tag:blogger.com,1999:blog-879226427561095229.post3961540715195736316..comments2024-02-10T20:36:43.004+11:00Comments on Duae Quartunciae: New improved number system!sylashttp://www.blogger.com/profile/10594421176931832170noreply@blogger.comBlogger14125tag:blogger.com,1999:blog-879226427561095229.post-77079581332690053442007-07-29T11:20:00.000+10:002007-07-29T11:20:00.000+10:00Wikipedia has balanced ternary http://en.wikipedia...Wikipedia has balanced ternary http://en.wikipedia.org/wiki/Balanced_ternary and http://en.wikipedia.org/wiki/Negabinary "Negative numerical bases were first considered by Vittorio Grunwald in his work Giornale di Matematiche di Battaglini, published in 1885. Grunwald gave algorithms for performing addition, subtraction, multiplication, division, root extraction, divisibility tests, and radix conversion. Negative bases were later independently rediscovered by A. J. Kempner in 1936 and Z. Pawlak and A. Wakulicz in 1959.<BR/><BR/>Negabinary was first implemented in computer hardware in the experimental Polish computers SKRZAT 1 and BINEG in 1950. Implementations since then have been rare." [Some references included.] <BR/>teal4twoAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-879226427561095229.post-42557870339938961522007-06-03T00:02:00.000+10:002007-06-03T00:02:00.000+10:00Thanks, stephen. I'll try to find it. If you have ...Thanks, stephen. I'll try to find it. If you have a more complete reference, I'd appreciate it!sylashttps://www.blogger.com/profile/10594421176931832170noreply@blogger.comtag:blogger.com,1999:blog-879226427561095229.post-66876204461707466322007-06-02T23:59:00.000+10:002007-06-02T23:59:00.000+10:00A delayed comment, but you may be pleased to know ...A delayed comment, but you may be pleased to know that this system has been around for a long time. I first saw it in a New Scientist article back in the late 70's. And if you look up "balanced ternary" notation, you'll discover it is the same system but in base 3, using the digits "-1", "0" and "1". Knuth uses it a lot.<BR/><BR/>SteveUnknownhttps://www.blogger.com/profile/01761160493212012560noreply@blogger.comtag:blogger.com,1999:blog-879226427561095229.post-75417519736695988572007-05-10T00:03:00.000+10:002007-05-10T00:03:00.000+10:00Hi Duo,Could you possible send me an email on this...Hi Duo,<BR/><BR/>Could you possible send me an email on this?<BR/><BR/>I will try to follow the post as much as possible as well, but need a little more demo on the tolerance properties.<BR/><BR/>Perhaps a small additional example on the -9 to 9 tolerance as well would be insightful for me also.<BR/><BR/>Also, any ideas as to when you might be posting some insight into other operations like squaring and square-roots?<BR/><BR/>I am also interesting to try and follow how squares are generated as well.<BR/><BR/>Thanks again for a quick post followup.<BR/><BR/>Cheers,<BR/>LonnieAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-879226427561095229.post-57299010495633860452007-05-09T23:48:00.000+10:002007-05-09T23:48:00.000+10:00Hey Lonnie; welcome to my corner of the web. Thank...Hey Lonnie; welcome to my corner of the web. Thanks for the interest you've shown in this!<BR/><BR/>You have the calculations with alpha and beta and gamma and delta correct.<BR/><BR/>The extra one you ask about is gamma plus delta. The answer is:<BR/>γ + δ = α3<BR/><BR/>This is analogous to 3 + 4, except with negative digits. 3 + 4 is seven, which is ten minus three. Reverse it for the negatives, and you get negative 10 plus three, or α3.<BR/><BR/>5 + 1 is 1δ (ten minus four).<BR/><BR/>I did include ε in the multiplication table. It's not strictly necessary, but was nice to show the symmetries.<BR/><BR/>The long division allows "tolerance" as long as we can use a few non-standard digits. That is, if you guess a digit incorrectly, then you can compensate in the next digit; but only by using something non-standard; such as 6 or more, or ε or less. As niket points out, you can also do this with conventional numbers, by extending 0 to 9 with a couple of small negative digits, and introducing digits with values ten, eleven and twelve.<BR/><BR/>After you obtain an answer, you may want to remove the non-standard digits by normalizing.<BR/><BR/>You ask about 122δδ1 / 3δγ and propose 4 as the first digit. I can't show easily how it would work in comments, as I don't have html tags to line it up nicely. But here's an attempt.<BR/><BR/>Pick 4 as the first digit.