Alright, let’s dive into some resources! I’ve put together a personally curated list of textbooks and other gems that I genuinely think are worth exploring. Full disclosure: this list is definitely shaped by my own experiences – some are treasured recommendations from professors I deeply respect, while others are simply books I’ve found incredibly enjoyable and insightful on my own journey. You’ll also find links to other helpful bits like insightful blog posts or handy tutorials for that extra layer of understanding.

But here’s a crucial piece of advice before you jump in: one of the most important things you can do is cultivate your own “mathematical taste.” Seriously! What makes one textbook a page-turner for one person might feel like a slog to another. Some of us thrive on a narrative, intuitive approach (think the classic Feynman style), while others find the straightforward clarity of a theorem-corollary-proof structure the most illuminating. The key is figuring out what resonates with you!

Mathematics

Single-Variable Calculus and Real Analysis

For those starting their calculus journey, a classic text that lays out the fundamentals in a clear and accessible manner is Thomas’ Calculus. It’s widely regarded as a standard choice for introductory calculus courses, and for good reason. The book is known for its intuitive explanations, wealth of examples, and its focus on helping students grasp the core concepts that underpin calculus.

If you’re after a text that bridges the gap to more formal mathematical thinking, Spivak’s Calculus is another excellent option. It was penned during a time when the distinction between “calculus” and “real analysis” wasn’t as sharply drawn as it is today. Consequently, Spivak offers a rigorous treatment of calculus, perfect for those who want to engage with the subject with an early taste of real analysis-style rigor.

Venturing into the realm of real analysis itself, the often-mentioned classic is Walter Rudin’s Principles of Mathematical Analysis, affectionately known as Baby Rudin. This book provides a concise and famously rigorous treatment of the subject, making it a valuable reference to accompany any real analysis course. However, be prepared for its characteristic terseness. Rudin’s writing style is exceptionally concise and minimalist, sometimes to the point of being quite challenging for those new to the material. The narrative depth it might lack can often be supplemented by exploring other excellent books on real analysis, such as Understanding Analysis by Stephen Abbott or Real Mathematical Analysis by Charles Pugh.

Another notable and more contemporary treatment of real analysis comes from Terence Tao with his Analysis I & II. These volumes offer a comprehensive, honors-level approach. A key feature of Tao’s style here is that many lemmas and corollaries are left as exercises for the reader. For someone already seasoned in analysis (who can often “eyeball” the logic of a proof), this can be an engaging way to learn. For a beginner, however, this can become quite frustrating. It’s also worth noting that Tao’s presentation is quite “barebones”—it focuses laser-like on what’s essential for an undergraduate analysis textbook, with minimal detours.

This is where, for many, Charles Pugh’s Real Mathematical Analysis truly shines and offers a different kind of experience. I’d particularly highlight Pugh for his signature intuitive approach and his fantastic (and sometimes incredibly tough!) problem sets. You’ll find exercise after exercise brimming with exciting ideas from the rich “buffet” of analysis. It’s a rewarding read not just for building rigor, but for mathematical culture as well. That said, Pugh expects a fair degree of self-sufficiency in proving things; there isn’t a lot of hand-holding. A practical tip: with the sheer volume of problems, you’ll likely want to be selective with your choices.

If you aren’t yet comfortable with constructing proofs independently, then diving straight into Pugh might be a very steep climb. In that situation, Stephen Abbott’s Understanding Analysis often proves to be a more accommodating starting point. It’s an excellent book that builds understanding effectively without demanding the same level of prior proof-writing fluency that Pugh presumes.

Linear Algebra

Moving from the continuous world of calculus and analysis, let’s shift gears to the structured realm of linear algebra—another cornerstone of a solid mathematical education. Navigating the landscape of linear algebra textbooks can be interesting, as different authors champion quite different pedagogical approaches.

When it comes to learning linear algebra, a popular modern choice is Sheldon Axler’s Linear Algebra Done Right. It’s well-known for its unique approach, often emphasizing linear operators and vector spaces from the get-go, and typically deferring the introduction of determinants. Many students and instructors find it a fine and insightful text. Ironically, though, I must admit I personally find myself gravitating more towards Sergei Treil’s Linear Algebra Done Wrong. While the title is a bit tongue-in-cheek (it’s an excellent book!), its more traditional approach, perhaps with a different emphasis on topics like matrix decompositions earlier on, often resonates more with me for building intuition or tackling certain applications.

For a truly amazing and comprehensive treatment, you really can’t go wrong with Linear Algebra by Friedberg, Insel, and Spence. This book is a powerhouse, widely respected for its thoroughness, a good balance between abstract theory and concrete computations, and clear explanations of fundamental concepts. It’s a standard recommendation in many universities for a reason and serves as an excellent, in-depth resource that will serve you well.

Now, if you’re seeking what might be called the “Rudin of linear algebra”—a text that embodies a certain demanding rigor, elegance, and perhaps a touch of uncompromising austerity—then Serge Lang’s Linear Algebra is the one to look towards. Much like Rudin’s Principles of Mathematical Analysis in the analysis world, Lang’s book is known for its concise, abstract, and highly theoretical approach. It can be incredibly rewarding for developing a deep, structural understanding, but it’s also famously challenging and expects a lot from the reader. It’s a text that truly solidifies your understanding if you can fully engage with its distinctive style.

