The American Institute of Mathematics (AIM) seeks to encourage the adoption of open source and open access mathematics textbooks. The AIM Editorial Board has developed evaluation criteria to identify the books that are suitable for use in traditional university courses. The Editorial Board maintains a list of Approved Textbooks which have been judged to meet these criteria.
Algebra: Abstract and Concrete, by Frederick Goodman, provides a thorough introduction to "modern'' or "abstract'' algebra at a level suitable for upper-level undergraduates and beginning graduate students. The book addresses the conventional topics: groups, rings, fields, and linear algebra, with symmetry as a unifying theme.
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The content of this book is traditional for a first course in abstract algebra at the junior or senior level. It may be used for either one or two semesters. The exercises include both computational and theoretical and there are a number of applications. Hints or short answers are given to most problems but not fully written solutions.
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Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them. Due to the comprehensive nature of the material, this book is offered in three volumes for flexibility and efficiency.
Click on the volume number to see each textbook and its ancillary resources.
Volume 1 covers functions, limits, derivatives, and integration.
Volume 2 covers integration, differential equations, sequences and series, and parametric equations and polar coordinates.
Volume 3 covers parametric equations and polar coordinates, vectors, functions of several variables, multiple integration, and second-order differential equations.
The book covers the standard material in a calculus course for science and engineering. The size of the book is such that an instructor does not have to skip sections in order to fit the material into the typical course schedule. The single variable material is contained in eleven chapters beginning with analytic geometry and ending with sequences and series. The multivariable material consists of five chapters and includes with the vector calculus of in two and three dimensions through the divergence theorem. The book ends with a final chapter on differential equations.
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Where many texts present a general theory of calculus followed by substantial collections of worked examples, we instead pose problems or situations, consider possibilities, and then ask students to investigate and explore. Following key activities or examples, the presentation normally includes some overall perspective and a brief synopsis of general trends or properties, followed by formal statements of rules or theorems. While we often offer a plausibility argument for such results, rarely do we include formal proofs.
The peer-reviewed textbooks below also provide ancillary resources such as PowerPoint slides, test banks, sample syllabus, etc.
Click on the book cover to access multiple formats of the textbook.
Prealgebra is designed to meet scope and sequence requirements for a one-semester prealgebra course. The book’s organization makes it easy to adapt to a variety of course syllabi. The text introduces the fundamental concepts of algebra while addressing the needs of students with diverse backgrounds and learning styles. Each topic builds upon previously developed material to demonstrate the cohesiveness and structure of mathematics.
College Algebra provides a comprehensive exploration of algebraic principles and meets scope and sequence requirements for a typical introductory algebra course. The modular approach and richness of content ensure that the book meets the needs of a variety of courses. College Algebra offers a wealth of examples with detailed, conceptual explanations, building a strong foundation in the material before asking students to apply what they’ve learned.
Precalculus is adaptable and designed to fit the needs of a variety of precalculus courses. It is a comprehensive text that covers more ground than a typical one- or two-semester college-level precalculus course. The content is organized by clearly-defined learning objectives and includes worked examples that demonstrate problem-solving approaches in an accessible way.
Applied Combinatorics, by Mitchel T. Keller and William T. Trotter
Applied Combinatorics is an open-source textbook for a course covering the fundamental enumeration techniques (permutations, combinations, subsets, pigeon hole principle), recursion and mathematical induction, more advanced enumeration techniques (inclusion-exclusion, generating functions, recurrence relations, Polyá theory), discrete structures (graphs, digraphs, posets, interval orders), and discrete optimization (minimum weight spanning trees, shortest paths, network flows). There are also chapters introducing discrete probability, Ramsey theory, combinatorial applications of network flows, and a few other nuggets of discrete mathematics Instructor resources are available upon request.
A First Course in Complex Analysis
Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka
For many of our students, Complex Analysis is their first rigorous analysis (if not mathematics) class they take, and this book reflect this very much. We tried to rely on as few concepts from real analysis as possible. In particular, series and sequences are treated from scratch, which has the (maybe disadvantageous) consequence that power series are introduced late in the course. The goal our book works toward is the Residue Theorem, including some nontraditional applications from both continuous and discrete mathematics.
Important license note: Copyright 2002-2017 by the authors. All rights reserved. This book may be freely reproduced and distributed, provided that it is reproduced in its entirety from the most recent version. This book may not be altered in any way, except for changes in format required for printing or other distribution, without the permission of the authors.
