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The classical stuff is great: * Geometry and the imagination by Hilbert and Cohn-Vossen * 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'm in love with the tone of these articles, serious and playful at the same time.
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“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
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As a programmer I really wish math notation was more rigorous: less ambiguity, more explicit typing, no implicit variables, etc. So much of it would never pass code review. We programmers figured out that code should be optimized for readability, not writtability ; I wish mathematicians did too.
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Plus one for this!
I bought two copies of the referenced book…and for the exact same reasons; I’m a programmer and being able to explain my algorithms using mathematical notation helps validate a program as well as troubleshoot a program…
An oldie but goodie is “Mathematics for the million”
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This is different from the other answers, but it does answer your question: When I was a kid I had tons of math and logic puzzle books. Two I remember specifically are "Aha! Insight" and "Aha! Gotcha" by Martin Gardner. Decades later, when a math problem comes up in my work, I have an apparently unusual ability to cut to the heart of it ("by symmetry, we must have X" or "looking at this extreme case, we must have Y" or "this looks like a special case of Z" sort of things) instead of starting by soldiering through equations, and I credit a lot of that to all the puzzle-solving I did as a kid.
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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 context-less 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/philosophy-mathematics/
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Reading Euclid's Elements and Newton's Principia really helped me get an intuitive feel for geometry and calculus. They may not be entirely easy (at least the second) without some commentary, but well worth the study.
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While it's laudable that you sought those texts and profited from them, I worry about what others might take away from this. When I was young I knew some geniuses who highly spoke of Principia and how it gave them great insights. And the teenager me said, okay cool, I'll have a go! The problem is that it's in Latin and quite impenetrable. We have some geniuses here and they would no doubt be able to take away a lot from these texts, but for you normals out there: don't optimize too much, you're quite alright in taking the normal approach of just taking a class at a community college, doing the exercises the teacher assigns, etc.
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I didn't read Euclid in Greek or Newton in Latin; there are quite good translations available - even free! In general I find that if someone is insisting that you study the philosophy of someone in their original language, they don't have a good enough translation yet.
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It might make sense to read a translation in a language you understand. Many of the books that are considered classics are specifically because they ARE accessible. That doesn't necessarily mean that they are easy, but there is a big difference between reading Euclid and learning how to create mathematical proofs, and taking a class focused on calculating the area of various shapes or determine angles. I haven't read Mathematical Principles of Natural Philosophy (the English title) but I have read Euclid and it definitely doesn't require a genius to understand. here is an online edition with great illustrations: https://www.c82.net/euclid/
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History of mathematics as well, it will give you a very subtle entry into the minds of mathematicians and the motivation behind their theorems. This will surely make you more appreciative of subjects and concepts you are learning.
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Absolutely, and there are some really interesting personalities in the history of mathematics. Newton and Galois come to mind.
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In that vein I highly recommend Foundations and Fundamental Concepts of Mathematics by Howard Eves. I think it might be a little dated, but it gives an amazing overview of the most important developments in mathematics that were relevant at the time. It's less focused on practice (though there are some problems) and more on the history and motivation behind the ideas. This book introduced me to axiomatics, non-Euclidean geometry, quaternions, and abstract algebra in my senior year of high-school.
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Is that entire course just wading through the wreckage wrought by Gödel?
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1. Principles of Mathematical Analysis by Walter Rudin (baby Rudin) - I'd studied real analysis in the past, but this book is direct and rigorous and provided a good framework to move forward into things like functional analysis in a way that I was not prepared for with other books. 2. Differential Equations and Dynamical Systems by Lawrence Perko - Solidified for me how dynamic systems behaved and were solved. Very much helped my understanding of control theory as well. 3. A Concise Introduction to the Theory of Integration by Daniel Stroock - Helped solidify concepts related to Lebesgue integration and a rigorous formulation of the divergence theorem in high dimensions. 4. Convex Functional Analysis by Kurdilla and Zabarankin - Filled in a lot of random holes missing in my functional analysis knowledge. Provides a rigorous formulation of when an optimization formulation contains an infimum and whether it can be attained. Prior to this point, I often conflated the two.
