The Best Free Mathematics Textbooks Available Online

Why Free Textbooks Matter

Mathematics textbooks are expensive. A single hardcover can cost $50–$150, and a student taking several courses per semester can easily spend hundreds of dollars on books. Fortunately, many excellent mathematics textbooks are available for free — either because their authors have made them freely available, because they are published under open licenses, or because classic texts have entered the public domain.

This guide lists the best free mathematics textbooks organized by subject. Every link points to a legally free version made available by the author or publisher.

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Linear Algebra

Jim Hefferon — Linear Algebra

Linear Algebra by Jim Hefferon is a complete, freely available textbook that covers:

• Vector spaces, linear independence, bases
• Linear maps and matrix representations
• Determinants
• Eigenvalues and eigenvectors
• Jordan normal form

The book is well-written with many examples and exercises. It is published under a free license and available as a PDF with LaTeX source.

Sheldon Axler — Linear Algebra Done Right

While the print edition is not free, Sheldon Axler has made Linear Algebra Done Right freely available as an open-access book through Springer. This text takes a distinctive approach by de-emphasizing determinants and instead building the theory through linear maps and operator theory.

David Cherney, Tom Denton, Rohit Thomas, Andrew Waldron — Linear Algebra

This freely available textbook from UC Davis provides a clear, modern introduction with applications to data science and engineering alongside the pure theory.

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Real Analysis

Jiří Lebl — Basic Analysis: Introduction to Real Analysis

Basic Analysis by Jiří Lebl is available for free online and covers:

• The real number system and completeness
• Sequences and series
• Continuity and differentiability
• The Riemann integral
• Sequences of functions and uniform convergence
• Metric spaces (in Volume II)

The writing is clear and accessible, with many exercises ranging from routine to challenging. This is an excellent free alternative to textbooks like Rudin or Abbott.

William Trench — Introduction to Real Analysis

Introduction to Real Analysis by William Trench is freely available through Trinity University's digital commons. It covers standard real analysis topics with careful proofs and plentiful exercises.

Allen Hatcher — Real Analysis Lecture Notes

While primarily known for his topology text, Allen Hatcher provides free measure theory and integration notes that serve as an excellent bridge from basic analysis to measure theory.

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Abstract Algebra

Tom Judson — Abstract Algebra: Theory and Applications

Abstract Algebra by Tom Judson is one of the best free algebra textbooks available. It covers:

• Groups: definitions, subgroups, quotient groups, isomorphism theorems
• Rings: ideals, quotient rings, polynomial rings
• Fields: extensions, splitting fields, Galois theory
• Additional topics: lattices, coding theory, cryptography

The book is published under the GFDL license and is available as an interactive online textbook, PDF, and even in print.

Robert Ash — Abstract Algebra: The Basic Graduate Year

Robert Ash's algebra text is freely available from the author's website. It covers graduate-level algebra including:

• Group theory (Sylow theorems, solvable groups)
• Ring theory (Noetherian rings, Dedekind domains)
• Module theory
• Field theory and Galois theory

This is an excellent resource for students preparing for qualifying exams.

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Topology

Allen Hatcher — Algebraic Topology

Algebraic Topology by Allen Hatcher is the standard free reference for algebraic topology. It covers:

• The fundamental group and covering spaces
• Homology theory
• Cohomology theory
• Homotopy theory (additional topics)

Hatcher's writing is geometric and intuitive, with many illustrations. This book is used in graduate courses worldwide and has earned its status as a modern classic.

Sidney Morris — Topology Without Tears

Topology Without Tears by Sidney Morris is a free introduction to general (point-set) topology that covers:

• Topological spaces and continuous functions
• Compactness, connectedness, and separation axioms
• Quotient spaces
• The Tychonoff theorem

The text is designed for self-study and is especially accessible for students encountering topology for the first time.

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Number Theory

William Stein — Elementary Number Theory: Primes, Congruences, and Secrets

This textbook by William Stein is freely available and covers elementary number theory with a computational perspective. Topics include:

• Prime numbers and the fundamental theorem of arithmetic
• Congruences and modular arithmetic
• Quadratic reciprocity
• Continued fractions
• Elliptic curves (an introduction)

The computational emphasis makes it a natural companion for students who also use SageMath.

Andrew Granville — Number Theory Revealed

Andrew Granville has made early versions of his number theory textbook available for free through the American Mathematical Society. Check Granville's personal website for lecture notes covering analytic and algebraic number theory topics.

