Operators, Delimiters, and Math Formatting - Basic Math Typesetting Course

Common Operators

Binary Operators

Command: + | Output: $ + $ | Command: \times | Output: $ \times $
Command: - | Output: $ – $ | Command: \div | Output: $ \div $
Command: \pm | Output: $ \pm $ | Command: \mp | Output: $ \mp $
Command: \cdot | Output: $ \cdot $ | Command: \ast | Output: $ \ast $
Command: \cap | Output: $ \cap $ | Command: \cup | Output: $ \cup $
Command: \wedge | Output: $ \wedge $ | Command: \vee | Output: $ \vee $
Command: \otimes | Output: $ \otimes $ | Command: \oplus | Output: $ \oplus $

Relations
Command: = | Output: $ = $ | Command: \neq | Output: $ \neq $
Command: < | Output: $ < $ | Command: > | Output: $ > $
Command: \leq | Output: $ \leq $ | Command: \geq | Output: $ \geq $
Command: \ll | Output: $ \ll $ | Command: \gg | Output: $ \gg $
Command: \approx | Output: $ \approx $ | Command: \equiv | Output: $ \equiv $
Command: \sim | Output: $ \sim $ | Command: \simeq | Output: $ \simeq $
Command: \in | Output: $ \in $ | Command: \notin | Output: $ \notin $
Command: \subset | Output: $ \subset $ | Command: \supset | Output: $ \supset $
Command: \subseteq | Output: $ \subseteq $ | Command: \supseteq | Output: $ \supseteq $

Named Functions

Use predefined commands for function names so they appear upright (not italic):

% WRONG: sin is italic
$sin(x)$

% RIGHT: sin is upright
$\sin(x)$, $\cos(x)$,
$\tan(x)$, $\log(x)$,
$\ln(x)$, $\exp(x)$,
$\lim_{n\to\infty}$,
$\max$, $\min$, $\sup$

$ sin(x) $ (wrong — italic) $ \sin(x) $, $ \cos(x) $, $ \tan(x) $, $ \log(x) $, $ \ln(x) $, $ \exp(x) $, $ \lim_{n\to\infty} $, $ \max $, $ \min $, $ \sup $

Sums, Products, Integrals, and Limits

These “big operators” behave differently in inline and display modes:

Inline: $\sum_{i=1}^{n} i^2$

Display:
<!--M98-->

<!--M99-->

<!--M100-->

<!--M101-->

Inline: $ \sum_{i=1}^{n} i^2 $

Display:

$$ \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} $$ $$ \int_0^\infty e^{-x}\,dx = 1 $$ $$ \prod_{k=1}^{n} k = n! $$ $$ \lim_{x \to 0} \frac{\sin x}{x} = 1 $$

Tip: In display mode, limits go above and below. In inline mode, they go to the side (to avoid stretching line height). To force display-style limits inline, use \displaystyle: $ \displaystyle\sum_{i=1}^{n} $. But use this sparingly — it disrupts line spacing.

Delimiters

Parentheses, brackets, and braces should scale with their content:

% Fixed size (bad)
$(\frac{a}{b})$

% Auto-scaling (good)
$\left(\frac{a}{b}\right)$

% Other delimiters:
$\left[\frac{a}{b}\right]$
$\left\{\frac{a}{b}\right\}$
$\left|\frac{a}{b}\right|$
$\left\langle x \right\rangle$

$ (\frac{a}{b}) $ (bad) $ \left(\frac{a}{b}\right) $ (good) $ \left[\frac{a}{b}\right] $ $ \left\{\frac{a}{b}\right\} $ $ \left|\frac{a}{b}\right| $ $ \left\langle x \right\rangle $

\left and \right must always appear as a pair. If you need only one side, use a dot for the invisible delimiter:

\left. \frac{d}{dx} x^n \right|_{x=1} = n

renders as $ \left. \frac{d}{dx} x^n \right|_{x=1} = n $

Manual Sizing

Sometimes \left/\right produces awkward sizes. Manual sizing gives you control:

Command: \big( | Size: $ \big( $
Command: \Big( | Size: $ \Big( $
Command: \bigg( | Size: $ \bigg( $
Command: \Bigg( | Size: $ \Bigg( $
Math Spacing

