Book list · Open textbooks · Updated September 2026
Free Math Textbooksfrom University Mathematicians
Textbooks that mathematicians give away on their own pages, sorted by subject. From high school algebra to graduate analysis, and every link goes to the author's or the university's own site.
- Books
- 125
- Subjects
- 13
- Levels
- 3
- Cost
- $0
Many mathematicians who teach at universities eventually turn years of lecture notes into a book and put that book on their own page for free. Some of these books also come out in print from a publisher while the PDF stays on the author's site, and some never go to print at all and live only online.
I collect those books here, subject by subject. I have taught mathematics at every level from elementary school through advanced calculus, so when I add a book I mostly ask whether a student could pick it up and work through it alone, and under each one I note its level, its format and a short comment.
If you would rather follow a path than browse a shelf, the self-learner's roadmap puts thirty books in reading order, and the best books for learning proofs cover the step most students find hardest. For a gentler way into calculus, there is also Calculus Made Easy.
The list grows as I add books, and right now it holds 125 of them.
For the 19 books that also have a print edition, I include an Amazon link. These are affiliate links, so I earn a small commission on purchases made through them, and the price of the book stays the same.
The atlas
Thirteen subjects, one shelf
Tap a subject to filter the list. You can pick more than one.
Algebra
From high school algebra to graduate abstract algebra. Groups, rings and fields are here, along with introductory algebra for high school and college.
16books
- 01.01AdvancedWeb
Abstract Algebra: The Basic Graduate Year
Robert B. AshUniversity of Illinois
Ash wrote this book for graduate algebra courses and arranges groups, rings, modules and Galois theory so they fit into a single academic year.
- 01.02UndergraduatePDF
Abstract Algebra: Theory and Applications
Thomas W. JudsonStephen F. Austin State University
An open-licensed book for undergraduate abstract algebra that has been updated for years. The chapters end with Sage exercises, so groups can be explored on a computer as well as on paper.
- 01.03High schoolWeb
Advanced Algebra II
Kenny FelderRaleigh Charter High School
Algebra II material from Kenny Felder, a high school teacher in North Carolina. It is split into concept notes, homework and a teacher's guide.
- 01.04High schoolWeb
Algebra 1
Andrew Gloag, Anne GloagHigh Tech High, San Diego
CK-12's algebra book for grades 9 and 10. It covers the first year of high school algebra from equations to functions.
- 01.05UndergraduatePDF
Algebra: Abstract and Concrete
Frederick M. GoodmanUniversity of Iowa
Frederick Goodman's algebra book starts from symmetry and concrete examples and lets the abstraction grow out of them. It covers groups, rings, fields and Galois theory for an advanced undergraduate or beginning graduate course, and began as a commercial textbook before Goodman made it free.
- 01.06UndergraduateWeb
College Algebra
Paul DawkinsLamar University
Dawkins's notes for filling the algebra gaps before calculus. Equations, functions, and exponential and logarithmic functions, all with worked examples.
- 01.07High schoolWeb
Core Mathematics
Katherine YoshiwaraLos Angeles Pierce College
The groundwork beneath Yoshiwara's algebra series. It reviews the arithmetic and basic algebra skills a student needs before Elementary Algebra, and it is the place to start for anyone whose foundations feel shaky.
- 01.08AdvancedWeb
A Course in Commutative Algebra
Robert B. AshUniversity of Illinois
Graduate lecture notes on commutative algebra. They gather the ring and module theory needed before algebraic geometry.
- 01.09UndergraduateWeb
Elementary Abstract Algebra
W. Edwin ClarkUniversity of South Florida
W. Edwin Clark's introductory notes on abstract algebra. Short, plain and enough for a first semester course.
- 01.10High schoolPDF
Elementary Algebra
Katherine YoshiwaraLos Angeles Pierce College
The first book in Yoshiwara's algebra series. It starts with variables, equations and graphs of lines, and it is the step to take before Intermediate Algebra for anyone starting algebra from scratch.
- 01.11UndergraduatePDF
Elements of Abstract and Linear Algebra
Edwin H. ConnellUniversity of Miami
Edwin Connell's notes combining abstract and linear algebra in one book. It is written densely and rewards reading every sentence.
- 01.12High schoolPDF
Intermediate Algebra: Functions and Graphs
Katherine YoshiwaraLos Angeles Pierce College
Yoshiwara's intermediate algebra book. It comes one step before Modeling, Functions, and Graphs and covers linear, quadratic, exponential and logarithmic functions, again with examples built from real situations.
- 01.13UndergraduatePDF
Modeling, Functions, and Graphs
Katherine YoshiwaraLos Angeles Pierce College
Katherine Yoshiwara's college algebra book. It teaches functions and graphs through models built from real data and turns precalculus algebra from a list of formulas into a tool for solving problems.
