Based on the r/learnmath thread "Full Guide to Self-Studying Math" by u/RecipeBeneficial6378 (who self-studied math & physics for ~2.5 years via an MIT-Challenge-style project), including corrections and additions from the comment section — most notably the textbook-focused critique by u/Reshi86.
Time estimates assume 10–15 hrs/week. Double the pace if studying full-time. Total path: ~3–5 years part-time to Phase 4; most people should stop wherever their goal is met.
The #1 lesson from the thread's comments: videos build intuition, but you only learn math by doing a LOT of problems. The most upvoted critique (u/Reshi86) warned that a video-only path "would lead learners to dead ends." Every subject below follows the same three-layer method:
- Intuition layer — a visual/overview series (3Blue1Brown-style). Watch fast, don't take notes, just get the shape of the subject.
- Lecture layer — a full video course (Professor Leonard, MIT OCW, etc.). Pause and attempt every example before the lecturer solves it.
- Problem layer — textbook exercises + MIT OCW problem sets/recitations/past exams. This is where learning actually happens. Aim for 2–3 hours of problems per 1 hour of video.
Daily workflow that works: 25% watching/reading new material, 60% solving problems, 15% reviewing yesterday's mistakes and redoing failed problems from a week ago (spaced repetition on problems, not flashcards).
Rules:
- Don't move on until you pass the subject's exit checklist (given below for each subject).
- Keep an error log: every problem you get wrong goes in a notebook with why you got it wrong. Redo it 3 days later.
- If stuck > 30–45 min on one problem, look at the solution, close it, and rewrite it from scratch the next day.
- Within a phase, you can run 2 subjects in parallel (one computational + one conceptual). Never 3.
Skip only if you can comfortably manipulate fractions, exponents, negative numbers, and simple equations. When in doubt, spend 2 weeks here — gaps at this level poison everything above.
Time: 2–8 weeks
Topics to master: integer & fraction arithmetic, decimals & percents, ratios and proportions, order of operations, exponent rules, radicals, negative numbers, basic geometry (area, perimeter, angles), reading word problems.
Resources:
- Professor Leonard — Pre-Algebra playlist (video) — recommended in the comments for anyone with weak basics who finds Khan Academy boring
- Khan Academy Arithmetic/Pre-Algebra — good for the practice exercises even if you watch Leonard for lectures
Exit checklist: you can simplify any expression with fractions/exponents/parentheses without a calculator, and solve percent/ratio word problems reliably.
Time: 4–9 months total
Topics to master: linear equations & inequalities, systems of equations, polynomials & factoring, rational expressions, radicals & rational exponents, quadratic equations (factoring, completing the square, quadratic formula), functions & graphs, exponential & logarithmic basics, word-problem modeling.
Resources:
- Professor Leonard — Intermediate Algebra (video): simple, motivated, tons of worked examples — playlist
- Prof Rob Bob — Algebra 1 & 2 (video): best for the problem-solving side — Algebra 1 · Algebra 2
- Khan Academy — Algebra Foundations (video): best for conceptual "clicks," builds algebra from its origins — playlist
How to study: Leonard for main lectures; Khan for anything that doesn't click conceptually; Rob Bob when you need more worked examples of a problem type. Do every Khan Academy exercise set to mastery.
Exit checklist: solve any quadratic three different ways; graph a function and its transformations from the equation alone; set up and solve a mixture/rate/work word problem without help.
Topics to master: unit circle (cold, from memory), radian/degree fluency, the six trig functions & their graphs, inverse trig functions, trig identities (Pythagorean, sum/difference, double/half angle), solving trig equations, law of sines & cosines, applications (triangles, periodic motion).
Resources:
- Professor Leonard — Trigonometry (video): maps the whole journey, connects each lesson to the next — playlist
- Khan Academy — Trigonometry (video): great at connecting ideas — playlist
Exit checklist: reproduce the unit circle in under 2 minutes; prove 10 identities in a row without hints; solve trig equations with multiple solutions on a given interval.
Topics to master: function composition & inverses, polynomial & rational function behavior (end behavior, asymptotes, sketching), exponentials & logs in depth, sequences & series, conic sections, parametric equations, polar coordinates, vectors & basic matrices, intro limits.
