Ask HN: Math books that made you considerably higher at math?

The classical stuff is great:
* Geometry and the imagination by Hilbert and CohnVossen * Methods of mathematical physics by Courant and Hilbert * A comprehensive introduction to differential geometry by Spivak (and its little brothers Calculus and Calculus on manifolds) * Fourier Analysis by Körner * Arnold’s books on ODE, PDE and mathematical physics are breathtakingly beautiful. * The shape of space by Weeks * Solid Shape by Koenderink * Analyse fonctionnelle by Brézis * Tristan Needhams “visual” books about complex analysis and differential forms * Information theory, inference, and learning algorithms by MacKay (great book about probability, plus you can download the .tex source and read the funny comments of the author) And finally, a very old website which is full of mathematical jewels with an incredibly fresh and clear treatment: https://mathpages.com/ …I am in love with the tone of those articles, severe and playful on the identical time.


Norman Wildberger’s YouTube channels are the most thorough I’ve seen ( https://www.youtube.com/@njwildberger and https://www.youtube.com/@WildEggmathematicscourses ).
There are lots of of movies, organised in playlists, from undergraduate lectures ( https://www.youtube.com/playlist?list=PL55C7C83781CF4316 ) and analysis seminars ( https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF ) all the way in which to primary fundamentals like how to consider counting (e.g. https://www.youtube.com/watch?v=PukipOTiD4&list=PL5A714C94D… ) The explanation I discover them fascinating is that Wildberger would not agree with a number of the typical approaches, particularly with using infinity and taking limits. This leads him down attentiongrabbing paths (e.g. Rational Trigonometry and Algebraic Calculus), which (a) present the course of of arithmetic (exploring, making definitions, increase in numerous instructions, and so on.), while (b) remaining principally grounded and approachable (e.g. no appeals to inscrutable lemmas from summary analysis areas). For instance, he is lately been making movies about “multisets” (laptop scientists would name them Luggage), their arithmetic (the place “including” is union, and “multiplying” is pairwise/cartesian product of the weather), and the way this generalises: from an algebra containing solely empty baggage (trivial, however selfconsistent; behaves like zero), to baggage of zeros (behaves like pure quantity arithmetic), to baggage of pure numbers (behaves like polynomial arithmetic), to baggage of polynomials (behaves like polynomials in arbitrarilymany variables) https://www.youtube.com/watch?v=4xoF2SRp194


I should mention focusing on doing a few interesting problems, rather than many not so interesting ones, is also one way to help yourself understand more deeply.


Do you know a good path or book that’s suitable for that?
I’ve been using Professor Leonard’s Youtube video series[1] mostly, along with some of those “workbook” type books by Chris McMullen, and a variety of books with titles like “1001 solved problems in $SUBJECT”, “The Humongous Book of $SUBJECT problems”, and the like. The nice thing about Professor Leonard is that he has videos on everything starting from prealgebra, middleschool math, up through Differential Equations. Note that his diffeq class isn’t quite complete but he just announced he’s about to start recording new videos to finish that, and he’s also going to be starting a Linear Algebra sequence. And he’s a great lecturer who does a really good job of explaining things and making them understandable. I also use Khan Academy sometimes, and stuff on Youtube from The Math Sorcerer[2]. Oh, and of course there is 3blue1brown[3], whose videos are also useful. And for Linear Algebra I’ve been using Gilbert Strang’s OCW videos[4] on Youtube. FWIW, I’ve evolved the way I study math, and what I do now works for me, even though it’s 100% not the way you’d ordinarily see suggested. That is, I watch math videos fairly passively and don’t work problems at the same time and treat it like being in a class perse. I used to do the thing of treating it like a class, pausing the video to work examples, and whatnot, and that does work. But it’s very slow and tedious. Now, I just watch the videos, acknowledging that I won’t absorb everything and that I also need to work problems for longterm retention. So now what I do is watch passively to a certain point (which I determine fairly subjectively) then I stop with the videos for a while, pick up a textbook or one of those “workbook” type books I mentioned earlier, and work problems for a while. Then I review the parts that I find myself struggling with. I’m also just now starting to add “creating Anki cards” as something I do during that second pass. Once I start getting a decent Anki deck built up, I’ll be reviewing that regularly as well to help build retention. I only create cards for things that seem amenable to rote memorization, and TBH, I’m still working on figuring out what things are best to include, and how to structure those cards. What I don’t intend to do is include specific problems where all I’d be doing is memorizing the answer to a problem. So far it’s just formulas and things are are very obvious candidates to be memorized, and “algorithm” things like the “chain rule” from calculus, and similar. [1]: https://www.youtube.com/@ProfessorLeonard [2]: https://www.youtube.com/@TheMathSorcerer [3]: https://www.youtube.com/c/3blue1brown [4]: https://www.youtube.com/playlist?list=PLE7DDD91010BC51F8