<BR/><BR/>Then 4 * 3δγ = 103β<BR/><BR/>Subtract 122δ - 103β = 2αβ<BR/><BR/>Carry down the next digit, and you have a new division 2αβδ1 / 3δγ<BR/><BR/>Now the next digit has to be pretty big, and this is where tolerance comes in. We can pick a non-standard digit, large enough to cover the shortfall in our previous guess.<BR/><BR/>3 goes into 2α more than 6 times; and since the next digit is after the 3 is negative, we probably want something more than 6 even. Pick 7. Of course, you are going to need to remember the times table for any additional non-standard digits. Fortunately I do remember my seven times table, so here we go...<BR/><BR/>Multiply 3δγ by 7 to have 2β0α<BR/>Subtract 2αβδ - 2β0α = 1βγ<BR/>Carry the final digit, and then the next division step is 1βγ / 3δγ<BR/><BR/>The next digit is 3, and we have the answer now with only positive digits!<BR/><BR/>The final tableau looks like this, using underscores to help with spacing.<BR/><BR/>_________473<BR/>____--------<BR/>3δγ_)_122δδ1<BR/>______103β<BR/>______----<BR/>_______2αβδ<BR/>_______2β0α<BR/>_______----<BR/>________1βγ1<BR/>________1βγ1<BR/>________----<BR/>___________0sylashttps://www.blogger.com/profile/10594421176931832170noreply@blogger.comtag:blogger.com,1999:blog-879226427561095229.post-81038171193284461692007-05-09T22:39:00.000+10:002007-05-09T22:39:00.000+10:00Greetings All,I am working to get a good feel for ...Greetings All,<BR/><BR/>I am working to get a good feel for the "extended" Base-10 number system as I think that it exhibits some nice properties that are of great benefit.<BR/><BR/>currently, I have a feel for conversion of numbers B10 to EB10 and back, but do not have a complete feel for some of the things that happen in addition and subtractions.<BR/><BR/>For Example:<BR/><BR/>alpha + beta = gamma<BR/>alpha - beta = 1<BR/><BR/>but,<BR/><BR/>gamma - delta = 1 (????)<BR/>gamma + delta = ????<BR/><BR/>and is,<BR/><BR/>5 + 1 = alpha or ( 1 alpha )<BR/><BR/>Additionally, on the multiplication table you have included (eplison) which brings in symmetry but in your division problem, you are not using it and only going from (-4 to 5).<BR/><BR/>could you please explain a little more about this and how using additional tokens enhances the "tolerance" in such operations as division.<BR/><BR/>In the division problem presented, I can see how things progress although I am still working on understand the addition operations which should come in time, and you mentioned that finding the first digit is harder, but that we can basically ignore all but the most significant digit at a time.<BR/><BR/>I am wondering what would have happened if you tried to divide this instead:<BR/> <BR/><BR/>122δδ1 / 3δγ <BR/><BR/>and started off by choosing a 4 since 3*4 = 12,<BR/><BR/>122δδ1 / 3δγ = 4??<BR/><BR/>then would the system still be "self-correcting" as I really like that property and would like to understand it better in t his system.<BR/><BR/>Sorry for the long post but I find this system to be very exciting and think that I will be able to use it a lot in the future.<BR/><BR/>Thanks,<BR/>LonnieAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-879226427561095229.post-58992053778622467522007-05-08T22:04:00.000+10:002007-05-08T22:04:00.000+10:00Greetings All,I recently read the article from Duo...Greetings All,<BR/><BR/>I recently read the article from Duo Quartuncia about the "New Improved Number System" and find that it interesting while trying to get a good feel for some of the properties that this improvement enhances to the base-10 number system.<BR/><BR/>It would be good for you to post more simple examples of base operations to illuminate the various operation in comparrison to the base-10 systems.<BR/><BR/>From what I have read, I like what I have seen very much and it may ( or may not in the worse case) bring light to some research that I am doing in being able to look at number systems in a different way.<BR/><BR/>I would also be very interested in reading any additional papers on this subjust that may be about as well as eagerly looking forward to your up and coming post relating to higher operations like "squaring" and higher powers as well as the "roots".<BR/><BR/>Additionally, if you (Duo) or any one else on the blog has some more information (and examples) on this very interesting topic then I would really like to hear from you since I am stiull trying to get use to the system.<BR/><BR/>I can be reached at:<BR/><BR/>lonnie (at) outstep (dot) com<BR/><BR/>Great job and continued success.