Typesetting Math in Texts

21 Nov 2015

Basic math

Whenever you typeset mathematical notation, it needs to have “Math” style. For example: If $a$ is an integer, then $2a+1$ is odd.

Superscripts and subscripts are created using the characters ^ and _, respectively: $x^2+y^2=1$ and $a_n=0$. It is fine to have both on a single letter: $x_0^2$.

If the superscript [or subscript] is more than a single character, enclose the superscript in curly braces: $e^{-x}$.

Greek letters are typed using commands such as \gamma ($\gamma)$ and \Gamma ($\Gamma$).

Named mathematics operators are usually typeset in roman. Most of the standards are already available. Some examples: $\det A$, $\cos\pi$, and $\log(1-x)$.

Displayed equations

When an equation becomes too large to run in-line, you display it in a “Math” paragraph by itself.

$$f(x) = 5x^{10}-9x^9 + 77x^8 + 12x^7 + 4x^6 - 8x^5 + 7x^4 + x^3 -2x^2 + 3x + 11.$$

The \begin{aligned}...\end{aligned} environment is superb for lining up equations.

$$\begin{aligned} (x-y)^2 &= (x-y)(x-y) \\ &= x^2 -yx - xy + y^2 \\ &= x^2 -2xy +y^2. \end{aligned}$$ $$\begin{aligned} 3x-y&=0 & 2a+b &= 4 \\ x+y &=1 & a-3b &=10 \end{aligned}$$

To insert ordinary text inside of mathematics mode, use \text:

$$f(x) = \frac{x}{x-1} \text{ for $x\not=1$}.$$

This is the $3^{\text{rd}}$ time I’ve asked for my money back.

The \begin{cases}...\end{cases} environment is perfect for defining functions piecewise:

$$|x| = \begin{cases} x & \text{when $x \ge 0$ and} \\ -x & \text{otherwise.} \end{cases}$$

Relations and operations

• Equality-like: $x=2$, $x \not= 3$, $x \cong y$, $x \propto y$, $y\sim z$, $N \approx M$, $y \asymp z$, $P \equiv Q$.

• Order: $x < y$, $y \le z$, $z \ge 0$, $x \preceq y$, $y\succ z$, $A \subseteq B$, $B \supset Z$.

• Arrows: $x \to y$, $y\gets x$, $A \Rightarrow B$, $A \iff B$, $x \mapsto f(x)$, $A \Longleftarrow B$.

• Set stuff: $x \in A$, $b \notin C$, $A \ni x$. Use \notin rather than \not\in. $A \cup B$, $X \cap Y$, $A \setminus B = \emptyset$.

• Arithmetic: $3+4$, $5-6$, $7\cdot 8 = 7\times8$, $3\div6=\frac{1}{2}$, $f\circ g$, $A \oplus B$, $v \otimes w$.

• Mod: As a binary operation, use \bmod: $x \bmod N$. As a relation use \mod, \pmod, or \pod:

  $$\begin{aligned} x &\cong y \mod 10 \\ x &\cong y \pmod{10} \\ x &\cong y \pod{10} \end{aligned}$$

• Calculus: $\partial F/\partial x$, $\nabla g$.

Use the right dots

Do not type three periods; instead use \cdots between operations and \ldots in lists: $x_1 + x_2 + \cdots + x_n$ and $(x_1,x_2,\ldots,x_n)$.

Built up structures

• Fractions: $\frac{1}{2}$, $\frac{x-1}{x-2}$.

• Binomial coefficients: $\binom{n}{2}$.

• Sums and products. Do not use \Sigma and \Pi.

  $$\sum_{k=0}^\infty \frac{x^k}{k!} \not= \prod_{j=1}^{10} \frac{j}{j+1}.$$ $$\bigcup_{k=0}^\infty A_k \qquad \bigoplus_{j=1}^\infty V_j$$

• Integrals:

  $$\int_0^1 x^2 \, dx$$

  The extra bit of space before the $dx$ term is created with the \, command.

