daon
Thank you for trying to further my mathematical education, which sort of came to a screaching halt with the rational numbers. In my one-semester course on abstract algebra, we studied extending natural numbers to integers and rationals, but were stopped dead at reals with the comment that we were too uneducated (perhaps too dense) to grasp them. I have been with Bishop Berkeley all these years: the reals are a matter of faith with me.
I must admit that I am having difficulty following parts of your argument. My fault I am sure.
I have no difficulty following you that mathematicians have decided that it is useful and possible to extend the notation of exponentiation and radicals beyond their original meanings. That kind of extension has happened many times in mathematics.
Nor would it surprise me to learn that the extension means that the laws of exponents must be re-stated in such a way that they reduce to the familiar laws of exponents when restricted to non-negative reals, but that the familiar laws do not apply generally in the extended realm of discourse. I think you have implied that. Is that correct?
I am guessing that the natural log function has also been extended in conjunction with the extension of exponents. Otherwise it makes no sense to say:
\(\displaystyle x^{(1/a)} = e^{(1/a) * ln(x)}\ if\ a\ is\ an\ integer\ and\ x\ is\ any\ complex\ number\) unless ln(x) has been defined for all x. However, the statement is virtuallly self-evident for positive real x so I can see why that may be an interesting route for extension.
Am I with you so far?
I am further guessing that the extension of the natural log function takes the form of a recursive definition where \(\displaystyle \ln(x) = [\ln(|x|] + i\theta\ for\ all\ x.\)
How am I doing so far? I see what is happening with the positive reals. Theta is zero so the whole thing reduces to the familiar. I see what is happening (without saying that I could justify it or even really understand the rationale) for the negative reals: we are rotating in the complex plane to flip from the positive portion of the real line to the negative portion of the real line.
I also get that you are saying that it is a matter of convention that, when one root out of several is to be identified, the principal complex root is to be identified (although I gather from pka that the convention is not universally accepted). But in the case at hand, we have an equation to be solved. It has four solutions, two real, two complex. (I admit I completely overlooked the two complex solutions and so gave an incomplete answer.)
I say \(\displaystyle x^2 = 4\ entails\ x = \pm \sqrt{4} = \pm 2\)
I simply do not see that \(\displaystyle x^2 = 4\ entails\ x = 2\)
Moreover, if as a matter of convention, a single answer must be given, then you say it should be the principal complex root, which means both 20.032 and 19.968 are incorrect answers.
Or if a real answer is required, why is 20.032 to be preferred over 19.968?
Finally, if by the conventions of notation, \(\displaystyle Solve\ for\ z\ where\ (\dfrac{z}{4} - 5)^{(2/3)} = 0.04\) requires a single solution
how does one ask for all the solutions?