tag:blogger.com,1999:blog-20046017.post5021472841440942280..comments2011-04-25T03:41:07.527-05:00burton mackenziehttp://www.blogger.com/profile/13139594737135732882noreply@blogger.comBlogger12125tag:blogger.com,1999:blog-20046017.post-24224369523425663662010-09-14T18:48:32.150-05:002010-09-14T18:48:32.150-05:00@Scott, good question.<br /><br />In base golden ratio (1.618...), the digits are 0 and 1 because there can be no two.<br /><br />In base e (2.718...), you could have a 0, 1, and a 2. <br /><br />It is hard to conceive of irrational digits, I must admit! :)burton mackenziehttp://www.blogger.com/profile/13139594737135732882noreply@blogger.comtag:blogger.com,1999:blog-20046017.post-7775307916745027882010-02-05T09:54:34.089-06:002010-02-05T09:54:34.089-06:00In base 2, there are 2 digits, 0 and 1. In base 10, there are 10 digits, 0-9. In base e, there are ... e digits? What does a fractional digit look like? How do you represent any numbers?Scotthttp://www.blogger.com/profile/09696813570058056053noreply@blogger.comtag:blogger.com,1999:blog-20046017.post-47612209891167365272009-07-16T01:41:29.803-05:002009-07-16T01:41:29.803-05:00I never thought about determining the efficiency of a base in this way. I&#39;ll have to give it some thought.<br /><br />Thus far, my thoughts on efficiency have been to consider cognitive utility. To begin with, I&#39;m interested in the relevance of &quot;the crow&quot; (a term in philosophy pertaining to the maximum number of units/concretes the human mind can simultaneously hold, which serves as the basis for a great deal of conclusions in epistemology), which for most people is 7 plus or minus 2. This already leads me to believe that cognitively, 7 is an efficient base.<br /><br />I think it&#39;s also relevant that 7 is a prime number. I suspect (though I haven&#39;t worked anything out) that you would be able to do all sorts of neat tricks with a prime base, relying on a lot of the facts that make the integers mod a prime number a field.<br /><br />What are your thoughts on this?Arthurhttp://www.blogger.com/profile/06659782577361297120noreply@blogger.comtag:blogger.com,1999:blog-20046017.post-29301985736277985712009-06-28T10:56:19.935-05:002009-06-28T10:56:19.935-05:00You might have originally seen this here: http://www.americanscientist.org/issues/id.3268,y.0,no.,content.true,page.6,css.print/issue.aspx<br /> I&#39;ve always loved the concept.chttp://www.blogger.com/profile/12084323599276620563noreply@blogger.comtag:blogger.com,1999:blog-20046017.post-47225995166247311322009-04-14T00:46:00.000-05:002009-04-14T00:46:00.000-05:00Interesting idea, though I think the metric you are using is maybe a little bit simplistic. It might be good for computing applications where you are representing numbers as arrays for example. But if you are thinking more organically, perhaps about a time when we still wrote numbers on paper from time to time, I think a different metric would be preferable. One idea is essentially the number of pen strokes required to write a number. For example the number 10 is fairly simple to write in our base 10 system -- 1 straight line and 1 circle. If we were to write it in binary, we would have 1010 which requires twice as much work to write. Of course it is less obvious how one can decide how to optimally design written numbers. Certainly if we went to higher base systems, we would have much more complicated symbols. I think the number of symbols required to represent a given base would probably not be simply proportional to the size of the base (if it were your metric would be essentially sufficient). Maybe base 10 turns out to be a nice compromise where the symbols are still quite basic but we don't require too many digits per number.Jeff Qhttp://www.blogger.com/profile/04136138962106192173noreply@blogger.comtag:blogger.com,1999:blog-20046017.post-54651215974973046142008-12-19T01:34:00.000-06:002008-12-19T01:34:00.000-06:00Thanks, Brian. Lately I've been interested in hardware implementations in base golden ratio. It has some interesting characteristics for power operations, but right now it looks like the translation to and from would diminish any computational advantage from using it. <BR/><BR/>I'd be happy to hear back on any hardware base-e implementations you muse on/design/build!burton mackenziehttp://www.blogger.com/profile/13139594737135732882noreply@blogger.comtag:blogger.com,1999:blog-20046017.post-59164239468157597082008-12-12T18:24:00.000-06:002008-12-12T18:24:00.000-06:00All this great work you're doing and I've been casually mouthing off for years that 12 is the better base. I almost joined a dozenal society (I think they called it). Ha!<BR/><BR/>For social, human use I'm sticking with lecturing about 12 to anyone who will listen, for its everyday computational value. With its factor edge over 10 it would be handy in a market, for example. Speaking of handy, there is also, still an organic aid with three parts on each of our four fingers and a thumb to point for counting, making 12 even handier than using both to count 10. There are also not too many digits in common values, as there would be in base 60, balancing simplicity in common fractional math. There is a more apparent symmetry in some basic calculations useful in math education, for 12 more than 10.