23 Mathematics, Calculus and Statistics

3D Interactive Plots for Multivariate Calculus

Authors: Dr. Na Yu

Description: An open textbook created to improve both teaching and learning vital concepts and techniques in multivariable calculus, one of the fundamental courses across the undergraduate curriculum in science and engineering. The goals of this resource are to help learners develop their geometric intuition about abstract and complex mathematical concepts (e.g., partial derivatives, multiple integrals, vector fields), and train them to make connections between concepts visually (e.g., connecting “vectors” in mathematics with “magnitude” and “direction” in physics) in order to more fully understand engineering, physics and mathematical problems (e.g., differential equations) in their subsequent STEM coursework.

Abstract Algebra: Theory and Applications

Description: Tom Judson’s Abstract Algebra: Theory and Applications is an open source textbook designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. Its strengths include a wide range of exercises, both computational and theoretical, plus many nontrivial applications. The 2020 Annual Edition is now available. Electronic editions have been updated.

First, there is no cost to acquire this text, and you are under no obligation whatsoever to compensate or donate to the author or publisher. So in this most basic sense, it is a free textbook. Therefore you can also make as many copies as you like, ensuring that the book will never go out-of-print. You may modify copies of the book for your own use – for example, you may wish to change to a prefered notation for certain objects or add a few new sections. There is a copyright on the book, and subsequently it is licensed with a GNU Free Documentation License (GFDL). It is this combination that allows the author to give you greater freedoms in how you use the text, thus liberating it from some of the antiquated notions of copyright that apply to books in physical form. The main caveat is that if you make modifications and then distribute a modified version, you are required to again apply the GFDL license to the result so that others may benefit from your modifications.

Active Calculus Single Variable

Description: This textbook was written by math professors from an American university.

Includes: interactive exercises and solutions, a lab manual, and other instructor resources.

Active Calculus Multivariable

Description: This textbook is meant to complement the Single Variable edition of Active Calculus.

Includes: exercises and solutions.

Advanced Data Analysis from and Elementary  Point of View

Author: Cosma Rohilla Shalizi

Description: This is a draft textbook on data analysis methods, intended for a one-semester course for advance undergraduate students who have already taken classes in probability, mathematical statistics, and linear regression. It began as the lecture notes for 36-402 at Carnegie Mellon University.

Supplementary materials:  These include the following

Algebra and Trignometry

Description: Published by OpenStax College, Algebra and Trigonometry provides a comprehensive and multi-layered exploration of algebraic principles. The text is suitable for a typical introductory Algebra & Trigonometry course, and was developed to be used flexibly. The modular approach and the richness of content ensures that the book meets the needs of a variety of programs. Algebra and Trigonometry guides and supports students with differing levels of preparation and experience with mathematics. Ideas are presented as clearly as possible, and progress to more complex understandings with considerable reinforcement along the way. A wealth of examples – usually several dozen per chapter – offer detailed, conceptual explanations, in order to build in students a strong, cumulative foundation in the material before asking them to apply what they’ve learned.

A Gentle Introduction to the Art of Mathematics

Author: Joseph E. Fields

Description: 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. It is written in an informal, conversational style with a large number of interesting examples and exercises, so that a student learns to write proofs while working on engaging problems.

Supplementary materials: Video lectures, exercise workbook and hints and solutions manual; latex source code

APEX Calculus

Description: This text, or portions of it, has been adopted at a considerable number of American institutions to teach calculus, according to their website. These institutions include the Virginia Military Institute, where the author is an associate professor of mathematics.

Includes: interactive graphics, exercises and solutions.

Applied Discrete Structures

Authors: Al Doerr, Ken Levasseur

Description: Applied Discrete Stuctures is a free open content textbook. You can essentially share it with anyone as long as you leave the Creative Commons license in place. See a more precise legal description below. 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. The following are available for download:

The text is available in Version 3.7, May 2020.  Source code for the current version is available at https://github.com/klevasseur/ads

One can view the textbook online at http://faculty.uml.edu/klevasseur/ads/index-ads.html

Includes: Exercises with short answers to odd problems, some concepts illustrated with Sage enabled web pages, blog page, Wiki, practice exams and videos.

Applied Statistics with R

Editors: Alex Stepanov, David Unger, James Balamuta

Contributors: Daniel McQuillan, Mason Rubenstein, Yuhang Wang, Zhao Liu, Jinfeng Xiao, Somu Palaniappan, Michael Hung-Yiu Chan, Eloise Rosen, Kiomars Nassiri, Jeff Gerlach, Brandon Ching, Ray Fix, Tyler Kim, Yeongho Kim, Elmar Langholz, Thai Duy Cuong Nguyen, Junyoung Kim, Sezgin Kucukcoban, Tony Ma, Radu Manolescu, Dileep Pasumarthi, Sihun Wang, Joseph Wilson, Yingkui Lin, Andy Siddall, Nishant Balepur, Durga Krovi, Raj Krishnan, Ed Pureza, Siddharth Singh, Schillaci Mcinnis, Ivan Valdes Castillo.

Description: This book was originally (and currently) designed for use with STAT 420, Methods of Applied Statistics, at the University of Illinois at Urbana-Champaign. It may certainly be used elsewhere, but any references to “this course” in this book specifically refer to STAT 420.

