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I Can You Define Pure vs. Applied Math?

  1. May 17, 2016 #1
    I know in Calculus there is a pure vs. applied calc. distinction. I took Applied Calculus last year, but not the pure form. I've been checking out my university's math catalog and they bring up an applied math vs. pure math track.

    I can see that applied math is math that is used in specific settings (chemistry, business, biology, etc.), but I'm still a little uncertain about what is meant by pure math.

    Can someone explain the distinction.

    Also, does the applied vs. pure distinction only occur when you get to Calculus? What about arithmetic, algebra, pre-calc/trig., geometry, etc. that we learned in high school? Is all the stuff prior to calculus neither applied, nor pure?

  2. jcsd
  3. May 17, 2016 #2
    Hi bbwb:

    I think different mathematicians will have different answers. Here is my take.

    Pure math is mostly about proving theorems. Examples of areas that are mostly pure math are topology and number theory.

    Applied math is mostly about solving practical problems (typically in science and engineering) using mathematical methods. Examples of areas that are mostly applied math are differential equations and linear algebra.

    Hope this helps.

  4. May 17, 2016 #3


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    In my experiences this is a typical German distinction. Meanwhile I regard it as a kind of sickness. There have been real camps at my university and each member of one camp used to look down on the members of the other one. The pure mathematicians were sick to hear the question "what for" which they usually hadn't a satisfactory answer to and the applied mathematicians were sick of being seen as "calculators".
    PF as a rather good example which shows that this distinction is artificial. To do serious physics you certainly will need knowledge of both.
  5. May 18, 2016 #4
    This is an awesome answer. Sadly, I don't think it is a typical German distinction, it seems to be everywhere. I do agree the distinction is completely artificial. I think that any pure mathematician should have an idea of how pure mathematics is applied in a concrete situation. For example, even a basic knowledge of quantum mechanics reveals a lot about an otherwise pure topic like functional analysis. I believe the converse is true as well.

    The work of the pure and the applied mathematician is both interesting and should both be respected. The only distinction is that the pure doesn't care about applications, while the applied always has some application in mind. This does not mean that the pure should not know how his work might be applied. Similarly, the applied mathematician should be aware of any pure advances in his field.
  6. May 18, 2016 #5


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    In some respects, the distinction between "applied math" and "pure math" is artificial, with some disciplines starting out as pure mathematics, but becoming applied at a later time. A classic example of this comes from G. H. Hardy's book, "A Mathematician's Apology."

    From the wiki page, https://en.wikipedia.org/wiki/A_Mathematician's_Apology:
  7. May 18, 2016 #6


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    In addition to the emphasis on proofs, pure math will head toward abstraction. That allows theorems and proofs to concentrate on how much can be proven with the fewest assumptions, trying to get to the root cause of fundamental truths. But it leads them away from specific practical applications. A pure mathematician might be happy if he could prove that a numerical algorithm gave an answer in 100 million function evaluations, but an applied mathematician would not be happy.
  8. May 19, 2016 #7


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    Pure math is applied math for the math :oldlaugh:
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