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Abstract Algebra or Real Analysis

  1. Sep 2, 2008 #1
    Which one should I take first? Does it help to take one before the other?
  2. jcsd
  3. Sep 2, 2008 #2


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    Depends on your interests.
    Have you developed any interests yet?
    The two subject matters are largely unrelated. As long as you are comfortable with logical thinking and proofs as in the normal upper division linear algebra you should be fine.

    Maybe taking one before the other might be beneficial in the sense that you would have gained more experience with the mathematical way of thinking by the time you take the second class.
  4. Sep 2, 2008 #3
    Both courses approach mathematical abstraction in different ways, so in my opinion, it doesn't really matter which you take first. Some find Algebra to be trickier than Analysis, while others, like myself, feel differently.

    At my university, the standard is to take Abstract Algebra first, and then Analysis. It may be different at other schools. I don't think you should worry about it too much, and try to take whatever makes your schedule prettier.
  5. Sep 2, 2008 #4
    Not really. But I am majoring in applied mathematics. So I guess the question is: which would be more useful for my major?
  6. Sep 2, 2008 #5
    Real analysis is likely more applicable. And my school requires real analysis before abstract algebra.
  7. Sep 2, 2008 #6
    Also, do I need to know anything for real analysis or algebra besides proof skills? I know that my school has prereqs, but sometimes not everything is used.
  8. Sep 3, 2008 #7
    For Analysis and Algebra, it would be helpful to know basic set theory, 1-1 functions, etc. I assume most of it will be covered in the first few weeks.

    Algebra also uses some concepts of number theory, such as divisibility, greatest common divisors, etc. The first few chapters of an elementary number theory book would suffice, but again, I am sure this will be reviewed/taught in the beginning weeks of the course.
  9. Sep 3, 2008 #8


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    Here's an idea: real analysis is probably more important to an applied mathematician, so you want to take algebra first so that when you come to real analysis, you will have more mathematical maturity and real analysis will sink in more smoothly.
  10. Sep 3, 2008 #9
    Well, I'm taking a second course in linear algebra during fall. So would I be "mature" enough to take analysis during winter? Or would you still suggest that I take abstract algebra during winter and analysis during spring?
  11. Sep 3, 2008 #10


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    Sure, you can never be too "mature"...

    But of course there are other factors to consider. For instance, surely real analysis is a prerequisite to analysis 2 and 3 and PDE, and maybe other course that you want to take... So taking real analysis so late, will you be able to take all these courses before completing your degree?
  12. Oct 8, 2009 #11

    I have taken real analysis and abstract algebra

    i will say this...

    real analysis is easier than abstract algebra. Abstract algebra was the single most horrific class I ever suffered through in college. The last thing I remember learning was a field extension where essentially you were adding a "pseudo-square root of 2" to the integers which meant you could really force square root of 2 to be an integer? god help us all...

    Real analysis is just another continuation of the horrificness

    If you want to take these classes, it is really essential you have toned your logic skills before hand by taking at least one course on logic because both courses are 100% theorem-proof based and the proofs are more than just "prove 4 is an even number" so just make sure you have some understanding of mathematical theory

    that being said, nothing will ever make you think more logically other than classes like these... i can say they were the scorn of my college existence, but if there's one thing that made me think different as a result of study its mathematical theory like this

    I recommend you talk to the profs of those classes beforehand tell them your doubts, your likes, your major, all of this. See if they can advice you more

    Good luck!

    oh yeah and I can't stress enough how difficult these courses are... I am not joking.

    JUst to add...

    I have never heard of an analysis course being required for something like PDE, ADE, Linear, anything like that... actually anything at all. Many grad mathematics programs don't even require you've had more than one course on analysis. These courses like PDE aren't going to be any more abstract than linear algebra... they will rely on the fact that you can understand reading theorems and proofs but not that you are actually able to write them. They are still applied and you shouldn't have any problems in them without analysis

    Further... I think one semester of linear algebra isn't *quite* enouh to prepare you for a heavy duty course like abstract algebra or real analysis. YOu really need a specific course in mathematical theory that prepares you for it. In linear algebra, the more abstract methods of mathematics are introduced, but it is still strictly an applied course. I'm sure your college offers something like "fundamentals of mathematical logic" or "intro to mathematical reasoning" or something like that. You will pretty much cover everything you would cover in a discrete mathematics course only in a theorem-proof way instead of merely studying things. For example, you will go over basic things like, whats it mean to be a one-to-one or onto, but it will be given in a 100% mathematical logic way and then you will learn how to prove that something is onto or 1-1.

