On this page, I will be curating a list of personally recommended books for amateur and professional mathematicians, as well as any layman who is enthusiastic about mathematics. Books are sorted by mathematical topic.

The Amazon links to the books will be provided, but feel free to peruse whichever book store you can find, online or (if lucky) physically, for the book you are looking for. Please direct any feedback to my email address located at the homepage (click the "Back to home" button above).

- A First Course in Abstract Algebra by John B. Fraleigh
- A Book of Abstract Algebra by Charles C. Pinter
- Introduction to Lie Algebras and Representation Theory by James E. Humphreys

** Notes:** This book was a prescribed textbook for my second-year and third-year abstract linear algebra course. This book can be seen as being sectioned in three part; the first part, consisting of four chapters, covers group theory quite thoroughly. The second part, consisting of three chapters, covers the fundamentals of ring theory, with the last two chapters forming the final part of the book, covering Galois theory in its detail. Personally, I found the group theory part to be too theoretically intensive, preferring the Pinter book for this section; however, the ring theory and Galois theory parts of the section are very well-presented and I exceptionally liked how the latter was discussed quite precisely. I would conclude by adding that the formatting of the book is quite elegant!

** Notes:** I purchased this book while on holiday in Singapore at the famous Kinokuniya bookstore, and to be honest I do not regret getting this book at all! It is very similar in format, style and content to the Fraleigh book; while the Galois theory was not as thoroughly discussed as in Fraleigh, the presentation of the group theory framework in this book is exceptional and as a simpleton I found this part of the book much more appealing to read than the Fraleigh book. While the formatting of the book leaves a lot to be desired (it can be a bit of a slog to read, though the chapters are short), the content of the book more than makes up for most of the deficiencies of the book.

** Notes:** I was recommended this book by my doctoral advisor as a good introduction to Lie algebras and representation theory; currently I am using it to develop my Lie theory playlist. The book presents a nice introduction to Lie algebras, a natural extension of linear algebra and the cross product, and takes advantage of the linear algebraic concept to intuitively introduce Lie algebras through some particular examples. Most of the main ideas in Lie algebras, namely that of solvable Lie algebras, semisimple and simple Lie algebras, and root systems, are covered and there is also an added section discussing representation theory in some detail. The book may be pricey (I did have to fork over AU$90 just to get it!), but for someone interersted in Lie theory this is an invaluable resource that is well worth the money.

- Understanding Analysis by Stephen Abbott
- Schaum's Outlines of Complex Variables by Murray R. Spiegel et al.

** Notes:** I used this book in my second-year real analysis course back when I was an undergraduate at Monash University in Melbourne. It is a very rigorous introduction to analysis, which starts from a chapter dedicated to understanding the real numbers, proceeding to limits and convergence through sequences and series and then dealing with important topics like continuity, differentiability and integration. This book is fairly advanced for an enthusiast or an amateur, since it gets quite involved fairly quickly, but this is in a fair few universities a prescribed text for the standard treatment of real analysis, and thus I would recommend it on the basis of its quality rather than its access.

** Notes:** I refer to this book at times when I tutored second-year complex analysis at UNSW. Like most of the books in the series, it is by no means a rigorous or detailed exposition of the subject matter, but it is a great resource for any student or professional to have when it comes to reviewing concepts of complex analysis, which is summarised very succinctly and precisely, as well as having a boatload of practice questions on particular topics and some questions are presented with detailed worked solutions, which is a great added benefit to the beginner or the undergraduate. It contains most standard topics in complex analysis, though the second-year complex analysis course I tutored did not cover conformal mappings. Any book in the Schaum's Outlines series is definitely worth a pick-up, but this particular book is a gem!

- Metric Affine Geometry by Ernst Snapper & Robert J. Troyer
- Divine Propotions: Rational Trigonometry to Universal Geometry by Norman J. Wildberger

** Notes:** This book came to my attention as a result of doing a literature review on the field of rational trigonometry. The presentation of this book is clear; the first two sections introduce respectively synthetic affine geometry and inner product spaces, with the last section unifying the first two sections to set up what is called metric affine geometry, which is affine geometry over a quadratic form. This book is especially beneficial for the geometric-minded, especially those who have approached geometry from the view of rational trigonometry and universal geometry. The last section introduces the notion of squared distance similar to that to rational trigonometry, though this section ultimately becomes short due to the lack of depth in understanding geometry from this perspective.

** Notes:** This book provides the groundwork for the controversial topic of rational trigonometry, which aims to replace linear notions of geometry with quadratic notions, which supposedly makes the study of trigonometry more dependent on algebraic methods. The first chapter of the book is to be avoided at all costs, though the rest of the book can be quite a pleasure to read. Contrary to the author's opinions, the book is definitely not accessible to the average amateur; the second chapter goes straight into a discussion of concepts in abstract algebra, which will seem foreign to the untrained student. The presentation of the book is beyond comparison and the general content, while probably not new, is still interesting to know and understand; however, the views of the author, made clearly apparent quite early on, is to be taken at best with a grain of salt.

- Elementary Linear Algebra with Applications by Howard Anton & Chris Rorres
- A Text Book of Vector Algebra by Shanti Narayan

** Notes:** I used this book in my second-year linear algebra course back when I was an undergraduate at Monash University in Melbourne. This book is definitely not the best for anyone wanting to start off in linear algebra, but what sets this book apart from the rest is how much this book emphasises the applications of linear algebra. It may be best for you to peruse the other texts on this list for a better treatment of the theoretical aspects of linear algebra, but this book is not only chock full of questions but also applications of using linear algebra in the real world, ranging from Markov chains to least squares, and graph theory to cryptography; there is one full chapter dedicated to just the various applications of linear algebra in the real world! Definitely a must-have if you want to see the real-world applications of linear algebra.

** Notes:** This choice of book is definitely one that is way out of left field, but it a a highly informative text-book with a whole range of ideas and applications. My doctoral advisor had found this book on a total whim, but I am glad that he introduced this book; produced in India, it is not a common choice of book but the quality of the content more than makes up for it. The overall presentation of the text leaves a lot to be desired, though it is quite similar to more traditional presentations of mathematics; however, given the breadth of examples and applications to the basic ideas of linear algebra it is extremely hard for me to not recommend any budding student to read such an insightful book.

- Introduction to Probability (2nd edition) by Dimitri P. Bertsekas & John N. Tsiksiklis

** Notes:** This book was recommended reading for my second-year probability theory course and my third-year random processes course at Monash University in Melbourne. This book is probably the most comprehensive and rigorous introduction to probability theory out there, at least in my opinion; the book starts out with the bare basics of set theory and probability theory, which assumes little from the reader, and progresses nicely to talk about discrete and continuous random variables, Bernoulli, Poisson and Markov processes, and concluding with a set of limit results, e.g. Chebyshev and Markov inequality, and the Central Limit Theorem. It may be quite pricey to have on your shelf, but it is definitely worth the price tag!

- Introduction to Topology (3rd edition) by Bert Mendelson

** Notes:** I am currently using this book to create my Topology playlist on my YouTube channel. The structure of the book is excellent, in the sense that topology is approached from the perspective of basic set theory as opposed to other more specific treatments; in saying that, the book does build up intuition of general topology by considering specific cases of topologies through metric spaces. Admittedly, I have not read that particular chapter yet as I only plan to cover metric spaces at a later point; however, the book very nicely introduces the basic idea of topological spaces, with a focus eventually on connectedness and compactness. This book is definitely a good read for a person learning topology for the first time.

Last updated: 5 October 2021