- 01.14High schoolPDF
Open Resources for Community College Algebra (ORCCA)
Portland Community College facultyPortland Community College
An open algebra book written jointly by the mathematics faculty at Portland Community College. It covers all of high school algebra, from linear equations to functions and quadratics, and the exercises in the online version are checked instantly.
- 01.15UndergraduatePDF
Precalculus
David H. Collingwood, K. David Prince, Matthew M. ConroyUniversity of Washington
The textbook for Math 120 at the University of Washington. It builds the functions, modeling and trigonometry a student needs before calculus through realistic problems rather than drills, and the department keeps it current, so this link points to the 2025 edition.
- 01.16UndergraduateWeb
Precalculus
Carl Stitz, Jeff ZeagerLakeland Community College, Lorain County Community College
Carl Stitz and Jeff Zeager's precalculus book, one of the most widely used free texts for the course. It is long and thorough, covering college algebra and trigonometry in full with plenty of exercises and answers, and this link opens the interactive Ximera edition.
Linear Algebra
Matrices, vector spaces and eigenvalues. I keep it separate because it is the common language of nearly every field from engineering to machine learning.
6books
- 02.01UndergraduateWeb
Elementary Linear Algebra
Keith MatthewsUniversity of Queensland
Keith Matthews's linear algebra lecture notes from the University of Queensland. Short, direct and full of worked examples.
- 02.02UndergraduateWeb
A First Course in Linear Algebra
Robert A. BeezerUniversity of Puget Sound
Rob Beezer's open-licensed linear algebra book. Every theorem comes with a complete proof, and the book can be read alongside Sage computations.
- 02.03UndergraduateWeb
Introduction to Matrix Algebra
Autar K. KawUniversity of South Florida
Autar Kaw's matrix algebra book for engineering students, written as preparation for a numerical methods course.
- 02.04UndergraduateWeb
Linear Algebra
Jim HefferonSaint Michael's College
Jim Hefferon's linear algebra book starts from Gauss's method, and the solutions to all of its exercises are a separate free book.
- 02.05UndergraduateWeb
Linear Algebra
Paul DawkinsLamar University
Dawkins's linear algebra notes. Matrices, determinants, vector spaces and eigenvalues, again in the worked-example format.
- 02.06UndergraduateWeb
Linear Algebra: An Interactive Introduction
Anna Davis, Paul ZachlinThe Ohio State University, Lakeland Community College
Anna Davis and Paul Zachlin's linear algebra book, built to be read on screen. GeoGebra interactives let readers drag vectors and watch transformations act on the plane, and the full text runs on Ximera with exercises that check answers as you go.
Calculus
Single and multivariable calculus. Next to the classic books built on limits there are two that build calculus on infinitesimals.
28books
- 03.01UndergraduateWeb
Active Calculus
Matt BoelkinsGrand Valley State University
Matt Boelkins wrote this book to be worked through rather than read. Each section opens with a preview activity and moves through activities the student does before the theory is summarized, and the online version includes exercises that check answers as you go.
- 03.02UndergraduateWeb
APEX Calculus
Gregory HartmanVirginia Military Institute
Gregory Hartman's open calculus book, written with a group of contributors and covering the full sequence from limits to vector calculus. The University of Lethbridge hosts this online edition, and the book reads like a conventional textbook for anyone who wants one that happens to be free.
- 03.03UndergraduateWeb
Business Calculus with Excel
Mike May, S.J., Anneke BartSaint Louis University
Mike May and Anneke Bart teach calculus to business students through spreadsheets. Rates of change, optimization and marginal analysis are worked out in Excel, the tool these students will use at work, with less weight on symbolic manipulation.
- 03.04UndergraduateWeb
Calculus
Gilbert StrangMIT
Gilbert Strang's calculus book. MIT OpenCourseWare offers the whole book as chapter PDFs, together with Strang's short video lectures.
- 03.05UndergraduatePDF
Calculus
David GuichardWhitman College
David Guichard's open-licensed calculus book from Whitman College. It covers single and multivariable calculus, and many universities adapt it for their own courses.
- 03.06UndergraduateWeb
Calculus for Beginners and Artists
Daniel J. KleitmanMIT
A short book in which Daniel Kleitman of MIT explains calculus to readers who are wary of mathematics. He presents derivatives and integrals conversationally, without piling up formulas.
- 03.07UndergraduateWeb
Calculus I
Paul DawkinsLamar University
Known as Paul's Online Notes, and nearly every calculus student in the US runs into them at some point. Calculus I covers limits, derivatives and an introduction to integrals, with worked problems next to every topic.
