Mathematical Typesetting Basics¶

Topic: LaTeX Lesson: 4 of 16 Prerequisites: Lesson 3 (Text Formatting) Objective: Master inline and display math modes, Greek letters, subscripts/superscripts, fractions, roots, operators, delimiters, and mathematical symbols

Introduction to Math Mode¶

LaTeX's mathematical typesetting is one of its most powerful features. Unlike regular text, mathematical notation requires special formatting rules, spacing, and symbols.

Two Math Modes¶

Inline math: Mathematics within a line of text
Display math: Mathematics on its own line, centered

The same commands work in both modes, but display math provides more vertical space and larger symbols.

Inline Math¶

Inline math is used for mathematical expressions within a paragraph.

Dollar Sign Notation¶

The traditional way:

The quadratic formula is $ax^2 + bx + c = 0$ where $a \neq 0$.

Einstein's famous equation is $E = mc^2$.

Output:

The quadratic formula is ax² + bx + c = 0 where a ≠ 0. Einstein's famous equation is E = mc².

Parenthesis Notation (Recommended)¶

LaTeX2ε provides an alternative:

The quadratic formula is \(ax^2 + bx + c = 0\) where \(a \neq 0\).

Why $...$ is better: - More explicit (clear begin/end markers) - Better error messages when you forget to close - Consistent with display math \[...\]

Both styles work, but $...$ is recommended for new documents.

Display Math¶

Display math creates a centered equation on its own line.

Double Dollar Signs (Avoid)¶

The old TeX way:

$$
E = mc^2
$$

Problems with $$...$$: - Plain TeX syntax, not LaTeX - Inconsistent spacing - Doesn't work well with some packages

Bracket Notation (Recommended)¶

The LaTeX way:

\[
E = mc^2
\]

This is the preferred method for unnumbered display equations.

Equation Environment¶

For numbered equations:

\begin{equation}
E = mc^2
\end{equation}

Output:

E = mc²    (1)

The equation number can be referenced (covered in later lessons).

Unnumbered Equation¶

\begin{equation*}
E = mc^2
\end{equation*}

Note: Requires amsmath package for the * variant.

The amsmath Package¶

The amsmath package is essential for serious mathematical typesetting.

Always include in preamble:

\usepackage{amsmath}

Benefits: - Enhanced equation environments - Better spacing - Multi-line equations - Matrix environments - Mathematical operators - And much more...

Additional math packages:

\usepackage{amsmath}    % Enhanced math
\usepackage{amssymb}    % Additional symbols (requires amsfonts)
\usepackage{amsthm}     % Theorem environments
\usepackage{mathtools}  % Extensions to amsmath

Greek Letters¶

Greek letters are fundamental in mathematics and science.

Lowercase Greek¶

$\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$, $\zeta$, $\eta$, $\theta$

$\iota$, $\kappa$, $\lambda$, $\mu$, $\nu$, $\xi$, $\pi$, $\rho$

$\sigma$, $\tau$, $\upsilon$, $\phi$, $\chi$, $\psi$, $\omega$

Output:

α, β, γ, δ, ε, ζ, η, θ ι, κ, λ, μ, ν, ξ, π, ρ σ, τ, υ, φ, χ, ψ, ω

Variants:

$\epsilon$ vs $\varepsilon$    % ε vs ϵ
$\theta$ vs $\vartheta$        % θ vs ϑ
$\pi$ vs $\varpi$              % π vs ϖ
$\rho$ vs $\varrho$            % ρ vs ϱ
$\sigma$ vs $\varsigma$        % σ vs ς
$\phi$ vs $\varphi$            % φ vs φ

Uppercase Greek¶

$\Gamma$, $\Delta$, $\Theta$, $\Lambda$, $\Xi$, $\Pi$, $\Sigma$

$\Upsilon$, $\Phi$, $\Psi$, $\Omega$

Output:

Γ, Δ, Θ, Λ, Ξ, Π, Σ Υ, Φ, Ψ, Ω

Note: Some uppercase Greek letters look like Latin letters, so they use the Latin alphabet: - A (Alpha) → A - B (Beta) → B - E (Epsilon) → E - etc.

Usage Example¶

The standard deviation is $\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2}$
where $\mu$ is the mean.

The wave function $\Psi$ satisfies Schrödinger's equation.