This book is under active development. When possible, it would be best to always access the text online to be sure you are using the most up-to-date version. Also, the html version provides additional features such as changing text size, font, and colors. If you are in need of a local copy, a pdf version is continuously maintained, however, because a pdf uses pages, the formatting may not be as functional. (In other words, the author needs to go back and spend some time working on the pdf formatting.) The source code for the text is available from Github.

Basic Analysis: Introduction to Real Analysis

Description: This free online textbook (OER more formally) is a course in undergraduate real analysis (somewhere it is called “advanced calculus”). The book is meant both for a basic course for students who do not necessarily wish to go to graduate school, but also as a more advanced course that also covers topics such as metric spaces and should prepare students for graduate study. A prerequisite for the course is a basic proof course.  An advanced course could be two semesters long with some of the second-semester topics such as multivariable differential calculus, path integrals, and the multivariable integral using the second volume. There are more topics than can be covered in two semesters, and it can also be reading for beginning graduate students to refresh their analysis or fill in some of the holes. This text was originally developed by Jiri Lebl in 2012 (Volume I).  Both Volume I and II were updated in June 2020. The following are available for download:

Includes: Exercises with no solutions.

Bayesian Data Analysis

Authors: Andrew Gelman, John Carlin, Hal Stern, David Dunson, Aki Vehtari, and Donald Rubin.

Description: This book is intended to have three roles and to serve three associated audiences: an introductory text on Bayesian inference starting from first principles, a graduate text on effective current approaches to Bayesian modeling and computation in statistics and related fields, and a handbook of Bayesian methods in applied statistics for general users of and researchers in applied statistics. Although introductory in its early sections, the book is definitely not elementary in the sense of a first text in statistics. The mathematics used in our book is basic probability and statistics, elementary calculus, and linear algebra. A review of probability notation is given in Chapter 1 along with a more detailed list of topics assumed to have been studied.

The practical orientation of the book means that the reader’s previous experience in probability, statistics, and linear algebra should ideally have included strong computational components.To write an introductory text alone would leave many readers with only a taste of the conceptual elements but no guidance for venturing into genuine practical applications, beyond those where Bayesian methods agree essentially with standard non-Bayesian analyses.On the other hand, we feel it would be a mistake to present the advanced methods with-out first introducing the basic concepts from our data-analytic perspective. Furthermore,due to the nature of applied statistics, a text on current Bayesian methodology would be incomplete without a variety of worked examples drawn from real applications.

To avoid cluttering the main narrative,there are bibliographic notes at the end of each chapter and references at the end of the book.Examples of real statistical analyses appear throughout the book, and we hope thereby to give an applied flavor to the entire development. Indeed, given the conceptual simplicity of the Bayesian approach, it is only in the intricacy of specific applications that novelty arises. Non-Bayesian approaches dominated statistical theory and practice for most of the  last century, but the last few decades have seen a reemergence of Bayesian methods. This has been driven more by the availability of new computational techniques than by what many would see as the theoretical and logical advantages of Bayesian thinking.In our treatment of Bayesian inference, we focus on practice rather than philosophy.

Note: This books is available for download and non-commercial purposes.

Supplementary materials: Video lectures, slides, chapter notes, code examples (R, Python, Matlab/Octave), software, data sets, solutions to some exercises

Book of Proof

Description: This book is an introduction to the standard methods of proving mathematical theorems. It has been approved by the American Institute of Mathematics’ Open Textbook Initiative. Also see the Mathematical Association of America Math DL review (of the 1st edition) and the Amazon reviews. An adoptions list is here. This book was originally developed by Richard Hammack in 2018. Please note that the Creative Commons License for this text does not permit altering of content for anything other than personal use.  A pdf version of the book is available for download from the book site.

Includes: Syllabus, calendar, sample tests and workbook.


Description: Originally published through traditional means, this textbook is now available as an open educational resource. This text includes single and multivariable calculus and was written by a professor of mathematics at the Massachusetts Institute of Technology.

Includes: an instructor’s manual, a student study guide with exercises and select solutions, and links to a calculus video series by the author.

Calculus: Early Transcendentals

Description: Calculus: Early Transcendentals, originally by D. Guichard, has been redesigned by the Lyryx editorial team. Substantial portions of the content, examples, and diagrams have been redeveloped, with additional contributions provided by experienced and practicing instructors. This approachable textbook provides a comprehensive understanding of the necessary techniques and concepts of the typical Calculus course sequence, and is suitable for the standard Calculus I, II and III courses. The latex source can be requested from Lyryx Learning.

Includes: Exercises and answers, slides. Lyryx Learning also has some online homework available.


Calculus for the Life Sciences: A Modelling Approach Volume 1

Description: This textbook was written by two professors at Iowa State University with the goal introducing students to scientific modeling. This course aims to teach students the necessary skills and concepts from a traditional physical sciences course, while explaining the applications of calculus to the life sciences.

Includes: exercises, solutions to select problems, and MATLAB exercises.

Calculus for the Life Sciences: A Modeling Approach Volume 2

Authors: James L. Cornette and Ralph A. Ackerman

Description: Our writing is based on three premises. First, life sciences students are motivated by and respond well to actual data related to real life sciences problems. Second, the ultimate goal of calculus in the life sciences primarily involves modeling living systems with difference and differential equations. Understanding the concepts of derivative and integral are crucial, but the ability to compute a large array of derivatives and integrals is of secondary importance. Third, the depth of calculus for life sciences students should be comparable to that of the traditional physics and engineering calculus course; else life sciences students will be short changed and their faculty will advise them to take the ‘best’ (engineering) course.