<BR/>LonnieAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-879226427561095229.post-10682223489884540922007-05-05T08:05:00.000+10:002007-05-05T08:05:00.000+10:00Niket, you are completely correct! The flexibility...Niket, you are completely correct! The flexibility in division is basically a tolerance of additional digits. I recognize this point in the second comment. Using digits for ten, eleven and twelve gives you the same capacity in decimal for continuing the tableau after having chosen the earlier digit one too small.sylashttps://www.blogger.com/profile/10594421176931832170noreply@blogger.comtag:blogger.com,1999:blog-879226427561095229.post-88924742439547773472007-05-05T07:36:00.000+10:002007-05-05T07:36:00.000+10:00I agree with the carry-over part making the job mu...I agree with the carry-over part making the job much easier. But you might be cheating with division. To compare apples with apples, you ought to add alpha, beta,... to the original decimal system as well. That way, even if you guessed incorrect, it would be fine because a number equivalent of 10 or 11... (or -1, -2,...) would come next... just as in your new system.<BR/><BR/>In fact, some of the "vedic math" tricks, developed close to a millenia and half ago, allows one to do something similar to compute difficult problems rather quickly.Nikethttps://www.blogger.com/profile/14882163077938014472noreply@blogger.comtag:blogger.com,1999:blog-879226427561095229.post-59047210552741053152007-05-02T18:29:00.000+10:002007-05-02T18:29:00.000+10:00Quite right on the square roots. In fact, you've a...Quite right on the square roots. In fact, you've anticipated a followup post that I have been planning.sylashttps://www.blogger.com/profile/10594421176931832170noreply@blogger.comtag:blogger.com,1999:blog-879226427561095229.post-40788996234515363952007-05-02T15:53:00.000+10:002007-05-02T15:53:00.000+10:00I haven't been seen on arc for ages either. I don...I haven't been seen on arc for ages either. I don't think they've found a new moderator.<BR/><BR/>The advantage that I can see of using the full range -9 to 9 means that the existing investment in multiplication tables isn't wasted.<BR/><BR/>I just did some doodlings, and I can report that the usual pencil-and-paper method for computing square roots is also much better in this system, thanks to the self-adjusting nature, much like with long division.Pseudonymhttps://www.blogger.com/profile/04272326070593532463noreply@blogger.comtag:blogger.com,1999:blog-879226427561095229.post-52381310928796418632007-05-02T15:52:00.000+10:002007-05-02T15:52:00.000+10:00Fascinating. I may have to show this to some of my...Fascinating. I may have to show this to some of my family; they would be quite interested.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-879226427561095229.post-13179587788613186542007-05-02T14:23:00.000+10:002007-05-02T14:23:00.000+10:00Hail and well met! It's a real pleasure to hear fr...Hail and well met! It's a real pleasure to hear from you again! I've not been on arc now for years.<BR/><BR/>I'm also delighted that someone else gets it with this number system. I've cheated a bit calling it a new number system. It works best as a convenient extension to the existing number system. In different contexts there can be advantages in manual calculation methods to using any of the digits from -9 to 9; or even beyond in principle.<BR/><BR/>As a party trick in those limited contexts where the trick is actually appreciated, you need to be able to express the numbers vebally. I just put a "b" (buh) sound at the front of any negated digit. For example:<BR/><BR/><I>"Ten b'three squared is fifty b'one". Ten b'two squared is one hundred and b'forty four"</I>sylashttps://www.blogger.com/profile/10594421176931832170noreply@blogger.comtag:blogger.com,1999:blog-879226427561095229.post-41826493149241370892007-05-02T14:04:00.000+10:002007-05-02T14:04:00.000+10:00I love it! Move over negabinary and ternary...Wha...I love it! Move over negabinary and ternary...<BR/><BR/>What I like the most is that it's easy to transform from our usual positive-digit-base-10 to negative digits and back again. I do on-paper arithmetic a lot, and I think I'll be using this now, though I'll add (-5) at least, and use positive digits for normal forms.<BR/><BR/>By the way... G'day, Chris. It's been a while.Pseudonymhttps://www.blogger.com/profile/04272326070593532463noreply@blogger.com