 & 12 \\ \hdashline
 & 1 & -1 & 0 & -1 & 0 & 0 & 4 & \\ \hline
 s_1 & 0 & 1 & 1 & 0 & 0 & 0 & 8 & \\
 x_1 & 1 & -1 & 0 & -1 & 0 & 1 & 4 & \\
 s_3 & 0 & 2 & 0 & 2 & 1 & -1 & 8 & \\ \hdashline
 & 0 & 0 & 0 & 0 & 0 & -1 & 0 &
\end{array}

\begin{array}{rrrrrrr|rr} & x_1 & x_2 & s_1 & s_2 & s_3 & w & & \text{ratio} \\ \hline s_1 & 0 & 1 & 1 & 0 & 0 & 0 & 8 & - \\ w & 1^* & -1 & 0 & -1 & 0 & 1 & 4 & 4 \\ s_3 & 1 & 1 & 0 & 0 & 1 & 0 & 12 & 12 \\ \hdashline & 1 & -1 & 0 & -1 & 0 & 0 & 4 & \\ \hline s_1 & 0 & 1 & 1 & 0 & 0 & 0 & 8 & \\ x_1 & 1 & -1 & 0 & -1 & 0 & 1 & 4 & \\ s_3 & 0 & 2 & 0 & 2 & 1 & -1 & 8 & \\ \hdashline & 0 & 0 & 0 & 0 & 0 & -1 & 0 & \end{array}

Dual simplex tableau

\begin{array}{rrrrrrrr|r}
 & x_1 & x_2 & x_3 & x_4 & x_5 & x_6 & x_7 & \\ \hline
 x_4 & 0 & -3 & 7 & 1 & 0 & 0 & 2 & 2M -4 \\
 x_5 & 0 & -9 & 0 & 0 & 1 & 0 & -1 & -M -3 \\
 x_6 & 0 & 6 & -1 & 0 & 0 & 1 & -4^* & -4M +8 \\
 x_1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & M \\ \hline
 & 0 & 1 & 1 & 0 & 0 & 0 & 2 & 2M \\
\text{ratio} & & & 1 & & & & 1/2 &
\end{array}

\begin{array}{rrrrrrrr|r} & x_1 & x_2 & x_3 & x_4 & x_5 & x_6 & x_7 & \\ \hline x_4 & 0 & -3 & 7 & 1 & 0 & 0 & 2 & 2M -4 \\ x_5 & 0 & -9 & 0 & 0 & 1 & 0 & -1 & -M -3 \\ x_6 & 0 & 6 & -1 & 0 & 0 & 1 & -4^* & -4M +8 \\ x_1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & M \\ \hline & 0 & 1 & 1 & 0 & 0 & 0 & 2 & 2M \\ \text{ratio} & & & 1 & & & & 1/2 & \end{array}

It can be stacked up to give a theoretical illustration of what happens in the upcoming steps.

\begin{array}{rrrrrrr|r} & x_1 & x_2 & x_3 & s_1 & s_2 & s_3 & \\ \hline s_1 & -2 & 0 & -2 & 1 & 0 & 0 & -60 \\ s_2 & -2 & -4^* & -5 & 0 & 1 & 0 & -70 \\ s_3 & 0 & -3 & -1 & 0 & 0 & 1 & -27 \\ \hdashline & 8 & 10 & 25 & 0 & 0 & 0 & 0 \\ \text{ratio} & -4 & -5/2 & -5 & & & & \\ \hline s_1 & -2^* & 0 & -2 & 1 & 0 & 0 & -60 \\ x_2 & 1/2 & 1 & 5/4 & 0 & -1/4 & 0 & 35/2 \\ s_3 & 3/2 & 0 & 11/4 & 0 & -3/4 & 1 & 51/2 \\ \hdashline & 3 & 0 & 25/2 & 0 & 5/2 & 0 & -175 \\ \text{ratio} & -3/2 & & 25/4 & & & & \\ \hline x_1 & 1 & 0 & 1 & -1/2 & 0 & 0 & 30 \\ x_2 & 0 & 1 & 3/4 & 1/4 & -1/4 & 0 & 5/2 \\ s_3 & 0 & 0 & 5/4 & 3/4 & -3/4^* & 1 & -39/2 \\ \hdashline & 0 & 0 & 19/2 & 3/2 & 5/2 & 0 & -265 \\ \text{ratio} & & & & & \dots & & \\ \hline x_1 & 1 & 0 & 1 & -1/2 & 0 & 0 & 30 \\ x_2 & 0 & 1 & 1/3 & 0 & 0 & -1/3 & 9 \\ s_2 & 0 & 0 & -5/3 & -1 & 1 & -4/3 & 26 \\ \hdashline & 0 & 0 & 41/3 & 4 & 0 & 10/3 & -330 \end{array}

