🚀 go-pugleaf

There has been a lot of discussion recently in the symbolic math
newsgroup on how to simplify surds. I want to make another point in
the context of REDUCE.

I have been looking for some time into the question of complex
arithmetic and found something which I think is an inconsistency in
the current model. To demonstrate, consider the expression

(-27)**(1/3)

How should this be simplified?

In a real domain, -3 is a reasonable answer. In a complex domain,
however, I argue that it is not, for the following reason:

In a complex domain, there are three possible answers, corresponding
to the three roots of unity:

-3

3 * (1 + sqrt(3)*I)/2

3 * (1 - sqrt(3)*I)/2

The conventional definition of the exponential (see, for example,
Abramowitz/Stegun) uses the second of these as the principal branch.

The same applies to all surds of the form

(negative real number) ** (1/(odd integer))

One might try to argue that one can always choose the principal branch
of these surds in a way so that, e.g., (-27)**(1/3) can be simplified
to (-3), even in a complex domain. Unfortunately, this is not
possible: this choice would result in 

(-1)**(1/(2n+1)), n= 1,2,....

to tend to -1 as n goes to infinity, whereas continuity of ** requires
that it tends to 1 as n goes to infinity.

Therefore I propose that REDUCE should in future simplify surds like
(-27)**(1/3) to (-3) only if complex is switched off.

If complex is switched on, REDUCE should select the principal branch, e.g.,

(-27)**(1/3) 

should be simplified to 

3 * (1 + sqrt(3)*I)/2

and so on.

I would appreciate comments on this proposal.

   Rainer Schoepf
   Konrad-Zuse-Zentrum 
    fuer Informationstechnik Berlin
   Heilbronner Strasse 10    
   1000 Berlin 31
   Federal Republic of Germany
   <Schoepf@sc.ZIB-Berlin.de> or <Schoepf@sc.ZIB-Berlin.dbp.de>