- 03.08UndergraduateWeb
Calculus I
Jerrold E. Marsden, Alan WeinsteinCaltech, University of California, Berkeley
The first of three volumes by Jerrold Marsden and Alan Weinstein, published by Springer and now free through Caltech's library. It covers limits, derivatives and their applications and the first steps of integration, carefully and with plenty of worked examples, from two mathematicians better known for their work in mechanics and geometry.
- 03.09UndergraduateWeb
Calculus II
Paul DawkinsLamar University
The second course in the same series. Integration techniques, sequences and series are here, and Dawkins works through every method step by step.
- 03.10UndergraduateWeb
Calculus II
Jerrold E. Marsden, Alan WeinsteinCaltech, University of California, Berkeley
The second Marsden and Weinstein volume picks up where the first ends, with techniques and applications of integration, followed by infinite series and Taylor expansions. The same careful style and worked examples carry over.
- 03.11UndergraduateWeb
Calculus III
Paul DawkinsLamar University
The multivariable part of the series. Partial derivatives, multiple integrals and vector calculus in Dawkins's usual worked-example format.
- 03.12UndergraduateWeb
Calculus III
Jerrold E. Marsden, Alan WeinsteinCaltech, University of California, Berkeley
The third and last Marsden and Weinstein volume moves into several variables, with vectors, partial derivatives, multiple integrals and vector calculus, and completes the series.
- 03.13UndergraduatePDF
Calculus in Context
James Callahan et al.Smith College, Five College Calculus Project
A book from the Five College Calculus Project in western Massachusetts that puts modeling first. It opens with a model of how an epidemic spreads, introduces the derivative as the language of change inside that model, and brings in differential equations and computation from the start rather than at the end.
- 03.14UndergraduateWeb
The Calculus of Functions of Several Variables
Dan SloughterFurman University
Sloughter's book on multivariable calculus. A short, clean treatment from vectors to multiple integrals.
- 03.15UndergraduateWeb
Calculus with Applications
Daniel J. KleitmanMIT
An online book written for MIT's 18.013A. Kleitman teaches the topics through small applets and spreadsheets embedded in the pages.
- 03.16High schoolWeb
CK-12 Single Variable Calculus
Multiple authorsCK-12, grades 11–12
Single variable calculus written for high school students who want to start calculus before college.
- 03.17UndergraduatePDF
CLP-1 Differential Calculus
Joel Feldman, Andrew Rechnitzer, Elyse YeagerUniversity of British Columbia
The first volume of the CLP series, written for the calculus courses at the University of British Columbia. It covers limits, derivatives and their applications, and like every volume in the series it has a companion problem book with hints and full solutions.
- 03.18UndergraduatePDF
CLP-2 Integral Calculus
Joel Feldman, Andrew Rechnitzer, Elyse YeagerUniversity of British Columbia
The second CLP volume, on integration techniques, applications of the integral, and sequences and series. The authors write in a relaxed voice and explain why a technique works before asking the reader to use it.
- 03.19UndergraduatePDF
CLP-3 Multivariable Calculus
Joel Feldman, Andrew Rechnitzer, Elyse YeagerUniversity of British Columbia
The third CLP volume takes calculus into several variables, with vectors and geometry in space, partial derivatives, optimization and multiple integrals.
- 03.20UndergraduatePDF
CLP-4 Vector Calculus
Joel Feldman, Andrew Rechnitzer, Elyse YeagerUniversity of British Columbia
The last CLP volume covers curves, vector fields, line and surface integrals, and the theorems of Green, Gauss and Stokes that tie them together.
- 03.21UndergraduateWeb
Difference Equations to Differential Equations
Dan SloughterFurman University
An unusual book that enters calculus through sequences and difference equations. Sloughter builds derivatives and differential equations on that discrete foundation.
- 03.22UndergraduatePDF
Elementary Calculus
Michael CorralSchoolcraft College
A compact single variable calculus book by Michael Corral of Schoolcraft College, who has also written free books on trigonometry and vector calculus. It keeps the explanations short and clear and leaves the rest of the work to the exercises.
- 03.23UndergraduateWeb
Elementary Calculus: An Infinitesimal Approach
H. Jerome KeislerUniversity of Wisconsin–Madison
A calculus book built on infinitesimals rather than limits. Keisler does this through Abraham Robinson's nonstandard analysis and puts Leibniz's intuition on solid ground.
- 03.24UndergraduatePDF
The Method of Lagrange Multipliers
William F. TrenchTrinity University
Trench's short note on the method of Lagrange multipliers. It shows with proof why constrained optimization works the way it does.
- 03.25UndergraduateWeb
Multivariable Calculus
George Cain, James HerodGeorgia Institute of Technology
George Cain and James Herod's notes on multivariable calculus, from vectors to curves, surfaces and Stokes's theorem.
- 03.26UndergraduatePDF
A Summary of Calculus
Karl Heinz DovermannUniversity of Hawaiʻi
Karl Heinz Dovermann's summary of calculus. Useful for seeing everything in one file when studying for an exam or refreshing calculus years later.
