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.