Chapter 1 · LaTeX

Everything below is a live playground — edit the left side, the math re-renders instantly. You cannot break anything; errors show in red and vanish when fixed. Ten topics, from first symbol to publication-grade formulas — roughly 4 hours, best split over several days (the streak system will thank you).
🎯 New: each topic ends with a “write this in LaTeX” challenge — we show the math, you reproduce it, the page tells you instantly when your version matches.

Topics 1 Mental model 2 Symbols 3 Real formulas 4 Brackets & multi-line 5 Matrices 6 Documents 7 Decorations 8 Calculus 9 Fine-tuning 10 Grand formulas ✅ Self-exam

🧠 Before you start (30 seconds)Try to answer these from intuition — being wrong is fine, guessing first is proven to make the real answers stick better:

How do you think LaTeX makes a fraction?

You guessed something like frac? It's \frac{a}{b} — command, then two brace arguments. Details in Topic 1.

What might the symbol for 'infinity' be called?

\infty — most symbols are just their English name with a backslash. Topic 2 has the greatest hits.

Guess: how would LaTeX tell a fraction's top from its bottom?

Braces: first {top} then {bottom}. Order and braces are the whole grammar — Topic 3.

1What LaTeX is (and the 60-second mental model)

LaTeX is not a text editor — it's a language: you type plain text with commands like \frac{a}{b}, and a program turns it into perfectly typeset output. Every command starts with a backslash \, arguments go in braces { }. That's 90% of the mental model.

EASYYour first formula — edit it!

This playground renders as you type. Change the 2 to a 3. Break it. Fix it.

💡 ^ makes a superscript (exponent). _ makes a subscript.

EASYSubscripts and superscripts

💡 Braces group things: try x_{10} vs x_10 to see why braces matter.

LEVEL UPCommands take arguments in braces

💡 \frac{top}{bottom} takes TWO arguments. \sqrt takes an optional [n] for nth roots.

LEVEL UPNesting: commands inside commands

💡 Any argument can itself contain commands — that's how complex formulas are built.

🎯 Can you write this? (superscripts + subscripts)

a_n = a_1 r^{n-1}

Topic 1 completes automatically — read to the end and clear its exercises +15 XP

2Greek letters & the symbols mathematicians actually use

Every Greek letter is a command: \alpha → α. Capitalize the command for capitals: \Delta → Δ. You'll memorize the ~15 you use, and look up the rest.

EASYThe greatest hits

💡 Try \Omega, \phi, \varphi, \epsilon, \varepsilon.

EASYComparison & arrows

💡 \le ≤, \ge ≥, \ne ≠, \approx ≈, \to →, \Rightarrow ⇒, \infty ∞.

LEVEL UPSets and logic

💡 \mathbb{R} = blackboard bold. \in / \notin = set membership. \; adds a little space.

LEVEL UPOperators that size themselves

💡 On \sum and \prod, _ and ^ become the lower/upper limits automatically.

🎯 Your turn — an infinite sum with Greek letters:

\sum_{k=1}^{\infty} \frac{\lambda^k}{k!} = e^{\lambda} - 1

Topic 2 completes automatically — read to the end and clear its exercises +15 XP

3Fractions, roots, integrals, limits — building real formulas

Real formulas are just the pieces you've seen, combined. The skill is reading a formula outside-in: find the outermost structure first, then fill the slots.

EASYThe quadratic formula, step by step

Outer structure: a fraction. Top slot: -b \pm \sqrt{...}. Bottom slot: 2a.

💡 \pm gives the ± sign.

EASYA definite integral

💡 \int works like \sum: _ lower limit, ^ upper limit. \, is a thin space before dx.

LEVEL UPLimits and derivatives

💡 \lim_{...} puts its argument underneath in display mode.

LEVEL UPPutting it all together

💡 Read it outside-in: an integral, with limits, of a product of two powers.