- 03.27UndergraduatePDF
Vector Calculus
Michael CorralSchoolcraft College
Corral's book for the third semester of calculus. It covers vectors in space, functions of several variables, multiple integrals, and line and surface integrals through Stokes's theorem, in the same short, clear style as his other books.
- 03.28UndergraduateWeb
Yet Another Calculus Text
Dan SloughterFurman University
Sloughter's short calculus book built on infinitesimals, for readers who want Keisler's approach in a shorter text.
Analysis
Real and complex analysis, where everything taken for granted in calculus finally gets proved.
12books
- 04.01AdvancedPDF
Advanced Calculus
Lynn Loomis, Shlomo SternbergHarvard University
The book used for years in Harvard's Math 55. Despite the title it covers linear algebra, Banach spaces, differential forms and manifolds, which makes it advanced analysis in everything but name.
- 04.02UndergraduatePDF
Basic Analysis: Introduction to Real Analysis
Jiří LeblOklahoma State University
One of the most widely used open books for a first course in real analysis. Lebl writes it to be taught from and keeps it updated.
- 04.03UndergraduateWeb
Complex Analysis
George CainGeorgia Institute of Technology
George Cain's short complex analysis notes from Georgia Tech. The content of a one-semester course, from complex numbers to the Cauchy integral formula.
- 04.04UndergraduatePDF
Complex Analysis
Russell Howell, John MathewsWestmont College, California State University, Fullerton
A complex analysis book for students in mathematics and engineering alike, from Russell Howell and the late John Mathews. It began as a commercial textbook, pays close attention to applications such as conformal mapping and fluid flow, and is now free on its own site.
- 04.05UndergraduateWeb
Complex Variables
Robert B. AshUniversity of Illinois
Ash's book on complex variables. It explains the residue theorem, conformal mappings and analytic continuation in plain language.
- 04.06UndergraduatePDF
Elementary Real Analysis
Brian S. Thomson, Judith B. Bruckner, Andrew M. BrucknerSimon Fraser University, University of California, Santa Barbara
A long and careful real analysis text that began as a commercial textbook before its three authors made it free. It covers sequences, series, continuity, differentiation and integration in full detail, with more exercises and more attention to why definitions are shaped the way they are than most books at this level.
- 04.07UndergraduatePDF
A First Course in Complex Analysis
Matthias Beck, Gerald Marchesi, Dennis PixtonSan Francisco State University, Binghamton University
An open book for a first course in complex analysis that its three authors have developed over many years. It stays short, and every chapter ends with plenty of exercises.
- 04.08UndergraduatePDF
Functions Defined by Improper Integrals
William F. TrenchTrinity University
Trench's short note on functions defined by improper integrals. It opens up the convergence questions behind topics like the gamma function and the Laplace transform.
- 04.09UndergraduateWeb
How We Got from There to Here: A Story of Real Analysis
Robert Rogers, Eugene BomanSUNY Fredonia, Penn State Harrisburg
Robert Rogers and Eugene Boman tell real analysis as the story of how it was needed. The book follows the problems with power series and Fourier series that forced mathematicians to make calculus rigorous, and asks readers to work through those problems themselves before the modern definitions arrive.
- 04.10UndergraduatePDF
Introduction to Real Analysis
William F. TrenchTrinity University
Trench's real analysis book. It covers sequences, series, continuity and integration at the undergraduate level with full proofs.
- 04.11UndergraduatePDF
Mathematical Analysis I
Elias ZakonUniversity of Windsor
The analysis book that Zakon's Basic Concepts of Mathematics was written to prepare for. It brings in metric and normed spaces early and treats limits, continuity and differentiation in that general setting, rigorous throughout and free from the same publisher.
- 04.12UndergraduateWeb
Real Variables with Basic Metric Space Topology
Robert B. AshUniversity of Illinois
Ash's book on real variables. It takes analysis one step further by bringing in the basics of metric space topology.
Differential Equations
Ordinary differential equations, boundary value problems and dynamical systems.
8books
- 05.01UndergraduateWeb
Differential Equations
Paul DawkinsLamar University
Dawkins's notes on differential equations. Everything an engineering student needs, from first-order equations to Laplace transforms and Fourier series, with worked examples.
- 05.02UndergraduateWeb
Differential Equations: An Interactive Introduction
William F. TrenchTrinity University
An interactive edition of Trench's Elementary Differential Equations, the same book that sits a few entries up in PDF form. On Ximera the text is broken into short sections with exercises that check answers as you go, which suits a student working through it alone.
- 05.03AdvancedWeb
Dynamical Systems
Shlomo SternbergHarvard University
Shlomo Sternberg's book on dynamical systems, running from iterated functions to chaos and Newton's method at an advanced undergraduate level.
