1. To see how any of the formulas were made in any question or answer, including this one, use the "edit" link to view the complete source. To quickly see the source of a single expression, right-click on it and choose "Show Math As > TeX Commands".
(Note that in some browsers, such as Firefox, the MathJax right-click menu that contains this command will be obscured by the browser's own right-click menu. Click somewhere outside the main browser canvas -- such as in the address bar -- to dismiss the browser menu and reveal the MathJax one behind it).
2. For inline formulas, enclose the formula in `$...$`. For displayed formulas, use `$$...$$`. These render differently: $\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{2}$ (inline) or $$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{2}\tag{displayed}$$
3. For **Greek letters**, use `\alpha`, `\beta`, …, `\omega`: $\alpha, \beta, … \omega$. For uppercase, use `\Gamma`, `\Delta`, …, `\Omega`: $\Gamma, \Delta, …, \Omega$.
4. For **superscripts and subscripts**, use `^` and `_`. For example, `x_i^2`: $x_i^2$.
5. By default, 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: $10^10$. But `10^{10}` gives what you probably wanted: $10^{10}$. Use curly braces to delimit a formula to which a superscript or subscript applies: `x^5^6` is an error; `{x^y}^z` is ${x^y}^z$, and `x^{y^z}` is $x^{y^z}$. Observe the difference between `x_i^2` $x_i^2$ and `x_{i^2}` $x_{i^2}$.
6. **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 `(\frac12)` the parentheses will be too small: $(\frac12)$. Using `\left(`…`\right)` will make the sizes adjust automatically to the formula they enclose: `\left(\frac12\right)` is $\left(\frac12\right)$.
`\left` and`\right` apply to all the following sorts of parentheses: `(` and `)` $(x)$, `[` and `]` $[x]$, `\{` and `\}` $\lbrace x \rbrace$, `|` $|x|$, `\langle` and `\rangle` $\langle x \rangle$, `\lceil` and `\rceil` $\lceil x \rceil$, and `\lfloor` and `\rfloor` $\lfloor x \rfloor$. There are also invisible parentheses, denoted by `.`: `\left.\frac12\right\rbrace` is $\left.\frac12\right\rbrace$.
7. **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` $\iint$.
8. **Fractions** There are two ways to make these. `\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 ${a+1\over b+1}$.
9. **Fonts**
* Use `\mathbb` or `\Bbb` for "blackboard bold": $\mathbb{CHNQRZ}$.
* Use `\mathbf` for boldface: $\mathbf{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ $\mathbf{abcdefghijklmnopqrstuvwxyz}$.
* Use `\mathtt` for "typewriter" font: $\mathtt{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ $\mathtt{abcdefghijklmnopqrstuvwxyz}$.
* Use `\mathrm` for roman font: $\mathrm{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ $\mathrm{abcdefghijklmnopqrstuvwxyz}$.
* Use `\mathcal` for "calligraphic" letters: $\mathcal{ ABCDEFGHIJKLMNOPQRSTUVWXYZ}$
* Use `\mathscr` for script letters: $\mathscr{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$
* Use `\mathfrak` for "Fraktur" (old German style) letters: $\mathfrak{ABCDEFGHIJKLMNOPQRSTUVWXYZ} \mathfrak{abcdefghijklmnopqrstuvwxyz}$.
10. **Radical signs** 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.
11. 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}` $$\lim_{x\to 0}$$
12. There are a very large number of special symbols and notations, too many to list here; see [this shorter listing](http://pic.plover.com/MISC/symbols.pdf), or [this exhaustive listing](http://mirror.math.ku.edu/tex-archive/info/symbols/comprehensive/symbols-a4.pdf). 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 \rightarrow \leftarrow \Rightarrow \Leftarrow \mapsto` $\to\, \rightarrow\, \leftarrow\, \Rightarrow\, \Leftarrow\, \mapsto$
* `\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 \cong \equiv \prec` $\approx\, \sim \, \cong\, \equiv\, \prec$.
* `\infty \aleph_0` $\infty\, \aleph_0$ `\nabla \partial` $\nabla\, \partial$ `\Im \Re` $\Im\, \Re$
* For modular equivalence, use `\pmod` like this: `a\equiv b\pmod n` $a\equiv b\pmod n$.
* `\ldots` is the dots in $a_1, a_2, \ldots ,a_n$ `\cdots` is the dots in $a_1+a_2+\cdots+a_n$
* Some Greek letters have variant forms:
`\epsilon \varepsilon` $\epsilon\, \varepsilon$, `\phi \varphi` $\phi\, \varphi$, and others. Script lowercase l is `\ell` $\ell$.
[Detexify](http://detexify.kirelabs.org/classify.html) 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 are a good place to start.
13. **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 `\,` for a thin space $a\,b$; `\;` 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{…}`.
14. **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` $\overrightarrow{xy}$. For dots, as in $\frac d{dx}x\dot x = \dot x^2 + x\ddot x$, use `\dot` and `\ddot`.
15. Special characters used for MathJax interpreting can be escaped using the `\` character: `\$` $\$$, `\{` $\{$, `\_` $\_$, etc.
(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.