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 released under a Creative Commons Attribution-ShareAlike 4.0 International licence. This means that you are welcome to use, download and share the textbook, provided that attribution is given to the author (Clive Newstead) and that it is released under the same licence.
The TeX source of the textbook can be found on Gitlab.
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. 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 study
All mathematics textbooks, including this one, should be read slowly: it is important to understand each line or paragraph before moving onto the next, and this takes time. Patience and persistence are absolutely essential.
It is impossible to learn mathematics well by reading alone, so readers should take care to complete the in-chapter exercises as they appear, and to complete a selection of the end-of-chapter exercises as a means of solidifying what you have read.
The textbook need not be read from cover to cover: sections can be skipped and covered out of order. See the preface of the textbook for a table of prerequisites.
Using the textbook to teach
If you are using An Infinite Descent into Pure Mathematics to teach a course, please contact teaching@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. 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 feedback@infinitedescent.org.
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