On Nov 23, 2007 3:53 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I've often wondered if it is somehow a mistake for mathematics to distinguish between i and -i.
There is absolutely no mathematical way to distinguish between the two. Perhaps they should only be referred to as a pair, and never one at a time?
The question is how you would manage to do this. If I'm considering some statement like "For all x, e^x is real" I would want to say it's false, and be able to give a value of x as a counterexample. But you couldn't do that unless you could refer to a single number x that is not real. Perhaps you could formalize the complex numbers in some way in which you only quantified over pairs of numbers, rather than individual numbers. But how would you make sense of an expression like "x * (y + z)" when each of {x, y, z} referred to a pair of complex conjugate numbers? It would end up referring not to pair of numbers, but to a set of up to 8 numbers, in 4 conjugate pairs. To put it another way, if you can't make statements like "A degree n polynomial has exactly n roots, counting multiplicity" in a language, then it's the wrong language for talking about the complex numbers. I don't quite see a way to make this statement in a language that only lets me talk about conjugate pairs of numbers, since the roots of a polynomial with complex coerfficients need not be in complex pairs.
Opinions?
--Dan
P.S. In a faintly related vein, if we define i^z := exp(pi*z*i/2), then iterating this function on the starting value of z = i approaches a limit L of approx. 0.6528812343931018 + i 0.3675743023883531 (or so says my C program).
Hmmm, Mathematica seems to give a rather different answer. (Can someone please recompute; my program iterated i^z on the starting value z = i forty times before the new value was within 10^(-9) of the last one, but maybe my computation was done in by roundoff error.)
In any case:
1. Is it possible that this limit L of towers of exponentiated i's can be identified as some familiar number?
2. In any case, what can be said about its number-theoretic properties? (In a sense L^(1/L) = i, so apparently Gelfond-Schneider implies L can't be an algebraic irrational.)
--Dan
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com