In our text, mathematical modeling and difference and differential equations lead, closely follow, and extend the elements of calculus. Chapter one introduces mathematical modeling in which students write descriptions of some observed processes and from these descriptions derive first order linear difference equations whose solutions can be compared with the observed data. In chapters in which the derivatives of algebraic, exponential, or trigonometric functions are defined, biologically motivated differential equations and their solutions are included. The chapter on partial derivatives includes a section on the diffusion partial differential equation. There are two chapters on non-linear difference equations and on systems of two difference equations and two chapters on differential equations and on systems of differential equation.

Calculus: OpenStax Volumes 1-3

Volume 1 concepts: functions, limits, derivatives, and integration

Volume 2 concepts: integration, differential equations, sequences and series, and parametric equations and polar coordinates

Volume 3 concepts: parametric equations and polar coordinates, vectors, functions of several variables, multiple integration, and second-order differential equations

Includes: Exercises and answers. Volume 1 has been faculty-reviewed, adopted, is accessible, and has ancillary resources. Volume 2 has been adopted, has ancillary resources, and is accessible. Volume 3 is accessible and has ancillary resources. Adopt and adapt these volumes at the links above on Volumes 1, 2, and 3, respectively.

Calculus Two

Description: Calculus Two  is a Computer-Aided Problem-Solving Approach Using SageMath written by Charles Bergeron and other open source creators. This is a text for a second undergraduate Calculus course. This book includes applications from many disciplines, and integrates the use of the open-source computer algebra system SageMath throughout. This free online book should be usable as a stand-alone textbook or as a supplementary resource.

You can use, print, duplicate, share this book as much as you want. You can base your own book on it and reuse parts if you keep the license the same. If you plan to use it commercially (sell it for more than duplicating cost), then you need to contact me and we will work something out.

This work is based upon the open-source books Community Calculus by David Guichard et al. and Mooculus and Mooculus: Sequences and Series by Jim Fowler and Bart Snapp.

Major changes include:

  • Creating a full chapter on linear algebra topics.
  • Strengthening the presentation of topics that prepare students for a course in Differential Equations, including hyperbolic functions and complex numbers.
  • Inserting instruction and exercises using the computer algebra system SageMath.
  • Adding applications in healthcare, chemistry and pharmacy.

Further content was adapted from these open-source resources:

I am grateful to all authors listed on the Attributions page, and indeed all open-source creators, for making their materials available. Without this publishing model, I probably would not have been able to to realize this project. What a nice way for us to collaborate! I am also grateful to the authors listed on the Further Reading page, for influencing my thinking on this subject, and its presentation.

The following are available for this resource:

  1. Download the textbook as a single PDF file, Spring 2017 edition (typeset on March 6, 2017)
  2. Progress packet (lists all learning activities for all class meeting times during the course)
  3. Syllabus for MAT 211

Includes: Exercises with solutions to selected exercises, example solutions using Sagemath computations, syllabus, progress packet, videos.


Author: Joy Morris

Description: This free undergraduate text book provides an introduction to enumeration, graph theory, and design theory. It is aimed at upper-level undergraduate students and the exercises expect some mathematical sophistication, including a reasonable ability to construct proofs. The text is designed to be used in an undergraduate course, but could be suitable for independent study by a student with some mathematical background and understanding of proofs. It does not assume any background knowledge of combinatorics. This text is written by Joy Morris of the University of Lethbridge.

The book is being released online with a Creative Commons license (Attribution-NonCommercial-ShareAlike 2.0). Although not in final form, it has already been used as a textbook for several semesters by 2 different instructors at the University of Lethbridge. The following are available:

  1. Download book in PDF format, June 2017 version (approximately 250 pages and 1.2 MB)
  2. Latex source available by contacting the author: joy.morris@uleth.ca

Includes: Exercises with solutions to selected exercises.

Community Calculus (Single, Multivariable)

Description: This online compilation of four calculus textbooks is frequently updated and has been used in courses. It has been positively reviewed by the Mathematical Association of America. It also covers early and late transcendentals.

Includes: exercises, solutions, and WeBWorK problem sets.

Contemporary Calculus

Description: This textbook was created as part of the Washington State Colleges’ Open Course Library Project, which received funding through the Gates Foundation. This text has been used by thousands of students within Washington State and was developed by a professor at Bellevue College.

Includes: exercises, solutions to odd-numbered problems, and other resources.

Development Versions, Source Code with Revision Control

Source code is available in a git repository, hosted at GitHub, and authored with PreTeXt. Prior to the 2015 edition, the source code was LaTeX and was primarily hosted at bitbucket.org.

Periodically a current snapshot of the book as a PDF is available here. These are not archived, nor tagged in the source code repository. They simply accumulate fixes or additions made to annual official releases. If there is nothing listed here, then use the current Annual Edition.

The absolute most recent version is available as source code through a git repository at github.com, the specific address is https://github.com/twjudson/aata. Links below to recent source code archives come from github.com, so the current version links below are the most up-to-date version.

