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?
(In reply to
re(2): The Specific Problem (fleshed out a bit more) by Richard)
Nothing much to worry here, we just have to get used to the fact that exponentiation a^b is not a unique operation. Even for real a and b it is so: a^b is only unique if b is an integer. If b = p/q with coprime p and q, then there are precisely q "versions" of a^b. And if b is irrational then there is an infinite (countable) number of choices. However, when we write things like Pi^Pi = 36.46... we always exclude the other (complex) choices as we are trained to consider the realvalued choice only.
I'd like to cite Fredericos posting her, because it nicely illustrates the problem for b=2 without even considering complex number:
1 = (1)^1 =* ((1)^2)^(1/2) = 1^(1/2) =# 1
Ambiguity comes into play for the equations =* and =#, the other equal signs are fine. The equation =* chooses 1^(1/2) = 1, whereas equation =# chooses 1^(1/2) = 1. Due to the different choices, we end up with different results. No surprise here. Same for the original puzzle.
Finally, if we want to keep our good old (a^b)^c = a^(bc) without worrying about ambiguity, I am afraid we have to stick with real a, b, c and positive a and the convention that a power is always a positive, real number.

Posted by JLo
on 20060828 09:59:09 