- 05.04UndergraduatePDF
Elementary Differential Equations
William F. TrenchTrinity University
William Trench's differential equations book from Trinity University. It was first published in print, and Trench later made it free.
- 05.05UndergraduatePDF
Elementary Differential Equations with Boundary Value Problems
William F. TrenchTrinity University
The same book extended with boundary value problems and Fourier series. The step toward partial differential equations is in these added chapters.
- 05.06UndergraduatePDF
Notes on Diffy Qs: Differential Equations for Engineers
Jiří LeblOklahoma State University
Jiří Lebl's notes for the first differential equations course taken by engineering students, widely known as Notes on Diffy Qs. It covers first-order equations, higher-order linear equations, systems, Fourier series and a first look at partial differential equations, and many universities use it as their textbook.
- 05.07UndergraduatePDF
The Ordinary Differential Equations Project
Thomas W. JudsonStephen F. Austin State University
Thomas Judson's open book for a first course in ordinary differential equations, from the author of Abstract Algebra: Theory and Applications. It treats equations analytically, qualitatively and numerically side by side, with modeling and phase plane analysis throughout and Sage for the computations.
- 05.08UndergraduatePDF
Student Solutions Manual
William F. TrenchTrinity University
The student solutions manual for Trench's two differential equations books. Used alongside them, it makes studying alone much easier.
Number Theory
From divisibility to algebraic number theory and cryptography.
6books
- 06.01AdvancedPDF
A Computational Introduction to Number Theory and Algebra
Victor ShoupNew York University
A book that teaches number theory and algebra through algorithms and cryptography. Shoup keeps a free PDF of the printed edition on his own page.
- 06.02AdvancedWeb
A Course in Algebraic Number Theory
Robert B. AshUniversity of Illinois
A graduate introduction to algebraic number theory. Dedekind domains, the ideal class group and ramification, turned from Ash's lecture notes into a book.
- 06.03UndergraduateWeb
Elementary Number Theory
W. Edwin ClarkUniversity of South Florida
Clark's introductory notes on number theory. Divisibility, congruences and primes, sized for a one-semester course.
- 06.04UndergraduatePDF
Elementary Number Theory: Primes, Congruences, and Secrets
William SteinUniversity of Washington
William Stein, the creator of SageMath, wrote this short number theory book for Springer and keeps the PDF free. It moves from primes and congruences to public-key cryptography, quadratic reciprocity, continued fractions and elliptic curves, with Sage computations throughout.
- 06.05UndergraduateWeb
Introduction to the Theory of Numbers
Leo MoserMcGill University
A short number theory book from Leo Moser's lectures. A quick introduction to the distribution of primes and a few classic problems.
- 06.06UndergraduatePDF
Number Theory: In Context and Interactive
Karl-Dieter CrismanGordon College
Karl-Dieter Crisman's number theory book treats computation as part of the subject, using SageMath to test conjectures and look at patterns before proving them. It runs from congruences and primitive roots through cryptography to the distribution of primes, and the online edition has live Sage cells, while this link is the PDF.
Discrete Mathematics and Combinatorics
Counting, graphs, generating functions and the mathematics of computer science.
10books
- 07.01AdvancedWeb
A = B
Marko Petkovšek, Herbert S. Wilf, Doron ZeilbergerUniversity of Pennsylvania, Rutgers University
A book that hands the proof of combinatorial identities to the computer. With the methods described here, an identity involving hypergeometric sums can be verified algorithmically, and the foreword is by Donald Knuth.
- 07.02UndergraduateWeb
Algebra and Analysis for Computer Science
Jean GallierUniversity of Pennsylvania
Jean Gallier's collection for computer science students. It gathers the algebra and analysis needed before reading about machine learning and optimization.
- 07.03UndergraduatePDF
Applied Combinatorics
Mitchel T. Keller, William T. TrotterGeorgia Institute of Technology
The textbook for the combinatorics course at Georgia Tech, by Mitchel Keller and William Trotter. It covers counting, graph theory, partially ordered sets, generating functions, recurrences, probabilistic methods and network flows, and it opens many chapters with a problem worth thinking about before the theory arrives.
- 07.04UndergraduateWeb
Applied Discrete Structures
Alan Doerr, Kenneth LevasseurUniversity of Massachusetts Lowell
Al Doerr and Ken Levasseur's discrete mathematics book, written with computer science students in mind. It runs from sets, logic and counting through graphs and trees to algebraic structures like groups and Boolean algebras, and SageMath examples are woven into the text.
- 07.05UndergraduatePDF
Combinatorics Through Guided Discovery
Kenneth P. BogartDartmouth College
Kenneth Bogart's combinatorics book is a sequence of problems rather than a sequence of explanations. Readers work through carefully ordered questions and discover the counting principles, generating functions and group actions for themselves, which makes it a natural fit for an inquiry-based course.