If you would like to participate in contributing to the book, you will want to use git. Then you can fork the repository on GitHub, or create a local copy of the source code by doing the following in a terminal window:

  1. Set your working directory to a place where a new directory named aata can live.
  2. Issue the command: git clone https://github.com/twjudson/aata.git

Data Analysis with SAS: An Open Textbook

Author: Jerry Brunner

Description: This is the home page for Data Analysis: An Open Textbook. The text is free, and deliberately modeled on open source software. The source code is mostly LaTeX, and the compiled binary “program” is a PDF file. The pictures in the text are pdf files, included in the document using the LaTeX graphicx package. Most of them were produced with R. The R code appears as comment statements in the LaTeX source. At least one additional graphics file is in the open and modifiable SVG (Scalable Vector Graphics) format produced by the Open Office drawing program.

There are also (temporarily, I trust) three sections in that were produced using WriteNow, an obsolete and proprietory Mac word processing program. The pdfs of these parts (Section 5.9 on interactions in regression, Section 7.4 on nested designs and random effects, and Chapter 8 on selecting sample size) are brought directly into the document using the \texttt{pdfpages} package. I have translated the material from WriteNow to OpenOffice.org. The results are pretty rough, but the Openoffice.org documents are provided below in lieu of source code.

You are welcome to use this text under the conditions of the GNU Free Document License.

Supplementary materials: Latex source, graphics files,

Differential Calculus for the Life Sciences

Description: This textbook was written by a math professor at the University of British Columbia and is currently used as a textbook for introductory calculus courses for life sciences undergraduates.

Includes: exercises and solutions to selected problems.

Differential Equations

Description: A first course on differential equations, aimed at engineering students. The prerequisite for the course is the basic calculus sequence. This free online book (OER more formally) 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). This book was originally developed by Jiri Lebl in 2012. The current version was updated July 21st, 2020. The following are available for download:

Includes:  Exercises (740) with some solutions (247), homework exercises using the WebWork system.

Differential Equations

Description: This is a differential equations book including Linear Algebra Topics And Computer-Aided Problem-Solving Using Maxima or SageMath by Charles Bergeron, Jiri Lebl, and other open-source creators.

This is a text for a first undergraduate Differential Equations course. It does not assume previous coverage of Linear Algebra. This book includes applications from many disciplines, and integrates the use of the open-source computer algebra system Maxima or SageMath throughout. This free online book should be usable as a stand-alone textbook or as a supplementary resource.

This work is heavily based upon the open-source book Notes on Diffy Qs: Differential Equations for Engineers by Jiri Lebl. That book has been selected as an Approved Textbook in the American Institute of Mathematics Open Textbook Initiative.

Major changes include:

  • Stripping the chapters on PDEs and nonlinear systems.
  • Creating a full chapter on linear algebra topics.
  • Strengthening the presentation of review topics, including hyperbolic functions and complex numbers.
  • Inserting instruction and exercises using the computer algebra system Maxima or SageMath.
  • Adding applications in healthcare, chemistry and pharmacy.

Further content was adapted from these open-source resources:

The following are available for this resource:

  1. Download the textbook as a single PDF file Fall 2017 edition (typeset on July 6, 2017)
  2. Download the LaTeX source folder as a zipped file (9.0 MB)

Includes: Syllabus, progress packet, easy guide to Sage for Differential Equations, example solutions using Maxima, videos

Discover Linear Algebra

Description: This text is written by Jeremy Sylvestre of the University of Alberta.

Discover Linear Algebra is an open-access linear algebra textbook that uses a discovery-based approach to introduce students to this beautiful subject. The philosophy of this treatment is to allow the undeniable core ideas and patterns of linear algebra to reveal themselves to the student.

Each chapter begins with a set of guided-discovery activities suitable for use as in-class group activities, as pre-class preparatory explorations, or for self-study. The exposition in the remainder of each chapter reflects and expands upon these introductory explorations, beginning with an informal Concepts section, followed by a section of Examples, and ending with a more formal Theory section of theorems and proofs. Please see the book’s preface (two-semester version; one-semester version) for a little more detail on the organization and pedagogical approach of the book.

Discover Linear Algebra is free as in “freedom” — released under the GNU Free Documentation License (GFDL), you are free to use, copy, redistribute, and/or modify this textbook. (Though that freedom comes with some responsibilities; see the full text of the GFDL, included as an appendix to the textbook.) If you wish to make use of this work under a different license, please contact the author using the contact info below. The following are available:

  1. Discover Linear Algebra – Winter 2019 Edition: Two-semester version
  2. Discover Linear Algebra:  A First Course in Linear Algebra – Winter 2020 Edition: One-semester version
  3. Github repository for the PreTeXt document source code: github.com/jjrsylvestre/dla
  4. Author contact: Jeremy Sylvestre, University of Alberta, Augustana Campus, jeremy.sylvestre@ualberta.ca

Includes: Guided discovery activities, HTML version.

Discrete Mathematics: An Open Introduction

Description: Discrete Mathematics: An Open Introduction is a free, open source textbook appropriate for a first or second year undergraduate course for math majors, especially those who will go on to teach. Since Spring 2013, the book has been used as the primary textbook or a supplemental resource at more than 75 colleges and universities around the world (see the partial adoptions list). The text is endorsed by the American Institute of Mathematics’ Open Textbook Initiative and is well reviewed on the Open Textbook Library. The author of this text is Ocar Levin.

