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Ask HN: Independent Math Study

20 points by devin 17y ago

I never thought I'd be asking this. Math was never my "strong suit", but over the last year I've really grown to enjoy it as I learn more. I've taken Calc. I in a fairly demanding college environment, and am planning on continuing with Calc. II and Linear Algebra. My question to HN is: How does one go about doing self-study in Math? It seems, of all the sciences, to be especially difficult to tackle without the built-in support of the classroom. I assume that like most things, it just takes a lot of hard work and study, but I'm curious if anyone out there has a rough plan for tackling a reasonably rich understanding of mathematics on their own. Sites, materials, etc. are appreciated. Thanks!

rms 17y ago
I don't have a complete answer for you, but I linked to this book a few days ago. It's pretty good. http://www.math.wisc.edu/~keisler/calc.html http://www.math.wisc.edu/~keisler/calc.html Elementary Calculus: An Infinitesimal Approach for a mathematically rigorous course in infinitesimal calculus. I think it is much more intuitive than typical limit calculus.
- devin 17y ago
  Fancy that! I'm at the UW right now. I have been considering a few different routes. Philosophy, Classics, Music, Computer Science, or Math. I'd really like to go for a Math undergrad with a minor in one of the other subjects, but I'm going to need a supplement during the summer to catch up with some of the other math guys at the UW. Lots of competition.
- jibiki 17y ago
  There are vast sections of mathematics which cannot be understood without first understanding limits. There are very few areas of mathematics which require understanding infinitesimals.
  - rms 17y ago
    Sure. I just think the infinitesimal is an interesting approach.
mlLK 17y ago
Checkout this thread, http://news.ycombinator.com/item?id=108723 http://news.ycombinator.com/item?id=108723
mark-t 17y ago
To be honest, calculus isn't that important for mathematicians, but if you want to study mathematics seriously, I'd suggest picking up a rigorous text like Rudin's or Apostol's. It will be difficult. You'll have to read most of it several times. That's perfectly fine; the point is that it will help you learn to think like a mathematician does. Now, on the other hand, linear algebra is almost universally important and is probably easier for a programmer to grasp. I would also suggest picking up a Number Theory or Combinatorics text; they're practically useless, but they're fun and interesting, they'll give you a better idea of what mathematicians do, and you don't need much education to get into them. My usual advice for building skills is to work on contest problems. See if you can find some AMC12 problems. If those are too easy, you can work your way up. AIME and Putnam would be good next steps (those can be found here: http://web.archive.org/web/20080205091131/http://www.kalva.demon.co.uk/index.html http://web.archive.org/web/20080205091131/http://www.kalva.d... ).
- aneesh 17y ago
  Saying the Putnam is a next step from AMC12 problems is like saying the NBA is a next step from pickup basketball with friends in middle school! There are people who can do Putnam problems for fun, but those people generally know who they are already.
  - albertni 17y ago
    Solving problems 1-4 on each day of the Putnam with an "unlimited" amount of time is not a ridiculous expectation. Putnam's difficulty is partly due to its time format.
  - mark-t 17y ago
    Putnam problems are not considerably harder than AIME problems, and AIME is definitely the next step from AMC12. Anybody who can solve a few AIME problems can certainly solve A1 on the Putnam of almost every year.
- TriinT 17y ago
  I would claim that Calculus isn't that important for engineers / scientists / programmers either. Real Analysis is important if one needs to understand thing deeper. In the real world, problems can't be solved analytically... and many of the tools one learns in Calculus are kind of useless. I think Linear Algebra is much, much more important than Calculus. Linear Algebra is the arithmetic of higher mathematics, like Bellman said.
sdesol 17y ago
It'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 is greater than 0, you'll learn to look at things differently. In studying real analysis, you are almost learning how to walk again. Everything that you have taken for granted as being obvious in the past will now have to be proven. And by going through these exercises, you'll learn the importance of truly understanding what you are doing.
- CamperBob 17y ago
  What's a good real analysis text these days? Is Spivak's calculus book still a common favorite?
  - sdesol 17y ago
    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.
  - krepsj 17y ago
    I quite liked Stewart's Calculus. http://www.stewartcalculus.com/ http://www.stewartcalculus.com/ And it is really helpful to use some CAS (like Maple, Octave or Maxima) to visualize problems.
wheels 17y ago
I've learned a lot of math that's beyond the scope of what I had in college. I usually find that it works best when it's on the way to something that I'm trying to do or understand. I never really tried to learn math for the sake of math -- I wanted to understand quantum computing algorithms, recommender systems and graph clustering -- and had to fill in the gaps so that the papers in the fields made sense.
streblo 17y ago
You should take a look at a book called The Road to Reality by Roger Penrose. While it's geared more towards physics, this book has proven to me to be the most enlightening mathematics text I've ever read. Admittedly I'm only about 10 chapters in - it's a very dense book, and you'd do well to go through it slowly. But, if you're interested in math, this book will blow your mind. http://www.amazon.com/Road-Reality-Complete-Guide-Universe/dp/0679776311/ref=sr_1_1?ie=UTF8&s=books&qid=1247988835&sr=8-1 http://www.amazon.com/Road-Reality-Complete-Guide-Universe/d...
Herring 17y ago
Always use more than 1 text, always do the problems, & always keep up a steady pace. I haven't found anything else to be really important.
- boryas 17y ago
  This is really good advice. Also, remember that it means nothing beyond what it says, all you really have to work from are the definitions and the theorems. :)
jonsen 17y ago
First 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.
- BrentRitterbeck 17y ago
  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.
kqr2 17y ago
Book recommendation: Princeton Companion to Mathematics It's a good way to skim a lot of different mathematical topics for further exploration. http://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809/ http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...
cool-RR 17y ago
I self-studied math for 2 years. I just attended lectures without officially enrolling to the university. I also did about half of the homework problems given in these courses (My math-student friend was envious of me: I could choose the interesting questions out of the homework paper, and ditch the boring ones!) Some of my studies I also did with books, video lectures, and articles I found on the internet.
gtani 17y ago
You could do a purely applied approach, look at some Data Mining books, like Witten/Franke and the Weka java framwork (there's quite a few good books, check amazon reviews, ) and the assortment of methods that are applied from basic logit/probits, through clustering, SVM, neural, evolutionary programming, .
yannis 17y ago
You could try MIT's free courses!
BrentRitterbeck 17y ago
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.
- krepsj 17y ago
  And in addition -- it greatly enhances ones ability of abstract thinking. At least in my case it was true :-)
secret 17y ago
I really recommend http://www.mathxl.com 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.
mping 17y ago
My 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.

