Entering The Second Half of Mathematics

[Improvisation]

Good evening guys,

I now announce that I am entering the second *hard* half of mathematics, in college!
Courses I have finished:
Calculus I
Calculus II
Calculus III
Differential Equations
Linear Algebra
Foundations of Mathematics
Operations Research

I am registered for two math courses next semester:
Complex Analysis
Theory of Numbers

Now, as you can see, the courses I am registered for are certainly different than the courses I have been in (or are they?) and I am honestly quite worried about what the semester will hold. I haven't really been involved in any proof-heavy mathematics course instead of the introduction to proofs ones. What should I expect, and how can I best prepare myself to do well? I hope to one day get a PhD in mathematics.

Note: I have a sciency background, especially in biology, chemistry, and physics.

 

[R136a1]

First you need to relax. The semester that is coming will be difficult, but also fun. In comparison to proof-based mathematics, calculus and stuff is boring as hell.

What to expect is an enormous amount of proofs and theorems. Being able to digest those is crucial. Don't be scared if you go through your textbook reeeeeaally slow, that's normal. Also don't be scared if the author or professor finds things obvious that you don't, that's also normal.

Here's my method of going through math texts, and it apparently works for me:
1) Read the examples and theorems. Don't bother to verify anything and don't read the proofs.
2) Study everything carefully. Verify all examples and proofs. Be sure to understand everything. Don't worry if you don't see the big picture yet.
3) Try to expand a bit on the material: does the theorem have interesting limiting cases? Does the converse of the theorem hold? Why would the theorem be useful? Try to describe the theorem and proof by just giving the ideas instead of the technical details. Where are the hypotheses of the theorem used in the proof? What previous theorems are used? What techniques are used and seem useful for later?
4) Close the book and write down the theorems and proofs from scratch. Only look if you're really stuck. Keep doing it until you know it very well.
5) Do exercises. This step is by far the most important one. It is only by doing this that you'll obtain mastery!

 

[Hercuflea]

Complex analysis is tough. I haven't had number theory but I've heard it's not so bad. However I did what you are doing and I took complex analysis before taking real analysis. It's not necessary to have Real before Complex, but it would help. I'd suggest dropping complex and signing up for real analysis or "advanced calculus" as some universities call it.

How well did you do in your earlier math classes? Did you like calculus and differential equations? Did you find yourself wanting to learn the "why" of all the derivatives and integrals, or are you more interested in learning the applications?