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Ask HN: Math books that made you significantly better at math?

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Re: Ask HN: Math books that made you significantly better at math?

#311

Post by Jonas4321 »

nextos wrote: post #251
jsenn wrote: post #156

* The Art of Probability by Hamming. An opinionated, slightly quirky text on probability. Unlike the text used in my university course its explanations were clear and rigourous without being pedantic. The exercises were both interesting and enlightening. The only book in this list that taught skills I've actually used in the real world. * Calculus by Spivak. This was used in my intro calculus course in university. It…

These are good recommendations, but I think beginners tend to burn out due to the lack of a structured program and/or exercise solutions if they are trying to study on their own. The simplest structured program I can think of that satisfies both is: * Basic Mathematics by Lang. Covers basic algebra and geometry at high school level. Then one of these two, depending on your interests, or both: * Vector Calculus, Linea…

Meh, Hubbard and Hubbard is good, but it certainly does not take you through single-variable calculus. For example, the following topics are assumed and not treated in any detail:

- power series and analytic functions

- the various properties of exponentials, logarithms, trigonometric functions, etc.

- L'Hopital's rule

- Integration by parts

You should definitely work through something like Spivak or Tao in addition to Hubbard.

Re: Ask HN: Math books that made you significantly better at math?

#312

Post by oogway8020 »

bob_theslob646 wrote: post #289
nextos wrote: post #251

Earlier quoted context omitted.

These are good recommendations, but I think beginners tend to burn out due to the lack of a structured program and/or exercise solutions if they are trying to study on their own. The simplest structured program I can think of that satisfies both is: * Basic Mathematics by Lang. Covers basic algebra and geometry at high school level. Then one of these two, depending on your interests, or both: * Vector Calculus, Linea…

These are great. Thanks for sharing. Do you happen to have any on probability or statistics and or both?

There is a large number of html format books on bookdown.org, mostly related to Data Science and R, lots of Bayesian too. There are some more theoretical math books there, one of which I found to be very well written: Theory of Distributions by Peter K. Dunn

Re: Ask HN: Math books that made you significantly better at math?

#313

Post by sn9 »

dysoco wrote: post #286
sn9 wrote: post #261

Earlier quoted context omitted.

For a case study in how specifically to do this, see this post about how Cal Newport studied discrete mathematics in college: https://www.calnewport.com/blog/2008/11/25/case-study-how-i-...

I usually enjoy Newport but... this was rather underwhelming? It's basically "do a lot of proofs and study consistently not just 48hs before the exam". Most if not all of my math and some comp-sci courses would require this, they were very proof-heavy, specially Algebra, Logic and Graph Theory, and there was no way you could even pass just studying for a week after let alone 48hs before.

I mean it's underwhelming if you've done it before, but for people who are unsure about how to succeed in a proof-heavy course, seeing it laid out like this in concrete steps is helpful.

You've outgrown being in the target audience for it, but that doesn't mean the audience wouldn't find it helpful.

Re: Ask HN: Math books that made you significantly better at math?

#314

Post by nextos »

Jonas4321 wrote: post #311
nextos wrote: post #251

Earlier quoted context omitted.

These are good recommendations, but I think beginners tend to burn out due to the lack of a structured program and/or exercise solutions if they are trying to study on their own. The simplest structured program I can think of that satisfies both is: * Basic Mathematics by Lang. Covers basic algebra and geometry at high school level. Then one of these two, depending on your interests, or both: * Vector Calculus, Linea…

Meh, Hubbard and Hubbard is good, but it certainly does not take you through single-variable calculus. For example, the following topics are assumed and not treated in any detail: - power series and analytic functions - the various properties of exponentials, logarithms, trigonometric functions, etc. - L'Hopital's rule - Integration by parts You should definitely work through something like Spivak or Tao in addition…

I don't think this is a big concern. Hubbard discusses L'Hôpital and integration by parts, but it doesn't place a great deal of emphasis on them because the way topics are presented is a bit unusual.

If this is a concern, you can always use another text. I don't think this is a problem. Some freshman courses jump straight into Hubbard.

Re: Ask HN: Math books that made you significantly better at math?

#315

Post by fatneckbeard »

fn-mote wrote: post #255
chriswarbo wrote: post #248

Earlier quoted context omitted.