Out of this list, the books I am familiar with, are great (HilbertCourant, Spivak, Korner’s books). At the same time, even with extensive mathematical training, I haven’t read them from start to finish. I wouldn’t even like to say “read”. For someone who’s not used to mathematical reading, some of these books require careful study. That means generating examples to understand results (theorems), trying your own conjectures, proving things yourself etc. Over time, one becomes familiar with most/all the material in a book but the knowledge might have been acquired through various books (and courses) over time.
Also, mathematics is a massive field. The first question would be what kinds of mathematics would you like to get better at. There are great books in analysis. If you are starting out with a solid calculus knowledge, try Abbott’s Understanding Analysis [1] or Duren’s Invitation to Classical Analysis [2]. For asymptotic methods in PDEs, try Bender and Orszag [3] which is a wonderful book. But again, this might not be your cup of tea at all and there are more abstract or formal books like Rudin’s. If you want to approach fields without a lot of machinery, graph theory books by Bollobas are great (but difficult). See his Modern Graph Theory book [4] as an example. For linear algebra, one of my favorites (but it was after I already learned the subject) is Trefethen’s Numerical Linear Algebra book [5]. Another beautiful topic is at the intersection of linear algebra and combinatorics. See Babai and Frankl’s lectures freely available online. Then there are wonderful topics in geometry. A massive mountain to climb would be algebraic geometry. For one starting point, see [6]. Differential geometry (Spivak’s multivolume work or Needham’s differential forms book) is another wonderful area. I would recommend Crane’s discrete differential geometry course at Carnegie Mellon [7] if you want a concrete introduction. You might want to demystify a topic you have heard about. E.g. Galois theory and the unsolvability of quintic equations. You could look at [8] which guides your way through wonderful problems. We haven’t even touched huge swathes of mathematics including anything topological or number theory. Even within the topics mentioned above, once you start, your journey will take a life of its own and you’ll encounter multiple books and papers opening up new subfields. The only approach that worked well for me in the past was to get completely consumed by what one topic one was studying. This meant not getting distracted by multiple topics. Once one enters the workforce, this is very hard (or at least has been for me). Without knowing someone, it’s hard to recommend anything but the advantage with topics like graph theory and combinatorics is that one needs less machinery (as opposed to something like algebraic geometry). These fields lead you to interesting problems very rapidly and one can wrestle with them parttime. [1] https://www.amazon.com/UnderstandingAnalysisUndergraduate… [2] https://www.amazon.com/InvitationClassicalAnalysisApplied… [3] https://www.amazon.com/AdvancedMathematicalMethodsScienti… [4] https://www.amazon.com/ModernGraphTheoryGraduateMathemat… [5] https://www.amazon.com/NumericalLinearAlgebraLloydTrefet… [6] https://www.amazon.com/AlgebraicGeometryApproachMathemati… [7] https://www.cs.cmu.edu/~kmcrane/Projects/DDG/ [8] https://www.amazon.com/ThroughExercisesSpringerUndergradu…


“Mathematical Notation: A Guide for Engineers and Scientists”[0] really changed my abilities with being able to read papers and decipher what was going on. I had university math experience but it was a long time ago. When I started reading papers for algorithms later in my career I couldn’t get past the notation. Once the symbols are explained, as a programmer, I was able to grok so much more. This should be on everyone’s shelf.
[0] https://a.co/d/gQmDIo7


