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Hardest upper undergraduate pure maths subjects?

  1. Jun 27, 2007 #1
    Which undergraduate pure maths subject do people find the most challenging?

    I thought I might just list the three major areas, analysis, algebra, topology.
  2. jcsd
  3. Jun 27, 2007 #2
    I don't have an answer, but I'm interested to see people's answers as well...
  4. Jun 27, 2007 #3
    In my classes, some how, the physicists always say the one they're in is hardest :smile:

    Honestly though, a lot of factors come into play. I would say, who's taught all the courses up until those counts big time, and more importantly your interests.

    For myself analysis has been hardest, and that's the exact opposite of what is common at my school. Most people as undergraduates have a harder time with their first algebra. Algebra comes quick to me - mostly because I love using that type of thinking.
  5. Jun 27, 2007 #4


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    The hardest one is probably the one in which you got the lowest grade.

    I got A's in each of them, so I don't really know. I would say that Analysis felt like the hardest.
  6. Jun 27, 2007 #5
    Topology felt hardest, but that's probably because I actually took classes in Analysis and Algebra (got A's) but tried to learn topology on my own (still not quite sure I've even got the fundamentals down!)
  7. Jun 27, 2007 #6
    Quality of teaching is definitely a big factor in how hard a class seems. Teaching aside though, I think analysis was the one that clicked the least with me. Each subject has a different way that you have to approach the proofs, a different set of skills, a unique mindset. For me, getting the hang of epsilon-delta proofs and all that jazz took the longest. But then again, it was the first of those 3 that I took, so maybe I just gained more confidence after that.
  8. Jun 27, 2007 #7
    Definitely analysis. It's hard to get used to that kind of thinking.

    Slowly but surely, all the topics from that class are starting to make sense. Especially as I've seen them applied in other fields, like statistics.
  9. Jun 27, 2007 #8
    Real Analysis as in Measure Theory is probably the most difficult undergrad math course, on average.
  10. Jun 27, 2007 #9


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    I have to say the hardest part of my undergraduate course was a module on Differential Geometry. I'm not sure which particular category that fits into though.
  11. Jun 27, 2007 #10
    Logic. nothing will test your abstraction ability more than logic, hands down.
  12. Jun 27, 2007 #11
    I was thinking 3 year level maths subjects.

    In my uni they teach differential geometry at honours or 4th year level. They also have algebraic topology at that level as well. Would you call the latter algebra or topology? I have heard that that course was the hardest in his studies by one academic.

    As for me, analysis is hard also. I hate it when I think about it geometrically which means things get smaller and tighter at the exact places where the interesting maths is. The notion of infinity arises in analysis so you somehow have to fit that into your picture. So it's hard to picture it. I think my geometic approach may not be the best. That is why they use epsilon and delta.

    I haven't done proper topology yet. From what I have seen, it is mainly set theory which can get you a bit if notions of infinity arises.

    Algebra seems clearest. I feel it is the best maths subject to learn deductive reasoning. Once you have it then things become easier. Also learning the definitions seem hard at first as they may be less intuitive than definitions in analysis and topology where you can use pictures to think about things more.
    Last edited: Jun 27, 2007
  13. Jun 27, 2007 #12


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    From what I have learned from it so far, I'd call it topology since it deals with topological issues and not algebra. Algebra is just used to solve problems.
  14. Jun 28, 2007 #13
    Looks like not much votes from our resident maths experts at PF. You could also intepret the question as which is the least easiest if you found them all easy.
  15. Jun 28, 2007 #14


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    i say analysis, but it may be rudins fault, since the analysis profs mostly use his book, and it is one of the least user friendly of all maths books. i.e. analysts are among the few remaining profs who take pride in presenting the material in a way that pleases themselves and not the student.

    but i think pivoxa expressed my intrinsic reason perfectly about the subject getting smaller and tighter exactly where the action is. after years of practice, it seems the soloution is to stop short, accept that fact, and just be content that for every epsilon, all the rest of the action takes place inside an epsilon disc one has no need to imagine the center of.

    for me topology was eaisest, algebra next, and analysis hardest. somehow algebraic topology is considered hard too, but the differential version of topology is much easier. to get into algebraic or differential topology without tears, anything by andrew wallace is recommended.

    so again maybe it depends on the presentation. milnor is also great. for analysis i like dieudonne, even though he tries purposely to make it harder by drawing absolutely no pictures. but maybe his intent was to force the reader to draw them.

    riesz nagy is also excellent but old fashioned. but then i have given all these recommendations in my mathematician thread, which of course is now too long for anyone to read.
  16. Jun 28, 2007 #15
    The hardest math class I took was combinatorics/graph theory.
  17. Jun 28, 2007 #16
    So you took classes in all three areas in the option but still found combinatorics/graph the hardest?
  18. Jun 28, 2007 #17
    What do you think of Spivak's Calculus as a book to learn analysis from? It looks basic enough but it is a bit wordy I found. However the other extreme might be Rudin's book which is not wordy enough? Is that its major problem? Or does it skip too much stuff? I probably need one that has minimal words but lays out the maths completely and fully without any abbreviations. I wonder if such a book exists?

    Even though a lot of people think analysis is difficult. I tend to get the picture that it is a 'lower order' subject compared to algebra and topology. The latter two seems more prestigous than analysis? Is analysis less abstract than the latter two?
    Last edited: Jun 28, 2007
  19. Jun 28, 2007 #18
    Yes and yes. I didn't like the subject and I think that's why I had a hard time.
  20. Jun 28, 2007 #19
    I think Spivak's Calculus is an amazing book - but I don't like the Bourbaki method that Rudin sticks to. I think, though, that Rudin would be great if you've already read Spivak's for the problems he has. I think, the "beauty" of the terse bourbaki method is mostly due to the fact you're suppose to fill in all the gaps and do all the problems.

    A more modern book is Bartle's. Though, I would save some money, and go for an older 2nd edition, b/c IMO it's written better.

    Do yourself a favor, throwout that word 'prestige'. Just chuck it, forget it even exists. It's a silly game, played by insecure people. Study what you love and never listen to the "prestige".

    And though, I may never be an analyst, Terrance Tao is one. And analysis is one of the oldest most developed forms of modern mathematics. It's impossible to prove which is used the most, but it appears that analysis may be the most useful. So I say it deserves the same respect as any math, whether that be Stats or Category Theory or Mathematical Physics.

    Down with this hampering by prestige. What do you like?

    edit: all math is abstract if you take it far enough
    Last edited: Jun 28, 2007
  21. Jun 28, 2007 #20


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    Although Spivak is a great book, it is definitely NOT suitable as an analysis text. For example, it contains no metric space theory. You're better off getting a real analysis text (pun not intended :smile:). I second mathwonk's remarks about Rudin; it's only good if you've seen all the material before, and want a slick reference. Some of my favorite analysis texts are: Intro to Topology and Modern Analysis (Simmons), Elements of Real Analysis (Bartle) [note: there is also Intro to Real Analysis by Barte; this is not the same book], Elements of Integration (Bartle), and Functional Analysis: An Intro to Banach Space Theory (Morrison).

    And no: analysis is definitely not less abstract than algebra or topology. Actually, the subjects get intertwined down the road, e.g. Banach algebras are algebras that are complete normed vector spaces (i.e. Banach spaces), we can define a measure (the Haar measure) on the Borel algebra generated by the compact subsets of a locally compact topological group, etc.

    As for me, the two hardest math courses I've taken were probability and graph theory.
    Last edited: Jun 28, 2007
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