Subscripts and Superscripts¶

Superscripts (Exponents)¶

Use ^:

$x^2$, $x^3$, $x^{10}$, $x^{n+1}$

$2^{2^{2^2}}$  % Nested exponents

$e^{i\pi} = -1$  % Euler's identity

Important: Use braces {} when the superscript is more than one character:

$x^2$      % Correct
$x^10$     % Wrong! Only the 1 is superscript
$x^{10}$   % Correct

Subscripts¶

Use _:

$x_1$, $x_2$, $x_i$, $x_{i,j}$

$a_0, a_1, a_2, \ldots, a_n$

Combining Subscripts and Superscripts¶

$x_i^2$, $x^2_i$  % Order doesn't matter

$x_{i,j}^{(k)}$   % Multiple levels

$\sum_{i=1}^{n} x_i^2$  % Summation with limits

Primes¶

For derivatives:

$f'(x)$      % First derivative (f prime)
$f''(x)$     % Second derivative
$f'''(x)$    % Third derivative
$f^{(4)}(x)$ % Fourth derivative (better notation)

$x'$, $y'$, $z'$  % Primes on variables

Fractions¶

Basic Fractions¶

$\frac{1}{2}$, $\frac{a}{b}$, $\frac{x + y}{x - y}$

\[
\frac{dy}{dx} = \frac{f(x + h) - f(x)}{h}
\]

Output (display):

dy     f(x + h) - f(x)
── = ─────────────────
dx          h

Nested Fractions¶

\[
\frac{1}{1 + \frac{1}{2}}
\]

% Complex nested fraction
\[
\frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{2}}}}
\]

Display Style Fractions¶

In inline math, fractions are smaller. Force display style:

Inline: $\frac{1}{2}$ vs $\dfrac{1}{2}$  % \dfrac forces display style

Display: \[\tfrac{1}{2}\]  % \tfrac forces text (inline) style

Commands: - \dfrac{}{} Display style fraction (larger) - \tfrac{}{} Text style fraction (smaller) - \frac{}{} Adapts to context

When to use: - \dfrac in inline math when readability is important - \tfrac in display math when space is tight

Binomial Coefficients¶

$\binom{n}{k}$ = $\frac{n!}{k!(n-k)!}$

\[
\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}
\]

Output:

(n choose k) = n! / (k!(n-k)!)

Roots¶

Square Roots¶

$\sqrt{2}$, $\sqrt{x}$, $\sqrt{x^2 + y^2}$

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

nth Roots¶

$\sqrt[3]{8} = 2$  % Cube root

$\sqrt[n]{x}$      % nth root

$\sqrt[4]{16} = 2$ % Fourth root

Nested Roots¶

\[
\sqrt{1 + \sqrt{2 + \sqrt{3}}}
\]

\[
\sqrt{x + \sqrt{x + \sqrt{x + \cdots}}}
\]

Common Mathematical Operators¶

Summation¶

% Inline
$\sum_{i=1}^{n} x_i$

% Display
\[
\sum_{i=1}^{n} x_i = x_1 + x_2 + \cdots + x_n
\]

% Multiple indices
\[
\sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij}
\]

Product¶

$\prod_{i=1}^{n} x_i = x_1 \cdot x_2 \cdot \ldots \cdot x_n$

\[
n! = \prod_{i=1}^{n} i
\]

Integrals¶

% Simple integral
$\int f(x) \, dx$

% Definite integral
\[
\int_{0}^{\infty} e^{-x} \, dx = 1
\]

% Multiple integrals
\[
\iint_{D} f(x,y) \, dx \, dy
\]

\[
\iiint_{V} f(x,y,z) \, dx \, dy \, dz
\]

Integral variants: - \int Integral - \iint Double integral - \iiint Triple integral - \oint Contour integral - \oiint Surface integral

Limits¶

% Inline
$\lim_{x \to 0} \frac{\sin x}{x} = 1$

% Display
\[
\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e
\]

% One-sided limits
$\lim_{x \to 0^+} f(x)$  % Limit from right
$\lim_{x \to 0^-} f(x)$  % Limit from left

Other Large Operators¶

\[
\bigcup_{i=1}^{n} A_i  % Union
\]

\[
\bigcap_{i=1}^{n} A_i  % Intersection
\]

\[
\bigoplus_{i=1}^{n} V_i  % Direct sum
\]

\[
\coprod_{i=1}^{n} A_i  % Coproduct
\]

Delimiters¶

Delimiters are brackets, parentheses, and braces that enclose expressions.

Basic Delimiters¶

$(x + y)$          % Parentheses
$[x + y]$          % Square brackets
$\{x + y\}$        % Curly braces (escaped!)
$|x|$              % Absolute value bars
$\|x\|$            % Double bars (norm)
$\langle x \rangle$ % Angle brackets

Automatic Sizing with \left and \right¶

For properly sized delimiters:

% Too small (manual)
$(\frac{1}{2})$

% Automatic sizing
$\left(\frac{1}{2}\right)$

% More examples
\[
\left[ \sum_{i=1}^{n} x_i^2 \right]
\]

\[
\left\{ x \in \mathbb{R} : x^2 < 1 \right\}
\]

Important: \left and \right must be paired, even if one is invisible:

% Right delimiter only
\[
\left. \frac{dy}{dx} \right|_{x=0}
\]

The \left. creates an invisible left delimiter.