A one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence. This free online book (e-book in webspeak) should be usable as a stand-alone textbook or as a companion to a course using another book such as Edwards and Penney, Differential Equations and Boundary Value Problems: Computing and Modeling or Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems (section correspondence to these two is given).
License: This work is dual licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License and Creative Commons Attribution-Share Alike 4.0 License.
These texts are appropriate for a first course in differential equations for one or two semesters. There are more than 2000 exercises, and the student manual has solutions for most of the even numbered ones.
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This textbook contains the content of a two semester course in discrete structures, which is typically a second-year course for students in computer science or mathematics, but it does not have a calculus prerequisite. The material for the first semester is in chapters 1-10 and includes logic, set theory, functions, relations, recursion, graphs, trees, and elementary combinatorics. The second semester material in chapters 11-16 deals with algebraic structures: binary operations, groups, matrix algebra, Boolean algebra, monoids and automata, rings and fields.
This open source textbook is being used at the University of Northern Colorado in a discrete mathematics course taken primarily by math majors, many of whom plan to become secondary teachers. This text can also be used in a bridge course or introduction to proofs. The major topics are introduced with Investigate! activities designed to get students more actively involved and which are suitable for inquiry based learning.
Discrete Mathematics: An Open Introduction by Oscar Levin is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
From the preface:
Geometry with an Introduction to Cosmic Topology offers an introduction to non-Euclidean geometry through the lens of questions that have ignited the imagination of stargazers since antiquity. What is the shape of the universe? Does the universe have an edge? Is it infinitely big? This text is intended for undergraduate mathematics and physics majors who have completed a multivariable calculus course and are ready for a course that practices the habits of thought needed in advanced courses of the undergraduate mathematics curriculum. The text is also particularly suited to independent study, with essays and other discussions complementing the mathematical content in several sections.
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Joseph E. Fields
This book is designed for the transition course between calculus and differential equations and the upper division mathematics courses with an emphasis on proof and abstraction. The book has been used by the author and several other faculty at Southern Connecticut State University. There are nine chapters and more than enough material for a semester course. Student reviews are favorable.
GIAM is licensed under the GNU Free Documentation License version 1.3.
Mathematical Reasoning: Writing and Proof is designed to be a text for the ﬁrst course in the college mathematics curriculum that introduces students to the processes of constructing and writing proofs and focuses on the formal development of mathematics.
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Andrew M. Bruckner, Brian S. Thomson, Judith B. Bruckner
This is a book about mathematics appreciation via discovery, rather than about practical mathematics. It considers several problems that don't appear to be amenable to ordinary arithmetic, algebraic or geometric techniques. It then guides the reader through the process of discovering the solution to each problem, using creative methods and simple techniques that arise naturally. It also indicates how each solution leads to new questions, provides a bit of history of the problem, and discusses a few related problems of current interest that have not yet been solved.
From the American Institute of Mathematics: "This book has the standard content of a course for science, math, and engineering students that follows calculus. A semester of calculus is the explicit prerequisite, but most students would have three semesters of calculus and for them some of the beginning sections of the book can be skipped. Each chapter ends with three or four applications of that chapter’s subject."
License: This book is licensed under both the GNU Free Documentation License and the Creative Commons Attribution-ShareAlike 2.5 License,
From the American Institute of Mathematics: "Brown University has two introductory linear algebra courses. This text is used in the honors course that emphasizes proofs. The book’s title suggests that it is not the typical approach to linear algebra even among those books that are more theoretical. For example, the concept of a basis is treated as more fundamental than the concept of linear independence, and linear transformations are introduced before solving systems of linear equations. Especially noteworthy is the motivation and development of determinants."
License: This book is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
A Friendly Introduction to Mathematical Logic, by Christopher C. Leary and Lars Kristiansen
At the intersection of mathematics, computer science, and philosophy, mathematical logic examines the power and limitations of formal mathematical thinking. In this expansion of Leary’s user-friendly 1st edition, readers with no previous study in the field are introduced to the basics of model theory, proof theory, and computability theory. The text is designed to be used either in an upper division undergraduate classroom, or for self study. Updating the 1st Edition’s treatment of languages, structures, and deductions, leading to rigorous proofs of Gödel’s First and Second Incompleteness Theorems, the expanded 2nd Edition includes a new introduction to incompleteness through computability as well as solutions to selected exercises.
From the American Institute of Mathematics:
Using a clear and informal approach, this book introduces readers to a rigorous understanding of mathematical analysis and presents challenging math concepts as clearly as possible. This book is intended for those who want to gain an understanding of mathematical analysis and challenging mathematical concepts.
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From the author website: