Let's "prove" that every complex number
z is real.
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?
...exponentiation of complex number x and y is not unique. It is defined as x^y:=e^ln(x)y. If the natural logarithm ln was welldefined, this expression would be welldefined because exponentiation with real basis (in this case e) can be defined via the wellknown taylor expansion of the exp function f(x)=e^x which works for every complex x.
The natural logarithm unfortunately is not uniquely defined because ln(y) is basically defined as the solution of e^x=y, which has in fact an infinite number of solutions. This BTW is also the reason why there are two square roots of every complex number except 0.

Posted by JLo
on 20060824 19:24:54 