Duality

A picture is worth a thousand words.

$$ \require{extpfeil} % produce extensible horizontal arrows \begin{array}{ccc} % arrange LPPs % first row % first LPP \begin{array}{ll} \max & z = c^T x \\ \text{s.t.} & A x \le b \\ & x \ge 0 \end{array} & \xtofrom{\text{duality}} & % second LPP \begin{array}{ll} \min & v = b^T y \\ \text{s.t.} & A^T y \ge c \\ & y \ge 0 \end{array} \\ ({\cal PC}) & & ({\cal DC}) \\ \text{add } {\Large \downharpoonleft} \text{slack var} & & \text{minus } {\Large \downharpoonright} \text{surplus var}\\ % Change to your favorite arrow style % % second row % third LPP \begin{array}{ll} \max & z = c^T x \\ \text{s.t.} & A x + s = b \\ & x,s \ge 0 \end{array} & \xtofrom[\text{some steps skipped}]{\text{duality}} & % fourth LPP \begin{array}{ll} \min & v = b^T y \\ \text{s.t.} & A^T y - t = c \\ & y,t \ge 0 \end{array} \\ ({\cal PS}) & & ({\cal DS}) % \end{array} $$

• Mr Pie: It must have taken more than a thousand words to write that picture though :D
• mmesser314: How many words is a broken link worth?

Answer 30

Units

While $\LaTeX$ has packages that format units, MathJax does not. For visual consistency, one should format units within the same string of MathJax code as the value to which it corresponds, separating the value and unit with \ (space-backslash-space) since the BIPM recommends a small space between the value and units. In addition, follow the below conventions for formatting values and units:

Decimal Separator & Digit Separation

Following the conventions of the English-speaking world, a period . should be used to separate the decimal part of a number from the integral part, not a comma , as is common in some languages. This is because commas are already reserved for separating mathematical notation such as arguments of multivariate functions, elements of a set, and the coordinates of ordered tuples.

No punctuation should be used to separate multiples of three digits on either side of the decimal separator; instead, a small space rendered by ,円 should be used on both sides of the decimal marker when the string of digits consists of more than four or five digits. For example,

• 4321.1234 4321ドル.1234$
• 54,321円.123,45円 54ドル,321円.123,45円$
• 0.56789 0ドル.56789$
• 0.567,89円 0ドル.567,89円$

If you use a decimal separator, you should include a digit on both sides of the separator, even if the digit is simply 0ドル$.

Powers of 10ドル$

It is preferable to write scientific or engineering notation like this: 4.15\times10^{n} 4ドル.15 \times10^{n}$. The spacing around \times $\times$ is taken care of on its own, so there is no need to insert the spacing manually.

Nevertheless, if necessary, use an upright variant of the letter ‘E’ or ‘e’ to indicate order of magnitude, such as

• \mathrm{E},6円 $\mathrm{E},6円$
• \scriptsize{\mathrm{E}},円\normalsize{6} $\scriptsize{\mathrm{E}},円\normalsize{6}$
• \mathrm{e},6円 $\mathrm{e},6円$

A small space on either side is perfectly fine and recommended.

