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I'm writing a report on the philosophy of pure mathematical knowledge, and how pure mathematics isn't what people commonly consider it, as non-applied. I want to see if anyone has an example of, or can make up a simple mathematical proof/theorem/example, in the realm of pure mathematics, that actually has an application, although one that cannot be seen a first sight.

I want this to be a brief thing that I can add in the report and explain,

I appreciate your help,

hariq
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    Such as number-theory and its applications in cryptogaphy? – Hagen von Eitzen Sep 24 '15 at 19:27
  • You may want to check out https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem, particularly the "illustrations" part. I believe the Brouwer Fixed Point theorem also has applications in economics, but I don't know enough about that to really say. – Ben Sheller Sep 24 '15 at 19:40
  • Cyclic redundancy checks are a nice example of some concepts from abstract algebra with a surprising real world application - well let's treat the data bits as the coefficients of a polynomial ... – Rob Arthan Sep 24 '15 at 19:46
  • I recall an application of the Brouwer Fixed Point theorem in an explanation of a cause of heart fibrillation,There was a Scientific American article a very long time ago.I'm sorry I can't recall more – DanielWainfleet Sep 24 '15 at 22:42
  • @HagenvonEitzen Sure, it could be anything in pure mathematics. I simply want to demonstrate a small theorem or something that has applicability, to make a bigger point on the essay that I am doing. This shouldn't be something overly complicated, but significant. I really appreciate your help. – hariq Sep 25 '15 at 01:55
  • Interesting comments, I appreciate all of your help. I really actually want to see something that is like for example, a small simple equation for finding all the even numbers up to any certain number, and then explain how this theorem (I want to show a small theorem) can be used in cryptography or to generate astronmical patters. (I know this is an absurd example, I couldnt think of anything else) All I want is a small mathematical theorem that can easily be made up. Thanks again. – hariq Sep 25 '15 at 01:55
  • If graph theory is enough pure for you, there are multiple theorems that are indirectly used in practice. For example in the USA the National Resident Matching Program uses matching theory results to assign medical school students to hospitals—although the current algorithm isn't pure anymore, it is still based on stable matchings and related results. – dtldarek Sep 25 '15 at 09:34
  • Also, if I were to name one concept that exemplifies how much pure math is useful (besides obvious things like numbers, etc.), it would be the Fourier transform (not cryptography, because its impact gets diluted between all the small results involved). – dtldarek Sep 25 '15 at 09:41
  • @dtldarek This is quite interesting. I guess I'm not rephrasing my question too well. I really want to add something like this (please note that this proof makes no sense whatsoever probably - but I'm just putting it down to make a point, not being a mathematician) So, say x+y+z is a pure mathematical formula that allows us to estimate or find any even number. Now, I see a small proof, whatever that would be, and then I state how this can be used in information security. I don't necessarily want anything big. I just want to see a small formula with a proof and go from there. – hariq Sep 25 '15 at 16:08
  • @dtldarek Thanks for your kind and great help though!!! – hariq Sep 25 '15 at 16:11
  • If you are fixed on even/odd numbers: in many cities the houses/buildings are numbered with even numbers on one side of the street and odd on the other, in such case $2n$ and $2n+1$ are the formulas for these numbers. – dtldarek Sep 25 '15 at 19:39
  • @dtldarek Can you firstly explain this a bit more? Additionally, does this constitute pure math? In my paper, I want to say: "Pure Mathematics once thought useless, indeed have a strong element of applicability, though it may not be realized initially. For example, cities the houses/buildings are numbered with even numbers on one side of the street and odd on the other. I want to explain what you suggested, but do you think this would fit my paper. Finally, can you explain this a bit more? Thanks! – hariq Oct 23 '15 at 13:55
  • @hariq Unfortunately, any example that I know of at the level of simplicity you require is trivial, and wouldn't be taken seriously. The even/odd numbers can be thought of as "pure math" in the context of group $\mathbb{Z}_2$, but this not even a tip of an iceberg. One of the reasons why people considered pure math useless, is that the useful bits are often a bit more complicated and we needed time to discover their applications. – dtldarek Oct 23 '15 at 21:35
  • On the other hand, there are a few places where you can see a glimpse of pure math, like the matching theory I mentioned, or group testing which uses coding theory. If you are not familiar, the latter is, in my opinion, quite illuminating. – dtldarek Oct 23 '15 at 21:41

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