This 3rd edition brings many improvements, including nearly 100 new exercises, a new section on trees in the graph theory chapter, and improved exposition throughout. Previous editions will continue to be available indefinitely. A few times a year, the text is updated with a new “printing” to correct errors. See the errata list for more information.

New for Fall 2019: Online homework sets are available through Edfinity or as WeBWorK sets from the author. Additional exercises have been added since Spring 2020.

Get the book

The entire book is available for free as an interactive online ebook. This should work well on all screen sizes, including smart phones. Hints and solutions to examples and exercises are hidden but easily revealed by clicking on their links. Some exercises also allow you to enter and check your work, so you can try multiple times without spoiling the answer.

For offline use, a free pdf version, suitable for reading on a tablet or computer, is available for download. This should be searchable and easy to navigate using embedded links. Hints and solutions (when available) can be accessed by clicking on the exercise number, and clicking on the number of the hint or solution will bring you back to the exercise.

If you prefer a physical copy, an inexpensive print version of the text is available on Amazon. This should be cheaper than printing the entire book and binding it yourself. Page numbers match the pdf version.

Includes: The following come with this text:

  • 473 exercises, including 275 with solutions and another 109 with hints. Exercises range from easy to quite involved, with many problems suitable for homework.
  • Investigate! activities throughout the text to support active, inquiry based learning.
  • A full index and list of symbols.
  • Consistent and helpful page layout and formatting (i.e., examples are easy to identify, important definitions and theorems in boxes, etc.).

A First Course in Linear Algebra

Description: A First Course in Linear Algebra is an introductory textbook designed for university sophomores and juniors. Typically such a student will have taken calculus, but this is not a prerequisite. The book begins with systems of linear equations, then covers matrix algebra, before taking up finite-dimensional vector spaces in full generality. The final chapter covers matrix representations of linear transformations, through diagonalization, change of basis and Jordan canonical form. Along the way, determinants and eigenvalues get fair time. There is a comprehensive online edition and PDF versions are available to download for printing or on-screen viewing.

This textbook has more freedom than most (but see some exceptions). First, there is no cost to acquire this text, and you are under no obligation whatsoever to compensate or donate to the author or publisher. So in this most basic sense, it is a free textbook. Therefore you can also make as many copies as you like, ensuring that the book will never go out-of-print. You may modify copies of the book for your own use – for example, you may wish to change to a prefered notation for certain objects or add a few new sections. I have applied a copyright to the book, and subsequently licensed it with a GNU Free Documentation License (GFDL). It is this combination that allows me to give you greater freedoms in how you use the text, thus liberating it from some of the antiquated notions of copyright that apply to books in physical form. The main caveat is that if you make modifications and then distribute a modified version, you are required to again apply the GFDL license to the result so that others may benefit from your modifications.

A First Course in Linear Algebra: Lyryx

Description: A First Course in Linear Algebra, originally by K. Kuttler, has been redesigned by the Lyryx editorial team as a first course for the general students who have an understanding of basic high school algebra and intend to be users of linear algebra methods in their profession, from business \& economics to science students. This version was revised by the Lyryx editorial team in 2017.

Includes: Exercises with selected answers. Slides and question bank along with the source are available on request.

First-Semester Abstract Algebra: A Structural Approach

Author: Jessica K. Sklar

Description: A note on algebra: At its most basic level, abstract algebra is the study of structures. Just as an architect may examine buildings or an anthropologist societal hierarchies, an algebraist explores the nature of sets equipped with binary operations that satisfy certain properties. While these structures may not seem at first to be very important, they are at the heart of most, if not all, mathematical endeavors. On an elemental level, they allow us to solve systems of equations; on a more global-level, they are behind some of our most important cryptographic systems. We even use them implicitly when telling time!

Our focus in this book is the study of algebraic structures called groups. Along the way, we will explore rigorous mathematical notions of similarity and difference: When can we consider two objects to be more or less “the same”? When are they fundamentally different? For instance, consider two houses that have exactly the same construction, but are painted different colors. Are they the same house? No. But viewed structurally (as opposed to aesthetically) they are the same. This means that if we know certain information about one of the houses (say, how far the bathroom is from the kitchen) we know the same information about the other house. However, knowing that the first house is painted yellow does not tell us anything about the second house’s color. We explore an analogous idea in mathematics, namely, the concept of isomorphism.

Throughout, we provide readers with many mathematical proofs, as well as specific examples demonstrating more general ideas.

Forecasting: Principles and Practice

Authors: Robert J. Hyndman and George Athanasopoulos

Description: This textbook is intended to provide a comprehensive introduction to forecasting methods and to present enough information about each method for readers to be able to use them sensibly. We don’t attempt to give a thorough discussion of the theoretical details behind each method, although the references at the end of each chapter will fill in many of those details.

The book is written for three audiences: (1) people finding themselves doing forecasting in business when they may not have had any formal training in the area; (2) undergraduate students studying business; (3) MBA students doing a forecasting elective. We use it ourselves for masters students and third-year undergraduate students at Monash University, Australia.

For most sections, we only assume that readers are familiar with introductory statistics, and with high-school algebra. There are a couple of sections that also require knowledge of matrices, but these are flagged.