It can be made to be. Dijkstra came up with a nice and rigorous notation he used for his own proofs[1]. That page also includes some slightly spicy takes on why things are as they are. I agree that this is an area where the broader mathematical field has much to learn from computing science. The unforgiving nature of computing automata really drove that innovation. Meanwhile one can afford to be sloppy when one is trying to convince some other mathematician with a sky high IQ.
[1] https://www.cs.utexas.edu/users/EWD/transcriptions/EWD13xx/E…


Wow, I wish I had known about this book (and had a license to Mathematica) when I was in college. I always got hung up on the notation and my inability to visualize the concept.


I would not call myself great at math – I struggled with it in school, in fact – but in recent years I’ve begun “correcting” my lack of mathematical knowledge. The single best decision I’ve made is to first start with the philosophy of mathematics. Maybe it’s because my background is in philosophy, but I also think that for certain people like myself, understanding what math is makes me far more interested in understanding how it works, rather than just doing contextless calculations using formulas I don’t know the history or deeper purpose of. When I learned math in school, it was entirely cut off from any of these deeper questions.
Here’s a good starting point for philosophy of mathematics https://plato.stanford.edu/entries/philosophymathematics/


Elements was a school textbook for 2000 years, up until ~100 years ago. It’s a fine book to use for selfstudy.
Edit: Also, to state the obvious, it’s been translated into English


In my experience, the best way to get better at math is to do a lot of it. Find some book that’s “good enough” for some topic you’re interested, and work many many problems from the book. You’ll learn about the topic, but more importantly you’ll learn problem solving skills. I recommend working the problem until you’re sure the answer is right — in grad school problem sets didn’t have answers you could check, and full understanding was necessary to get the problem sets correct.
For me, a watershed book was Introduction to Analysis by Rosenlicht [1]. Proofbased, very “mathy”, small and compact (so to speak) but with a massive scope. A great introduction to a really important topic, and it’ll put your brain through its paces. Again, I recommend working nearly every problem. [1] https://www.amazon.com/IntroductionAnalysisDoverBooksMat…


The best resource I’ve found is this random, somewhat obscure website (though I’ve learned that it has grown in popularity) called Paul’s Online Notes. The professor has a real knack of pedagogy, and the problems are perfectly structured in terms of their difficulty. His explanations are clear and without jargon, and it goes from algebra to diff eq.
A note: this isn’t a resource for higherlevel, proof based maths. It will give you a solid foundation and a pragmatic understanding to build upon. Very useful for STEM. Link: https://tutorial.math.lamar.edu


I have to second this. It’s very well written and presents a clear view of what Linear Algebra is. Although it might be best used as a second book in Linear Algebra (depending on your preparation).


I second this strongly, you can’t get better at math just by reading books. You need to hone your problem solving skills, you need to fight with the problems, have the mindset of a warrior, a conqueror, only then you’ll get the juice out of it and have a clear understanding of the subject. I’ll suggest starting with Concrete Mathematics by Donald Knuth, it’s a beautiful book that catches the essence of mathematics.
Art of problem solving(https://www.amazon.in/ArtProblemSolvingBasics/dp/09773045…) can also be a fantastic begin, particularly should you don’t have a lot expertise.


Measure and Category by John Oxtoby. This book studies duality results between different notions of “small” sets in measure theory and topology. It’s the first (and to some extent the only) math book where things just clicked and I didn’t feel like I was drowning in a sea of notation and ideas. Here are some more thoughts on it: https://bcmullins.github.io/TopBooks2019.


