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Imaginary numbers past calculus level

  1. Oct 25, 2005 #1
    My teacher in the charter school I go to wanted to be a mathematition. He said calculus was no problem for him and he got past vector calculus, although he cant remember because it was so long ago.

    He said he got stuck on imaginary numbers past the caluclus level and this made him quit is dream

    Is this a problem for anyone else?
    Could someone give me some examples of the problems?
  2. jcsd
  3. Oct 25, 2005 #2


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    He's probably referring to what covered in a complex analysis course (or maybe a "complex variables" course). Have a lookie:


    just a sampling of things you'd meet in such a course and a few applications.

    I wouldn't say this stuff kills would be mathematicians dreams any more than the other courses that are typical at this level, real analysis, abstract algebra, etc. Not just the topics but also the depth and rigor expected frightens some away.
  4. Oct 25, 2005 #3


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    Certainly I would not consider Complex Analysis to be as hard as Analysis or Abstract Algebra.
  5. Oct 25, 2005 #4
    *Nothing* is as hard as Abstract Algebra...
  6. Oct 25, 2005 #5
    Wait, isnt abstract algebra just algebra like in high school?
  7. Oct 25, 2005 #6


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    Not at all. Search around for the terms "group", "ring", and "field" to get a taste of the objects studied in abstract algebra.
  8. Oct 26, 2005 #7
    Wow, absolutely not!
  9. Oct 26, 2005 #8


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    Not unless you went to a heck of a high school!

    (I was tempted to answer "it's just more abstract.")
  10. Oct 26, 2005 #9


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    I'll second that one. I *HATE* that class.
  11. Nov 1, 2005 #10
    in abstract algebra, you add paintings.

    (didn't come up with that--a friend did, so all credit to her, etc.)

    i take that next semester. :frown:

    i had complex variables, and i loved it. my avg. was actually over a 100. :eek:
  12. Nov 3, 2005 #11
    QT: Penrose's book that came out earlier this year starts off with a pretty good primer on imaginary number calculations, before moving on to quantum physics.

    Give it a read, and see for yourself if "i" math is likely to give you any sticking points. Either way, (if your screen name is any indication) you can go on to enjoy the rest of the book.
  13. Nov 3, 2005 #12


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    i am puzzled that people dislike abstract algebra. my opinion is that dislike for a subject is usually due to poor presentation.

    there is a new book out by a famous mathematician, perhaps manin, on algebra that might help.

    i rather liked allan clark's little book. a fundamental concept in abstract algebra is group theory which is merely the study of symmetry.

    symmetry occurs in many subjects, certainly physics, and is very useful at simplifying them. here is an example of a problem solved instantly by symmetry and groups: a friend once challenged me to rpove that the rgassman variety parametrizing all subspaces of a given dimension in a vector space of some larger dimension formed a manifold. I remarked that it the object was symmetrical, hence since there must be a non singular point somewhere (i.e. a "manifold" point), then by symmetry every point is a manifold point, qed.

    as for complex numbers in calculus, surely more peopl have been drawn into mathematics by this beautiful topic than otherwise. to appreciate it, it suffices to have some familiarity with path integration.

    i.e. imagine the definition of log(z) for complex z. as the integral of dw/w along a path reaching from 1 to z. obviously the path may not pass through w=0, and the integral around the unit circle equals 2i pi, so the value of the integral and hence of the log, depends on the choice of the path.

    however any two paths which surround the origin the same number of times in the same direction yield the same value.

    Riemann then devised a way to cover the punctured plane with many copies of a surface, on which there exist enough different points over each point z, so that the log function can take every different value at a different point!

    to me this is the basic phenomenon that makes complex analysis interesting, the problem of "analytic continuation" or extension of analytic functions into larger and larger domains.
    Last edited: Nov 3, 2005
  14. Nov 3, 2005 #13
    Could be.

    I should point out, I haven't taken a course in it yet, I just picked up Lang's Algebra and tried to read it. That might be why I found it hard.
  15. Nov 3, 2005 #14


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    lang is a wise guy, but a brilliant teacher for some people. if you mean his graduate algebra, that is the wrong place to begin. try mike artin's algebra.
  16. Nov 3, 2005 #15
    Complex Anaylsis is a lot of fun and the basis for many of the special functions you use in physics.

    On a side note abstract algebra is just a hell of a lot of proofs and is not taught in a fun manner. To me the concepts are what made it abstract like isomorphic and ring and stuff just did not make sense when I was an undergrad.
  17. Nov 3, 2005 #16


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    i was told years ago that much of theoretical physics was essentially group representations, the representation of abstract groups into matrix groups, i.e. abstract algebra.

    an isomorphism is a representation that preserves all the structure you are interested in.
  18. Nov 3, 2005 #17
    Sadly, I can't find Artin.
  19. Nov 3, 2005 #18
    It's been ages since I studied math. I was told the difference between algebra and analysis is that in analysis you get a definition followed by a bunch of theorems. In algebra, you get a bunch of definitions followed by a theorem.
  20. Nov 4, 2005 #19
    I had a short intro into abstract algebra in High School. We talked about groups, rings and fields, with some examples and simple theorems.

    Complex Analysis is, strangly enough, easier than real analysis I've always found (and most people I know agree with this). It would seem strange that anyone would quit a career in math for this.

    That's not true. Just define a group (and maybe a cyclic group and a quotient/sub group) and you can keep going and going and going... aah, memories :-)
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