An Infinite Descent into Pure Mathematics by Clive Newstead is a free and open-source introductory pure mathematics textbook with a focus on logical reasoning, proof-writing and problem-solving. It is licensed under CC BY-SA 4.0.
The book is still in the pre-publication (Version 0.x) stage. It is suitable for use in most introductory undergraduate-level courses on mathematical proof, but some of the later sections lack exercises and a few other gaps need to be filled.
The TeX source of the textbook can be found on Codeberg.
Version 0.7 can be purchased as a 7"×10" paperback (ISBN 978-1-950215-99-7) from IngramSpark (UK/USA only) or Lulu (ships worldwide).
What's in the textbook
The main body of the textbook is broken into two parts:
- Part I covers the core concepts of pure mathematics—logic, sets, functions, induction, relations and infinity—and builds a system of strategies for creating and writing mathematical proofs;
- Part II introduces several specific topics within pure mathematics, including number theory, combinatorics, real analysis, infinite cardinality, and probability theory.
The appendices include guidance for proof-writing, hints and solutions to selected exercises, technical details that would distract from the big picture if covered in the main text, and typesetting mathematical documents using LaTeX.
Who the textbook is for
This is an undergraduate-level textbook intended for use in a guided study environment—typically a college or university course under the guidance of a professor or teacher, or possibly in a tutoring setting. While it can be used for self-study, it is not designed specifically for that purpose.
The reader is assumed to have a basic background in mathematics at the level of elementary algebra, arithmetic and geometry: in order to get the most out of the textbook, you should feel comfortable manipulating equations and inequalities involving variables, addition, subtraction, multiplication, division and exponentiation.
Using the textbook to teach
If you are using An Infinite Descent into Pure Mathematics to teach a course, please get in touch (cnewstead@infinitedescent.org)—I would love to hear from you. I have made no effort to promote the textbook, so I find it very interesting to hear how it is being used, and I am happy to answer teaching-related questions.
Course instructors using Infinite Descent are strongly encouraged to either use the stable version, or download a static copy of the most recent unstable version and distribute that to their students.
Please be aware that solutions to in-chapter exercises are (or will soon) be in the appendix, so questions for homework assignments should be selected from the end-of-chapter exercises. The decision to exclude solutions from the end-of-chapter exercises is a conscious one; there is no separate instructor solution manual.
Feedback and errata
If you have any feedback about the textbook, or if you discover any errors or typos, please contact cnewstead@infinitedescent.org.
Versions of the textbook
The unstable PDF is updated in real-time, meaning that the content can change (sometimes significantly) from one day to the next. Every year or two, a new stable version is released, whose content will not change and is more suitable for use in a teaching setting.
The stable versions released to date are archived below, with static links and a brief description of the major changes since the previous stable version.
- v0.7 (): Page size changed from 6"×9" to 7"×10"; Ch 0 material revised to include more about sets and prioritise content that will be useful in future sections; treatment of intersection and union changed; appendices tidied up; many typos fixed; replaced thmtools with ntheorem to fix bug with reference links in the PDF; redesigned theorem environments.
- v0.6 (): Definition of [n] changed from {1, 2, ..., n} to {0, 1, ..., n-1} and proofs involving [n] updated; proof by infinite descent added to strong induction section; many typos fixed.
- v0.5 (): Organisation of finite and infinite sets material changed; typos fixed; exercises added.
- v0.4 (): Organisation of sets and functions material changed; typos fixed; exercises added.
- v0.3 (): Chapters divided into Part I and Part II; proof-writing appendix added; sections on real numbers, alternating sums and structural induction written; end-of-chapter exercises introduced; theorem environment symbols added; many more minor changes.
- v0.2 (): Page size changed from 8½"×11" to 6"×9"; Ch 0 created; logic material overhauled to emphasise inference rules and proof strategies; finite and infinite sets material moved to a new chapter; various other smaller structural changes.
- v0.1 (): Initial stable version (earlier versions were unstable and not numbered).
- Original lecture notes (). These are the lecture notes that evolved into what the textbook is today.
Website last updated: . The book's old website, infinitedescent.xyz, now redirects here.