- 07.06UndergraduatePDF
Counting Rocks! An Introduction to Combinatorics
Henry Adams, Kelly Emmrich, Maria Gillespie, Shannon Golden, Rachel PriesColorado State University
The text half of Colorado State's introductory combinatorics course, which pairs each section with one of 47 short lecture videos on the department's YouTube channel. It covers counting, proof techniques, recurrences and generating functions, then turns to graph theory, and most chapters end with an investigation that leads students through a harder problem. The title is a nod to the Rocky Mountains near campus.
- 07.07UndergraduatePDF
Discrete Mathematics: An Open Introduction
Oscar LevinUniversity of Northern Colorado
Oscar Levin's discrete math book covers counting, graphs, logic and proof techniques, and its online version has interactive exercises that check answers instantly.
- 07.08UndergraduateWeb
Foundations of Combinatorics with Applications
Edward A. Bender, S. Gill WilliamsonUniversity of California, San Diego
A fuller combinatorics text from the same UC San Diego pair behind Lectures in Discrete Mathematics. It covers counting, graphs and trees, recursion and generating functions, and ties them to applications in computer science, and Dover also publishes it in print.
- 07.09AdvancedWeb
generatingfunctionology
Herbert S. WilfUniversity of Pennsylvania
A classic on generating functions. Wilf compares a generating function to a clothesline on which a sequence of numbers is hung for display, and the rest of the book is written with the same ease.
- 07.10UndergraduateWeb
Lectures in Discrete Mathematics
Edward A. Bender, S. Gill WilliamsonUniversity of California, San Diego
Lecture notes by Edward Bender and Gill Williamson of UC San Diego for a discrete mathematics course. They cover counting, functions, graphs, induction and recursion with an eye toward computer science, written by two combinatorialists who have taught the subject for decades.
Proofs and Foundations
Books for the move from computing answers to writing proofs, the transition most students find hardest in their first years of college.
6books
- 08.01UndergraduateWeb
Basic Concepts of Mathematics
Elias ZakonUniversity of Windsor
Sets, logic and the axiomatic construction of the real numbers. Zakon wrote it as preparation for his own analysis book, and it is a good place to firm up the ground before moving into proof-based mathematics.
- 08.02UndergraduatePDF
Book of Proof
Richard HammackVirginia Commonwealth University
If I had to pick one book from this list for someone starting to write proofs, I would pick this one. It covers logic, sets, induction and functions with plenty of examples, and while the print edition is inexpensive, the PDF is free on the author's page.
- 08.03UndergraduateWeb
A Gentle Introduction to the Art of Mathematics
Joseph E. FieldsSouthern Connecticut State University
Joseph Fields's book for the transition course into proof-based mathematics. It moves through logic, sets, proof techniques, relations, functions and cardinality at an unhurried pace, which suits a student meeting proofs for the first time.
- 08.04UndergraduatePDF
An Introduction to Proof via Inquiry-Based Learning
Dana C. ErnstNorthern Arizona University
Dana Ernst's book for an inquiry-based introduction to proof. It gives definitions, theorems and problems but no proofs, so the students write every proof themselves and present them to the class, which is how the course is meant to be taught.
- 08.05UndergraduatePDF
Mathematical Reasoning: Writing and Proof
Ted SundstromGrand Valley State University
Ted Sundstrom's transition-to-proof book puts as much weight on writing as on logic. Each section opens with preview activities, and the book keeps asking students to explain their reasoning in clear sentences, treating a proof as a piece of writing meant for a reader.
- 08.06UndergraduateWeb
Proofs and Concepts: The Fundamentals of Abstract Mathematics
Dave Witte Morris, Joy MorrisUniversity of Lethbridge
A transition book into proof-based mathematics. It uses logic, sets and functions as tools for learning to write proofs.
Geometry and Trigonometry
High school geometry and trigonometry, plus the geometry behind computer-aided design.
5books
- 09.01High schoolWeb
CK-12 Geometry
Multiple authorsCK-12, grades 10–12
A high school geometry book written to the California standards. It covers the basic proofs of Euclidean geometry and transformations.
- 09.02High schoolWeb
CK-12 Trigonometry
Multiple authorsCK-12, grades 10–12
The second edition of CK-12's high school trigonometry, written by several authors and running from the unit circle to trigonometric identities.
- 09.03AdvancedWeb
Computational Geometry
Nicholas M. Patrikalakis, Takashi MaekawaMIT
A book that grew out of MIT's computational geometry course. It covers computing the intersections, distances and curvature of curves and surfaces in computer-aided design.
