If you wish to move beyond the level of learning methods to solve a very specific class of problems (like Calculus I/II/III teaches, no offence/looking down one's nose is intended), you'll need to eventually learn to write proofs. A good book to get you over the initial hurdles is Daniel Velleman's How to Prove It.
Ask HN: Independent Math Study
31–40 of 40 posts
Re: Ask HN: Independent Math Study
#32First make sure you have a solid operational foundation on the basics. Advanced topics will feel so much easier. For that I can recommend Discrete Mathematics and its Applications by Kenneth H. Rosen. Optionally supplemented by Student's Solutions Guide for more elaborate answers to exercises. Do as many exercises as possible.
I second Discrete Mathematics and its Applications. This book was the book I used in the first class that required a substantial amount of proof writing. A majority of it was easily tackled within a six week course.
Re: Ask HN: Independent Math Study
#33It's been ages since my last university math class (I was a I math major), so I can't point you to any reference material, but I can say the following. If you really want to improve your problem solving skills, I would highly recommend studying real analysis. What you get out of this will go a long way to making you a better problem solver. The reason why I say this is when you have to so something like prove why 1 i…
What's a good real analysis text these days? Is Spivak's calculus book still a common favorite?
I wouldn't know as it has been over 10 years since I looked at a math textbook. The thing about math is it doesn't change. Well at least the basics so it's safe to say any text that you find in the library would be a good source.
Where different text books may deviate from one another is how they prove a theorem. Like programming, you can usually get the same results by going down different paths. Some paths are more efficient than others, but that is predicated by what you know.
If you are just learning, the best thing to do is find textbooks with answer keys to assignments. Also with the advent of google and such, I would have to imagine you can probably find answers to a lot of the questions that would be posed in these text books so answer keys may not be all that important now.
Re: Ask HN: Independent Math Study
#34I really recommend http://www.mathxl.com . I've used it for calc 3 and linear algebra in place of physically being in those classes. It will walk you through examples and keeps track of your weakness to review later. It seems to be powered by Mathematica, from what I can tell.
Re: Ask HN: Independent Math Study
#35Earlier quoted context omitted.
No. Many pure math classes require no (or very little) calculus. Abstract algebra, number theory, combinatorics, and graph theory certainly fall into this category. Topology does, too, depending on which area you study and what you consider calculus. Sure, there are obviously fields that do rely heavily on calculus, as well as certain branches in the above fields, but my point was that it's nowhere near universally n…
This is terrible, wrongheaded advice. It's like Pablo Picasso, in the middle of his Blue Period, trying to convince younger painters that red isn't a useful color for serious artists. If you want to study graph theory or combinatorics [1], then calculus will be pretty much useless to you, and you'll naturally go years without using it. Calculus is also useless in some situations in abstract algebra (which are said to…
That post does not contain any advice. It contains facts, none of which were contradicted by your post (a typical property of facts). I never told him not to study calculus. Given our current education system, that would be impossible anyway. I guided him more toward real analysis and suggested some other areas of math that might interest him. Since his only background is high school math, I felt it would be best to introduce him to something where proofs play a central role. If he can't stand that, then he probably shouldn't go into math.
Re: Ask HN: Independent Math Study
#36If you wish to move beyond the level of learning methods to solve a very specific class of problems (like Calculus I/II/III teaches, no offence/looking down one's nose is intended), you'll need to eventually learn to write proofs. A good book to get you over the initial hurdles is Daniel Velleman's How to Prove It .
And in addition -- it greatly enhances ones ability of abstract thinking. At least in my case it was true :-)
Re: Ask HN: Independent Math Study
#37It's been ages since my last university math class (I was a I math major), so I can't point you to any reference material, but I can say the following. If you really want to improve your problem solving skills, I would highly recommend studying real analysis. What you get out of this will go a long way to making you a better problem solver. The reason why I say this is when you have to so something like prove why 1 i…
What's a good real analysis text these days? Is Spivak's calculus book still a common favorite?
I quite liked Stewart's Calculus. http://www.stewartcalculus.com/
And it is really helpful to use some CAS (like Maple, Octave or Maxima) to visualize problems.
Re: Ask HN: Independent Math Study
#38Earlier quoted context omitted.
> calculus isn't that important for mathematicians That's a joke, right?
No. Many pure math classes require no (or very little) calculus. Abstract algebra, number theory, combinatorics, and graph theory certainly fall into this category. Topology does, too, depending on which area you study and what you consider calculus. Sure, there are obviously fields that do rely heavily on calculus, as well as certain branches in the above fields, but my point was that it's nowhere near universally n…
> calculus isn't that important for mathematicians
> Many pure math classes require no (or very little) calculus.
These are not the same thing (hence my confusion.) Your initial comment seemed to indicate that nobody does analysis anymore, which is just not true at all (look at the most recent fields medal.)
Re: Ask HN: Independent Math Study
#39Earlier quoted context omitted.
No. Many pure math classes require no (or very little) calculus. Abstract algebra, number theory, combinatorics, and graph theory certainly fall into this category. Topology does, too, depending on which area you study and what you consider calculus. Sure, there are obviously fields that do rely heavily on calculus, as well as certain branches in the above fields, but my point was that it's nowhere near universally n…
> calculus isn't that important for mathematicians > Many pure math classes require no (or very little) calculus. These are not the same thing (hence my confusion.) Your initial comment seemed to indicate that nobody does analysis anymore, which is just not true at all (look at the most recent fields medal.)
Nope. I'm well aware that people still study analysis. I just meant that one doesn't necessarily need to learn calculus before taking the plunge into serious mathematics. Even in analysis, there's quite a bit you can do without knowing the stuff from a standard calculus class (though it certainly helps).
Re: Ask HN: Independent Math Study
#40My advice to you is to find a really good book and go with the book program. I passed many college classes just by studying hard with a good book.