Manual Sizing¶

Fine control over delimiter size:

( \big( \Big( \bigg( \Bigg(

% Example
\[
\Bigg( \bigg( \Big( \big( ( x ) \big) \Big) \bigg) \Bigg)
\]

When to use manual sizing: - When automatic sizing is too large - For consistent sizing across multiple equations - For stylistic preferences

Common Delimiter Pairs¶

\left( x \right)           % Parentheses
\left[ x \right]           % Brackets
\left\{ x \right\}         % Braces
\left| x \right|           % Absolute value
\left\| x \right\|         % Norm
\left\langle x \right\rangle  % Angles
\left\lfloor x \right\rfloor  % Floor
\left\lceil x \right\rceil    % Ceiling

Dots (Ellipsis)¶

Different types of dots for different contexts:

% Centered dots (multiplication, etc.)
$a \cdot b \cdot c$
$x_1 \cdot x_2 \cdots x_n$

% Low dots (sequences, lists)
$a_1, a_2, \ldots, a_n$

% Vertical dots (matrices)
\[
\begin{matrix}
a_{11} \\
\vdots \\
a_{n1}
\end{matrix}
\]

% Diagonal dots (matrices)
\[
\begin{matrix}
a_{11} & & \\
& \ddots & \\
& & a_{nn}
\end{matrix}
\]

Commands: - \cdots Centered dots (···) - \ldots Low dots (...) - \vdots Vertical dots (⋮) - \ddots Diagonal dots (⋱)

Text in Math Mode¶

Sometimes you need words within mathematics:

% Wrong way (spacing is off)
$x is positive$

% Correct way
$x \text{ is positive}$

% Another example
\[
f(x) = \begin{cases}
x^2 & \text{if } x \geq 0 \\
-x^2 & \text{if } x < 0
\end{cases}
\]

Commands: - \text{...} Normal text (adapts to surrounding style) - \textrm{...} Roman text - \textit{...} Italic text - \textbf{...} Bold text

For single letters in upright (non-italic):

$\mathrm{d}x$  % Upright d for differential
$\mathrm{e}^x$ % Upright e for Euler's number

Spacing in Math Mode¶

LaTeX handles spacing automatically, but sometimes you need manual control:

% No space
$xy$

% Thin space
$x\,y$

% Medium space
$x\:y$

% Thick space
$x\;y$

% Quad space (1em)
$x\quad y$

% Double quad (2em)
$x\qquad y$

% Negative space
$x\!y$

Common uses:

$\int f(x) \, dx$          % Thin space before dx
$f(x) = 0 \quad \text{if}$ % Quad for text separation
$e^{i\pi} \!+ 1 = 0$       % Negative space for tightening

Common Mathematical Symbols¶

Relational Operators¶

$x < y$         % Less than
$x > y$         % Greater than
$x \leq y$      % Less than or equal
$x \geq y$      % Greater than or equal
$x = y$         % Equals
$x \neq y$      % Not equal
$x \equiv y$    % Equivalent
$x \approx y$   % Approximately equal
$x \sim y$      % Similar to
$x \cong y$     % Congruent to
$x \propto y$   % Proportional to

Set Operators¶

$x \in A$              % Element of
$x \notin A$           % Not an element of
$A \subset B$          % Subset
$A \subseteq B$        % Subset or equal
$A \supset B$          % Superset
$A \supseteq B$        % Superset or equal
$A \cup B$             % Union
$A \cap B$             % Intersection
$A \setminus B$        % Set difference
$\emptyset$            % Empty set
$\mathbb{N}$           % Natural numbers
$\mathbb{Z}$           % Integers
$\mathbb{Q}$           % Rationals
$\mathbb{R}$           % Reals
$\mathbb{C}$           % Complex numbers

Logic Operators¶

$\land$         % And
$\lor$          % Or
$\lnot$         % Not
$\forall$       % For all
$\exists$       % There exists
$\nexists$      % Does not exist
$\implies$      % Implies
$\iff$          % If and only if

Arrows¶

$\rightarrow$ or $\to$        % Right arrow
$\leftarrow$ or $\gets$       % Left arrow
$\leftrightarrow$             % Left-right arrow
$\Rightarrow$                 % Double right arrow (implies)
$\Leftarrow$                  % Double left arrow
$\Leftrightarrow$             % Double left-right (iff)
$\mapsto$                     % Maps to
$\longmapsto$                 % Long maps to
$\uparrow$                    % Up arrow
$\downarrow$                  % Down arrow
$\updownarrow$                % Up-down arrow