Resources:
- Khan Academy — Precalculus (video): gap-free, ground-up — playlist
- Professor Leonard — Precalculus (video) — playlist
OP's sequence: Khan Academy → Professor Leonard.
Also mentioned in comments as easy-to-follow alternatives for this whole phase: Prof Richard Delaware (UMKC, YouTube) and Jason Gibson's math courses.
Exit checklist: sketch a rational function (asymptotes, intercepts, holes) by hand; solve exponential/log equations fluently; explain intuitively what a limit is. This is the gate to calculus — be honest with yourself here.
Time: 12–20 months total (subjects 4–5 are the priority; 6–8 order is flexible)
Topics to master:
- Calc I: limits & continuity, derivative definition and rules, implicit differentiation, related rates, optimization, curve sketching, Mean Value Theorem, Riemann sums → definite integral, Fundamental Theorem of Calculus, u-substitution
- Calc II: integration techniques (parts, trig sub, partial fractions), improper integrals, applications (volume, arc length, work), sequences & series, convergence tests, Taylor & power series, parametric/polar calculus
- Calc III: vectors & vector functions, partial derivatives, gradient, multiple integrals, change of variables/Jacobian, vector fields, line & surface integrals, Green's/Stokes'/Divergence theorems
Resources:
- 3Blue1Brown — Essence of Calculus (video): start here; minimal prerequisites, maximum "aha" — playlist
- Professor Leonard — Calc I, II, III (video): step-by-step through tons of examples — playlists
- The Math Sorcerer — Calculus lectures (video): leaves almost no gaps — playlist
- MIT OCW — 18.01 & 18.02 (video + problem sets + exams with solutions): deep, assumes solid prerequisites; the problem sets are the real prize — 18.01 · 18.02
⚠️ One commenter found MIT 18.01 too hard to follow as a first exposure — that's what Leonard is for. MIT comes second.
📚 Textbooks (from comments — OP's guide had none for calculus, the top critique):
- Calculus — Spivak (with its solutions manual): "you will not find a better treatment of single variable calculus." Rigorous — really a gentle intro to analysis. Do it alongside or after Leonard, not instead.
- Introduction to Calculus and Analysis, Vol. 1 & 2 — Courant: Vol. 1 = rigorous single-variable (can run parallel to Spivak), Vol. 2 = the recommended rigorous multivariable treatment (Spivak has no real multivariable book).
- (If Spivak is too steep at first, any standard text — Stewart, Thomas — supplies computational drill; return to Spivak before Real Analysis.)
Sequence: 3Blue1Brown (1–2 weeks) → Leonard Calc I + Math Sorcerer + drill problems → Leonard Calc II → MIT 18.01 problem sets & past exams as your test → Spivak → Leonard Calc III → MIT 18.02 problem sets → Courant Vol. 2 (optional rigor).
Exit checklist: score ≥70% on a past MIT 18.01 and 18.02 final under timed conditions; compute any standard integral without table lookup; state and use FTC, MVT, and Stokes' theorem on real problems.
Topics to master: systems & Gaussian elimination, matrix algebra, LU factorization, vector spaces & subspaces, linear independence/basis/dimension, the four fundamental subspaces, linear transformations, orthogonality & projections, Gram–Schmidt, least squares, determinants, eigenvalues & eigenvectors, diagonalization, symmetric matrices & spectral theorem, SVD, positive definiteness.
Resources:
- 3Blue1Brown — Essence of Linear Algebra (video): the canonical intuition-builder — playlist
- Gilbert Strang — MIT 18.06 (video + recitations + decades of problem sets and exams with solutions) — lectures · recitations
📚 Textbooks:
- Linear Algebra Done Right — Axler: proof-based, mathematician's perspective; determinant-free until the end (OP's pick for the second pass)
- Linear Algebra Done Wrong — Treil (free PDF): a commenter argued it's "10-fold better" than Axler — free, so try both and keep the one that fits
- Linear Algebra — Hoffman & Kunze (from comments): fully formal treatment that includes determinants properly
Sequence: 3Blue1Brown → Strang lectures + recitations + every problem set → (after the proofs bridge below) Axler or Treil as a second, proof-based pass.
Exit checklist: pass a past 18.06 final ≥70%; explain the four fundamental subspaces and rank-nullity in your own words; diagonalize a matrix and explain when you can't; (second pass) prove basic statements about linear maps without peeking.