I was reading this book, when the ideas of function spaces, functions as vectors, functions as elements of vector spaces, functional analysis clicked on me.
I am not sure if this book is particularly good or better than other books. (Well, it still looks like a very gentle introduction to the topic.) But as per your question, this was the book at the right time for me. “Fourier Series and Orthogonal Functions” by Harry S. Davis. https://www.amazon.com/dp/0486659739/


I think you guys might find this list I found long ago very useful when deciding on a mathematics book you want to read.
This is an introduction written by the original author of the list: “Somehow I became the canonical undergraduate source for bibliographical references, so I thought I would leave a list behind before I graduated. I list the books I have found useful in my wanderings through mathematics (in a few cases, those I found especially unuseful), and give short descriptions and comparisons within each category. I hope that this list may serve as a useful “road map” to other undergraduates picking their way through Eckhart Library. In the end, of course, you must explore on your own; but the list may save you a few days wasted reading books at the wrong level or with the wrong emphasis. The list is biased in two senses. One, it is light on foundations and applied areas, and heavy (especially in the advanced section) on geometry and topology; this is a consequence of my interests. I welcome additions from people interested in other fields. Two, and more seriously, I am an honorstrack student and the list reflects that. I don’t list any “regular” analysis or algebra texts, for instance, because I really dislike the ones I’ve seen. If you are a 203 student looking for an alternative to the awful pink book (Marsden/Hoffman), you will find a few here; they are all much clearer, better books, but none are nearly as gentle. I know that banging one’s head against a more difficult text is not a realistic option for most students in this position. On the other hand, reading mathematics can’t be taught, and it has to be learned sometime. Maybe it’s better to get used to frustration as a way of life sooner, rather than later. I don’t know.” – by original author. [List] https://www.ocf.berkeley.edu/~abhishek/chicmath.htm


> Discrete Mathematics with Applications
by Susanna S. Epp Fantastic book for Discrete Mathematics with lucid explanation and good exercises. The other one would be concrete mathematics.


If I were a student, eager to solve math problems and become better at math, no book would be better than any other book at presenting and guiding me to problems that would improve my understanding?


A Programmer’s Introduction to Mathematics https://pimbook.org/
It introduces math from a mathematician’s standpoint (full with proofs, and so on.) moderately than rote memorization and workouts, however it does so from the attitude of a programmer.


There exist only two kinds of modern mathematics books: ones which you cannot read beyond the first page and ones which you cannot read beyond the first sentence. — Chen Ning Yang


”Road to reality” by Roger Penrose is an interesting book as a refresher and review if the content is otherwise within familiar territory.


The Art of Problem Solving: https://artofproblemsolving.com/store/book/aopsvol1
Sure, it’s focused in the direction of center and highschool college students. Sure, I learn it and (extra importantly) labored by many of the issues in my mid30’s. It’s nice if, like me, you coasted/crammed by your early arithmetic training and by no means felt such as you dialed within the fundamentals. Additionally it is nice if, like me, you wanted some penonpaper observe and didn’t know the place to begin.


Had to do a Calculus course in uni despite not having taken any calc or precalc in high school. “Precalculus Mathematics in a Nutshell” and “Calculus Made Easy” were complete lifesavers.


A book of abstract algebra – Charles C. Pinter. Each chapter is a few pages of explanation, and the rest you solve yourself by doing exercises that introduce aspects of the theory step by step.


* 4Manifolds and Kirby Calculus by Andras I. Stipsicz and Robert E. Gompf
* Differential Manifolds by Antoni A. Kosinski * Introduction to Smooth Manifolds by John M. Lee


If you are into numerical optimization, a nice source of intersting problems and examples (that e.g. contradict the intuition) can be found in
Mathematical Tapas: Volume 1 and Vol. 2.


Significantly better I don’t know but when I was a child I was given Der Zahlenteufel. Ein Kopfkissenbuch für alle, die Angst vor der Mathematik haben (The Number Devil) and I liked it very much


I had a print of Euclid’s Elements as a kid.
My mom was really into mathematical proofs and I being a huge loser kid with no friends naturally took to this book as well.


