Let's "prove" that every complex number z
If z=0 it's obvious. For all other complex numbers z=r*e^(θi), where r is a real number, and i=√-1.
Now, z= r*e^(θi)= r*(e^(2πi))^(θ/2π). Now as we know that e^(2πi)=1 we can write z =r*(1)^(θ/2π) → z=r.
What's wrong with this?
(In reply to re: The General Problem
by Federico Kereki)
Depends on what your definition of ^(1/2) is. It is possible to define
the 1/2 power to make (-1)^1=((-1)^2)^(1/2). In fact, there are ways to
define things so that z^(a*b)=(z^a)^b for complex values of z, a, and b
as long as the definition is being followed. You cannot, however, just
define the powers however you feel like, or absurd results like the
alleged one of this problem will be obtained. The question is,
then, how does one for complex values define z^w in such a way that the
exponent law z^(a*b)=(z^a)^b holds?
Posted by Richard
on 2006-08-24 17:32:19