Topics to master: logic & quantifiers, proof techniques (direct, contrapositive, contradiction, induction — this doubles as proof training), sets & functions & relations, counting/permutations/combinations, pigeonhole, inclusion–exclusion, recurrences, graph basics, modular arithmetic, basic probability on finite sets.
Resources:
- TrevTutor — Discrete Math 1 & 2 (video): simple, clear, example-heavy — DM1 · DM2
- Mathemaniac — Deep Dive into Combinatorics (video): lovely visuals, combinatorics intuition (not comprehensive) — playlist
- MIT — Mathematics for Computer Science (video + large, varied problem sets; free course textbook PDF on OCW) — playlist
Sequence: TrevTutor → Mathemaniac → MIT course with its problem sets.
Exit checklist: write a correct induction proof unaided; solve counting problems mixing combinations, inclusion–exclusion, and pigeonhole; this subject overlaps heavily with the proofs bridge — treat it as a warm-up for pure math.
Topics to master: sample spaces & axioms, conditioning & Bayes, independence, discrete & continuous random variables, expectation/variance/covariance, common distributions, joint distributions, law of large numbers, central limit theorem, Markov chains (intro), estimators, confidence intervals, hypothesis testing, maximum likelihood, linear regression.
Resources:
- MIT 6.041 — Probabilistic Systems Analysis (John Tsitsiklis) (video): superb intuition and analogies; excellent recitations + problems — playlist
- MIT 18.650 — Statistics for Applications (Philippe Rigollet) (video): motivated, formal, application-rich — playlist
Sequence: Tsitsiklis (probability) → Rigollet (statistics). Rigollet assumes calculus + linear algebra + Tsitsiklis-level probability.
Exit checklist: solve Bayes-rule word problems cold; derive expectation/variance of the standard distributions; explain the CLT and use it; run and interpret a hypothesis test by hand.
Topics to master: first-order ODEs (separable, linear, exact), existence/uniqueness (statement), second-order linear with constant coefficients, undetermined coefficients & variation of parameters, oscillations/resonance, Laplace transforms, systems of ODEs & eigenvalue methods, phase portraits & qualitative analysis, intro numerical methods (Euler).
Resources:
- 3Blue1Brown — Differential Equations (video): intuitive overview first — playlist
- The Math Sorcerer — ODE lectures (video): many example variations, frequent concept reviews — playlist
- Professor Leonard — ODE lectures (video) — playlist
- MIT 18.03 (lectures + recitations; OP: go to older versions of the course for the biggest problem archive) — lectures · recitations
Sequence: 3Blue1Brown → Leonard + Math Sorcerer → MIT 18.03 problems & recitations.
Exit checklist: classify and solve any first/second-order ODE type above on sight; solve a system via eigenvalues and sketch its phase portrait; solve an IVP with Laplace transforms.
The comment section's strongest addition: "I question anyone's ability to dive into any of the pure mathematics topics without a study of set theory and proof writing." Everything after this point is proof-based; this phase is the difference between reading proofs and writing them.
Topics to master: propositional & predicate logic, quantifier manipulation, direct proof, contrapositive, contradiction, induction (weak, strong, structural), sets & set operations & proofs about sets, relations & equivalence classes, functions (injective/surjective/bijective) with proofs, cardinality & countability, basic proof style/writing.
Resources:
- 📚 Book of Proof — Hammack (free online, with odd-numbered solutions) — the recommended starting point; do essentially every exercise
- 📚 How to Prove It — Velleman — the alternative, slightly more thorough on logic
- Bonus: chapter 1 of Munkres' Topology is a solid set theory treatment you'll reuse in Phase 3
How to study: this phase is 90% writing. For every theorem, close the book and reprove it. Have your proofs "graded" harshly — compare word-for-word against the book's solutions and note every unjustified step.
Exit checklist: prove √2 is irrational, that there are infinitely many primes, and that ℚ is countable but ℝ is not — all from memory, cleanly written; comfortably negate statements with nested quantifiers.
Time: 18–30 months if you do all of it. Priority order for most people: Real Analysis and Abstract Algebra first (they gate Phase 4), then Topology, then the rest by interest.
