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?
The solution to this problem is that 1^(fractional power) is not in general a real number, whereas the last step in your argument assumes that we select the real root.
For example, take z=i so that r=1 and ?=?/2. We can write z=i=r*(1)^(?/2?)=1^(1/4). However, it is incorrect to infer from this that i=1 because i is also a fourth root of 1: i^4 = 1.
Your argument is correct up until the last step (? z=r ) and you have shown that z/r=1^(?/2?) i.e. z/r is always a root of unity.

Posted by Danny
on 20070106 22:29:02 