    So while linear is great to have I can't stress how much easier your life will be if you take a pre-req course on mathematical logic/analysis/thinking, whatever its listed as. YOu can go in without a course like this, but chances are all ready high to crash and burn in a course like abstract algebra so you want to make the chances better that you will succeed!

    Good luck!

    btw both these courses are what persuaded me to pursue my masters in computer science rather than mathematics :P

    one more thing...

    real analysis is easier than abstract algebra. I truly believe this and stand by it. If I had it to do again, I would have taken them in the reverse order. I think once you get to the maturity that you can take one course, you can take the other. the question isn't are you mature enough for one over the other its whether you are mature enough for either yet. You should start practicing now on your own time writing proofs. If you don't want to take a pre req course at least check a book from the library where you will write formal proofs in all number of ways (contradiction, contrapositive, direct..) and go beyond the scope of linear algebra.

    Best wishes again ;D
    Last edited: Oct 8, 2009
  13. Oct 8, 2009 #12
    I find the ideas of analysis much more intuitive than esoteric rings and fields. In my opinion, analysis or advanced calculus would be a better introduction to higher-level math than algebra. That is of course if it's an intro analysis course that starts with epsilon-delta definitions and stuff. As a freshman, I once rented a book from the library entitled "Intro Real Analysis" that dove strait into topological spaces in the first chapter. Man, was I lost...ha!

    That's all an opinion of course, you may be different.
  14. Oct 8, 2009 #13
    No I completely agree with you, abstract alg. is a better intro to higher math than real analysis, but what I'm saying is that neither is good to jump into... i think to get the maturity you should take a general introduction to mathematical logic, something more than the way linear algebra or PDE makes you think. and then i kind of think at that point, you are ready to try either. I think it also depends on how the course is taught. yeah if you just started on toplogy in real analysis you should have a course below that like abstract algebra first for sure

    both are disasterous though lol
  15. Oct 8, 2009 #14
    Well, depending on your school, the linear algebra class may be the introduction to mathematical proofs, etc. class. I know that proving things like 1-1, onto, proof of subspacey-ness, abstraction to function spaces, etc. were the bread and butter of my linear algebra class.

    Which one is more intuitive will vary from person to person. I find abstract algebra much more intuitive and down-to-earth than analysis. I would recommend reading the first chapter or 2 of a given introductory text to both and figuring out which you like more from there.
  16. Oct 8, 2009 #15
    I guess every college is different. I didn't even know there were linear algebra courses that were set up in that way ! :D

    I agree it comes down to the person whether real analysis or abstract algebra will be harder but i think another important factor is what text book you are working out of.

    My real analysis text was great but my abstract algebra text was so horrible that i was very miserable throughout the class. No clarification on anything really. My poor professor was about to strangle me from all the questions that semester :D

    So I think a good text can make all the difference in many situations

    maybe we can recommend some books? I used to Bartle/sherbert real analysis book, I can't remember what was my modern algebra text
    ... but I'll look when I'm at home and post it as a bad book to have as a study companion... anyone can recommend abstract algebra books for people to supplement their courses with in case someone gets a crappy text that should be of some help to others!

    I think the evil author of my abstract algebra text was by the name of Goodman... rrrr if you are studying a course with the goodman "abstract algebra : stressing symmetry" text... i recommend you get another book on the side
    Last edited: Oct 8, 2009
  17. Oct 8, 2009 #16


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    I think analysis is technically a harder subject than algebra because of all the limiting arguments and epsilon/delta manipulation that can often obfuscate what's going on as you read the proofs (and for the same reason, it's harder to generate your own proofs).

