3 ms·

I love analytic/holomorphic functions, they're so smooth and beautiful.

• lmkg 13y ago
  The more I studied holomorphic functions, the more I felt that the real numbers were fundamentally incomplete without the imaginary numbers to accompany them. In just the reals, there are tons of ways for functions to misbehave and become irritating to manage, and there's no real rhyme or reason to explain it. In the complex numbers, the 'true' structure of a function is revealed: well-behaved functions are well-behaved everywhere and in all respects, and if a function misbehaves it's clear where and how.
  • evanb 13y ago
    Some people use similar reasoning to "explain" why quantum mechanics uses complex numbers. http://www.scottaaronson.com/democritus/lec9.html http://www.scottaaronson.com/democritus/lec9.html (scroll down to "Real vs. Complex Numbers", about 2/3rds of the way through the page)
    • jjoonathan 13y ago
      Awesome, I was about to go dig up the link but you saved me the trouble. I found some relevant discussion on stackexchange, but I haven't digested it thoroughly enough to have an opinion yet. http://physics.stackexchange.com/questions/32422/qm-without-complex-numbers http://physics.stackexchange.com/questions/32422/qm-without-...
    • nileshtrivedi 13y ago
      This was an amazing read. I love explanations like this which are based on pure concepts that require just a little bit of bending the mind, but then explain so much more.
  • jjoonathan 13y ago
    I got the same feeling learning linear algebra from Axler. He always introduced proofs using complex numbers and only subsequently dealt with the case of real numbers. The real-number proofs invariably devolved into a rat's nest of edge cases that, if one looked closely, accomplished their goals only by inelegantly re-inventing the concept of complex numbers. Admittedly some of the differences in elegance came directly from the fundamental theorem of algebra, but not all of them.
  • deleted 13y ago
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