More than that: no Real numbers, no pi, no square root of 2, no sine/cosine, etc. It's similar to 'reverse mathematics' (trying to find the minimum set of assumptions required to prove a known result)

The square root of 2 does not require any limiting process. It's the hypotenuse of a right triangle with legs of length one.

no computer can calculate that exact distance, which is kind of Wildberger's point.

Infinities are very interesting but the non-infinite maths have kind of got neglected over the past 100 years. I had to memorize Laplace transforms in college but never heard of Fairey sequences until I watched his videos.

People get upset at him but he's basically just having fun seeing how far you can go in Math without infinity. It's quite interesting to a certain audience (like myself).

Re: Ask HN: Math books that made you significantly better at math?

#316

Post by Jonas4321 »

nextos wrote: post #314
Jonas4321 wrote: post #311

Earlier quoted context omitted.

Meh, Hubbard and Hubbard is good, but it certainly does not take you through single-variable calculus. For example, the following topics are assumed and not treated in any detail: - power series and analytic functions - the various properties of exponentials, logarithms, trigonometric functions, etc. - L'Hopital's rule - Integration by parts You should definitely work through something like Spivak or Tao in addition…

I don't think this is a big concern. Hubbard discusses L'Hôpital and integration by parts, but it doesn't place a great deal of emphasis on them because the way topics are presented is a bit unusual. If this is a concern, you can always use another text. I don't think this is a problem. Some freshman courses jump straight into Hubbard.

I disagree. L'Hôpital's rule is not discussed, it is simply presented without proof. And I think it's very important to carefully construct logarithms, exponentials and trigonometric functions, e.g. through power series - else the properties of these functions are just arbitrary axioms. Don't Hubbard and Hubbard claim themselves that the reader should be familiar with power series (somewhere in the chapter on Taylor series)?

Don't get me wrong, I really like the book, but I don't believe it replaces a book on single-variable calculus. I think the authors would agree.

Re: Ask HN: Math books that made you significantly better at math?

#317

Post by doetoe »

enriquto wrote: post #49

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 * So…

I will be checking out the ones I don't know, because the ones I do (Weeks, Arnold's classical mechanics and Spivak's differential geometry) are fantastic IMHO

Re: Ask HN: Math books that made you significantly better at math?

#318

Post by veqq »

bombcar wrote: post #102
krmboya wrote: post #63

Earlier quoted context omitted.

I've been flirting with the idea of working through all of Leonard Euler's publications (as a life goal). Many of them are still not translated from Latin, so there's a possibility I may have to learn it. Anyone knows how long it would take to learn enough latin to undertake such a task?

If you're trying to understand a domain work (such as Euler's) you could probably get a working knowledge in a month of strong study, a year of off-and-on. I bet you could start this week if you used machine translations as a crutch. I'd start working with a publication that exists in Latin and a good translation, so you can compare your work.

Use the book "Lingua Latina per se illustrata" to learn Latin. It's quite magical, you just start reading Latin which is comprehensible due to similarities to English and it stacks on this without using anything but Latin. It's also much faster and more thorough than other textbooks.

Re: Ask HN: Math books that made you significantly better at math?

#319

Post by debanjan16 »

empyrrhicist wrote: post #308
jay3ss wrote: post #298

Earlier quoted context omitted.

In my experience, nothing beats solving problems. Even if you get help via solution manuals. I went from being a B/A student to top of my classes (engineering) by solving many problems. I would do my homework and then go back before exams and redo the homework twice for a total of solving the problems 3 times. Never failed me. I actually managed to get a perfect score from a professor who wrote notoriously difficult…

Yep, a book with proofs and worked exercises is great for this because you can try to do the proof and then look at the solution to see if you got it reasonably correct. In my case the book covers a lot of stuff that I passively "know", but working problems has helped get be back to an active understanding.

Can you mention some books of this type?

Re: Ask HN: Math books that made you significantly better at math?

#320

Post by dev_0 »

jay3ss wrote: post #298
empyrrhicist wrote: post #276

Earlier quoted context omitted.

Yeah, I've been doing the same thing, but with the rule that I have to do "a math problem". Right now I'm going through a stochastic processes book a bit at a time that way and really enjoying it.

In my experience, nothing beats solving problems. Even if you get help via solution manuals. I went from being a B/A student to top of my classes (engineering) by solving many problems. I would do my homework and then go back before exams and redo the homework twice for a total of solving the problems 3 times. Never failed me. I actually managed to get a perfect score from a professor who wrote notoriously difficult…

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