Free Math Courses on Coursera: The 26 Worth Taking

FND 101

Introduction to Mathematical Thinking

Stanford University · Keith Devlin

School mathematics asks you to follow procedures. University mathematics asks you to understand why they work and to say so precisely. Devlin's course is built for exactly that gap: the logic hidden in ordinary sentences, the meaning of “for all” and “there exists,” and the first real proofs, which you write yourself and grade against other learners.

Take it if you are about to start a math degree, or you have always felt you could compute but not prove.

FND 102

Algebra: Elementary to Advanced

Johns Hopkins University · Joseph W. Cutrone

Equations, inequalities, functions, polynomials, exponentials and logarithms, rebuilt from the ground up in three short courses. Nothing here is new, which is the point: most people who struggle with calculus are really struggling with algebra they half learned years ago.

Take it if it has been a long time since school, or algebra never quite clicked.

FND 103

Precalculus through Data and Modelling

Johns Hopkins University · Joseph W. Cutrone

The functions calculus is about (linear, polynomial, exponential, logarithmic, trigonometric) taught through the job they do: describing real data. Each family arrives with a dataset it fits, so by the end a graph reads like a sentence.

Take it if you want to start calculus next month and need your functions solid first.

FND 104

Data Science Math Skills

Duke University · Daniel Egger, Paul Bendich

Set notation, the Cartesian plane, slopes, exponents, logarithms, sums with Σ and a first look at probability and Bayes' theorem. It is a short course and a gentle one, and with more than thirteen thousand reviews it is one of the most taken math courses on the platform.

Take it if symbols like Σ and log stop you reading, and you want them to stop stopping you.

CAL 201

Introduction to Calculus

The University of Sydney · David Easdown

A careful, old-fashioned first course: it starts with the precalculus you need, builds limits and derivatives honestly, and ends with integration and the fundamental theorem. Easdown explains slowly and draws everything, and the quizzes are hard enough to tell you whether you really understood.

Take it if you want one course that takes you from functions to integrals without skipping steps.

CAL 202

Calculus: Single Variable, Parts 1–5

University of Pennsylvania · Robert Ghrist

The most beautiful calculus lectures on the internet. Ghrist builds the whole subject around the Taylor series, so instead of a pile of rules you get one idea seen from five sides, drawn in his unmistakable hand-lettered style. It assumes you have seen some calculus before; it is the second look that makes it all make sense.

Take it if you passed calculus once and want to finally understand it.

• Part 1 · Functions
• Part 2 · Differentiation
• Part 3 · Integration
• Part 4 · Applications
• Part 5 · Discretization

CAL 203

Mathematics for Machine Learning: Multivariate Calculus

Imperial College London · Samuel J. Cooper, David Dye, A. Freddie Page

Partial derivatives, the Jacobian, the Hessian and the multivariable chain rule, put to work straight away on Newton–Raphson, gradient descent, fitting curves to data and the backpropagation that trains neural networks. Fast and visual, with short Python exercises.

Take it if you know one-variable calculus and want to see where many variables lead.

CAL 204

Vector Calculus for Engineers

The Hong Kong University of Science and Technology · Jeffrey R. Chasnov

Gradient, divergence and curl, then line and surface integrals, and finally the two great theorems of Gauss and Stokes, which turn out to be the same theorem twice. Chasnov ends with the Navier–Stokes and Maxwell equations, so you see the machinery do real physics.

Take it if electromagnetism or fluid mechanics is on your horizon.

LIN 211

Matrix Algebra for Engineers

The Hong Kong University of Science and Technology · Jeffrey R. Chasnov

The highest-rated course on this list, and it deserves it. Short, crisp lectures on matrices, systems of equations, vector spaces, determinants and eigenvalues, each followed by practice problems with full solutions. Chasnov writes on a clear board and never wastes a minute.

Take it if you want linear algebra done properly in two or three weeks.

LIN 212

Linear Algebra from Elementary to Advanced

Johns Hopkins University · Joseph W. Cutrone

The longest and most complete linear algebra on Coursera: systems and matrices, then determinants and eigenvectors, then orthogonality and diagonalization, with Markov chains among the applications. Slower than Chasnov, and more like a full university semester.

Take it if you want the whole course, theory included, not just the toolkit.

LIN 213

Mathematics for Machine Learning: Linear Algebra

Imperial College London · David Dye, Samuel J. Cooper, A. Freddie Page

Vectors as lists of data, matrices as transformations, a change of basis as a change of point of view, and eigenvectors as the directions a transformation leaves alone. It ends by building Google's PageRank in Python, which is the best possible reason to learn what an eigenvector is.

Take it if machine learning is your reason for learning linear algebra.

DIF 301

Differential Equations for Engineers

The Hong Kong University of Science and Technology · Jeffrey R. Chasnov

First- and second-order equations, resonance, series solutions, the Laplace transform, systems of equations and, in the last week, Fourier series and the diffusion and wave equations. A full university course in six compact weeks.

Take it if you have finished calculus and want the course physicists and engineers use most.

DIF 302

Differential Equations Part I: Basic Theory

KAIST · Kil Hyun Kwon

A classical, theory-first introduction from Korea's leading science university: first-order equations, linear equations of higher order and the methods for solving them, taught at a calm pace with worked examples. A good second voice next to Chasnov.