LaTeX handles most math spacing automatically, but sometimes you need manual adjustments:

Command: \, | Width: thin space (3/18 em) | Use Case: Before $ dx $: $ \int f(x)\,dx $
Command: \: | Width: medium space (4/18 em) | Use Case: Between related symbols
Command: \; | Width: thick space (5/18 em) | Use Case: Grouping in formulas
Command: \! | Width: negative thin space | Use Case: Tightening: $ \sqrt{\,x} $
Command: \quad | Width: 1 em | Use Case: Between equations
Command: \qquad | Width: 2 em | Use Case: Wider separation
The most common use: always put \, before the differential in integrals:

% Without thin space (cramped)
$\int f(x)dx$

% With thin space (correct)
$\int f(x)\,dx$

$ \int f(x)dx $ (cramped) vs. $ \int f(x)\,dx $ (correct)

Dots

Command: \ldots | Output: $ \ldots $ | Use: Low dots (lists: $ a_1, a_2, \ldots, a_n $)
Command: \cdots | Output: $ \cdots $ | Use: Centered dots (sums: $ a_1 + a_2 + \cdots + a_n $)
Command: \vdots | Output: $ \vdots $ | Use: Vertical dots (matrices)
Command: \ddots | Output: $ \ddots $ | Use: Diagonal dots (matrices)
Command: \dots | Output: context-dependent | Use: amsmath auto-selects the right dots

Tip: With amsmath loaded, \dots is smart: it produces low dots after commas and centered dots after $ + $, $ = $, etc. Use \dots by default and override with \ldots/\cdots only when needed.

Text in Math Mode

Text within math mode should use \text{...} (from amsmath):

% WRONG: "if" is italic
$x = 1 if n > 0$

% RIGHT: "if" is upright
$x = 1 \text{ if } n > 0$

<!--M102-->

$ x = 1 if n > 0 $ (wrong) $ x = 1 \text{ if } n > 0 $ (right) $$ f(x) = \begin{cases} x^2 & \text{if } x \geq 0 \\ -x^2 & \text{if } x < 0 \end{cases} $$

Accents and Decorations

Command: \hat | Output: $ \hat{a} $ | Command: \bar | Output: $ \bar{a} $
Command: \tilde | Output: $ \tilde{a} $ | Command: \vec | Output: $ \vec{a} $
Command: \dot | Output: $ \dot{a} $ | Command: \ddot | Output: $ \ddot{a} $
Command: \widehat | Output: $ \widehat{abc} $ | Command: \widetilde | Output: $ \widetilde{abc} $
Command: \overline | Output: $ \overline{ab} $ | Command: \underline | Output: $ \underline{ab} $
Command: \overbrace | Output: $ \overbrace{abc} $ | Command: \underbrace | Output: $ \underbrace{abc} $
Putting It All Together

Here are real-world examples combining the techniques from this chapter:

Euler’s identity:

<!--M103-->

$$ e^{i\pi} + 1 = 0 $$

Binomial theorem:

<!--M104-->

$$ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k $$

Cauchy-Schwarz inequality:

<!--M105-->

$$ \left| \sum_{i=1}^{n} a_i b_i \right|^2 \leq \left( \sum_{i=1}^{n} a_i^2 \right) \left( \sum_{i=1}^{n} b_i^2 \right) $$

Gaussian integral:

<!--M106-->

$$ \int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi} $$

Exercises

Typeset the following equations:
1. $ \displaystyle \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) – f(x)g'(x)}{[g(x)]^2} $ (Quotient rule)
2. $ \displaystyle \oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F}) \cdot d\vec{S} $ (Stokes’ theorem)
3. $ \displaystyle e = \sum_{n=0}^{\infty} \frac{1}{n!} $
Typeset a 3-case piecewise function using the cases environment.
Write the Taylor series expansion of $ \sin(x) $ around $ x = 0 $, using proper dots notation.

Find and fix three errors in this code:

 <!--M9-->
 $\int_0^1 f(x)dx$
 $sin(x) + cos(x)$

Common Operators

Binary Operators

Relations

Named Functions