"Robert Bonomi" wrote in message
communications...
Yes, you assumed that the function e^x is 1-to-1. And it is definitely
not.

Well, It _is_, for exponents without an imaginary component. The concept
of multiplying something by itself an imaginary number of times is utterly
lacking in any intuitive foundation.

Yes, think of expressions like 2^i as exp^(i *ln 2), then you can use your
intuition
a little more once you are adequately aquainted with the function e^x.



Once it was clarified that '2*pi' was angular measure, things clarified
It is true that 2*pi == 0 (plus 1 revolution), even though neither of
the multiplicands is zero.

What you REALLY mean is that: 2*pi radians ~ 0 radians!

0 and 2*Pi are complex numbers, different ones, which are over 6 units
apart in the complex plane. We save "=" for when the numbers coincide.


Quoting a line from an old comic strip -- where they got it absolutely
wrong (unintentionally!) in context: "How do you tell a pilot he's 360
degrees off-course?"

Again, The terminal side of angle of 0 degrees is the same of one of 360
degrees.
But, turning a gear or the steering wheel of your car 360 degrees or -360
degree or 0 degree
are 3 different things, right? I agree it probably doesn't matter how a
pilot turns west, if
west is the direction he is supposed to go! : )

Best,
Bill