Take it if you like your mathematics explained slowly and in order.

DIF 303

Numerical Methods for Engineers

The Hong Kong University of Science and Technology · Jeffrey R. Chasnov

What happens when the equation cannot be solved by hand, which is almost always. Root finding, interpolation, numerical integration and the methods that step differential equations forward in time, all programmed in MATLAB, with fractals, the Lorenz equations and a two-body orbit among the projects.

Take it if you have taken DIF 301 and want to make a computer do it.

DIS 221

Mathematical Thinking in Computer Science

UC San Diego · Alexander S. Kulikov, Michael Levin, Vladimir Podolskii

Proof by example, by contradiction and by induction, then invariants and the extremal principle, all learned through interactive puzzles you solve in the browser before anyone tells you the trick. It is the first course of UC San Diego's five-course discrete mathematics series, and the most fun way into proofs there is.

Take it if you learn better by playing with a problem than by reading about it.

DIS 222

Combinatorics and Probability

UC San Diego · Alexander S. Kulikov, Vladimir Podolskii

How to count without counting: the sum and product rules, permutations, combinations, the binomial theorem, and then probability built on top of counting, with a famous dice paradox and the Monty Hall problem along the way.

Take it if you can never remember when to use n! and when to use n choose k.

DIS 223

Graph Theory

UC San Diego · Alexander S. Kulikov, Vladimir Podolskii

Dots and lines, and everything they model: networks, maps, schedules, molecules. Paths and cycles, trees, Euler and Hamilton, coloring and matchings, with puzzles first and theorems second. The series' capstone, the Delivery Problem, turns all of it on the traveling salesman.

Take it if networks, maps or algorithms are what you find interesting.

DIS 224

Number Theory and Cryptography

UC San Diego · Alexander S. Kulikov, Michael Levin, Vladimir Podolskii

Divisibility, modular arithmetic, the Euclidean algorithm, Fermat's little theorem and the Chinese remainder theorem, assembled at the end into RSA, the code that protects every card payment you make. Then you break badly implemented RSA yourself.

Take it if you want to see the oldest mathematics guard the newest technology.

DIS 225

Introduction to Logic

Stanford University · Michael Genesereth

Propositional and relational logic as a formal language, with truth tables, formal proof systems and the resolution method computers use to reason. It is the most formal course on this list, and the one that makes the word “proof” mean something exact.

Take it if you want to know exactly what a proof is, down to the last symbol.

STA 231

Introduction to Statistics

Stanford University · Guenther Walther

Stanford's first statistics course: describing data, sampling and the normal approximation, confidence intervals, tests of significance, regression and the traps of multiple testing. Clear, compact and almost entirely free of formulas you have to memorize.

Take it if you read studies and want to know when to believe them.

STA 232

Bayesian Statistics: From Concept to Data Analysis

UC Santa Cruz · Herbert Lee

Probability as degree of belief, updated by evidence. Priors and posteriors, Bayes' theorem in its full form, and the conjugate models that make the arithmetic work by hand, compared side by side with the frequentist methods you may already know.

Take it if you have taken a statistics course and want the other half of the story.

STA 233

Mathematics for Machine Learning and Data Science

DeepLearning.AI · Luis Serrano

Linear algebra, calculus, and probability and statistics, each in one course, all taught by Luis Serrano, who may be the best visual explainer of mathematics working today. Every idea comes with a picture and a short Python lab. Coursera estimates three months at five hours a week.

Take it if you are heading into machine learning and want all its mathematics in one place.

XTR 401

Introduction to Complex Analysis

Wesleyan University · Petra Bonfert-Taylor

Calculus with complex numbers, which turns out to be more beautiful than the real thing. Along the way you draw Julia sets and the Mandelbrot set, meet conformal maps and Cauchy's theorem, and finish at the edge of the Riemann hypothesis. Rigorous, generous and visually stunning.

Take it if you have finished calculus and want to see why mathematicians fall in love with the subject.

XTR 402

Analytic Combinatorics

Princeton University · Robert Sedgewick

The hardest course on this list and the most rewarding. Sedgewick, co-author with Philippe Flajolet of the book that founded the field, shows how generating functions and complex analysis predict, with startling precision, how many structures of size n there are. Graduate-level material, taught by the person who wrote it down.

Take it if you have taken XTR 401 and want to see complex analysis count things.

XTR 403

Fibonacci Numbers and the Golden Ratio

The Hong Kong University of Science and Technology · Jeffrey R. Chasnov

Nine hours with one sequence. Fibonacci identities and their proofs, the golden ratio and Binet's formula, continued fractions, the golden spiral and the sunflower. A small, perfect course, and the best introduction to proof by induction disguised as a hobby.

Take it if you want a weekend course that is pure pleasure.

XTR 404

Game Theory

Stanford University & University of British Columbia · Matthew O. Jackson, Kevin Leyton-Brown, Yoav Shoham

The mathematics of strategy: dominant strategies, Nash equilibrium, mixed strategies, games in sequence, repeated games and games where players do not know everything. Three leading researchers teach it with real examples from auctions, traffic and the prisoner's dilemma.

Take it if economics, politics or poker is the part of the world you want to model.