Topics to master: construction/axioms of ℝ, sup/inf & completeness, sequences & convergence, Bolzano–Weierstrass, Cauchy sequences, series & convergence tests (rigorously), topology of ℝ (open/closed/compact sets), limits & continuity via ε–δ, uniform continuity, differentiation (MVT proved), Riemann integral (constructed), sequences & series of functions, uniform convergence, power series.
Resources:
- Francis Su — Real Analysis lectures (video): historical perspective, never skips steps, great for intuition — playlist
- Michael Penn — Real Analysis (video): one concept per video, full proof walk-throughs — playlist
📚 Textbooks:
- Understanding Analysis — Abbott: OP's favorite; teaches the thought process behind proofs.
⚠️ Comments: it's a bridge book (between Spivak and Rudin), not a complete treatment — do it first, not only. - Principles of Mathematical Analysis — Rudin: "the default book for undergraduate real analysis… you will not find a better treatment." Terse; the exercises are the course.
- Mathematical Analysis — Apostol: same ground as Rudin with more handholding, if Rudin's terseness is a wall.
Sequence: Abbott (with Su/Penn videos in parallel) → Rudin chapters 1–7 (or Apostol), doing the bulk of the exercises.
Exit checklist: produce ε–δ proofs unaided; prove Bolzano–Weierstrass and the Extreme Value Theorem from memory; determine (with proof) whether a given sequence of functions converges uniformly.
Topics to master: groups, subgroups, cyclic groups, permutation groups, cosets & Lagrange, normal subgroups & quotients, homomorphisms & isomorphism theorems, group actions, Sylow theorems (at least statements + use), rings, ideals & quotient rings, integral domains, polynomial rings, fields, field extensions, intro Galois theory (note: the video course below skips Galois — use the book).
Resources:
- 📚 Abstract Algebra: A Computational Introduction — Scherk: OP's favorite; lots of examples, intuition with rigor — your primary text
- Math Major — Abstract Algebra lectures (video): covers nearly everything except Galois theory; high production, thorough steps — playlist
- Harvard — Abstract Algebra (Benedict Gross) (video): great insights, formal proofs; sometimes calls non-trivial things "trivial" — playlist
- (Standard extra problem sources if needed: Dummit & Foote as a reference, Gallian for gentler exercises.)
Sequence (OP's): Scherk + Math Major together → Harvard lectures as the deeper second pass.
Exit checklist: prove Lagrange's theorem and the first isomorphism theorem from memory; classify all groups of order ≤ 8; work fluently in ℤ/nℤ, Sₙ, and polynomial rings; decide whether a given subset is a subgroup/ideal quickly.
Topics to master: topological spaces & bases, open/closed sets, closure/interior/boundary, continuity (topologically), homeomorphisms, subspace/product/quotient topologies, connectedness & path-connectedness, compactness (incl. Heine–Borel, Tychonoff statement), separation axioms & Hausdorff spaces, metric spaces & metrizability, convergence.
Resources:
- Fred Schuller — Topology videos (from Geometrical Anatomy of Theoretical Physics): OP: "the best professor I've encountered, period" — insightful, rigorous, unique perspectives — playlist
📚 Textbooks:
- Topology — Munkres: the standard undergrad text (from comments). The OP disliked its exposition but concedes the problems are good — which is what you need it for.
- General Topology — Willard: strong alternative (from comments)
- Schaum's Outline of General Topology: OP's pick — clear + lots of solved problems.
⚠️ Comments: it's an outline, not a textbook — use it as a supplement for problem volume.
Sequence: Schuller for lectures/intuition → Munkres (or Willard) as the text + Schaum's for extra solved problems.
Exit checklist: prove that continuous images of compact/connected sets are compact/connected; construct the quotient topology on a simple example (e.g., circle from interval); give counterexamples separating the major properties (compact ≠ closed+bounded in general, etc.).
Topics to master: complex numbers & geometry, holomorphic functions & Cauchy–Riemann equations, elementary functions, contour integration, Cauchy's theorem & integral formula, Taylor & Laurent series, singularities & residues, residue theorem & real-integral applications, argument principle, conformal maps (intro).
