I have written an introductory pure mathematics textbook, entitled *An infinite descent into pure mathematics*. It is currently in the form of lecture notes, which are in their fifth iteration. A table of contents can be found below. (Please be aware that the book is still a work in progress!)

*An infinite descent to pure mathematics* is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International Licence. This means that you are welcome to use, download and share the textbook, provided that credit is attributed to the author (Clive Newstead), that it is released under the same licence, and that it is not for commercial use.

**The textbook can be downloaded for free here.**

This textbook was born out of lecture notes I wrote for an introductory pure mathematics class at Carnegie Mellon University in summer 2015. I wanted to provide my students with a resource that emphasised not only the technical aspects of mathematics, but also the human aspects, particularly *communication* and *inquiry*—I was unable to find a resource that emphasised these aspects and also covered enough ground, so I decided to write my own. Particularly:

**LaTeX support.**The textbook contains a tutorial for typesetting mathematics using LaTeX. At any point when new mathematical notation is introduced, the corresponding LaTeX code is also presented. There is also an index of notation at the end of the book to assist in finding the desired LaTeX commands.**Writing guide.**The finalised version of the textbook will contain tutorials on writing mathematics, from as low a level as choosing which words and symbols to use in a clause within a sentence, to structuring a substantial mathematical document. This will be supplemented by suggestions for individual projects, in which students solve an interesting mathematical problem and write a mathematical paper to present their work—I implemented such projects when I taught in Summer 2017, and found it to be an incredible experience for both myself and my students.**Exercises.**Though many proofs and examples are provided in the textbook, a large quantity of material is delivered as exercises for the reader. I made a conscious decision*not*to provide an answer key for several reasons. First and foremost, the research on teaching and learning is clear that inquiry-based learning is more effective than reading worked examples. Second, this textbook is supposed to be used in conjunction with external support (be it peer support, a teacher, or an online Q&A forum such as Mathematics Stack Exchange)—students using the textbook to learn mathematics should communicate their thoughts with others in order to find the solutions to the exercises. Third, the textbook is also supposed to be an*instructional*tool, and the exercises can be used to design in-class activities, homework assignments, and quiz and examination problems.**Completeness.**I have tried to make sure that the material in the textbook is complete and coherent, in the sense that material later in the book depends only on material earlier in the book and that all details are included,*except*when the details obfuscate the intuition, in which case they are still included, but are relegated to an appendix.

To date, this book has been used to teach seven courses over five semesters Carnegie Mellon University:

**Fall 2017.**21-128*Mathematical Concepts and Proofs*and 15-151*Mathematical Foundations for Computer Science*, both instructed by Prof John Mackey.**Summer 2017.**15-151*Mathematical Foundations for Computer Science*, instructed by myself.**Spring 2017.**21-127*Concepts of Mathematics*, instructed by Prof David Offner.**Fall 2016.**21-128*Mathematical Concepts and Proofs*and 15-151*Mathematical Foundations for Computer Science*, both instructed by Prof John Mackey.**Summer 2015.**21-127*Concepts of Mathematics*, instructed by myself.

The following table of contents is subject to change.

**Chapter 1.**Mathematical reasoning**1.1.**Getting started**1.2.**Proof techniques**1.3.**Induction on the natural numbers

**Chapter 2.**Logic, sets and functions**2.1.**Symbolic logic**2.2.**Sets and set operations**2.3.**Functions

**Chapter 3.**Number theory**3.1.**Division**3.2.**Prime numbers**3.3.**Modular arithmetic

**Chapter 4.**Finite and infinite sets**4.1.**Functions revisited**4.2.**Counting principles**4.3.**Infinite sets

**Chapter 5.**Relations**5.1.**Relations**5.2.**Orders and lattices***5.3.**Well-foundedness and structural induction*

**Chapter 6.**Real analysis**6.1.**Inequalities**6.2.**Sequences and series****6.3.**Continuous functions**

**Chapter 7.**Discrete probability theory**7.1.**Discrete probability spaces**7.2.**Discrete random variables**7.3.**Expectation and variance*

**Chapter 8.**Additional topics**8.1.**Ring theory***8.2.**Ordinal and cardinal numbers****8.3.**Boolean algebra****8.4.**Complex numbers****8.5.**Asymptotics**

**Appendix A.**Mathematical writing**A.1.**Basic mathematical literacy****A.2.**Writing a proof****A.3.**Writing a mathematical paper**

**Appendix B.**Foundations**B.1.**Theories and models****B.2.**Set theoretic foundations****B.3.**Other foundational matters**

**Appendix C.**Typesetting mathematics in LaTeX

* Close to completion

** Not yet written