🎯 Write this Gaussian integral:

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

Topic 3 completes automatically — read to the end and clear its exercises +15 XP

4Brackets that grow & multi-line math

Plain parentheses stay small around tall content — ugly. \left( and \right) grow to fit. For multi-line derivations, aligned lines up the = signs: & marks the alignment column, \\ ends a line.

EASYSmall vs growing brackets

💡 Every \left needs a matching \right. Use \left. or \right. for an invisible side.

EASYAll the bracket families

💡 Braces must be escaped: \{ \}. Try \left\langle ... \right\rangle too.

LEVEL UPA two-line derivation with aligned

💡 & marks where lines align (before the =). \\ starts a new line.

LEVEL UPCases (piecewise functions)

💡 \text{...} switches to normal words inside math mode.

🎯 Recreate this piecewise definition:

f(x) = \begin{cases} x^2 & x \ge 0 \\ -x & x < 0 \end{cases}

Topic 4 completes automatically — read to the end and clear its exercises +15 XP

5Matrices & vectors

Matrices are grids: & separates columns, \\ separates rows. The environment name picks the brackets: pmatrix ( ), bmatrix [ ], vmatrix | |.

EASYA 2×2 matrix

💡 Change pmatrix to bmatrix or vmatrix and watch the brackets change.

EASYColumn vectors

💡 \vec{v} draws the arrow. One column = no & needed, just \\\\ between rows.

LEVEL UPMatrix × vector

💡 This is the 2D rotation matrix — you'll animate exactly this in the Manim chapter.

LEVEL UPA determinant with dots

💡 \cdots horizontal, \vdots vertical, \ddots diagonal dots.

🎯 Build this matrix equation:

\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}^n = \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix}

Topic 5 completes automatically — read to the end and clear its exercises +15 XP

6Complete documents (the part Manim does for you)

Everything so far was math mode — which is all Manim needs. But a standalone LaTeX document wraps that math in a small skeleton. Learn the skeleton once, mostly so error messages make sense.

EASYThe minimal document

This is a full compilable .tex file (paste into any LaTeX editor — this one isn't live because it's a document, not a formula):

\documentclass{article}
\begin{document}
Hello, \LaTeX!  Inline math: $E = mc^2$.
\end{document}

EASYInline vs display math

Euler proved that $e^{i\pi} + 1 = 0$   % inline: flows with text
\[ e^{i\pi} + 1 = 0 \]                  % display: own centered line

Same formula, two contexts. In Manim you'll only ever write the math part.

LEVEL UPSections, packages, labels

\documentclass{article}
\usepackage{amsmath}          % better math environments
\begin{document}
\section{The identity}
As shown in equation~\eqref{eq:euler}:
\begin{equation}\label{eq:euler}
  e^{i\pi} + 1 = 0
\end{equation}
\end{document}

LEVEL UPHow Manim uses this

# Manim writes the document FOR you. When you type:
MathTex(r"e^{i\pi} + 1 = 0")
# ...Manim generates a tiny document around your math,
# compiles it, and turns the result into animatable shapes.
# The r"" (raw string) stops Python from eating your backslashes!

This is why the math-mode skills above transfer 1:1 to Manim.

🎯 Final boss — everything at once:

\left( \sum_{i=1}^{n} x_i^2 \right)^{1/2} \le \sum_{i=1}^{n} |x_i|

Topic 6 completes automatically — read to the end and clear its exercises +15 XP

7Decorations: vectors, hats, braces that explain

Real math is annotated math: arrows over vectors, hats on estimates, braces that point at part of a formula and name it. These commands wrap around anything.

EASYAccents on single letters

💡 \dot / \ddot are time-derivatives in physics. \hat is 'estimate' in statistics.

EASYWide versions stretch over groups

💡 \overline stretches; \bar does not. Same idea: \widehat vs \hat.

LEVEL UPBraces that label parts of a formula

💡 _ after \underbrace puts the label BELOW; ^ after \overbrace puts it ABOVE.

LEVEL UPBoxing the result

💡 \boxed{...} draws a frame — perfect for final answers on slides.

