Math and Stats

# 106 Math

# Collections

American Institute of Mathematics – Approved Textbooks (Various CC licences)

This list groups open textbooks by course title. All the books have been judged to meet the evaluation criteria set by the AIM editorial board.

Applied Math and Science Education Repository (AMBER) (Various CC licences)

A collection of educational resources and services on topics related to applied math and science.

National Science Digital Library (Various CC licences)

A collection of high quality online educational resources for teaching and learning with emphasis on the sciences, technology, engineering, and mathematics.

PreTeXt (Various CC and open licences)

An uncomplicated XML vocabulary for authors of research articles, textbooks, and monographs. The best of DocBook, LaTeX, and HTML. Outputs: print, PDF, web, EPUB, Jupyter Notebooks.

# Courses

STEM Readiness (Various CC licences)

This course provides a refresher of a core skill related to STEM careers: Mathematics from arithmetic to beginning algebra. Students will be better prepared for success in post-secondary STEM-related technical programs and ultimately in STEM-related careers.

# Presentations/Slides

Graph Theory and Management (CC BY-SA)

Four sets of slides created by an instructor at the University of Hartford that cover four different sections of graph theory and management. These make up one unit or a little more than one fourth (five weeks) of the course.

- Graph Theory and Management Science 1 – Euler Paths and Circuits [PowerPoint]
- Graph Theory and Management Science 2- Fleury’s Algorithm and Eulerizing [PowerPoint]
- Graph Theory and Management Science 3-Hamilton Graphs and The Traveling Salesperson Problem [PowerPoint]
- Graph Theory and Management Science 4-Networks and Spanning Trees [PowerPoint]

The creator would like to keep track of who is using these slides. If you decide to use them in your course, please email Peggy Mitchell Beauregard to let her know.

# Supplementary Materials

Grasple (Various CC licences)

Curated open exercises and lessons on math and stats created by the community.

# Textbooks

## Textbook Collections

B.C. Open Textbook Collection: Math/Statistics (Various CC licences)

A collection of open textbooks on various topics relating to math and statistics.

Business Math I (CC By-NC-SA)

This book is aimed at university business students as an introduction to the mathematics required for the field of business. This textbook covers the fundamentals of business precalculus, finance, as well as the applications to general business management, human resources and the economy, marketing and accounting.

Mathematical Reasoning: Writing and Proof (CC BY-NC-SA)

For ﬁrst college mathematics course that introduces students to the processes of constructing and writing proofs and focuses on the formal development of mathematics.

Mathematics LibreTexts Library (CC BY-NC-SA)

A collection of open textbooks, assignments, and other educational resources on the topic of mathematics.

## Individual Titles

Advanced Problems in Mathematics: Preparing for University (CC BY)

This book is intended to help candidates prepare for entrance examinations in mathematics and scientific subjects, including STEP (Sixth Term Examination Paper). STEP is an examination used by Cambridge Colleges for conditional offers in mathematics. This book bridges the gap between school and university mathematics, and prepares students for an undergraduate mathematics course. The questions analysed in this book are all based on past STEP questions and each question is followed by a comment and a full solution. The comments direct the reader’s attention to key points and put the question in its true mathematical context. The solutions point students to the methodology required to address advanced mathematical problems critically and independently.

Abstract Algebra (GNU Free Documentation Licence)

This text is designed to teach the principles and theory of abstract algebra. The first half of the book presents group theory, through the Sylow theorems, with enough material for a semester-long course. The second-half is suitable for a second semester and presents rings, integral domains, Boolean algebras, vector spaces, and fields, concluding with Galois Theory.

Basic Concepts of Mathematics (CC BY-NC-ND)

This book helps the student complete the transition from purely manipulative to rigorous mathematics. The clear exposition covers many topics that are assumed by later courses but are often not covered with any depth or organization: basic set theory, induction, quantifiers, functions and relations, equivalence relations, properties of the real numbers (including consequences of the completeness axiom), fields, and basic properties of *n*-dimensional Euclidean spaces.