- 09.04UndergraduatePDF
Geometry with an Introduction to Cosmic Topology
Michael P. HitchmanLinfield University
Michael Hitchman builds Euclidean, hyperbolic and elliptic geometry from Möbius transformations of the complex plane, the way Felix Klein described geometry through its transformations. The book ends by asking what shape the universe might have, which is where the cosmic topology in the title comes in.
- 09.05High schoolPDF
Trigonometry
Katherine YoshiwaraLos Angeles Pierce College
Yoshiwara's trigonometry book. It covers angles, the unit circle and trigonometric functions through applications and models, as her algebra books do.
Probability
Probability theory, from first principles to random processes.
7books
- 10.01UndergraduateWeb
Applied Probability
Paul PfeifferRice University
Paul Pfeiffer's probability book for engineering students at Rice University. Most of the computations are done in MATLAB.
- 10.02High schoolWeb
Basic Probability and Statistics
Brenda MeeryHigh school teacher, Nova Scotia
An introductory book that Brenda Meery, a high school math teacher in Canada, wrote for CK-12. It is aimed at high school students meeting probability and statistics for the first time.
- 10.03UndergraduateWeb
Basic Probability Theory
Robert B. AshUniversity of Illinois
A probability book that needs only calculus and stays away from measure theory. Ash writes in solid mathematical language without making the subject heavier than it has to be.
- 10.04UndergraduatePDF
Introduction to Probability
Charles M. Grinstead, J. Laurie SnellSwarthmore College, Dartmouth College
A probability book by two authors. The American Mathematical Society publishes it in print, the authors also give it away for free, and computer simulations support the topics throughout.
- 10.05UndergraduateWeb
Introduction to Probability, Statistics, and Random Processes
Hossein Pishro-NikUniversity of Massachusetts Amherst
A book that puts probability, statistics and random processes on one site. Every chapter has solved problems and video lectures, and it is widely used by electrical engineering students.
- 10.06UndergraduateWeb
Probability for Data Science
Stanley H. ChanPurdue University
Stanley Chan's probability book written for students heading into data science and machine learning. It connects every topic to computation, with code in Python, MATLAB and Julia, and carries probability through to estimation, regression and hypothesis testing.
- 10.07UndergraduatePDF
Probability: Lectures and Labs
Mark HuberClaremont McKenna College
Mark Huber's probability course, with each lecture paired with a computer lab where students simulate what the lecture proved. The pairing gives a student two routes into every idea, one through the mathematics and one through the numbers a simulation produces.
Statistics and Data Science
From introductory statistics to experimental design and statistical learning.
13books
- 11.01UndergraduateWeb
Collaborative Statistics
Barbara Illowsky, Susan DeanDe Anza College
Written to be taught at De Anza College, the book lives on today as OpenStax's Introductory Statistics, and the link goes to that newer edition.
- 11.02AdvancedWeb
Design of Comparative Experiments
R. A. BaileyQueen Mary, University of London
R. A. Bailey's book on experimental design. It explains, with the statistical foundations, how an experiment should be set up before a single data point is collected.
- 11.03UndergraduateWeb
Electronic Statistics Textbook
T. Hill, P. LewickiStatSoft
An electronic book that grew out of a statistics software manual. It covers a wide range, from basic concepts to data mining methods, topic by topic like a reference.
- 11.04AdvancedPDF
The Elements of Statistical Learning
Trevor Hastie, Robert Tibshirani, Jerome FriedmanStanford University
The standard reference on statistical learning. It is the more mathematical, more advanced older sibling of An Introduction to Statistical Learning, and the three authors at Stanford give the whole book away for free.
- 11.05UndergraduateWeb
Introduction to Modern Statistics
Mine Çetinkaya-Rundel, Johanna HardinOpenIntro
An introductory statistics book that starts from data rather than formulas. Mine Çetinkaya-Rundel, a Turkish statistician known for her work in statistics education, and Johanna Hardin teach inference through randomization and simulation before the classical theory, and the companion materials use R.
- 11.06UndergraduateWeb
An Introduction to Statistical Learning
Gareth James, Daniela Witten, Trevor Hastie, Robert Tibshirani
The book most often recommended today for approaching machine learning from the statistics side. It comes in two editions, one with R and one with Python, and both are free on the official site.
- 11.07UndergraduateWeb
Introduction to Statistical Thought
Michael LavineUniversity of Massachusetts
Michael Lavine builds statistics on likelihood and Bayesian thinking rather than a list of formulas, and he works the examples in R.
- 11.08UndergraduateWeb
Introductory Statistics: Concepts, Models, and Applications
David W. StockburgerMissouri State University
David Stockburger's introductory statistics book. It explains concepts through models and is written for social science students.
- 11.09UndergraduateWeb
Lectures on Probability, Statistics and Econometrics
Marco TabogaBanca d'Italia
The probability, statistics and econometrics site of Marco Taboga, an economist at the Bank of Italy. Each topic moves through a short definition, a proof and solved exercises.