🎯 Label this sum with an underbrace:

\underbrace{\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots}_{= 1}

Topic 7 completes automatically — read to the end and clear its exercises +20 XP

8Calculus notation: derivatives, partials, big integrals

The notation of calculus is its own dialect: primes, Leibniz fractions, curly partials, double integrals, closed-loop integrals, and evaluation bars.

EASYThree ways to write a derivative

💡 ' is just an apostrophe. The Leibniz form is an ordinary \frac.

EASYPartial derivatives and the gradient

💡 \partial is the curly d. \nabla is the gradient triangle.

LEVEL UPBigger integrals

💡 \iint double, \iiint triple, \oint closed loop. \, adds the thin space before dx.

LEVEL UPEvaluation bars and limits underneath

💡 \left. is an invisible left wall so \right| can grow. \lim takes limits with _.

🎯 Write the heat equation:

\frac{\partial u}{\partial t} = \alpha \, \frac{\partial^2 u}{\partial x^2}

Topic 8 completes automatically — read to the end and clear its exercises +20 XP

9Fine-tuning: spacing, color, size — typesetting like a pro

The difference between 'renders' and 'beautiful' is spacing and emphasis. These are the knobs professionals turn — and in Manim they work exactly the same inside MathTex.

EASYThe spacing ladder

💡 From squeeze to gap: \! negative, \, thin, \; medium, \quad wide, \qquad double.

EASYWords and proper operator names

💡 \operatorname gives upright type + correct spacing — sin/cos/log already exist: \sin \log.

LEVEL UPColor for emphasis (works in KaTeX and Manim)

💡 \textcolor{name}{...}. In Manim you'd more often use set_color_by_tex.

LEVEL UPForcing display style in tight places

💡 \displaystyle makes things full-size; \textstyle compact. Useful inside fractions and tables.

🎯 Highlight the discriminant in orange:

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

Topic 9 completes automatically — read to the end and clear its exercises +20 XP

10Grand formulas: multi-line proofs & famous equations

Everything combines here. If you can typeset these, you can typeset anything you will ever put on a slide — this is the level of a published paper.

EASYA real chained derivation

💡 Three lines, one story. & always sits before the =.

LEVEL UPStacked conditions under a sum

💡 \substack stacks multiple lines under one \sum. \lceil\rceil are ceiling brackets.

LEVEL UPThe Fourier series — a formula with everything

💡 Fractions, subscripts, an infinite sum, growing brackets, operators — all four chapters of skills.

LEVEL UPMaxwell (integral form) — the boss fight

💡 Closed-loop integral, vectors, Leibniz derivative, double integral. You know every piece now.

🎯 The final exam — Euler's identity, derived:

e^{i\pi} + 1 = 0 \quad \text{because} \quad e^{i\theta} = \cos\theta + i\sin\theta

Topic 10 completes automatically — read to the end and clear its exercises +25 XP

Self-examination

No peeking at the topics above — that's the exam part. Solutions are one click away.

Which of these renders a fraction one-half?

\fraction{1}{2}1 \over{2} \frac\div{1}{2}\frac{1}{2}

What does the & symbol do inside an aligned or matrix environment?

Marks the column/alignment pointInserts extra spaceMakes the next character boldStarts a comment

Why must Python strings for Manim's MathTex be written as r\"...\"?

It's just traditionSo Python doesn't treat backslashes as escapes (\t, \n...)It renders fasterIt makes the text red

Write the LaTeX for: the sum from k = 0 to n of (n choose k) x^k = (1+x)^n. (Binomial coefficient is \binom{n}{k}.) Try it in any playground above first!

show solution

\sum_{k=0}^{n} \binom{n}{k} x^k = (1+x)^n

Typeset a 3×3 identity matrix with square brackets.

show solution

I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

Typeset the piecewise sign function: sgn(x) = -1 if x < 0, 0 if x = 0, 1 if x > 0.

show solution

\operatorname{sgn}(x) = \begin{cases} -1 & x < 0 \\ 0 & x = 0 \\ 1 & x > 0 \end{cases}