• Limits:

  $$\lim_{h\to0} \frac{\sin(x+h) - \sin(x)}{h} = \cos x .$$

  Also $\limsup_{n\to\infty} a_n$.

• Radicals: $\sqrt{3}$, $\sqrt[3]{12}$, $\sqrt{1+\sqrt{2}}$.

• Matrices:

  $$A = \left[\begin{matrix} 3 & 4 & 0 \\ 2 & -1 & \pi \end{matrix}\right] .$$

  A big matrix:

  $$D = \left[ \begin{matrix} \lambda_1 & 0 & 0 & \cdots & 0 \\ 0 & \lambda_2 & 0 & \cdots & 0 \\ 0 & 0 & \lambda_3 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & \lambda_n \end{matrix} \right].$$

Delimiters

• Parentheses and square brackets are easy: $(x-y)(x+y)$, $[3-x]$.

• For curly braces use \{ and \}: $\{x : 3x-1 \in A\}$.

• Absolute value: $\vert x-y\vert$, $\vert\vec{x} - \vec{y}\vert$.

• Floor and ceiling: $\lfloor \pi \rfloor = \lceil e \rceil$.

• To make delimiters grow so they are properly sized to contain their arguments, use \left and \right:

  $$\left[ \sum_{n=0}^\infty a_n x^n \right]^2 = \exp \left\{ - \frac{x^2}{2} \right\}$$

  Occasionally, it is useful to coerce a larger sized delimiters than \left/\right produce. Look at the two sides of this equation:

  $$\left((x_1+1)(x_2-1)\right) = \bigl((x_1+1)(x_2-1)\bigl).$$

  I think the right is better. Use \bigl, \Bigl, \biggl, and the matching \bigr, etc.

• Underbraces:

  $$\underbrace{1+1+\cdots+1}_{\text{$n$ times}} = n .$$

Styled and decorated letters

• Primes: $a'$, $b''$.

• Hats: $\bar a$, $\hat a$, $\vec a$, $\widehat{a_j}$.

• Vectors are often set in bold: $\mathbf{x}$.

• Calligraphic letters (for sets of sets): $\mathcal{A}$.

• Blackboard bold for number systems: $\mathbb{C}$.

The text above is based on a paper by Edward R. Scheinerman¹.

A few more examples from mathTeX tutorial².

$$e^x=\sum_{n=0}^\infty\frac{x^n}{n!}$$ $$e^x=\lim_{n\to\infty} \left(1+\frac xn\right)^n$$ $$\varepsilon = \sum_{i=1}^{n-1} \frac1{\Delta x} \int\limits_{x_i}^{x_{i+1}} \left\{ \frac1{\Delta x}\big[ (x_{i+1}-x)y_i^\ast+(x-x_i)y_{i+1}^\ast \big]-f(x) \right\}^2dx$$

Solution for quadratic:

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

Definition of derivative:

$$f^\prime(x)\ = \lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$

Continued fraction:

$$f=b_o+\frac{a_1}{b_1+\frac{a_2}{b_2+\frac{a_3}{b_3+a_4}}}$$

Demonstrating \left\{…\right. and accents.

$$\tilde y=\left\{ {\ddot x \mbox{ if $x$ odd}\atop\widehat{\bar x+1}\text{ if even}}\right.$$

Overbrace and underbrace:

$$\overbrace{a,...,a}^{\text{k a's}}, \underbrace{b,...,b}_{\text{l b's}}\hspace{10pt} \underbrace{\overbrace{a...a}^{\text{k a's}}, \overbrace{b...b}^{\text{l b's}}}_{\text{k+l elements}}$$

Illustrating array:

$$A\ =\ \left( \begin{array}{c|ccc} & 1 & 2 & 3 \\ \hline 1&a_{11}&a_{12}&a_{13} \\ 2&a_{21}&a_{22}&a_{23} \\ 3&a_{31}&a_{32}&a_{33} \end{array} \right)$$

See Wikibook on LaTeX for more examples.