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Probability and Statistics

Charles Grinstead and J. Laurie Snell — Introduction to Probability

Introduction to Probability by Grinstead and Snell is freely available from Dartmouth College. It provides a thorough introduction to probability theory with many examples and exercises.

Dmitry Panchenko — Introduction to Probability and Mathematical Statistics (Lecture Notes)

These MIT lecture notes are available through OpenCourseWare and cover both probability theory and statistical inference.

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Differential Equations

Jiří Lebl — Notes on Diffy Qs: Differential Equations for Engineers

Notes on Diffy Qs is a free, open-source textbook covering ordinary differential equations:

• First-order equations
• Higher-order linear equations
• Laplace transforms
• Systems of ODEs
• Power series solutions
• Fourier series and PDEs (introduction)

The writing is friendly and practical, with many worked examples.

William Trench — Elementary Differential Equations with Boundary Value Problems

This textbook by William Trench is freely available and covers the standard undergraduate ODE course with detailed solutions and exercises.

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Combinatorics and Discrete Mathematics

Oscar Levin — Discrete Mathematics: An Open Introduction

Discrete Mathematics by Oscar Levin is a free, open-source textbook covering:

• Counting and combinatorics
• Graph theory
• Logic and proof techniques
• Sequences and induction

It includes interactive exercises in its online version, making it especially useful for self-study.

Miklós Bóna — A Walk Through Combinatorics (Lecture Notes)

While the full textbook is not free, many excellent combinatorics lecture notes are freely available. Search for course notes from universities like MIT (18.211, 18.212) on OCW.

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Complex Analysis

Robert Ash and W.P. Novinger — Complex Variables

Complex Variables by Robert Ash is freely available and covers:

• Analytic functions and the Cauchy-Riemann equations
• Complex integration and Cauchy's theorem
• Power series and Laurent series
• Residue calculus
• Conformal mapping

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Category Theory

Emily Riehl — Category Theory in Context

Category Theory in Context by Emily Riehl is freely available as a PDF and is one of the best introductions to category theory. It covers categories, functors, natural transformations, limits, colimits, adjunctions, and monads.

Tom Leinster — Basic Category Theory

Basic Category Theory by Tom Leinster is available on arXiv and provides a concise, self-contained introduction to the subject.

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Mathematical Logic

Peter Smith — An Introduction to Formal Logic

An Introduction to Formal Logic by Peter Smith is freely available and covers propositional and predicate logic clearly and carefully.

Peter Smith — Gödel Without (Too Many) Tears

For students specifically interested in Gödel's incompleteness theorems, this guide is a freely available companion to more comprehensive treatments.

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Where to Find More Free Textbooks

Open Textbook Library

The Open Textbook Library at the University of Minnesota catalogs peer-reviewed open textbooks across all subjects, including a substantial mathematics section.

American Institute of Mathematics (AIM) Open Textbook Initiative

AIM's Open Textbook Initiative maintains a curated list of approved open-source mathematics textbooks, evaluated by experts for quality and completeness.

MIRI Research Guide

The Machine Intelligence Research Institute provides a curated reading list with many freely available mathematics texts, particularly in logic, set theory, and model theory.

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Summary Table

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Final Thoughts

There has never been a better time to be a mathematics student. The free textbooks listed above cover virtually every core subject in an undergraduate or early graduate mathematics curriculum. They are written by experts, carefully typeset, and available to anyone with an internet connection.

Use them wisely — and consider supporting the authors and open-access publishing initiatives that make this possible.

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References

• Jim Hefferon — Linear Algebra
• Sheldon Axler — Linear Algebra Done Right
• Jiří Lebl — Basic Analysis
• William Trench — Introduction to Real Analysis
• Allen Hatcher — Algebraic Topology
• Tom Judson — Abstract Algebra: Theory and Applications
• Robert Ash — Graduate Algebra
• Sidney Morris — Topology Without Tears
• William Stein — Elementary Number Theory
• Grinstead and Snell — Introduction to Probability
• Jiří Lebl — Notes on Diffy Qs
• Oscar Levin — Discrete Mathematics: An Open Introduction
• Robert Ash — Complex Variables
• Emily Riehl — Category Theory in Context
• Tom Leinster — Basic Category Theory
• Open Textbook Library — Mathematics
• AIM Open Textbook Initiative