Finally, for those looking to see linear algebra’s power in more advanced settings, pushing towards the interplay with calculus in higher dimensions and abstract spaces, texts such as Calculus on Vector Spaces by Cowin and Szczarba can offer a glimpse into that next level. These works explore how the concepts of calculus extend to vector spaces, often laying crucial groundwork for further studies in areas like differential geometry, functional analysis, and advanced mathematical physics. They can be quite illuminating after you’ve developed a solid grasp of core linear algebra principles.

Abstract Algebra

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Probability and Statistics

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History of Mathematics and Meta-Mathematics

This section does not have a specific aim, but roughly speaking those are books written by mathematicians about mathematics (either its practice or history).

First, an obvious recommendation would be G.H. Hardy’s “A Mathematician’s Apology”. In “A Mathematician’s Apology”, G.H. Hardy makes a compelling case — an “apology” in the classical sense of a defense, like Plato’s ‘Apology of Socrates’ — for mathematics being valuable for its own sake, beyond any utility. The cornerstone of his argument is the profound “beauty of mathematics.” Hardy eloquently describes this beauty, providing examples and criteria to understand it. Beyond this philosophical exploration, the book also includes a short autobiography, offering a personal window into the intellectual life of a working mathematician.

To quote the first passage:

It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.

Another book from a mathematician about mathematics would be “A Mathematician’s Lament” by Paul Lockhart, in which he meticulously deconstructs the standard school mathematics curriculum with a signature witty and ironic style I appreciate greatly.

To quote a passage:

Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depend heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood.

And this gem:

All metaphor aside, geometry class is by far the most mentally and emotionally destructive component of the entire K-12 mathematics curriculum. Other math courses may hide the beautiful bird, or put it in a cage, but in geometry class it is openly and cruelly tortured. (Apparently I am incapable of putting all metaphor aside.)

And this:

Could anything be more unattractive and inelegant? Could any argument be more obfuscatory and unreadable? This isn’t mathematics! A proof should be an epiphany from the Gods, not a coded message from the Pentagon. This is what comes from a misplaced sense of logical rigor: ugliness. The spirit of the argument has been buried under a heap of confusing formalism.

I really enjoy quoting Lockhart because of how amusing and insightful his writing is.

Other Topics and Outliers

A very interesting book I found and used to a large extent in high school was ‘An Infinitely Large Napkin’ by Evan Chen. It is a very good book spanning a wide range of topics, including linear & abstract algebra, algebraic & point-set topology, algebraic geometry, various flavors of analysis & measure theory, and more. It has a signature conversational style that makes it easy to read and understand, and it is a great resource for anyone interested in satisfying their curiosity about higher mathematics.

To avoid paraphrasing the intention of the textbook, I will cite the Preamble:

I’ll be eating a quick lunch with some friends of mine who are still in high school. They’ll ask me what I’ve been up to the last few weeks, and I’ll tell them that I’ve been learning category theory. They’ll ask me what category theory is about. I tell them it’s about abstracting things by looking at just the structure-preserving morphisms between them, rather than the objects themselves. I’ll try to give them the standard example Gp, but then I’ll realize that they don’t know what a homomorphism is. So then I’ll start trying to explain what a homomorphism is, but then I’ll remember that they haven’t learned what a group is. So then I’ll start trying to explain what a group is, but by the time I finish writing the group axioms on my napkin, they’ve already forgotten why I was talking about groups in the first place. And then it’s 1PM, people need to go places, and I can’t help but think: Man, if I had forty hours instead of forty minutes, I bet I could actually have explained this all. This book is my attempt at those forty hours.

In addition to such overviews, I would recommend spending time working through foundational material from established textbooks. This involves reading various textbooks and working through the majority of the exercises: it takes longer, but you build a solid, more sustainable foundation this way.

Computer Science

Algorithms and Data Structures

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Machine Learning

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Information Theory

Information theory being one of my fields of choice, I will pay special attention to this section. The classic textbook in information theory would be ‘Elements of Information Theory’ by Joy A. Thomas and Thomas M. Cover. Excellent coverage of all the fundamentals in information theory, and I found it relatively approachable. It does assume knowledge of probability and statistics, but that is to be expected.

To quote Chapter 1 (“Introduction and Preview”):

Information theory answers two fundamental questions in communication theory: what is the ultimate data compression (answer: the entropy H), and what is the ultimate transmission rate of communication (answer: the channel capacity C). For this reason some consider information theory to be a subset of communication theory. We will argue that it is much more. Indeed, it has fundamental contributions to make in statistical physics (thermodynamics), computer science (Kolmogorov complexity or algorithmic complexity), statistical inference (Occam’s Razor: “The simplest explanation is best”) and to probability and statistics (error rates for optimal hypothesis testing and estimation).

It would also be a good idea to supplement this with Claude Shannon’s (or even Nyquist’s, for culture) original paper(s), i.e. ‘A Mathematical Theory of Communication’ (1948). Shannon’s original paper is IMPRESSIVELY readable and comprehensive, even 50+ years later, and is a great supplement to a more modern book covering more ideas.

If you are looking for alternative forms of content, David MacKay has a good introductory lecture series on YouTube with a very good presentation style. Many enjoyed watching his lectures and following along with his book, which is quite readable. If you’re looking for a casual book on the side that is not as technical but covers the history of information theory and describes applications with conceptual examples, I would also recommend James Gleick’s book.

Compilers

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Electrical and Computer Engineering

Analog Signal Processing

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Communication Systems

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(Theoretical) Physics