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What is the most useful math?

  1. Sep 26, 2007 #1
    I am looking to study the most "useful" math. By useful, I mean math that is important in the real world. Probability I know is very important. But is Real Analysis and Abstract Algebra really necessary to study the more "useful" applied math?

    Is Boas book good to get most of the useful math you need?
  2. jcsd
  3. Sep 27, 2007 #2
    well, if by real life you mean day to day life, or working in a non-technical field, algebra should do just fine. If you're in a technical or numerical field, the answer is entirely dependent on your area of concentration. linear algebra, calculus, and differential equations provide a good base for a lot of engineering and science though
  4. Sep 27, 2007 #3


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    linear algebra and calculus.
  5. Sep 27, 2007 #4
    Complex analysis.

    Most useful? Arithmetic? I dunno. Boas is good, but I often find myself needing to look up stuff elsewhere for more detail. It's good in that it covers the basics of many techniques that are useful in physics, but I doubt it's all the math you'll ever need in the field.
  6. Sep 27, 2007 #5


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    Pick up an engineering maths book -- you'll find everything from calculus to linear algebra (as mathwonk suggested), through to differential equations and laplace/fourier transforms... without getting bogged down by proofs.
  7. Sep 27, 2007 #6


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    I would recommend the Boas text without hesitation. It will at least give you a flavor of the mathematics you might need, and then you can pursue it more with more detailed text if you so desire.

  8. Sep 27, 2007 #7


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    what boas book is being referred to? the classic carus monograph by ralph boas, primer of real functions?

    ok im guessing its mary boas' math for physics book. that does look comprehensive on the useful stuff.
    Last edited: Sep 27, 2007
  9. Sep 27, 2007 #8
    Boas - Mathematical Methods in the Physical Sciences. ISBN 0-471-19826-9
  10. Sep 27, 2007 #9


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  11. Sep 27, 2007 #10
    I have found Fourier series and differential equations particularly useful as a physics undergraduate. However those probably fall under Calculus.
  12. Sep 27, 2007 #11


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    the point is that linear problems are the most accessible, and differential calculus is the art of approximating non linear problems by linear ones.

    power series are also useful as a method of calculating approximations.
  13. Sep 27, 2007 #12
    You'd be surprised just how relevant multivariable calculus is to both micro and macro-economics. In finance, probability and calculus are very important, as well as linear algebra (at points).
  14. Sep 27, 2007 #13
  15. Sep 27, 2007 #14

    D H

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    Useful does not mean interesting or worthy of a math paper. What could be more useful than good old geometry and basic algebra? Much of the math used in the construction is little more than applied geometry. Much of the math used in accounting is little more than applied algebra.
  16. Sep 27, 2007 #15
    Trig, pythagorean theorem, and arithmetic.
  17. Sep 27, 2007 #16


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    I'm surprised no one's mentioned statistics yet!
  18. Sep 27, 2007 #17
    Pattern Recognition!!
    How can human live without recognising pattern? Our science is built upon some sort of patterns or order (jokingly)

    Honestly, if you are scientist, knowing ODE and Linear Algebra should get you quite far.
  19. Sep 27, 2007 #18
    Yeah, imo pattern recognition and stats are probably the most useful. Basically every field has stats and error analysis.
  20. Sep 28, 2007 #19

    Chris Hillman

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    Yes, you need graduate real/complex analysis plus algebra

    You mean the textbook on mathematical methods by Mary L. Boas? If so, I happen to think that is a very good book which should provide a solid background for junior year undergraduate mathematics.

    To answer your question, we might need more information about what your goals are.

    Naturally, I won't let lack of information stop me from trying to answer your question! Let's consider the prequisites for a handful of the most useful areas of "applicable mathematics" (applicable to physics, chemistry, biology, engineering, economics, you name it).

    Nothing in mathematics has proven more useful than information theory (see https://www.physicsforums.com/showthread.php?t=183900), and Shannon's information theory rests firmly upon ergodic theory, which rests upon probability theory, which rests upon measure theory, one of the most important topics in a good senior year or first year graduate analysis course. (Note too the prominent role played by group actions in my survey--- this is a topic in a good senior year or first year graduate course in algebra.) Very few things in mathematics have proven more useful than representation theory (see my post # 4 in https://www.physicsforums.com/showthread.php?t=185965), and this rests upon Lie theory, which rests upon graduate level analysis, algebra, and the theory of manifolds. And nothing in mathematics has proven more useful than the theory of differential equations, which --- you probably saw this coming!--- really requires first year graduate analysis (e.g. to understand integral transforms, operators on function spaces, harmonic analysis--- which is closely related to representation theory and even information theory, by the way). Most people who try to use DEs in model building encounter the problem of trying to solve nonlinear PDEs; here, the only really general tool is the symmetry analysis of the equation, which requires Lie theory.

    One huge area which is barely hinted at in Boas is combinatorics and graph theory. Indeed, the most important theorem in mathematics, the Szemeredi lemma, involves ergodic theory and graph theory. See http://www.arxiv.org/abs/math/0702396 (note the application of Szemeredi extends even to number theory) and then one of the best books published to date, Bollobas, Modern Graph Theory (which has almost no prerequisites).

    Ditto the others about statistics. See my post # 5 in https://www.physicsforums.com/showthread.php?p=1416394 and David Salsburg, The Lady Tasting Tea. I should point out that mathematical statistics is closely related to Shannon's information theory, for example via the Principle of Maximal Entropy. See http://www.math.uni-hamburg.de/home/gunesch/Entropy/stat.html And--- again, you probably saw this coming!--- statistics rests upon probability theory, and proper understanding of topics such as moments requires analysis, while proper understanding of factor analysis (a method for "lying with statistics") requires finite dimensional euclidean geometry and finite dimensional linear algebra, so you might as well go the limit and study Hilbert spaces for preparation.

    Pattern recognition can be viewed as part of information theory, but a relatively minor thread in the grand tapestry of applied mathematics. Kleinian geometry or exterior calculus, or computational algebraic geometry, or any items from a long list of further topics, would be far more important in the grand scheme of things.

    Summing up: I have discussed prerequisites for five of the most useful areas of "applicable mathematics": information theory, representation theory, differential equations, combinatorics/graph theory, and statistics. Mastering the most important techniques and results in any of these areas requires graduate level analysis (real and complex) and graduate level algebra. So plan on taking these!

    I don't think you need to study philosophy, unless you can find a good course on the philosophy of statistics which gets well into the titantic struggle between "frequentists" and "Bayesians". IMO, philosophy is useful, even essential, wherever mathematics meets the real world, but traditional undergraduate courses on the philosophy of mathematics are utterly useless to prepare you for any real-world philosophical conundrums you would be likely to encounter in the 21st century. IMO, philosophers are doing society and their own discipline a great disservice by failing (for the most part) to engage contempary mathematics. The reason is probably that philosophical analysis of modern mathematics requires a Ph.D. in mathematics as well as background in philosophy, so IMO almost no credible philosophy of modern mathematics or statistics yet exists. I have nominated this as perhaps the important challenge to scholars in the new century.
    Last edited: Sep 28, 2007
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