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I just learned (thanks to Harry Gindi's answer on MO and to Qiaochu Yuan's blog post on AoPS) that the chinese remainder theorem and Lagrange interpolation are really just two instances of the same thing. Similarly the method of partial fractions can be applied to rationals rather than polynomials. I find that seeing a method applied in different contexts, or just learning a connection that wasn't apparent helps me appreciate a deeper understanding of both.

So I ask, can you help me find more examples of this? Especially ones which you personally found inspiring.

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    Community wiki?2010-08-01
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    If you want to see a neat approach to the chinese remainder theorem check out page 14 of http://math.stanford.edu/~vakil/0708-216/216class0708.pdf2010-08-01
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    I removed the abstract algebra tag because I think it implies you are restricting answers, which I don't think you want to do.2010-08-01
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    @Akhil (not that I don't like gaining rep but) maybe you should edit it to make CW?..2010-08-01
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    @Grigory M: We can't edit to make it CW, the original poster has to (or maybe a ♦mod).2010-08-01
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    I think this question is too broad for a site like this. It is one step away from: "Tell me some interesting piece of mathematics."2010-08-02
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    How do I make it community wiki?2010-08-02
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    @muad: Edit the question and check the "Community Wiki" checkbox that's below and to the right of the question-editing box.2010-08-02

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