At the end of each chapter we provide a list of “further reading.” In general, these lists comprise suggested textbooks that provide a more advanced or detailed treatment of the subject. Where there is no suitable textbook, we suggest journal articles that provide more information.

We use R throughout the book and we intend students to learn how to forecast with R. R is free and available on almost every operating system. It is a wonderful tool for all statistical analysis, not just for forecasting. See the Using R appendix for instructions on installing and using R.

FREE: Federal Resources for Educational Excellence

Guide to Cultivating Complex Analysis

Description: The purpose of this book is to teach a one-semester graduate course in complex analysis for incoming graduate students. It was created to teach Math 5283 at Oklahoma State University. It is a natural first semester in a two semester sequence where the second semester could be several complex variables or perhaps harmonic analysis. It could perhaps be used for a more elementary two-semester sequence if the appendix (metric spaces, some basic analysis) is covered first, and all the optional bits of the main text are also covered. The text was originally developed by Jiri Lebl in 2020. The following are available for download:

Includes: Exercises (600) in the text.

Handbook of Biological Statistics

Author: John H. McDonald

Description: Welcome to the third edition of the Handbook of Biological Statistics! This online textbook evolved from a set of notes for my Biological Data Analysis class at the University of Delaware. My main goal in that class is to teach biology students how to choose the appropriate statistical test for a particular experiment, then apply that test and interpret the results. In my class and in this textbook, I spend relatively little time on the mathematical basis of the tests; for most biologists, statistics is just a useful tool, like a microscope, and knowing the detailed mathematical basis of a statistical test is as unimportant to most biologists as knowing which kinds of glass were used to make a microscope lens. Biologists in very statistics-intensive fields, such as ecology, epidemiology, and systematics, may find this handbook to be a bit superficial for their needs, just as a biologist using the latest techniques in 4-D, 3-photon confocal microscopy needs to know more about their microscope than someone who’s just counting the hairs on a fly’s back. But I hope that biologists in many fields will find this to be a useful introduction to statistics.

Supplementary materials: Spreadsheets, SAS code, R Companion for Handbook.

Introduction to Data Science

Author: Ron Sarafian

Description: This book accompanies the course I give at Ben-Gurion University, named “Introduction to Data Science”. This is an introductory-level, hands-on focused course, designed for students with basic background in statistics and econometrics, and without programming experience. It introduces students to different tools needed for building a data science pipeline, including data processing, analysis, visualization and modeling. The course is taught in R environment.

Many of the contents in this book are taken from BGU’s “R” course, given at the department of Industrial Engineering and Management.

The chapters in this book are arranged (roughly) according to the order of classes throughout the semester. Students are encouraged to go through the book during the lectures, and after class.

Introduction to Mathematics Analysis I

Authors: Beatriz Lafferriere, Gerardo Lafferriere, and Nguyen Mau Nam

Description: Our goal in this set of lecture notes is to provide students with a strong foundation in mathematical analysis. Such a foundation is crucial for future study of deeper topics of analysis. Students should be familiar with most of the concepts presented here after completing the calculus sequence. However, these concepts will be reinforced through rigorous proofs.

The lecture notes contain topics of real analysis usually covered in a 10-week course: the completeness axiom, sequences and convergence, continuity, and differentiation. The lecture notes also contain many well-selected exercises of various levels. Although these topics are written in a more abstract way compared with those available in some textbooks, teachers can choose to simplify them depending on the background of the students. For instance, rather than introducing the topology of the real line to students, related topological concepts can be replaced by more familiar concepts such as open and closed intervals. Some other topics such as lower and upper semicontinuity, differentiation of convex functions, and generalized differentiation of non-differentiable convex functions can be used as optional mathematical projects. In this way, the lecture notes are suitable for teaching students of different backgrounds.

The second edition includes a number of improvements based on recommendations from students and colleagues and on our own experience teaching the course over the last several years.

In this edition we streamlined the narrative in several sections, added more proofs, many examples worked out in detail, and numerous new exercises. In all we added over 50 examples in the main text and 100 exercises (counting parts).

Supplementary materials: video lectures

Introductory Statistics

Description: Introductory Statistics follows the scope and sequence of a one-semester, introduction to statistics course and is geared toward students majoring in fields other than math or engineering. This text assumes students have been exposed to intermediate algebra, and it focuses on the applications of statistical knowledge rather than the theory behind it. The foundation of this textbook is Collaborative Statistics, by Barbara Illowsky and Susan Dean, which has been widely adopted. Introductory Statistics includes innovations in art, terminology, and practical applications, all with a goal of increasing relevance and accessibility for students. We strove to make the discipline meaningful and memorable, so that students can draw a working knowledge from it that will enrich their future studies and help them make sense of the world around them. The text also includes Collaborative Exercises, integration with TI-83,83+,84+ Calculators, technology integration problems, and statistics labs.

Introductory Statistics: Saylor

Description: This peer-reviewed resource introduces statistical concepts, including: descriptive statistics, basic concepts of probability, discrete random variables, continuous random variables, sampling distributions, estimation, testing hypotheses, two-sample problems, correlation and regression, and chi-square and f-tests.

Includes: exercises and answers.

Khan Academy

Learning Statistics with jamovi

Authors: Danielle J. Navarro and David R. Foxcroft.