<BR/><BR/>For machine use though, also in the context of Joe's comment, I want to look into e. 2, 10, 16, or otherwise for machine calculations ought to be less efficient than base e in some cases, and machines are notably less confused with its use or less prejudiced than people ;) Should I find any practical implications, I will be sure to recall your insight. BGBrian Glanzhttp://www.blogger.com/profile/06971961344708864523noreply@blogger.comtag:blogger.com,1999:blog-20046017.post-81417240286962908992008-01-30T16:34:00.000-06:002008-01-30T16:34:00.000-06:00Great comment Joe. This metric for which is the optimum base is only that - a metric devised by humans to measure a specific trait of numeric representation. (i.e. optimal space consumed by digits*characters in a given base)<BR/><BR/>This metric has been around for over a hundred years (and I'm still looking for my original reference), but different needs would demonstrate different optimum bases for those specific needs. For instance, the Babylonians used base 60, which may be functionally optimal for math that doesn't use digits to the right of the radix point, but heavily uses fractions. In base 60, it is trivially easy to represent and work with the fractions 1/2, 1/3, 1/4, 1/5, 1/6, 1/10, 1/15, 1/20, 1/30, and 1/60. If that's what's important to you, base 60 is much more optimal.<BR/><BR/>Ternary is great for many things, but one of my favourites is for intuitively showing that Cantor's dust exists. Balanced ternary (-1, 0, +1) is wonderful for other applications.<BR/><BR/>I know you didn't mean complex numbers in your last paragraph, but Knuth explored using a complex base as well. In it, all positive and negative numbers are represented without using a sign. (not to mention the complex ones)<BR/><BR/>Anyway, as I'm sure you know, "optimal" is defined by what you need it to do. :-) Thanks for the comment.burton mackenziehttp://www.blogger.com/profile/13139594737135732882noreply@blogger.comtag:blogger.com,1999:blog-20046017.post-22016769673918074992008-01-14T11:17:00.000-06:002008-01-14T11:17:00.000-06:00I'd bet that something you should take into account is compressability/redundancy in the base, This is a fairly simple metric, so -- while interesting -- it doesn't really tell me a great deal about what really is the most optimal base. Part of the "niceness" of the base-ten metric is the fact that there are several numbers which allow for easy multiplication, 0 and 1, as well as the base-value itself , are fairly easy to multiply by in any metric. But in base 10, there are more numbers which are easily multiplied by, namely 2,5, and multiples thereof, are fairly easy to multiply, because they "look" the same. That is, 5555*5 is fairly simple to calculate quickly in base 10, in base 6, it's a bit harder. There was an article about this, though its location escapes me, the basic Idea is that there are two possiblities for bases which make them "better" or "worse" than others. The first is that composite numbers are better, because they have more "easy multiplication" numbers. for instance, base 12 is comparatively easier to multiply in than base 7, because base twelve has 2,3,4,6, and 8, plus the usual 0,1 and 12 as "easy" multipliers. The other camp says that prime bases are better, since they allow for more rational numbers to be expressed, or something like that. Basically, (no pun intended), the two options differ by who they're useful for, highly composite bases are good for computers and simple arithmetic, because multiplication-- a typically hard problem-- becomes easier for many cases; however, prime bases are better for mathematicians, since they allow numbers which are irrational repeating decimals to be written as irrational non-repeating decimals. This is also true of irrational bases, like e and the like. <BR/><BR/>It would be interesting to see a more complex metric similarly optimized. This is really a great example of how some simple calculus and a little intuition can go a long way in determining some interesting results.<BR/><BR/>Good stuff, keep it up!Joe Fredettehttp://www.blogger.com/profile/07170469679136904832noreply@blogger.comtag:blogger.com,1999:blog-20046017.post-2202262831453588962008-01-14T09:28:00.000-06:002008-01-14T09:28:00.000-06:00Tits?Alfredohttp://www.blogger.com/profile/17496633218879769276noreply@blogger.comtag:blogger.com,1999:blog-20046017.post-41956796749460129782008-01-12T04:44:00.000-06:002008-01-12T04:44:00.000-06:00Thanks, mohammed!<BR/>Since I posted this, I've been getting worked up about irrational bases (so far bases sqrt(2) and the cubic root of 3). I'll try and work them into a future blog post, too.burton mackenziehttp://www.blogger.com/profile/13139594737135732882noreply@blogger.comtag:blogger.com,1999:blog-20046017.post-33857540692238773202008-01-11T17:54:00.000-06:002008-01-11T17:54:00.000-06:00Great post. I've never really thought about this, but it does seem logical that e would be the optimal value.Mohammedhttp://www.blogger.com/profile/12361765428682658580noreply@blogger.com