The result from antibase2 is consistent with residue. #.#: i:5 3 4 5 6 7 0 1 2 3 4 5 (2^3)| i:5 3 4 5 6 7 0 1 2 3 4 5
On Tue, Dec 13, 2011 at 7:32 AM, Dan Bron <j...@bron.us> wrote: > I'm with you on the desire for simple comparison. But I can't quite see > where the trouble arises. > > Can you give an example of the difficulty in comparison (or derivation) > which arises from the negate-the-digits approach? > > -Dan > > -----Original Message----- > From: programming-boun...@jsoftware.com > [mailto:programming-boun...@jsoftware.com] On Behalf Of Marshall Lochbaum > Sent: Monday, December 12, 2011 7:58 PM > To: Programming forum > Subject: Re: [Jprogramming] How #: should have been designed > > I suppose my main objection to multiplying the entire base by a sign is > that it makes it more complicated to assess whether a number can be derived > from your version of antibase, and thus can be compared directly with > another via -: . Specifically, your version requires that a number either > consist of all 0s and 1s or all 0s and _1s; mine just has that digits after > the first are either 0 or 1. > However, that test doesn't distinguish between, for example, _1 1 0 0 and > _1 0 0, so you would additionally have to specify that if the first digit > is negative, the second cannot have to be zero, and that's just as messy. > There may be a way around that bit, though. > > Marshall > > On Mon, Dec 12, 2011 at 7:42 PM, Dan Bron <j...@bron.us> wrote: > > > Marshall wrote: > > > (negative MSB) version is mathematically more well-founded > > > > I am personally not qualified to assess the mathematical virtues of the > > different formulations. And I see (more than) enough mathematical > > horsepower on these Forums to be confident that a sensible conclusion > will > > be reached, irrespective of my (mostly uninformed) opinions. And I'll > > certainly be comfortable with that conclusion, whatever shape it takes. > > > > Plus, your solution is already gaining support, as Henry said: > > > > > I like your leading-digit-negative idea better too. > > > > But I'm still a bit reluctant, and the reason is along the lines of > > something Henry said earlier: > > > > > you can't tell how to interpret a bit-string without > > > knowing whether it is signed or unsigned. You can't > > > tell by looking at the bits. > > > > That is, sign is an extrinsic quality of a number. Put another way, an > > integer has two qualities: sign and magnitude*. So, since the digits of > a > > number are reserved for expressing magnitude, I think sign should be > > carried > > out-of-band (not cleverly encoded in those bits). > > > > Now, in a sense, the negative-MSB approach does use out-of-band > > information. > > In particular, the "negativity" of the MSB is out-of-band information, > that > > couldn't be carried in a true digit. That said, it has two drawbacks: > > > > 1. It singles out the MSB as "special". If one argues > > "Hey, the MSB _is_ special: did you see that 'M' in > > there?", my counterargument would be "well, yes, > > but that M-ness is already encoded in the bit's > > position, which is the correct -and only- way to > > do it". > > > > 2. It introduces a new order-of-magnitude, which wasn't > > there before. That is, in order to address a value > > on order r^N, we all of a sudden had to talk about > > values on the order of r^N+1 . Feels like a > > non-sequitor. > > > > Now, (1) is a fundamental aspect of the negative-MSB approach, but if > > Henry's right when he says: > > > > > Rather than prepending a leading _1, you could just change > > > the high-order 1 to _1 . > > > > then (2) is just an aspect of a particular formulation. And actually, > the > > critical issue (3) from my earlier message goes away too (i.e. the sign > of > > a > > value is unrelated to the number of its digits). But either I > > misunderstand > > him, or I can't get it to work: > > > > #: 5 > > 1 0 1 > > #. _1 0 1 NB. negate MSB, but don't change length of vector > > _3 > > > > But anyway, even if this, or something like it, worked, it would still > > offend my formed-in-first-grade arithmetical sensibilities. It just > > doesn't > > seem right for a numeric notation to say "First, you start really low, > then > > you add values in bits and pieces, til you get closer (but never all the > > way) to zero". > > > > -Dan > > > > * Or I guess you could say a real number has a quality and a quantity. > > > > > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
