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.
Best Answer
Galois Connections
Let's be honest, the correspondence between Galois groups and field extension is pretty hott. The first time I saw this I was duly impressed. However, about two years ago, I learned about universal covering spaces. Wow! I swear my understanding of covering spaces doubled when the prof told me that this was a "Galois correspondence for fundamental groups and covering spaces".
Again here is a link!