Description: Learning Statistics with jamovi covers the contents of an introductory statistics class, as typically taught to undergraduate psychology students. The book discusses how to get started in jamovi as well as giving an introduction to data manipulation. Descriptive statistics and graphing are followed by chapters on probability theory, sampling and estimation, and null hypothesis testing. The book covers the analysis of contingency tables, correlation, t-tests, regression, ANOVA and factor analysis.

Written in latex and published as a pdf file, for great design and easy access. The book is open source licensed and is free to download.

Supplementary materials: Data files, latex source downloadable from Github.

Learning Statistics with R

Author: Danielle Navarro

Description: Back in the grimdark pre-Snapchat era of humanity (i.e. early 2011), I started teaching an introductory statistics class for psychology students offered at the University of Adelaide, using the R statistical package as the primary tool. I wrote my own lecture notes for the class, which have now expanded to the point of effectively being a book. The book is freely available, and as of version 0.6 it is released under a creative commons licence (CC BY-SA 4.0). The following are available:

Linear Algebra

Author: Jim Hefferon

Description: This textbook has been positively reviewed by four math professors from American universities and the Mathematical Association of America.

Includes: exercises, solutions, a lab manual, lecture slides, videos and latex source code.

Linear Algebra

Description: A free linear algebra textbook and online resource written by David Cherney, Tom Denton, Rohit Thomas and Andrew Waldron. Edited by Katrina Glaeser and Travis Scrimshaw.

This “textbook” (+videos+WeBWorKs) is suitable for a sophomore level linear algebra course taught in about twenty-five lectures. It is designed both for engineering and science majors, but has enough abstraction to be useful for potential math majors. Our goal in writing it was to produce students who can perform computations with linear systems and also understand the concepts behind these computations. The text was last updated in 2016. The following are downloadable for this text:

  • the entire book can be downloaded as a single pdf file of 1.4 MB and 410 pages. (Last update: August 24, 2016)
  • The LaTeX source files can be downloaded as a single tarball. (Last update: January 25, 2019)
  • The WeBWorK files can be downloaded as a single tarball. (Last update: August 24, 2016)

Includes: There are Youtube videos linked to content throughout the book, problems at the end of each chapter, online homework using the WebWork system.

Linear Algebra with Applications

Author: Xinli Wang

Description: This open textbook is an adaptation of Linear Algebra with Applications by W. Keith Nicholson. The original book can be found and downloaded from Lyryx.com. Five topics are covered here: system of linear equations, matrix algebra, determinants and diagonalization, vector geometry and vector space. It’s suitable for beginners who are interested in learning linear algebra. Readers will have the opportunity to work through H5P elements embedded in every chapter to check their understanding of core concepts on their own.

Includes: H5P elements throughout the resource

Linear Algebra with Applications: Lyryx

Author: W. Keith Nicholson

Description: The first version of this textbook was published in 1986 as a traditional textbook. The current version from 2018 was published as an open educational resource for a traditional or advanced introduction to the topics within linear algebra.

Includes: instructor resources: solution manual to exercises and lecture slides.

Math Explorer Activity Database

Matrix Algebra with Computational Applications

Description: Matrix Algebra with Computational Applications is a collection of Open Educational Resource (OER) materials designed to introduce students to the use of Linear Algebra to solve real world problems. These materials were developed specifically for students and instructors working in a “flipped classroom” model that emphasizes hands-on problem solving activities during class meetings, with students watching lectures and completing readings and assignments outside of the classroom. The materials are organized into a semester long course with “pre-class” and “in-class” assignments. The “pre-class” assignments include readings, video lectures and coding projects (in Python), which students are expected to complete before attending class. The in-class assignments consist of hands-on individual and group activities intended to be completed during class. These in-class activities are supervised by the instructors, who actively answer questions and help guide the students in achieving the learning goals for the course.

Measure, Integration & Real Analysis

Author: Sheldon Axler

Description: This book seeks to provide students with a deep understanding of the definitions, examples, theorems, and proofs related to measure, integration, and real analysis. The content and level of this book fit well with the first-year graduate course on these topics at most American universities. This textbook features a reader-friendly style and format that will appeal to today’s students.

Supplementary material: Supplement for Measure, Integration & Real Analysis

MIT OpenCourseWare: Mathematics

Online Statistics Education

Editor: David M. Lane

Contributors: Georgette Baghdady, Evan Brott, Katie Bruton, Madeline Campbell, Patrick Connell, Tina Galante,
Paul Giguere, Rudy Guerra, Daniel Hatfield, Mikki Hebl, Robert F. Houser, W. Sloane Hoyle, Jo Jardina, Andrew Kennedy, Jennifer E. Konick, Alyssa Koomas, David Lane, Joan Lu, Hanqi Luo, Daniel Osherson, Lauren Pemberton,
Camille Peres, Anikó Sándor, David Scott, Alex Shabad, Zhihua Tang, Sebastian Thomas, Katherine Vasser, Heidi Ziemer, Emily Zitek.

Description: Online Statistics: An Interactive Multimedia Course of Study is a resource for learning and teaching introductory statistics. It contains material presented in textbook format and as video presentations. This resource features interactive demonstrations and simulations, case studies, and an analysis lab.
Supplementary materials: Instructor manual, PowerPoint slides, and additional questions.