    On the other hand, most students studying analysis have already taken a less-rigorous calculus sequence so there's no problem with understanding the motivation ("you're learning a rigorous version of calculus"), whereas on first encountering algebra it may not be at all clear why you're learning about these abstract things called groups, rings, and fields which seem to have nothing to do with the mathematics learned up to that point. But if you can deal with this abstraction and don't mind waiting to see the whole picture unfold, there's a lot less "dotting the i's and crossing the t's" grunge in most algebra proofs than in analysis.

    Both subjects are great, though, and every undergraduate should take both. There's almost no dependence between the two, so in principle you can take them in either order.

    Recommend some books? Absolutely. I used Bartle's "Elements of Real Analysis" in my first (honors) analysis course. I really like this book: it's challenging enough that you don't feel even slightly talked down to, but it's not quite as austere or slick as Rudin (and has a lot more, and better, exercises). Rudin is very beautiful and one of my favorite math books, but I think it's better as a reference after the fact than as a place to learn the material the first time around.

    Another good option for analysis at this level is Thomson/Bruckner/Bruckner, "Elementary Real Analysis," which is maybe a little gentler than Bartle's "Elements." I'm not very familiar with Bartle/Sherbert's "Introduction to Real Analysis" but it's well regarded and, as I understand it, a bit more leisurely and less ambitious than "Elements."

    I studied algebra after analysis, so had built some mathematical maturity by then. I found a lot of the introductory algebra books to be boring and not very challenging, e.g. Gallian. The course I took nominally used Dummit and Foote, but the professor followed his own notes and I hardly ever looked at the book as it seemed very longwinded and dry. I read part of Herstein's "Topics in Algebra" instead, and really liked it, particularly the group theory parts. That book would be my first recommendation to anyone new to the subject but not new to proof-oriented mathematics in general.
    Last edited: Oct 8, 2009
  18. Oct 8, 2009 #17
    algebra is generally considered easier than analysis. If your analysis course uses baby Rudin, I would suggest algebra before analysis. Since you are taking linear algebra, abstract algebra will be in a similar vein, and will prepare you more for analysis imo
  19. Oct 8, 2009 #18
    Dummit and Foote is probably the easiest book to work from, but the best intro is probably Michael Artin's book.

    I think that to be a good mathematician you need to be able to enjoy an abstract puzzle as much as one that has more concrete intuition behind it. You might start out in the concrete but you often have to delve into obscure areas with a good amount of understanding to get a proof you need.

    I liked both courses, they each had their own flavor.
  20. Oct 8, 2009 #19
    I'm talking both at once, along with my other Mathematical Computer Science course...and a Physics course too. The other two courses aren't much of high level so it's allright.

    Alike yourself i'm interested in Applied Mathematics too and proving isn't really my thing. However, these two courses are required for graduation so i really had no choice but to take them. I'm doing them together just so i can get it over with. After this i'm all applied and [i checked] none of those courses have abstract o analysis as prerequisites.

    regarding your qn on how similar they are...well, i find them different. Although both have fair deal of poofs, both are about different things (at least from a basic pov).

    Yes, you could do they both.

    and yes two linear algebra courses is good enough preparation.

    what do you need to know?

    abstact algebra:
    - poof methods (direct, contradiction, induction)
    - well ordering principle
    - modular mathematics
    - equivalence classes
    - gcd
    - partition
    - mapping

    - poof methods (direct, contradiction, induction)
    - set theory
    - inequality

    usually books have an introductory chapter on these so you'll get time to prepare.

    P.S. reviewing your CALC will help ALOT in analysis. In particular go over limit (epsilon definition), series & sequences (when it converges, diverges // tests: p-test, leb test, ratio, root, comparison test, limit comparison test, abs or conditional convergence), functions (graphs, composition, limit, continuity, circular, inverse, polynomial, rational function), differential (derivative, mean value theorem, taylor theorem), integration (riemann integral, properties of integral, fundamental theorem, techniques of integration). Above mentioned stuff are literally the topics you'll be doing in your analysis course....just more religiously than you did in Calc.
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