The Best Free Online Mathematics Courses
A curated guide to the best free online mathematics courses from MIT, Stanford, Harvard, and other top institutions, organized by subject and level, with advice on how to get the most out of online learning.
The Golden Age of Free Mathematics Education
We live in an extraordinary time for learning mathematics. Some of the best courses ever taught — at institutions like MIT, Stanford, Harvard, and Oxford — are freely available online. You can watch the same lectures, work through the same problem sets, and study the same materials as students at the world's top universities, all without paying tuition.
This guide organizes the best free courses by subject and level, with honest assessments of what each course offers and who it serves best.
Calculus
MIT 18.01: Single Variable Calculus (David Jerison)
This is a complete first-year calculus course covering:
- Limits, derivatives, and applications of differentiation
- Techniques of integration
- Taylor series and infinite series
The Scholar version includes recitation videos, worked examples, and exams with solutions. David Jerison is an excellent lecturer who balances rigor with intuition.
MIT 18.02: Multivariable Calculus (Denis Auroux)
Covers:
- Vectors, matrices, and parametric equations
- Partial derivatives and gradient
- Double and triple integrals
- Line integrals, surface integrals, and the classical theorems (Green's, Stokes', divergence)
Tip: The MIT Scholar courses are specifically designed for self-study, with more supplementary materials than the standard OCW versions. Always look for the "SC" (Scholar Course) versions.
Linear Algebra
MIT 18.06: Linear Algebra (Gilbert Strang)
This is perhaps the most famous mathematics course on the internet. Gilbert Strang's teaching style is legendary — intuitive, geometric, and deeply insightful. The course covers:
- Systems of linear equations and Gaussian elimination
- Vector spaces, column space, null space
- Orthogonality and least squares
- Eigenvalues and eigenvectors
- Positive definite matrices and the SVD
The full video lectures, problem sets, exams, and solutions are available.
MIT 18.065: Matrix Methods in Data Analysis (Gilbert Strang)
A more modern follow-up covering linear algebra's applications to data science, including SVD, PCA, neural networks, and optimization.
Essence of Linear Algebra (3Blue1Brown)
Essence of Linear Algebra on YouTube
Not a traditional course, but this video series provides the geometric intuition that complements any formal linear algebra course. Watch it alongside 18.06 for the best experience.
Real Analysis
MIT 18.100A: Real Analysis (Casey Rodriguez)
A full real analysis course covering:
- Construction of the real numbers
- Sequences, series, and convergence
- Continuity and differentiability
- The Riemann integral
- Sequences of functions
Video lectures, problem sets, and exams are available.
Harvey Mudd Real Analysis Lectures (Francis Su)
Francis Su's Real Analysis on YouTube
Francis Su's lectures are celebrated for their clarity and warmth. These 27 lectures cover a standard undergraduate real analysis course and are especially good for students who find the subject intimidating.
Especially recommended for self-study: Su's lectures have a conversational quality that makes them particularly well-suited for independent learning. He addresses common confusions and motivates every definition.
Abstract Algebra
Harvard E-222: Abstract Algebra (Benedict Gross)
Benedict Gross's Algebra Lectures on YouTube
Benedict Gross, a number theorist, teaches abstract algebra with exceptional clarity. The course covers groups, rings, fields, and Galois theory. These lectures are often cited as among the best freely available algebra courses.
Visual Group Theory (Nathan Carter)
For a more visual approach, these lectures accompany the textbook Visual Group Theory and emphasize geometric and visual understanding of group theory concepts.
Differential Equations
MIT 18.03: Differential Equations (Arthur Mattuck)
A thorough course covering:
- First-order ODEs (separation, integrating factors, exact equations)
- Second-order linear equations
- Laplace transforms
- Systems of ODEs and phase plane analysis
- Fourier series
Arthur Mattuck's lectures are engaging and well-paced.
MIT 18.03x: Introduction to Differential Equations (on edX)
A MOOC version with interactive exercises and structured learning paths, based on the MIT course.
Probability and Statistics
Harvard Statistics 110: Probability (Joe Blitzstein)
Joe Blitzstein's probability course is a joy. He teaches with vivid stories, clever examples, and deep insight. Topics include:
- Counting and combinatorics
- Conditional probability and Bayes' theorem
- Random variables and distributions
- Expectation and variance
- The law of large numbers and the central limit theorem
- Markov chains
The companion textbook Introduction to Probability by Blitzstein and Hwang is available through the course website.