- 11.10AdvancedWeb
Lectures on Statistics
Robert B. AshUniversity of Illinois
Ash's lecture notes on statistics. They present the mathematical side of estimation and hypothesis testing at the graduate level and keep it brief.
- 11.11UndergraduatePDF
Mathematics for Machine Learning
Marc Peter Deisenroth, A. Aldo Faisal, Cheng Soon OngUniversity College London, Imperial College London, CSIRO
The mathematics a student needs before reading machine learning papers, gathered in one book. The first half covers linear algebra, analytic geometry, matrix decompositions, vector calculus, probability and optimization, and the second half uses them to build linear regression, dimensionality reduction, density estimation and support vector machines. Cambridge University Press publishes it in print.
- 11.12UndergraduatePDF
Online Statistics Education: A Multimedia Course of Study
David M. LaneRice University
David Lane's introductory statistics course from Rice University. The website version has interactive simulations and demonstrations for nearly every concept, and this link is the PDF of the full text for readers who prefer to read offline.
- 11.13UndergraduateWeb
SticiGui: Statistics Tools for Internet and Classroom Instruction
Philip B. StarkUniversity of California, Berkeley
Philip Stark's introductory statistics book. The pages contain interactive tools, and readers learn the concepts by trying them out.
Applied and Engineering Mathematics
The mathematics that engineers, social scientists, economists and programmers use, game theory included.
5books
- 12.01UndergraduateWeb
Introduction to Game Theory: a Discovery Approach
Jennifer Firkins NordstromLinfield University
Jennifer Firkins Nordstrom's introduction to game theory teaches by discovery. Readers first play the games and search for strategies themselves, and the theory is built on those trials.
- 12.02UndergraduateWeb
Introduction to Social Network Methods
Robert A. Hanneman, Mark RiddleUniversity of California, Riverside
An introduction to social network analysis. It explains how to measure who is connected to whom with graphs and matrices, using the UCINET software.
- 12.03UndergraduateWeb
Math, Numerics, and Programming
Masayuki Yano et al.MIT
The textbook for MIT's numerical computation course for mechanical engineering students. It teaches mathematics, numerical methods and MATLAB programming together.
- 12.04AdvancedPDF
Mathematical Methods of Engineering Analysis
Erhan Çinlar, Robert J. VanderbeiPrinceton University
A textbook on engineering analysis by Erhan Çinlar and Robert Vanderbei of Princeton. Çinlar is a Turkish mathematician known for his work in probability theory, and the book presents the mathematics engineers use densely and with proofs.
- 12.05UndergraduateWeb
Tea Time Numerical Analysis
Leon Q. BrinSouthern Connecticut State University
Leon Brin's introduction to numerical analysis, written in a conversational voice meant to be read with a cup of tea. It covers root finding, interpolation, numerical differentiation and integration, and differential equations, with the algorithms implemented in Octave and MATLAB.
General and Popular Mathematics
Books written for readers outside the mathematics major.
3books
- 13.01UndergraduateWeb
Math Alive
Ingrid DaubechiesPrinceton University
A course Ingrid Daubechies designed for students outside mathematics. It explains the mathematics behind everyday technology, from digital audio and images to cryptography.
- 13.02UndergraduateWeb
Math in Society
David LippmanPierce College
David Lippman's open book for students majoring outside mathematics. Most of it needs no algebra, it gathers topics useful in daily life like voting systems, personal finance, statistics and graph theory, and each chapter can be read on its own.
- 13.03UndergraduateWeb
Mathematics for Elementary Teachers
Michelle ManesUniversity of Hawaiʻi at Mānoa
Michelle Manes wrote this for Math 111 at the University of Hawaiʻi, the course for future elementary school teachers. Rather than reteaching arithmetic, it has readers work through problems and explain why the familiar algorithms work, which is the understanding a teacher needs in front of a class.
Questions
About these free textbooks
Are these math textbooks really free and legal to download?
Yes. Every book on this list is published for free by its own author or university, and each link goes to that official page rather than to a mirror. Some of the books also have a print edition you can buy, but the free version on the author's site is the complete book.
Which free math textbook should I start with for self-study?
Pick by level first. High school readers can start in the algebra and geometry sections, first-year students in calculus and linear algebra, and anyone moving toward upper-level courses should read a proofs book before analysis or abstract algebra. Use the level filter to narrow the list.
What is the difference between the PDF and Web formats?
PDF books are single files you can download, print or read offline. Web books are read in the browser, usually chapter by chapter, and many of them include interactive exercises or links to code.
How are the books on this list chosen?
A book is added when it is written by a working mathematician, freely available from the author or the university, and clear enough for a student to work through alone. Each entry notes its level, its format and a short comment on who it suits.