Brief Calculus (CC BY-SA)

This short text is designed more for self-study or review than for classroom use; full solutions are given for nearly all the end-of-chapter problems. For a more traditional text designed for classroom use, see Fundamentals of Calculus (http://www.lightandmatter.com/fund/). The focus is mainly on integration and differentiation of functions of a single variable, although iterated integrals are discussed. Infinitesimals are used when appropriate, and are treated more rigorously than in old books like Thompson’s Calculus Made Easy, but in less detail than in Keisler’s Elementary Calculus: An Approach Using Infinitesimals. Numerical examples are given using the open-source computer algebra system Yacas, and Yacas is also used sometimes to cut down on the drudgery of symbolic techniques such as partial fractions. Proofs are given for all important results, but are often relegated to the back of the book, and the emphasis is on teaching the techniques of calculus rather than on abstract results.

Book of Proof (CC BY-NC-ND)

This book is an introduction to the standard methods of proving mathematical theorems.

Calculus (CC BY-NC-SA)

Published in 1991 by Wellesley-Cambridge Press, the book is a useful resource for educators and self-learners alike. It is well organized, covers single variable and multivariable calculus in depth, and is rich with applications. There is also an online Instructor’s Manual and a student Study Guide.

Combinatorics Through Guided Discovery (GNU Free Documentation Licence)

This book is an introduction to combinatorial mathematics, also known as combinatorics. The book focuses especially, but not exclusively, on the part of combinatorics that mathematicians refer to as “counting.” The book consists almost entirely of problems. From time to time there is a discussion that pulls together some of the things you have learned or introduces a new idea for you to work with. Many of the problems are designed to build up your intuition for how combinatorial mathematics works.

Community Calculus (CC BY-NC-SA)

This series of texts includes Single variable calculus, early transcendentals; Multivariable calculus, early trancendentals; Single variable calculus, late transcendentals; and Multivariable calculus, late transcendentals.

A Computational Introduction to Number Theory and Algebra (CC BY-NC-ND)

This books does not presuppose any previous background in number theory or algebra, but it quickly moves into material beyond the usual courses in math departments because of the emphasis on algorithms and computation. The chapter titles give an idea of the unusual flavor of this book, which has a number of topics that would be suitable for a senior level “advanced topics” course following a more traditional algebra or number theory course. The author writes that the book could “be used as a textbook in a graduate or upper-division undergraduate course on (computational) number theory and algebra, perhaps geared towards computer science students.”

Elementary Calculus (GNU Free Documentation Licence)

This is the first part (Calculus I) of a text on elementary calculus, designed for students who have completed courses in high-school algebra, geometry, and trigonometry. Though designed for college students, it could also be used in high schools. The traditional topics are covered, but the old idea of infinitesimals is resurrected, owing to its usefulness (especially in the sciences).

A First Course in Linear Algebra (GNU Free Documentation Licence)

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.

A Gentle Introduction to the Art of Mathematics (GNU Free Documentation Licence)

This text covers several topics in the foundations of mathematics (logic, sets, relations, functions and cardinality) and introduces the reader to many techniques of mathematical proof (direct, indirect, contradiction, contrapositive, mathematical induction, combinatorial proofs and magic). There are amusing quotations at the start of each chapter.

How We Got from There to Here: A Story of Real Analysis (CC BY-NC-SA)

This book covers the major topics typically addressed in an introductory undergraduate course in real analysis in their historical order. Written with the student in mind, the book provides guidance for transforming an intuitive understanding into rigorous mathematical arguments. For example, in addition to more traditional problems, major theorems are often stated and a proof is outlined. The student is then asked to fill in the missing details as a homework problem.

Introduction to Complex Numbers (CC BY)

This is an introduction to the mathematics of complex numbers, starting from the very basics of their definitions, up to proving theorems for polynomials. The text covers everything required of most first-year mathematics courses on complex numbers with proofs, where appropriate.

An Introduction to the Theory of Numbers (CC BY)

This book, which presupposes familiarity only with the most elementary concepts of arithmetic (divisibility properties, greatest common divisor, etc.), is an expanded version of a series of lectures for graduate students on elementary number theory. Topics include: Compositions and Partitions; Arithmetic Functions; Distribution of Primes; Irrational Numbers; Congruences; Diophantine Equations; Combinatorial Number Theory; and Geometry of Numbers. Three sections of problems (which include exercises as well as unsolved problems) complete the text.

Linear Algebra: A Course for Physicists and Engineers (CC BY-NC-ND)

This open textbook is meant for courses on space and atmospheric science, remote sensing, geographic information systems, meteorology, climate and satellite communications at UN-affiliated regional centers, various applications of the formal theory are discussed as well. These include differential equations, statistics, optimization and some engineering-motivated problems in physics.

