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.

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    $\begingroup$ I should mention that the contents of Real Analysis I and II differ greatly depending on the university (and some universities do not bother to split the course). It would be greatly helpful if you would mention which university you are taken the course from or perhaps an outline of the curriculum. $\endgroup$ – EuYu Nov 15 '12 at 3:11
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    $\begingroup$ If you're taking courses in a maths department then, perhaps, the course is more theoretical designed and less computational-applied. This makes all those horrible multiple integrals diminish (though they don't disappear...alas). The implicit/open function theorem is one of the most impressive theorems you will ever have the chance to see, and its complete version takes hours to write down, and more to completely understand. About Elem. Number Theory: it is going to be really elementary if you don't have some complex analysis background to tackle it. Perhaps will be good to wait... $\endgroup$ – DonAntonio Nov 15 '12 at 3:22
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    $\begingroup$ The best person to ask is probably the instructor of the course. As for taking this and Elementary Number Theory in the same semester, I see no particular reason not to: there is probably little in the way of interaction between the two. More advanced Number Theory can require quite a bit of analysis. $\endgroup$ – Robert Israel Nov 15 '12 at 3:27
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    $\begingroup$ @Alti Complex analysis is the complex number counter-part to real analysis, although many people consider it to be much nicer than real analysis. It is a beautiful field which you should definitely take a look at if you're interested in mathematics in general. As for your number theory course, there is nothing in there which would require a complex analysis background so it would be fine to take concurrently with Analysis II. $\endgroup$ – EuYu Nov 15 '12 at 3:31
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    $\begingroup$ What you wrote for Number Theory you can have without any complex analysis, which deals with functions of a complex variable (there's also multivariable complex analysis, of course), but don't worry by now about this: you won't need it for that course. $\endgroup$ – DonAntonio Nov 15 '12 at 3:33

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.

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