

Ask HN: Independent Math Study - devin

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.<p>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.<p>Thanks!
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mark-t
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.d...](http://web.archive.org/web/20080205091131/http://www.kalva.demon.co.uk/index.html)
).

~~~
aneesh
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
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.

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sdesol
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
What's a good real analysis text these days? Is Spivak's calculus book still a
common favorite?

~~~
sdesol
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.

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wheels
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.

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cool-RR
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.

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Herring
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
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. :)

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

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

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rms
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>

_Elementary Calculus: An Infinitesimal Approach_ for a mathematically rigorous
course in infinitesimal calculus. I think it is much more intuitive than
typical limit calculus.

~~~
jibiki
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
Sure. I just think the infinitesimal is an interesting approach.

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kqr2
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-
Timoth...](http://www.amazon.com/Princeton-Companion-Mathematics-Timothy-
Gowers/dp/0691118809/)

------
jonsen
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
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.

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gtani
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, .

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streblo
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/d...](http://www.amazon.com/Road-Reality-Complete-Guide-
Universe/dp/0679776311/ref=sr_1_1?ie=UTF8&s=books&qid=1247988835&sr=8-1)

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secret
I 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.

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mping
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.

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yannis
You could try MIT's free courses!