20 points by devin 17y ago

rms 17y ago

I don't have a complete answer for you, but I linked to this book a few days ago. It's pretty good. http://www.math.wisc.edu/~keisler/calc.html http://www.math.wisc.edu/~keisler/calc.html Elementary Calculus: An Infinitesimal Approach for a mathematically rigorous course in infinitesimal calculus. I think it is much more intuitive than typical limit calculus.

devin 17y ago
Fancy that! I'm at the UW right now. I have been considering a few different routes. Philosophy, Classics, Music, Computer Science, or Math. I'd really like to go for a Math undergrad with a minor in one of the other subjects, but I'm going to need a supplement during the summer to catch up with some of the other math guys at the UW. Lots of competition.
jibiki 17y ago
There are vast sections of mathematics which cannot be understood without first understanding limits. There are very few areas of mathematics which require understanding infinitesimals.
- rms 17y ago
  Sure. I just think the infinitesimal is an interesting approach.

mlLK 17y ago

Checkout this thread, http://news.ycombinator.com/item?id=108723 http://news.ycombinator.com/item?id=108723

mark-t 17y ago

To be honest, calculus isn't that important for mathematicians, but if you want to study mathematics seriously, I'd suggest picking up a rigorous text like Rudin's or Apostol's. It will be difficult. You'll have to read most of it several times. That's perfectly fine; the point is that it will help you learn to think like a mathematician does. Now, on the other hand, linear algebra is almost universally important and is probably easier for a programmer to grasp. I would also suggest picking up a Number Theory or Combinatorics text; they're practically useless, but they're fun and interesting, they'll give you a better idea of what mathematicians do, and you don't need much education to get into them. My usual advice for building skills is to work on contest problems. See if you can find some AMC12 problems. If those are too easy, you can work your way up. AIME and Putnam would be good next steps (those can be found here: http://web.archive.org/web/20080205091131/http://www.kalva.demon.co.uk/index.html http://web.archive.org/web/20080205091131/http://www.kalva.d... ).

