[math-fun] The Axiom of Choice for roots of z^2 + 1
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? 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
Dan Asimov wrote:
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.
You didn't tell what Mathematica gave. I hope it was approx 0.438283 + i 0.360592 which would then agree with the exact expression I give below.
(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?
Isn't it just 2 i/pi W(-i pi/2) where W denotes the principal branch of the Lambert W function? David
On Nov 23, 2007 1:53 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?
Ok, I'll show my ignorance here. How can we distinguish between 1 and -1 and not be able to distinguish between i and -i? Isn't the simple fact that they're distinct numbers (or distinct points in the complex plane) enough? What am I missing? Kerry
On Nov 23, 2007 7:25 PM, Kerry Mitchell <lkmitch@gmail.com> wrote:
Ok, I'll show my ignorance here. How can we distinguish between 1 and -1 and not be able to distinguish between i and -i? Isn't the simple fact that they're distinct numbers (or distinct points in the complex plane) enough? What am I missing?
Take any statement involving i and -i and interchange them -- the statement ought to still be true. For instance, 1/i = -i and 1/-i = i. Try the same with -1 and 1, and you'll have lots of trouble: for instance, 1*1 = 1 but -1 * -1 doesn't equal -1. --Joshua Zucker
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
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-- Andy.Latto@pobox.com
On 11/23/07, 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?
Opinions?
--Dan
"Pooh looked at his paws. He knew that one of them was the right, and he knew that when you had decided which one of them was the right, then the other one was the left, but he could never remember how to begin." [A.A.Milne (1926); http://www.everfree.ca/2007/01/pooh_on_your_shoe_kid.html] And, using elementary physics at least, there's also no way to tell the difference. But the existence of symmetry doesn't (necessarily) imply that it's a good idea to factor out the symmetry group! One situation where something analogous does routinely occur is classical projective geometry; and it's a significant nuisance when this discipline is applied to computer graphics, etc. [In practice it can be repaired by retaining the sign when normalising homogeneous coordinates --- easily overlooked in a coordinate system based on points, the only subspaces which happen not to be orientable.] Some authors have attempted to repair the omission --- see e.g. Jorge~Stolfi \sl Oriented Projective Geometry \rm Academic Press (1991); www.dcc.unicamp.br/~stolfi --- but the extra complexity doesn't engender any apparent mathematical interest, merely engineering practicality. Fred Lunnon
participants (6)
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Andy Latto -
Dan Asimov -
David W. Cantrell -
Fred lunnon -
Joshua Zucker -
Kerry Mitchell