Note: I am not sure if this is the correct place to ask these types of questions, so please let me know if I should remove my question.
I'm taking Real Analysis 1 this semester, and was thinking of taking the second part next semester, but I have heard that it is probably the hardest undergraduate math course. Is it that much harder than the first course?
Also, would it be a good idea to take Real Analysis 2 and Elementary Number Theory in one semester?
Thank you.
Here are the course descriptions:
Real Analysis 1:
Properties of the real number system; point set theory for the real line including the Bolzano-Weierstrass theorem; sequences, functions of one variable: limits and continuity, differentiability, Riemann integrability.
Real Analysis 2:
Includes the rigorous study of functions of two and more variables, partial differentiation and multiple integration. Special topics include: Taylor Series, Implicit Function Theorem, Weierstrass Approximation Theorem, Arzela-Ascoli Theorem.
Elementary Number Theory:
Properties of the integers, the division algorithm, Euclid's algorithm, Fermat's theorems, unique factorization of integers into primes, congruences, arithmetic functions, Diophantine equations, continued fractions, quadratic reciprocity.
Best Answer
From your description, all three courses look almost identical to what I took when I was undergrad. So yes analysis 2 will be more difficult and a lot more work like it demands more time you know reading, re-reading, and then re-re-reading theorems and proofs and lecture notes and the coming up with your own proofs and writing up the homework. And I think I agree with what you heard. Now that I think back, from all of the undergrad math courses I took, analysis 2 was the hardest. Number theory will be a piece of cake.
But remember being the "hardest" class doesn't mean that it is "hard" or impossible. In addition, the load also depends on how many other classes are you taking next semester. If these are the only two, then yeah for sure the load is doable. Otherwise if you want to take like five classes next semester including these two, then you should reconsider perhaps.
I never took both the same semester. I actually finished analysis sequence earlier and number theory was at the end right before graduation. But both were taken with a full time (3-4 courses every semester) load and I don't think it was that bad. Analysis took a LOT of time but number theory was like the easiest class I ever took. And FYI, this is only true for intro to undergrad number theory. It gets incredibly insane very fast if you ever want to take more number theory.