aneesh 17y ago
Saying the Putnam is a next step from AMC12 problems is like saying the NBA is a next step from pickup basketball with friends in middle school! There are people who can do Putnam problems for fun, but those people generally know who they are already.
- albertni 17y ago
  Solving problems 1-4 on each day of the Putnam with an "unlimited" amount of time is not a ridiculous expectation. Putnam's difficulty is partly due to its time format.
- mark-t 17y ago
  Putnam problems are not considerably harder than AIME problems, and AIME is definitely the next step from AMC12. Anybody who can solve a few AIME problems can certainly solve A1 on the Putnam of almost every year.
TriinT 17y ago
I would claim that Calculus isn't that important for engineers / scientists / programmers either. Real Analysis is important if one needs to understand thing deeper. In the real world, problems can't be solved analytically... and many of the tools one learns in Calculus are kind of useless. I think Linear Algebra is much, much more important than Calculus. Linear Algebra is the arithmetic of higher mathematics, like Bellman said.

sdesol 17y ago

It'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 is greater than 0, you'll learn to look at things differently. In studying real analysis, you are almost learning how to walk again. Everything that you have taken for granted as being obvious in the past will now have to be proven. And by going through these exercises, you'll learn the importance of truly understanding what you are doing.

CamperBob 17y ago
What's a good real analysis text these days? Is Spivak's calculus book still a common favorite?
- sdesol 17y ago
  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.
- krepsj 17y ago
  I quite liked Stewart's Calculus. http://www.stewartcalculus.com/ http://www.stewartcalculus.com/ And it is really helpful to use some CAS (like Maple, Octave or Maxima) to visualize problems.

wheels 17y ago

I've learned a lot of math that's beyond the scope of what I had in college. I usually find that it works best when it's on the way to something that I'm trying to do or understand. I never really tried to learn math for the sake of math -- I wanted to understand quantum computing algorithms, recommender systems and graph clustering -- and had to fill in the gaps so that the papers in the fields made sense.

streblo 17y ago

You should take a look at a book called The Road to Reality by Roger Penrose. While it's geared more towards physics, this book has proven to me to be the most enlightening mathematics text I've ever read. Admittedly I'm only about 10 chapters in - it's a very dense book, and you'd do well to go through it slowly. But, if you're interested in math, this book will blow your mind. http://www.amazon.com/Road-Reality-Complete-Guide-Universe/dp/0679776311/ref=sr_1_1?ie=UTF8&s=books&qid=1247988835&sr=8-1 http://www.amazon.com/Road-Reality-Complete-Guide-Universe/d...

Herring 17y ago

Always use more than 1 text, always do the problems, & always keep up a steady pace. I haven't found anything else to be really important.

boryas 17y ago
This is really good advice. Also, remember that it means nothing beyond what it says, all you really have to work from are the definitions and the theorems. :)

jonsen 17y ago

First 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.

BrentRitterbeck 17y ago
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.

kqr2 17y ago

Book recommendation: Princeton Companion to Mathematics It's a good way to skim a lot of different mathematical topics for further exploration. http://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809/ http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...

cool-RR 17y ago

I self-studied math for 2 years. I just attended lectures without officially enrolling to the university. I also did about half of the homework problems given in these courses (My math-student friend was envious of me: I could choose the interesting questions out of the homework paper, and ditch the boring ones!) Some of my studies I also did with books, video lectures, and articles I found on the internet.

gtani 17y ago

You could do a purely applied approach, look at some Data Mining books, like Witten/Franke and the Weka java framwork (there's quite a few good books, check amazon reviews, ) and the assortment of methods that are applied from basic logit/probits, through clustering, SVM, neural, evolutionary programming, .

yannis 17y ago

You could try MIT's free courses!

BrentRitterbeck 17y ago

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.

krepsj 17y ago
And in addition -- it greatly enhances ones ability of abstract thinking. At least in my case it was true :-)

secret 17y ago

I really recommend http://www.mathxl.com 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.

mping 17y ago

My 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.