Transition to (abstract) university mathematics.

[battousai]

Hello everyone. I am graduating high school in about a few days (yay!). I'm going to enter university this fall, majoring in math.

I just want a few advice on what to do this summer to prepare myself for college mathematics. How can I prepare myself to tackle abstract math, without any prior exposure to it?

Any books or other resources would be gladly appreciated. I think I pretty much have the entire summer to myself, so a book or two wouldn't be too much of a hassle.

 

[hitmeoff]

You probably will not encounter any abstract math on your first go round, unless you finished the Calculus Series and Elementary LA/DEs. Some schools have a class that on proof writing that you take sometime before you begin the abstract stuff. If your school doesn't, you may want to check out:

How to Prove It: A Structured Approach by Velleman
The Elements of Advanced Mathematic by Krantz

These books are all right. But I personally don't think you get much out of "intro to proofs" books/courses without really having some mathematical theme. These books and courses that use these books dabble a little in set theory, a little in number theory, a little bit of geometry, a little of this, a little of that, etc...As far as I am concerned, the only plus to these courses is that you start getting used to notation, you learn the alphabet/vocabulary of proofs.

You might do better trying to tackle something like Dudley's Elementary Number Theory. At least you get more than just "a little" number theory, but still simple enough that you can cover the book and understand the material while getting used to doing math differently than the typical "Step 1, Step 2, Step 3 and Answer" as you are probably used to in High School math.

If you are taking the Calc series, and certainly when you take Elementary Linear Algebra, make sure you go out of your way to tackle the proof problems, even if you professor doesn't assign them.

Junior/Senior level math usually hits people like a brick wall (of course, this always depends on: the school, the professor, the book used, etc etc) because they are not used to abstract thinking (they've been taught to be great calculators). It understandable, not everyone taking Calc I-III and Elem LA/DEs are going to be math majors and the school usually only has a couple of different flavors of those series (like Calc for Life Science Majors and Calc for Physical Science majors/ Engineers, and even if the school has this, Physical Science and Engineers don't necessarily need to know how to do proofs).

In any case, long post still long...you won't get good at "proofs" until you do them a lot, and even then, proofs are usually easy only when you really understand the material, and sometimes not even then. Just practice, practice, practice

 

[battousai]

btw, my math teacher gave me this book

https://www.amazon.com/dp/3540761780/?tag=pfamazon01-20

He says it presents a good foundation for higher mathematical topics. Any opinions about this book?

 

[hitmeoff]

Never seen this book.

It does seem to be in the same vain of some other "Intro to Higher Math" books. But I really do dig the topics that the book covers. Its an ambitious list of topics (to cover in any depth), but they are definitely REAL topics that you WILL run into in a full blown Analysis or Algebra class.

Really looks like an interesting book, I'd say stick with that!

 

[mal4mac]

Mathematics: A Very Short Introduction by Timothy Gowers - he very much concentrates on the *abstract* approach, and his ideal reader is you :)

 

[battousai]

Would self-studying number theory be productive for me?

 

[fourier jr]

btw, my math teacher gave me this book

https://www.amazon.com/dp/3540761780/?tag=pfamazon01-20

He says it presents a good foundation for higher mathematical topics. Any opinions about this book?

I'm with hitmeoff; I haven't seen that book but it looks like it covers a lot of stuff that would definitely be good to know before doing an analysis, topology or algebra course, since most of it is done with sets. I've always thought this one would be a good summer project also (& cheap too)
https://www.amazon.com/dp/0486616304/?tag=pfamazon01-20

re: number theory it probably wouldn't hurt, since you would get used to thinking about things that you do in algebra later but in a more abstract setting.

 

[battousai]

Should I also self study multivariable calculus over the summer? I will be taking the class this fall, but I was wondering if getting a head start would be good, or if my time would be best spent somewhere else.

 

[micromass]

I think it's better to learn something else. Calc III is quite easy and doesn't involve much proofs (in the USA). It's better to learn a proofy subject first.

Pick up a copy of "A book on abstract algebra" by Pinter, and work through the first few chapters. If you did this, then you should be ready for proofs and for abstract algebra (a notorious hard subject for many). The book I mentioned is quite a good one and is meant for people like you...

 

[battousai]

Thanks for the reply micromass (I'll be sure to pick up a copy sometime, it's got great reviews on Amazon too), and yes I am from the US

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On a side note, if I end up doing applied math, how much abstraction/proofs will it involve? I'm not averse to proofs at all, in fact I look forward to doing them. I was just wondering.

 

[micromass]

On a side note, if I end up doing applied math, how much abstraction/proofs will it involve? I'm not averse to proofs at all, in fact I look forward to doing them. I was just wondering.

Depends on the university really. With me, all the applied math courses had proofs. Of course, applied mathematics will have much less abstraction than pure mathematics, but it will still be proof-heavy...

 

[hitmeoff]

Depends on the university really. With me, all the applied math courses had proofs. Of course, applied mathematics will have much less abstraction than pure mathematics, but it will still be proof-heavy...

Same here, even our applied math classes are proof heavy. It's good you like doing proofs. As I said in some other thread before, I used to hate proofs and wondered why these upper level math classes were so different from Calc I-III which I smoked and felt so proud of myself for being able to do 3 page long computations of integrals and such. But then you realize that if math was just a matter of computation, then Mathematicians would be out of work, since computers as so much more efficient at doing computations.

 

[elfboy]

The proof is the process to arrive at the answer. It doesn't matter what kind of class or setting it is.

 

[battousai]

Depends on the university really. With me, all the applied math courses had proofs. Of course, applied mathematics will have much less abstraction than pure mathematics, but it will still be proof-heavy...

Will reading the abstract algebra book help me with applied math?

 

[micromass]

Will reading the abstract algebra book help me with applied math?

No, but it'll help you with any abstract algebra classes you'll take and it'll help you with proofs. It won't be useful in things like statistics for example..