[Jprogramming] How #: should have been designed

[Dan Bron]

Maybe we should explore a different track.  Forget all about the history of
computing hardware, and the advantages of one representation over another
when designing circuits.  In purely mathematical terms, what are the decimal
digits of the base-10 number -123 ?  

Working backwards:

   _123                                        NB.  begin
   -123                                        NB.  notation->operation
   _1 * 123                                    NB.  definition of -
   _1 * (1 * 10^2) + (2 * 10^1) + ( 3 * 10^0)  NB.  positional number
notation
   (_1 * 10^2) + (_2 * 10^1) + (_3 * 10^0)     NB.  distribution of * over +
   _1 _2 _3 +/ . * 10^2 1 0                    NB.  array simplification
   10 #. _1 _2 _3                              NB.  definition of #.

Thus, to satisfy the purposes of inversion, we should have: 
[all examples from here on are purely theoretical, and the 
results differ in the extant implementations of J]:

   10 #.^:_1: _123
_1 _2 _3   

And, analogously in base 2:

   2 #.^:_1: 2b_101                            NB.  2b_101 is J's notation
for -101 in base 2
_1 0 _1		
   #: 2b_101                                   NB.  #: <=> 2&#.^:_1:
_1 0 _1		
   #: _5                                       NB.  _5 <=> 2b_101 (numbers
are analytic)
_1 0 _1		

QED.* 

-Dan

*  Backwards compatibility notwithstanding

PS:  What implications does this have for modulus, as in _10 | 123 ?