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I have not yet become significantly better, and not a math book, but I recently read A mathematicians Lament by Paul Lockhart and it resonated so much with me that I plan to take another stab at math different from how it is taught in school. Waiting to get my hands on his book 'Measurement' and approach it more like art. If what he says is true, perhaps many who would have turned out great at math are locked out by how it's taught in school. For now, I have a test subject of one :)
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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 higher-level, 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
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Linear Algebra Done Right by Sheldon Axler for the following reasons: - I was revisiting a topic in greater depth, which is a common theme in university-level math courses. - It is a rigorous book, written in the style of definition, proposition, theorem, etc. - It was the first math book where the exercises don't just reinforce what you learned in the chapter, but teach you new material (another common theme in advanced math textbooks). - Linear Algebra is arguably the most important math subject these days.
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Linear Algebra Done Right by Sheldon Axler is indeed a good book if you are looking for a rigorous proof based book to learn linear algebra. Here [1] you can find Sheldon Axler himself explaining the topics of the book in his YouTube channel! How wonderful is that! Here [2] you can find the solutions to the exercises in the book. This [3] Lectures might help as well, among the books this course follow is Algebra Done Right. Good luck learning the subject of Linear Algebra you'll have fun doing so. [1] https://www.youtube.com/playlist?list=PLGAnmvB9m7zOBVCZBUUmS... [2] http://linearalgebras.com/ [3] http://nptel.ac.in/courses/111106051/
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I wonder how good you can get at maths just by casually reading books. You need to work on problems for hours and hours to get a grasp on the theories. Programming is different in the sense that it's something people routinely do as a hobby because it's quite fun and addictive. But maths? maybe if you have already strong foundations you can pick up a new topic and develop your culture. But I doubt one can get these foundations without actually graduating in maths as it's an extremely strong commitment.
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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 honors-track 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
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Probably the most elegant math book I have ever seen is Probabilty theory a graduate course by Achim Klenke. A very nice exposition into the abstract, measure theoretic prob. thoery (but it assumes some prior knowledge).
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Calculus Made Easy and Probability Through Problems. I'm not sure that I'd have gotten through either my university Calculus courses or Probability and Statistics without these two books. I used them as supplementary material to the course textbooks and homework. They both have a style that is approachable and helped me build an intuition for the material unlike anything else I found.
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I second this suggestion for Calculus Made Easy by Thompson. It's become a bit of a classic...was published in like 1915. Super unique approach to teaching calculus. It's an excellent supplement...lots of good insights. It may be particularly good for people who believe they're bad at math. His style may convince people otherwise. Also, Vibrations and Waves, by AP French. Granted, this is a physics book, but I appreciate his style so much. He makes use of a lot of geometric methods to solving problems. It definitly expanded my math horizons! His other books are good too.
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Rudin's Principles of Mathematical Analysis has a really special place in my heart. Chapter 3 is great- it's a great reference for derivations of a lot of fundamental identities about limits used in undergrad calculus. Chapter 4 is a great place to learn about topology for the first time. In general, it kicks up the mathematical rigor you're used to a notch. Seeing ">" defined as "not <" really blew my mind when I first read it! "<" is just something that satisfies some axioms, like anything else in math.
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Statistics by Freedman, Pisani and Purves. Don't know if I got better but loved the real world examples and cartoons. Does not have too many pre-requisites. Each section presents a tiny concept which is followed by plenty of exercises that have answers at the end. The furthest I got in a book in recent days, Math or not.
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Knuth's "Concrete Mathematics" is fantastic, precisely because it's very applicable.
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Mathematik für Ingenieure und Wissenschaftler I, II and III from Lothar Papula (in German). The solutions are detailed, making it perfect for self-studying. Book of Proof by Richard Hammack. A great introduction to proofs in mathematics. The book is available free online [0], but also I bought the physical version because I really enjoyed it. [0]https://jdhsmith.math.iastate.edu/class/BookOfProof.pdf
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I just started reading Book of Proof by Richard Hammack and I agree it's an amazing book
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During my undergraduate studies, I loved "Discrete Mathematics and Applications" by Kenneth Rosen. I really enjoyed reading through the various examples and biographies of famous mathematicians included in each chapter. For those looking to delve into discrete mathematics, I highly recommend the lecture notes from L. Lovasz and K. Vesztergombi (Yale University, Spring 1999) and from Eric Lehman, Tom Leighton, and Albert Meyer (MIT, 2010).
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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/Top-Books-2019.
