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Why do analysts care about complex numbers? watch

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    Why do analysts care about complex numbers? It's possible to think of lots of different operations between a couple of ordered pairs, and yet for some reason there is a great deal of focus on what happens when we define complex addition/multiplication. Surely there's some explanation other than that they are useful in algebra?

    Maybe because the complex addition/multiplication rules are the only ones which give closure of the reals (don't know if that's actually true, just guessing), which is presumably desirable since we want the fundamental theorem of algebra?

    I'm pretty certain that *any* field extension of the reals (that preserves commutativity) ends up being isomorphic to the complex numbers. So there aren't actually that many "interesting" possibilities for operations on ordered pairs.

    And of course complex differentiability turns out to be an incredibly powerful concept - far more so than real differentiability.
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Updated: February 5, 2010
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