Linear Algebra Done Wrong (CC BY-NC-ND)

This text is for a first linear algebra course for mathematically advanced students. It is intended for a student who, while not yet very familiar with abstract reasoning, is willing to study more rigorous mathematics that is presented in a “cookbook style” calculus type course. Besides being a first course in linear algebra it is also supposed to be a first course introducing a student to *rigorous* proof, formal definitions—in short, to the style of modern theoretical (abstract) mathematics.

Mathematical Analysis I (CC BY)

This text leads the student through the basic topics of Real Analysis. Topics include metric spaces, open and closed sets, convergent sequences, function limits and continuity, compact sets, sequences and series of functions, power series, differentiation and integration, Taylor’s theorem, total variation, rectifiable arcs, and sufficient conditions of integrability. Well over 500 exercises (many with extensive hints) assist students through the material.

Mathematical Reasoning: Writing and Proof (CC BY NC-SA)

This textbook is designed for the first course in the college mathematics curriculum that introduces students to the process of constructing and writing proofs.

Modelling, Functions, and Graphs: Algebra for College Students (GNU Free Documentation Licence)

All students, not just those headed for science and engineering, should develop a mathematical viewpoint, including critical thinking, problem-solving strategies, and estimation, in addition to computational skills. *Modeling, Functions and Graphs* employs a variety of applications to motivate mathematical thinking.

Number Theory: In Context and Interactive (CC BY-ND)

The book tackles all standard topics of modular arithmetic, congruences, and prime numbers, including quadratic reciprocity. In addition, there is significant coverage of various cryptographic issues, geometric connections, arithmetic functions, and basic analytic number theory, ending with a beginner’s introduction to the Riemann Hypothesis. Ordinarily this should be enough material for a semester course with no prerequisites other than a proof-transition experience and vaguely remembering some calculus.

Precalculus – College Algebra – Trigonometry (CC BY-NC-SA)

This text covers the following topics: Relations and Functions, Linear and Quadratic Functions, Polynomial Functions, Rational Functions, Further Topics in Functions, Exponential and Logarithmic Functions, Hooked on Conics, Systems of Equations and Matrices, Sequences and the Binomial Theorem, Foundations of Trigonometry, and Applications of Trigonometry.

Trigonometry (GNU Free Documentation Licence)

We have tried to make this edition of Trigonometry useful to students in a variety of programs. In addition to the Homework Problems, each Example in the book is followed by a similar Exercise for students to test their understanding. Each Section concludes with a Summary , a set of Study Questions, and a list of Skills to be addressed in the Homework. A Summary and a set of Review Problems follows each chapter.

Tea Time Numerical Analysis (CC BY-SA)

An introductory Numerical Analysis textbook designed to be a complete, one-semester textbook for mathematics classes.

Trigonometry (GNU Free Documentation Licence)

This is a text on elementary trigonometry, designed for students who have completed courses in high-school algebra and geometry. Though designed for college students, it could also be used in high schools. The traditional topics are covered, but a more geometrical approach is taken than usual. Also, some numerical methods (e.g. the secant method for solving trigonometric equations) are discussed. A brief tutorial on using Gnuplot to graph trigonometric functions is included.

Vector Calculus (GNU Free Documentation Licence)

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.

# Tools

Comparison of MyOpenMath and WeBWorK (CC BY)

A chart by Open Oregon Educational Resources comparing two open math platforms: MyOpenMath and WeBWork.

PhET Interactive Simulations (CC BY)

The PhET Interactive Simulations project at the University of Colorado Boulder creates free interactive math and science simulations. The site is available in a number of languages.

WeBWork – Mathematical Association of America (Open source)

WeBWorK is an open-source online homework system for math and science courses. WeBWorK is supported by the MAA and the NSF and comes with a National Problem Library (NPL) of over 20,000 homework problems. Problems in the NPL target most lower division undergraduate math courses and some advanced courses.

# Videos

Mathispower 4u Tutorials (CC BY-SA)

This site provides more than 6,000 free mini-lessons and example videos on different math topics.

### Media Attributions

- Canada Map Icon © Icons8 is licensed under a CC BY-ND (Attribution NoDerivatives) license