Motivating the complex numbers

- already seen
  maybe they've already seen 'em (and perhaps didn't know why to care
  then either), in which case you can just say: "so you know about
  complex numbers"

- formal
  You can just introduce them formally as
  $\bR[i]/(i^2=-1)$
  ...though for this I suggest discussing
  $\bQ[\sqrt{2}]:=\bQ[\alpha]/(\alpha^2=2)$
  first, as it's completely familiar and they -think- they understand
  it. (yes, you should do Galois theory in the very defining of this:
  note that you can switch it and its minus,
  analogous to ``you can't tell $i$ from $-i$!''
  [a clever student will say: sure, take the difference.
   Okay: $i-(-i)=2i$ -- but $(-i)-i=-2i=2(-i)$ -- so you
   really can't tell!])

- algebraic closure
  This is the obvious, and good, reason: complex numbers are
  algebraically closed

- quadratic/cubic/quartic formula
  This is the historical reason.

  The quadratic formula still makes sense even if the contents of the
  square root are negative! These were traditionally called
  ``impossible solutions'', and could be dismissed.

  However, in using the cubic and quartic formulae, even if all the
  answers are real, you still encounter complex numbers:
  you need to do manipulations with negative square roots and such!

  This feels really illegitimate, and with good reason:
  these are the kinds of calculations physicists today do, and they're
  often wrong! (and often smashingly -right-)
  (``oh, the infinities cancel, don't you worry'')

  There's something deeper going on, and you want to find out what.
  Putting in a $\sqrt{-1}$ allows you to make all these precise, but
  at the cost of doing ring theory (so that everything's consistent)!

- convergence of Taylor series
  Spivak's reason.
  Hard-core analysis reason, I swear.
  [OTOH, only -organic- way of doing it in a calc class
   w/o doing algebra. Which to me is a reason to do algebra in a calc
   class, duh. (IE, linear algebra and multivariable calc)]
  That is: $\arctan$ has no real poles, but has a finite radius of
  convergence: radius $1$ about $0$.
  What's going on???
  Is there some pole we don't see?
  
  Actually, life gets weirder: it has a radius of convergence
  $\sqrt{2}$ about $1$! More calculations show that it looks like it
  has a pole at distance $1$ ``above'' $0$.

  Once you look over the complexes, life is easy:
  it has poles at $\pm i$ (and in fact is something like
  $\log (x-i) - \log (x+i)$ or such).