MIT 6.041: Probabilistic Systems Analysis (John Tsitsiklis)
A more applications-oriented probability course with full video lectures, problem sets, and exams.
Topology
Topology (Tadashi Tokieda)
Tokieda's Topology Lectures on YouTube
Tadashi Tokieda's lectures from the African Institute for Mathematical Sciences (AIMS) are unconventional and brilliant. He develops topological ideas through hands-on experiments and physical intuition.
Algebraic Topology (Pierre Albin)
Pierre Albin's Algebraic Topology on YouTube
A more standard graduate-level course covering fundamental groups, covering spaces, and homology, following Hatcher's textbook.
Number Theory
MIT 18.781: Theory of Numbers (Abhinav Kumar)
Covers elementary number theory: divisibility, primes, congruences, quadratic reciprocity, and more.
Introduction to Number Theory (Michael Penn on YouTube)
Michael Penn has an extensive collection of number theory videos covering topics from elementary to analytic number theory. Search his YouTube channel for specific topics.
Advanced and Graduate-Level Courses
Functional Analysis (Frederic Schuller)
Frederic Schuller's Lectures on YouTube
Part of his "Geometric Anatomy of Theoretical Physics" series, these lectures cover topological spaces, Banach spaces, Hilbert spaces, and operator theory with exceptional clarity.
Algebraic Geometry (Ravi Vakil's notes)
While not a video course, Vakil's freely available notes The Rising Sea: Foundations of Algebraic Geometry are one of the most accessible introductions to modern algebraic geometry.
Commutative Algebra (MIT 18.705)
Lecture notes and problem sets for a graduate commutative algebra course.
Mathematics for Computer Science
MIT 6.042J: Mathematics for Computer Science
An excellent course covering:
- Proof techniques
- Graph theory
- Counting and generating functions
- Probability
- Recurrences
The full video lectures with Tom Leighton and Marten van Dijk are available, along with a free textbook.
How to Get the Most Out of Online Courses
1. Treat It Like a Real Course
Set a schedule, take notes, and do the problem sets. Watching lectures without working problems is almost useless for learning mathematics.
2. Do Every Problem Set
The problem sets are where the real learning happens. Spend serious time on them before looking at solutions.
Study strategy: After watching a lecture, close the video and try to write down the key definitions and theorem statements from memory. Then attempt the problems. Return to the lecture only for specific points where you are stuck.
3. Find a Study Partner
Online learning can be isolating. Find a friend, classmate, or online study group to discuss the material with. Mathematics Stack Exchange and Reddit's r/math community can help.
4. Supplement with a Textbook
Video lectures are great for initial exposure, but textbooks provide the level of detail needed for mastery. Pair each course with the recommended textbook.
5. Take Practice Exams Under Timed Conditions
MIT OCW courses often include past exams with solutions. Take these under timed conditions to test your understanding honestly.
Summary Table
| Subject | Recommended Course | Platform |
|---|---|---|
| Single Variable Calculus | MIT 18.01 (Jerison) | MIT OCW |
| Multivariable Calculus | MIT 18.02 (Auroux) | MIT OCW |
| Linear Algebra | MIT 18.06 (Strang) | MIT OCW |
| Real Analysis | Harvey Mudd (Su) or MIT 18.100A | YouTube / MIT OCW |
| Abstract Algebra | Harvard E-222 (Gross) | YouTube |
| Differential Equations | MIT 18.03 (Mattuck) | MIT OCW |
| Probability | Harvard Stat 110 (Blitzstein) | YouTube |
| Topology | Tokieda (AIMS) or Albin | YouTube |
| Number Theory | MIT 18.781 | MIT OCW |
| Math for CS | MIT 6.042J | MIT OCW |
Final Thoughts
The free courses listed here represent an extraordinary educational resource. They cover the entire undergraduate curriculum and much of the graduate curriculum in mathematics, taught by some of the best educators in the world.
The only thing these courses cannot provide is the discipline to complete them. That part is up to you.
References
- MIT OpenCourseWare — Mathematics
- MIT 18.06 Linear Algebra
- MIT 18.01 Single Variable Calculus
- MIT 18.02 Multivariable Calculus
- MIT 18.03 Differential Equations
- MIT 18.100A Real Analysis
- Harvard Statistics 110
- Francis Su's Real Analysis Lectures
- Benedict Gross's Abstract Algebra Lectures
- Essence of Linear Algebra — 3Blue1Brown
- Ravi Vakil — The Rising Sea
- edX — Mathematics Courses
- Coursera — Mathematics Courses