Single Units

The symbol of any unit—especially SI units—should follow the form \mathrm{u}. (I have this command saved under the keyboard shortcut usin on my devices.) For example,

• \mathrm{m} $\mathrm{m}$
• \mathrm{kg} $\mathrm{kg}$
• \mathrm{ft.} $\mathrm{ft.}$

Do not use a period with symbolic units; do use a period with abbreviated units.

Units with a Dot Multiplier

Multiplied units conjoined by a dot should follow the form \mathrm{u}\!\cdot\!\mathrm{v} $\mathrm{u}\!\cdot\!\mathrm{v}$. (I have this sequence of commands saved under the keyboard shortcut umul on my devices.) Because of how \cdot is designed (i.e., to separate numbers), the small negative space \! on either side maintains uniform spacing throughout the whole compound unit. For example,

• \mathrm{N}\!\cdot\!\mathrm{m} $\mathrm{N}\!\cdot\!\mathrm{m}$
• \mathrm{s}\!\cdot\!\mathrm{A} $\mathrm{s}\!\cdot\!\mathrm{A}$

Do not use \times $\times$ as a separator.

Units with a Solidus Separator

Divided units conjoined by a solidus should follow the form \left.\mathrm{u}\middle/\mathrm{v}\right. $\left.\mathrm{u}\middle/\mathrm{v}\right.$. (I have this sequence of commands saved under the keyboard shortcut udiv on my devices.) The extra markdown is to ensure that solidus stretches the entire height of the unit, especially when exponents are involved. For example,

• \left.\mathrm{J}\middle/\mathrm{s}\right. $\left.\mathrm{J}\middle/\mathrm{s}\right.$
• \left.\mathrm{m}\middle/\mathrm{s}^2\right. $\left.\mathrm{m}\middle/\mathrm{s}^2\right.$

You may include small negative spaces \! on either side of the solidus if you please.

Exponents

Exponents can be rendered with the standard MathJax markdown. The carat and number should immediately follow the closing brace of the mathrm{} argument. For example,

• \mathrm{m}^2 $\mathrm{m}^2$
• \left.\mathrm{m}\middle/\mathrm{s}^2\right. $\left.\mathrm{m}\middle/\mathrm{s}^2\right.$

Parentheses

Parentheses can also be rendered with standard MathJax markdown using \left( and \right) outside the argument of \mathrm. For example,

• \left.\mathrm{kg}\!\cdot\!\mathrm{m}^2\middle/\left(\mathrm{C}\!\cdot\!\mathrm{s}\right)\right. $\left.\mathrm{kg}\!\cdot\!\mathrm{m}^2\middle/\left(\mathrm{C}\!\cdot\!\mathrm{s}\right)\right.$

Exponents in Place of Separators

If you prefer to use no separators and only powers, separator each single \mathrm{} with a small space ,円 and use exponents as necessary. For example,

• \mathrm{m},円\mathrm{s}^{-2} $\mathrm{m},円\mathrm{s}^{-2}$
• \mathrm{s}^{-1},円\mathrm{mol} $\mathrm{s}^{-1},円\mathrm{mol}$

Examples in Context

\mu_0=4\pi\times10^{-7} \ \left.\mathrm{\mathrm{T}\!\cdot\!\mathrm{m}}\middle/\mathrm{A}\right.

$$\mu_0=4\pi\times10^{-7} \ \left.\mathrm{\mathrm{T}\!\cdot\!\mathrm{m}}\middle/\mathrm{A}\right.$$

180^\circ=\pi \ \mathrm{rad}

$180ドル^\circ=\pi \ \mathrm{rad}$$

N_A = 6.022\times10^{23} \ \mathrm{mol}^{-1}

$$N_A = 6.022\times10^{23} \ \mathrm{mol}^{-1}$$

• The Amplitwist: @SamuelMuldoon Please don't use \mathcal{MathJax} to write MathJax.