Abstract Algebra or Real Analysis?
Im taking Algebra next semester and Analysis in the fall and I was wondering which class you found the most beneficial. Overall, which class helped you gain more mathematical maturity and to help you write proofs better. If you guys can share your experiences with these classes I would appreciate it.
Thanks for your comments.
Comments
I had a better experience with Real Analysis. The proofs seemed more intuitive to me. The "limiting process" that takes place in a lot of analysis concepts can be elegant and slick. Algebra, on the other hand, is more rigid and coarse. A famous Algebraist once said, the key to being successful in Algebra is to find a neat trick, and then come up with a theorem and a proof which uses the trick. He was only halfway joking. However, oddly enough, in the graduate level equivalence of these classes I found Analysis took a turn for the obnoxiously technical which I disliked, and Algebra became much more useful and general.
As introduction level classes, I think they will both be beneficial and approachable. And you may find your tastes differ from mine. They are very different from each other and both have their own flavor, but they both require the same rigor and self motivation.
But, no one can argue, that these are two of the fundamental courses in undergraduate mathematical education. A course in Topology would be another foundational class (and the field I choose to specialize in).
My best advice is to think about the definitions and concepts until they make sense and you can articulate them to others. Don't worry too much about how people claim these courses are difficult; I'm sure you recall everyone telling you how hard "calculus 2" was, etc... It is not uncommon to hear people feeling sorry for themselves talking about how hard analysis is. Don't jump on this wagon. It is definitely manageable, but requires work and a dedication to understand the concepts. Some people have a tendency to resist anything that requires them to think for themselves. These are the courses that divide those who will go into mathematics and those who just had a talent for setting up and doing computational problems after seeing examples done for them. If you made it this far in your education, you have the background and the intelligence to pull these classes off.
There is no harm in your approach to take Algebra first. A good introductory Algebra course should run through the basic set theory and proof techniques that you need.
Good luck with your courses.
I would say both are beneficial. They go hand in hand. Algebra is a tool much like a hammer is to a carpenter. Analysis is it's mind. Without one or the other, nothing will be constructed.
often you will discover precis Algebra as a pre-req to authentic. for my area I trust that. If it have been me, i might take AA first. that is no longer an consumer-friendly problem, and it is often the stumbling block to math majors (that and stepped forward calculus). maximum suitable of success, and discover particularly some time to income!