Some favorites below. Books 03 are accessible. The remaining books are more difficult but I’d highly recommend them to math students.
0. Jan Gullberg, Mathematics, From the Birth of Numbers. A highly accessible popular survey on different branches of higher mathematics. I read this over the Summer between high school and starting my undergraduate degree. It’s what made me want to study math. Previously I’d wanted to be a guitar player, but had to find a new ambition after an injury left me unable to play. 1. The high school mathematics series by Israel Gelfand. Algebra, Trigonometry, The Method of Coordinates, and Functions and Graphs. I 2. George Polya. How to Solve it. A short book giving excellent high level advice on mathematical problem solving. 3. George E. Andrews, Number Theory. I worked through this freshman year contemporaneously with my first proof based class on simple logic and set theory. A very beautiful and accessible introduction to basic number theory. The combinatorial/geometric proofs of Fermat’s Little Theorem and Wilson’s Theorem are lovely. It also includes a very nice proof of Chebyshev’s theorem on the asymptotic density of primes and even the RogersRamanujan identities for integer partitions. 4. Vladimir Arnold, Ordinary Differential Equations: Undergrad ODE classes are often taught in a cookbook fashion and if so, don’t offer much enlightenment. This book explains what’s going on at geometrical level. I didn’t appreciate ODEs until I read this. See https://www.unimuenster.de/Physik.TP/~munsteg/arnold.html for Arnold’s views on instructing arithmetic. 5. E.C. Titchmarsh The Idea of Features: Really helpful by my undergraduate advisor as a result of he seen that I favored studying older books. It accommodates sections on advanced evaluation and actual evaluation with measure principle, however I’ve solely learn the advanced evaluation sections. It is not for everybody, if I recall appropriately, there may be not a single image, however it is rather energetic and has a variety of materials you will not discover in a normal advanced evaluation ebook, together with Dirichlet sequence. Wonderful as a complement to a normal advanced evaluation ebook. 6. George Polya. Arithmetic and Believable Reasoning. A wonderful growth on Polya’s concepts on The way to Resolve it. Whereas the aim is to hunt rigorous proofs, to get there it is highly effective to have the ability to suppose based mostly on instinct, heuristics, and believable reasoning. Loads of math exposition is theorem/proof based mostly and would not assist develop these abilities. In the same vein, see additionally Terence Tao’s traditional submit There’s extra to arithmetic than rigour and proofs https://terrytao.wordpress.com/careeradvice/theresmoreto…. 7. H.S.M Coxeter, An Introduction to Geometry. A ebook of very lovely classical geometry. One thing sometimes not touched on in any respect in a typical arithmetic curriculum.


The classic, How to Solve It by Polya.
A lot of the advice seems obvious in retrospect but being systematic about a problem solving framework is enormously helpful.


Algebra Baldor !
College Algebra Heineman Discrete Math Rosen ! Linear algebra D lay Calculus Stewart Nonlinear Dynamics Strogatz + Combinatorics Mazur + Statistics * ESLR +


From Mathematics to Generic Programming – Stepanov & Rose
Gödel, Escher, Bach: an Eternal Golden Braid – Hofstadter Euclid’s Elements


I highly recommend working through Claude Shannon’s Mathematical Theory of Communications [0]. It’s originally a paper but was later restructured as a book, in either form it works quite well.
The reason I recommend it is because it shows mathematical reasoning that is easy to follow and relevant to your daily life. It’s real math, but very easy to read through and understand. If your unfamiliar this paper is where the very idea of “bits” comes from. One of the most important things in the paper for nonmathematicians to see is that the definition Information Entropy is derived simply from the mathematical properties Shannon desires it to have. This is important because I find that one of the biggest questions people ask about mathematical formula and idea is “What does this mean? Why is it this way?” without realizing that math is really not engineering nor physics. When deriving his definition of Information, Shannon simply states that information should have the following x,y… properties and then goes on to show that the now standard definition of information meets all these criteria. In mathematics it is very often the case that only after an idea is created to we start realizing the applications. This is quite different than science where a model is only adopted if it correctly describes a physical process. Work through the paper and you will have worked through the mathematical underpinnings of the information age and will likely have understood most of it pretty well. 0. https://people.math.harvard.edu/~ctm/home/text/others/shanno…