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I'm so happy to hear that! I've always loved this little book (even if it's completely independent of the math needed for my work). In a similar spirit, but with a much more geometrical flavor, there is Evans-Gariepy.
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Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach It's a rigorous but chatty textbook in the style of Spivak but written by someone who is sensitive to applied maths. I would not have survived my astrophysics classes without it. (Not to mention it's where I first saw this really intuitive way of doing matrix multiplication: https://blogs.ams.org/mathgradblog/2015/10/19/matrix-multipl...)
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If I could pick one, that would be How to Solve it by George Polya.
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Any good book about the history of mathematics that will teach you a natural historical development of concepts to reach more generalizations. History of mathematics will give you a very subtle entry into the minds of mathematicians and the motivation behind their theorems. This will surely make you more appreciative of subjects and concepts you are learning.
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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.
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Not a book. But a course by Prof Keith Devlin on Coursera called Introduction to Mathematical Thinking.
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I jumped into the first analysis class (using baby Rudin) completely unprepared and this course saved me!
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Meta comment: might be good to add the level of mathematical maturity needed to enjoy the book.
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I'm coming from an applied math perspective. A few of my favorites (and ones I find myself regularly referring to) are: Matrix Analysis by Horn and Johnson (perhaps the best end-of-chapter problem sets of any math book I've encountered!) Matrix Computations by Golub and Van Loan Elements of Statistical Learning by Hastie, Friedman, Tibshirani Functional Analysis by Reed and Simon
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Since I haven't seen many discrete maths books, he's my list: Beginner: NL Biggs, Discrete Mathematics, Oxford University Press Intermediate: PJ Cameron, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press Advanced: JH van Lint & RM Wilson, A Course in Combinatorics, Cambridge University Press
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Also, Geometry for Programmers but only because I wrote it. I had to update my skills significantly while gathering material and doing all the experiments. Not sure if reading the book would have the same effect :-)
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A Programmer's Introduction to Mathematics https://pimbook.org/It introduces math from a mathematician's point of view (complete with proofs, etc.) rather than rote memorization and exercises, but it does so from the perspective of a programmer.
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”Road to reality” by Roger Penrose is an interesting book as a refresher and review if the content is otherwise within familiar territory.
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Calculus on Manifolds by Spivak. Brilliant. And relatively thin too.
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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.
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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
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The Art of Approximation gave me far more intuition than any class
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Some favorites below. Books 0-3 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
didn't have much mathematics background in high school, but working through these really solidified my grasp on the basics. 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 Rogers-Ramanujan 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.uni-muenster.de/Physik.TP/~munsteg/arnold.html for Arnold's views on teaching mathematics. 5. E.C. Titchmarsh The Theory of Functions: Recommended by my undergraduate advisor because he noticed that I liked reading older books. It contains sections on complex analysis and real analysis with measure theory, but I've only read the complex analysis sections. It's not for everyone, if I recall correctly, there is not a single picture, but it is very lively and has a lot of material you won't find in a standard complex analysis book, including Dirichlet series. Excellent as a supplement to a standard complex analysis book. 6. George Polya. Mathematics and Plausible Reasoning. An excellent expansion on Polya's ideas on How to Solve it. While the goal is to seek rigorous proofs, to get there it's powerful to be able to think based on intuition, heuristics, and plausible reasoning. A lot of math exposition is theorem/proof based and doesn't help develop these skills. In a similar vein, see also Terence Tao's classic post There's more to mathematics than rigour and proofs https://terrytao.wordpress.com/career-advice/theres-more-to-.... 7. H.S.M Coxeter, An Introduction to Geometry. A book of very beautiful classical geometry. Something typically not touched on at all in a typical mathematics curriculum.
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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.
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A lot of comments about textbooks that helped in specific topics but I don't think that answers the spirit of OP's question. Sure working through ANY Linear Algebra textbook is going to improve your Linear Algebra skills. In the spirit of OP's question: How to Solve it by G. Polya
Solving Mathematical Problems by Terrence Tao
Introduction to Mathematical Thinking by Keith Devlin Are all amazing, How to Solve it in particular is an all time classic.
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From Mathematics to Generic Programming - Stepanov & Rose Gödel, Escher, Bach: an Eternal Golden Braid - Hofstadter Euclid's Elements
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