The answer is always Elementary Number Theory by David Burton. I read this book since 7th grade.
Ask HN: Math books that made you significantly better at math?
131–140 of 323 posts
Re: Ask HN: Math books that made you significantly better at math?
#132The 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 * So…
I've had this idea of starting back at basics and relearning math from the beginning since I never "really" learned it besides memorizing and skirting my way through it in school. Do you know a good path or book that's suitable for that?
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 pre-algebra, middle-school math, up through Differential Equations. Note that his diff-eq 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 per-se. I used to do the thing of treating it like a class, pausing the video to work examples, and what-not, 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 long-term 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
Re: Ask HN: Math books that made you significantly better at math?
#133Math always seemed a bit arbitrary to me. Why and how did all those fields and branches develop, why are some so much more intuitive than others etc. What helped me cope with that challenge (and ultimately be a better learner / user of mathematics) was digging into the history of mathematics. Many great books in that genre but a very influential one for me was the Concise History of Mathematics by Dirk Jan Struik
>Math always seemed a bit arbitrary to me.
That's because math is fundamentally arbitrary. This realization came to me late in life. Math was always presented as some aspect objective reality. However over time I've come to understand it as software for human brain. Using this math or that to describe something is often simply a matter of taste! Very similar to programming, in fact. Your study of history gives context to the question of utility different maths "packages" for certain problems, but does not invalidate your original impression.
Re: Ask HN: Math books that made you significantly better at math?
#134The classic "Advanced Engineering Mathematics" by Erwin Kreyszig was absolutely 'good enough'. Maybe even more important - for me: it was one of the easiest books to follow and digest during my undergrad. See it as a solid base for other heavier/purer titles.
Re: Ask HN: Math books that made you significantly better at math?
#135The Art of Problem Solving: https://artofproblemsolving.com/store/book/aops-vol1
Yes, it is targeted towards middle and high school students. Yes, I read it and (more importantly) worked through most of the problems in my mid-30's. It is great if, like me, you coasted/crammed through your early mathematics education and never felt like you dialed in the fundamentals. It is also great if, like me, you needed some pen-on-paper practice and did not know where to start.
Re: Ask HN: Math books that made you significantly better at math?
#136Earlier quoted context omitted.
I've had this idea of starting back at basics and relearning math from the beginning since I never "really" learned it besides memorizing and skirting my way through it in school. Do you know a good path or book that's suitable for that?
I’m reading and like Thomas Garrity’s “All the mathematics you missed (but need for graduate school)” which is this but for people who did a bachelors degree but missed certain areas (or forgot them). https://www.amazon.co.uk/All-Math-You-Missed-Graduate/dp/100...
I bought and read it (more like, skimmed) and liked it a lot too.
Gives a bird's eye view of math very nicely. Even from a skimming it was very useful to help me understand the gaps I have, and the shape of those gaps, and partially filling them.
Re: Ask HN: Math books that made you significantly better at math?
#137Nonlinear Dynamics and Chaos by Strogatz
Chaos theory and deterministic systems are a fascinating vantage point for thinking about the dynamics of large computer systems. Thinking of them as stochastic systems is sometimes useful, but most of the systems are actually just operating in unstable periodic processes which are much closer to being a chaotic system rather than a stochastic system. This influences how I think about testing and debugging large distributed systems.
I will say, I'm not sure I could have learned it well without a class and a good professor. The author has a number of books though and is a professor at Cornell.
Re: Ask HN: Math books that made you significantly better at math?
#138Rudin'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 fi…
Wouldn't ">=" be "not <"?
Yup.
Re: Ask HN: Math books that made you significantly better at math?
#139Earlier quoted context omitted.
I've had this idea of starting back at basics and relearning math from the beginning since I never "really" learned it besides memorizing and skirting my way through it in school. Do you know a good path or book that's suitable for that?
I had the same idea. I bought myself a Remarkable 2 and signed up to Khan Academy. Now I'm revising algebra basics and I plan to go as advanced as Khan Academy lets me. I was really bad at maths in school (UK A Levels). But I'm a successful software developer today. I felt like knowing more advanced maths could make me a better developer and not feel intimidated by a lot of the things I see. I'm actually enjoying it…
If you don't mind the question how is the Remarkable helping you in this. Just to avoid the clutter of paper? Or does it some how OCR you're handwriting?
Re: Ask HN: Math books that made you significantly better at math?
#140Earlier quoted context omitted.
I've had this idea of starting back at basics and relearning math from the beginning since I never "really" learned it besides memorizing and skirting my way through it in school. Do you know a good path or book that's suitable for that?
I had the same idea. I bought myself a Remarkable 2 and signed up to Khan Academy. Now I'm revising algebra basics and I plan to go as advanced as Khan Academy lets me. I was really bad at maths in school (UK A Levels). But I'm a successful software developer today. I felt like knowing more advanced maths could make me a better developer and not feel intimidated by a lot of the things I see. I'm actually enjoying it…
Thinking of doing the same thing. The last math class I had was at 16, and the most advanced classes was on binary, so not very complex stuff.
I've mostly been winging it for another 16 years and seemingly picked up things here and there. But math is definitely been trial and error, and I definitely do not know the lingo in math, which I'm now starting to feel is a big disadvantage.
also: the remarkable 2 is great, we have one, but the screen broke and the refurbished replacement arrived with a screen that's not functioning correctly at all, making it a unusable device. Good reminder to reach out to them again.
Thanks!