Other Symbols¶

$\infty$        % Infinity
$\partial$      % Partial derivative
$\nabla$        % Nabla (gradient)
$\pm$           % Plus-minus
$\mp$           % Minus-plus
$\times$        % Times (cross product)
$\div$          % Division
$\cdot$         % Centered dot (multiplication)
$\circ$         % Circle (composition)
$\star$         % Star
$\dagger$       % Dagger
$\ddagger$      % Double dagger
$\perp$         % Perpendicular
$\parallel$     % Parallel
$\angle$        % Angle
$\triangle$     % Triangle

Complete Examples¶

Example 1: Quadratic Formula¶

\documentclass{article}
\usepackage{amsmath}

\begin{document}

The solutions to the quadratic equation $ax^2 + bx + c = 0$ are given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where $a \neq 0$.

\end{document}

Example 2: Calculus¶

The derivative of $f(x) = x^n$ is:
\[
\frac{d}{dx}(x^n) = nx^{n-1}
\]

The fundamental theorem of calculus states:
\[
\int_a^b f(x) \, dx = F(b) - F(a)
\]
where $F'(x) = f(x)$.

Example 3: Linear Algebra¶

For vectors $\vec{u}, \vec{v} \in \mathbb{R}^n$, the dot product is:
\[
\vec{u} \cdot \vec{v} = \sum_{i=1}^{n} u_i v_i
\]

The magnitude of a vector is:
\[
\|\vec{v}\| = \sqrt{\sum_{i=1}^{n} v_i^2}
\]

Example 4: Sets and Logic¶

For sets $A, B \subseteq X$:
\[
(A \cup B)^c = A^c \cap B^c \quad \text{(De Morgan's Law)}
\]

For all $x \in \mathbb{R}$:
\[
\forall \epsilon > 0, \; \exists \delta > 0 \text{ such that } |x - a| < \delta \implies |f(x) - f(a)| < \epsilon
\]

Exercises¶

Exercise 1: Basic Symbols¶

Write LaTeX code to produce: - α² + β² = γ² - x ∈ ℝ, y ∈ ℂ - A ⊆ B ⇒ A ∪ B = B

Exercise 2: Fractions and Roots¶

Typeset these expressions: - The fraction (x+1)/(x-1) - The square root of (a²+b²) - The cube root of 27 - A nested fraction: 1/(1+1/(1+1/2))

Exercise 3: Summations and Products¶

Write: - The sum from i=1 to n of i² - The product from k=1 to n of (1 + 1/k) - The double sum: ∑∑ aᵢⱼ

Exercise 4: Integrals¶

Typeset: - ∫₀^∞ e^(-x) dx - The double integral ∬_D f(x,y) dA - The contour integral ∮_C z dz

Exercise 5: Delimiters¶

Write these with proper delimiter sizing: - The absolute value |x| - The set {x ∈ ℝ : x² < 4} - A large fraction in parentheses: ((a+b)/(c+d)) - The evaluated derivative [dy/dx]ₓ₌₀

Exercise 6: Greek Letters¶

Create a table showing all lowercase and uppercase Greek letters with their LaTeX commands.

Exercise 7: Arrows and Relations¶

Write: - f: A → B - x ≤ y ⇒ f(x) ≤ f(y) - A ⇔ B - lim as x→∞ of f(x) = L

Exercise 8: Complex Expression¶

Typeset the Cauchy-Schwarz inequality: [ \left| \sum_{i=1}^{n} x_i y_i \right| \leq \sqrt{\sum_{i=1}^{n} x_i^2} \sqrt{\sum_{i=1}^{n} y_i^2} ]

Exercise 9: Piecewise Function¶

Create a piecewise function definition using proper math formatting.

Exercise 10: Real Document¶

Create a short document (1 page) explaining a mathematical concept (your choice) that includes: - At least 3 display equations - At least 5 inline math expressions - Greek letters, fractions, and roots - At least one summation or integral - Proper use of text within math mode

Summary¶

In this lesson, you learned:

Math modes: Inline $...$ or $...$ and display \[...\]
amsmath package: Essential for mathematical typesetting
Greek letters: Lowercase and uppercase, variants
Subscripts/superscripts: _ and ^, combining them
Fractions: \frac{}{}, \dfrac{}{}, \tfrac{}{}
Roots: \sqrt{}, \sqrt[n]{}
Operators: Sums, products, integrals, limits
Delimiters: Automatic \left...\right and manual sizing
Dots: \cdots, \ldots, \vdots, \ddots
Text in math: \text{}
Spacing: \,, \:, \;, \!, \quad, \qquad
Symbols: Relations, sets, logic, arrows

You now have the foundation for mathematical typesetting in LaTeX. In the next lessons, we'll explore advanced math environments (matrices, multi-line equations, alignment) and other essential features.

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