A practice space to learn LaTeX from scratch, experiment with live maths, and grow into clear writers of integrals, series, matrices, and proofs.
LaTeX is the tool mathematicians use to write clearly and precisely. This page is a dojo for mathematical writing:
People who work through these modules and use LaTeX regularly get strong very quickly.
For anyone new to LaTeX. Learn symbols, inline formulas, displayed equations, and basic notation that appears in almost every mathematics course.
A deeper walk through limits, integrals, sums, matrices, vectors, and piecewise definitions. Designed to sit next to your analysis, algebra, and ODE notes.
Full-document LaTeX: structuring articles, theorems and proofs, references, and Beamer slides for talks, projects, and competitions like the ODE–Integration Bee.
Start here if LaTeX is new for you. By the end of this module, you should be comfortable writing basic symbols in sentences and on separate lines.
Use $ ... $ when the maths lives inside a sentence, and
$$ ... $$ when the maths should sit on its own line.
Examples:
The function $f(x) = x^2$ is continuous.$$f(x) = x^2 + 1$$
Superscripts use ^, subscripts use _.
For more than one character, wrap them in curly braces {...}.
x^2, \alpha_n, a_{n+1}, x^{2n+1}
Most Greek letters are written with a backslash and their name:
\alpha, \beta, \gamma, \lambda, \pi, \sigma.
Some very common symbols:
\infty, \leq, \geq, \neq, \mathbb{R}, \mathbb{N}.
For analysis, ODE, and the ODE–Integration Bee: fractions, limits, integrals, and series.
Fractions: \frac{numerator}{denominator}
Roots: \sqrt{x}, and \sqrt[n]{x}.
$$\frac{1}{1+x^2}$$$$\sqrt{x^2+1},\quad \sqrt[n]{1+x}$$Some standard patterns:
$$\lim_{x \to 0} \frac{\sin x}{x} = 1$$$$f'(x) = \frac{d}{dx} f(x)$$$$\int_0^1 x^n\,dx$$
(note the thin space \, before dx)
Summations and products:
$$\sum_{n=0}^{\infty} a_n x^n$$$$\prod_{k=1}^{n} k = n!$$For vectors, matrices, eigenvalues, and inner products.
Use bold or arrows for vectors, and double bars for norms:
$\mathbf{v} = (v_1,\dots,v_n)$$\langle x, y \rangle = \sum_{i=1}^n x_i y_i$$\|x\| = \sqrt{\langle x, x \rangle}$
A common environment is pmatrix (parentheses matrix):
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}For writing clean proofs: quantifiers, implication, align, and piecewise definitions.
Some useful symbols:
\forall x \in \mathbb{R} (for all)\exists \varepsilon > 0 (there exists)A \implies B, \quad A \iff B
Use align to line up equal signs and make derivations easy to read:
\begin{align}
f(x) &= x^2 + 1 \\
&= (x+1)^2 - 2x
\end{align}
Piecewise functions are written with cases:
f(x) =
\begin{cases}
0, & x < 0, \\
x^2, & x \ge 0.
\end{cases}For full reports or project texts. This lab page cannot compile full PDFs, but you can copy these templates to Overleaf or your own LaTeX setup.
\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\begin{document}
\title{Sample Notes}
\author{Your Name}
\date{}
\maketitle
\section{Introduction}
This is where your text begins. Inline maths like $f(x)=x^2$ and
displayed equations like
\[
\int_0^1 x^n\,dx = \frac{1}{n+1}.
\]
\end{document}amsthm)\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\begin{theorem}
Let $f$ be continuous on $[a,b]$. Then $f$ is bounded.
\end{theorem}
\begin{proof}
Sketch your proof here, step by step.
\end{proof}Copy this into a full document to practise writing proper theorem/proof environments.
Type in the editor. The preview shows the rendered mathematics. Treat this as your sandbox for problems, notes, and solutions.
$...$ for inline maths and $$...$$ or \[...\] for displayed
equations.
Write as if you are preparing a solution for the Problems page.
\alpha, \beta, \gamma, \lambda, \pi\leq, \geq, \neq, \approx\in, \subset, \subseteq, \supset\mathbb{R}, \mathbb{N}, \mathbb{Z}\frac{a}{b}, \sqrt{x}, \sqrt[n]{x}\sum_{n=0}^{\infty}, \prod_{k=1}^n\int_a^b f(x)\,dx\begin{pmatrix} ... \end{pmatrix}\forall, \exists\implies, \iffA \cup B, A \cap B|A|, \#A (cardinality)\begin{align} ... \end{align}\begin{cases} ... \end{cases}\text{...} for words inside maths