Open Course Library

OpenStax (aka Connexions) Mathematics and Statistics

PhET: Math

Probability and Statistics – The Science of Uncertainty, Second Edition

Authors: Michael J. Evans and Jeffrey S. Rosenthal

Description: This book is an introductory text on probability and statistics, targeting students who have studied one year of calculus at the university level and are seeking an introduction to probability and statistics with mathematical content. Where possible, we provide mathematical details, and it is expected that students are seeking to gain some mastery over these, as well as to learn how to conduct data analyses. All the usual methodologies covered in a typical introductory course are introduced, as well as some of the theory that serves as their justification.

The text can be used with or without a statistical computer package. It is our opinion that students should see the importance of various computational techniques in applications, and the book attempts to do this. Accordingly, we feel that computational aspects of the subject, such as Monte Carlo, should be covered, even if a statistical package is not used. Almost any statistical package is suitable. A Computations appendix provides an introduction to the R language. This covers all aspects of the language needed to do the computations in the text. Furthermore, we have provided the R code for any of the more complicated computations. Students can use these examples as templates for problems that involve such computations, for example, using Gibbs sampling.

Also, we have provided, in a separate section of this appendix, Minitab code for those computations that are slightly involved, e.g., Gibbs sampling. No programming experience is required of students to do the problems. We have organized the exercises in the book into groups, as an aid to users. Exercises are suitable for all students and offer practice in applying the concepts discussed in a particular section. Problems require greater understanding, and a student can expect to spend more thinking time on these. If a problem is marked (MV), then it will require some facility with multivariable calculus beyond the first calculus course, although these problems are not necessarily hard. Challenges are problems that most students will find difficult; these are only for students who have no trouble with the Exercises and the Problems. There are also Computer Exercises and Computer Problems, where it is expected that students will make use of a statistical package in deriving solutions.

Note: We are pleased to now make the book available for free. If you are an instructor and would like a copy of the solutions manual please email one of the authors. The book is available as a single pdf file or by individual chapters.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.

Think Statistics: Probability and Statistics for Programmers

Author: Allen B. Downey

Description: Think Stats is an introduction to Probability and Statistics for Python programmers.

  • Think Stats emphasizes simple techniques you can use to explore real data sets and answer interesting questions. The book presents a case study using data from the National Institutes of Health. Readers are encouraged to work on a project with real datasets.
  • If you have basic skills in Python, you can use them to learn concepts in probability and statistics.Think Stats is based on a Python library for probability distributions (PMFs and CDFs). Many of the exercises use short programs to run experiments and help readers develop understanding.

This book is available under a Creative Commons license, which means that you are free to copy, distribute, and modify it, as long as you attribute the source and don’t use it for commercial purposes.

Supplementary resources: Latex source, code examples and solutions from Github.

Transition to Higher Mathematics: Structure and Proof

Authors: Bob A Dumas and John E McCarthy

Description: This book is written for students who have taken calculus and want to learn what “real mathematics” is. We hope you will find the material engaging and interesting, and that you will be encouraged to learn more advanced mathematics. This is the second edition of our text. It is intended for students who have taken a calculus course, and are interested in learning what higher mathematics is all about. It can be used as a textbook for an “Introduction to Proofs” course, or for self-study. Chapter 1: Preliminaries, Chapter 2: Relations, Chapter 3: Proofs, Chapter 4: Principles of Induction, Chapter 5: Limits, Chapter 6: Cardinality, Chapter 7: Divisibility, Chapter 8: The Real Numbers, Chapter 9: Complex Numbers. The last 4 chapters can also be used as independent introductions to four topics in mathematics: Cardinality; Divisibility; Real Numbers; Complex Numbers.

UBC Calculus Textbooks

Description: UBC has created its own sequence of open calculus texts to be used in the following courses:

  1. CLP-1 Differential Calculus, also available online using PreTeXt.
  2. CLP-2 Integral Calculus, also available online using PreTeXt.
  3. CLP-3 Multivariable Calculus, also available online using PreTeXt.
  4. CLP-4 Vector Calculus, also available online using PreTeXt.

The latex source from the first two of these texts has been made available from github (source for CLP-1, CLP-2, CLP-3 , CLP-4). Each of the texts also includes a problem book.

Includes: Problem book for each of the texts. Exercises and solutions included many of which were taken from old exams, midterm tests and quizzes.

Understanding Linear Algebra

Description: This is an open access textbook written by David Austin.

An HTML version and PDF version are now available at no cost. Besides being freely available, David Austin intends to make the PreTeXt source code available on GitHub in the near future.

Includes:  Activities for teaching Linear Algebra with downloadable latex source files, A blog for the text, embedded Sage calculations in HTML version

Vector Calculus

Description: This text was written by Michael Corral in 2013. This is a text on elementary multivariable calculus, designed for students who have completed courses in single-variable calculus. The traditional topics are covered: basic vector algebra; lines, planes and surfaces; vector-valued functions; functions of 2 or 3 variables; partial derivatives; optimization; multiple integrals; line and surface integrals.

The book also includes discussion of numerical methods: Newton’s method for optimization, and the Monte Carlo method for evaluating multiple integrals. There is a section dealing with applications to probability. Appendices include a proof of the right-hand rule for the cross product, and a short tutorial on using Gnuplot for graphing functions of 2 variables. The following are available for download:

Includes: Exercises with selected answers, some Java, Sage, Matlab/Octave programs illustrating specific concepts.



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