Resources (OP's full sequence):
- Welch Labs — Imaginary Numbers Are Real (video): OP's "favourite math playlist ever" — start here — playlist
- Mathemaniac — Complex Analysis (video): gorgeous visualizations, intuition — playlist
- Math Major (video): step-by-step proofs — playlist + qncubed3 (video): example-heavy, especially contour integration — playlist
- MIT OCW 18.04 — Complex Variables with Applications: tons of problems — course
Exit checklist: evaluate real integrals via residues; find Laurent expansions and classify singularities; state and apply Cauchy's integral formula without notes.
Topics to master: divisibility & gcd/Euclidean algorithm, primes & fundamental theorem of arithmetic, congruences, Chinese Remainder Theorem, Fermat's little theorem & Euler's theorem, primitive roots, quadratic residues & quadratic reciprocity, arithmetic functions, continued fractions (intro), Diophantine equations.
Resources:
- Michael Penn — Number Theory (video): OP: "the best Number Theory course I've come across"; high quality, no skipped steps — playlist
- MIT OCW 18.781 — Theory of Numbers: clever problem sets — course
Exit checklist: solve simultaneous congruences via CRT; prove Fermat/Euler theorems; compute Legendre symbols with reciprocity.
Topics to master: graphs/degrees/handshake lemma, paths & connectivity, trees & spanning trees, Eulerian & Hamiltonian graphs, bipartite graphs & matchings (Hall's theorem), planarity & Euler's formula, coloring & chromatic number, network flows (if CS-leaning), graph algorithms (BFS/DFS, Dijkstra, MST).
Resources (OP's sequence):
- Reducible (video): high-level intuition — video
- Wrath of Math — Graph Theory (video): depth, visuals, many examples — playlist
- William Fiset (video): graph theory + algorithms with code — best if you're CS-leaning — playlist
Exit checklist: prove Euler's formula and the handshake lemma; determine planarity/colorability of small graphs with justification; implement BFS/DFS if algorithm-focused.
Topics to master: classification (elliptic/parabolic/hyperbolic), heat/wave/Laplace equations, separation of variables, Fourier series & transforms, Sturm–Liouville theory, d'Alembert's solution, Green's functions (intro), boundary value problems; (rigorous track:) Sobolev spaces, weak solutions.
Resources (OP's sequence):
- Commutant — PDE playlist (video): uniquely intuitive; covers only the basics — playlist
- MIT 18.303 — Linear PDEs (notes + excellent problem variations) — course
- 📚 Partial Differential Equations — Evans: the gold standard.
⚠️ Requires Real Analysis and Measure Theory first — defer to after Phase 4's measure theory.
Exit checklist: solve heat/wave/Laplace on standard domains via separation of variables; compute Fourier series and use them; (rigorous track) define weak derivative and Sobolev space.
Prerequisites: Real Analysis + Abstract Algebra + Topology, done properly. A commenter doubted anyone could follow these lectures without real textbook grounding in Phase 3 — take that seriously. Time: 1.5–3 years for all of it; nearly everyone should pick a track instead:
- Analysis track: Measure Theory → Functional Analysis → Evans PDE
- Geometry/physics track: Differential Geometry (+ Measure Theory helps)
- Algebra/topology track: Algebraic Topology → Algebraic Geometry → Category Theory
Topics: σ-algebras, measures, Lebesgue measure, measurable functions, Lebesgue integral, convergence theorems (MCT, DCT, Fatou), Lᵖ spaces, product measures & Fubini, Radon–Nikodym.
- Fred Schuller — Measure Theory videos: OP: "Anyone trying to understand Measure Theory NEEDS to watch this" — makes a hard subject seamless — playlist
- (Pair with a problem source; standard choices: Folland's Real Analysis or Rudin's Real and Complex Analysis, chs. 1–8.)
Exit checklist: state and apply DCT/MCT/Fatou correctly; explain why the Lebesgue integral extends the Riemann one; prove basic measurability results.
Topics: normed & Banach spaces, Hilbert spaces & orthonormal bases, bounded operators, Hahn–Banach, open mapping & closed graph theorems, uniform boundedness, dual spaces & weak topologies, spectral theory (intro), compact operators.
- Fred Schuller — Functional Analysis (from his Quantum Mechanics course): "amazing and a must-watch" — playlist
- MIT — Functional Analysis (video + strong problem sets; OP found the lectures hard to follow — lean on the problems) — playlist
Sequence: Schuller → MIT problem sets.
Topics: smooth manifolds, tangent spaces, vector fields & flows, differential forms, integration on manifolds & Stokes, Riemannian metrics, connections & curvature, geodesics; (physics-leaning: bundles, Lie groups).
- Fred Schuller — Geometrical Anatomy of Theoretical Physics: OP: the most valuable thing is the perspectives; insight-dense throughout — playlist
- (Pair with a problem source; standard choices: Lee's Smooth Manifolds or do Carmo.)
Topics: homotopy & fundamental group, covering spaces, Van Kampen, simplicial/singular homology, exact sequences, Mayer–Vietoris, cohomology (intro).
- Pierre Albin — lectures (video): "one of the clearest professors I've found"; builds the subject ground-up with history; problem sets included (
⚠️ no solutions) — playlist - Princeton qualifying oral exams (text): real problems + how students answered — great for calibrating your level — archive
- (Standard free text: Hatcher's Algebraic Topology.)
Topics: affine & projective varieties, Nullstellensatz, morphisms, sheaves (intro), schemes (glimpse).
- University of Waterloo — lectures (video): great intuition;
⚠️ some lectures missing, few examples — playlist - Princeton qualifying exams (text, with solutions — good for pattern-spotting and procedural skill) — archive
Topics: categories/functors/natural transformations, universal properties, limits & colimits, adjunctions, Yoneda lemma.
Resources (OP's sequence — three intros, then depth):
- Oliver Lugg — A Sensible Introduction to Category Theory — video
- Eyesomorphic — Introduction to Category Theory: beautiful visuals — video
- Feynman's Chicken — Introduction to Category Theory: overview + basic proofs — video
- MIT — Category Theory lectures: clear, concise, real-world applications — playlist
Pre-Algebra → Algebra → Trigonometry → Precalculus
│
▼
Calculus ──→ ODE ──→ PDE (computational)
│
Linear Algebra ◄────────────┤
Discrete Math ◄────────────┤
Prob & Stats ◄────────────┘
│
Proofs & Set Theory (bridge)
│
┌──────────────┬─────────────────┼──────────────┬────────────┐
▼ ▼ ▼ ▼ ▼
Real Analysis Abstract Algebra Topology Complex Analysis Number Theory /
│ │ │ Graph Theory
▼ ▼ ▼
Measure Theory Alg. Geometry Alg. Topology
│
├──→ Functional Analysis
├──→ PDE (rigorous / Evans)
└──→ Differential Geometry
Goal: "calculus-literate" (engineering/data science baseline) — Phases 0–2 only: ~1.5–2.5 years part-time.
Goal: full pure-math undergrad equivalent — Phases 0–3: ~3–4 years part-time. Order within Phase 3: Real Analysis → Abstract Algebra → Topology → Complex Analysis → (Number Theory / Graph Theory / PDE by interest).
Goal: graduate prep — add one Phase 4 track (~1 more year).
A typical week (12 hrs):
| Day | Time | Activity |
|---|---|---|
| Mon | 2h | New lectures + notes |
| Tue | 2h | Problems on Monday's material |
| Wed | 2h | New lectures + notes |
| Thu | 2h | Problems on Wednesday's material |
| Sat | 3h | Long problem session (textbook/OCW sets) |
| Sun | 1h | Error-log review; redo old failed problems |
- Intuition → Lectures → Textbook problems, in that order, for every subject.
- Problems are the actual learning. Videos are the map, not the territory. The whole comment-section debate boiled down to this: the OP's guide was video-rich and textbook-poor, and the fix is pairing every course with a real text and doing its exercises. Older MIT OCW course versions often have the biggest problem archives.
- Don't skip the proofs bridge. Set theory + proof-writing (Hammack/Velleman) is the gate between computational and pure math — this was the comment section's most emphatic addition.
- Bridge books are fine, but finish the job. Abbott before Rudin, Schaum's alongside Munkres — not instead of.
- One trusted path per subject beats five open tabs. The OP's "optimal sequences" exist to prevent resource-hopping.
- Test yourself under exam conditions. Past MIT exams (18.01, 18.02, 18.03, 18.06) and Princeton quals are free calibration tools at every level.
- Expect it to take time. The OP spent ~2.5 years full-time on math + physics. A 72-year-old commenter is relearning trig after a 40-year gap; another is 9 courses from a math degree. Scale the plan to your hours and goal — the phases are checkpoints, not a race.