Talk:Optimal radix choice

Clarification on: "e has the lowest radix economy"
Layman here. I think in base 10; nevertheless I can understand how binary maps to base 10. I have no clue how $e$, an irrational, transcendental number, can form the basis of a numbering system. Could somebody fill in this table for $e$, and briefly explain...

It could be nice for those who need it ELI5 to have something similar in the article. Niubrad (talk) 10:00, 15 November 2017 (UTC)

https://en.wikipedia.org/wiki/Non-integer_representation#Base_e--2A02:A445:EB91:1:F9AD:ACA5:94E9:56F3 (talk) 09:33, 28 November 2019 (UTC)

Comment removed from article
The following text, added on 00:28, 6 June 2007 by, was recently [ removed] from the article. While it was clearly inappropriate in the article itself, I've chosen to copy it here to prevent it from being lost entirely.


 * This is the worst explanation of a mathematical concept that I have ever read on the wikipedia. I'm typing this comment here because this stub is so worthless, imo, that I don't feel that it deserves the respect of being spared vandalization, nor do I possess the care to correct and elaborate on the conceptualization contained here-in. Just fix it; and to whomever wrote this little fart of an explanation, I posit that you would feel better, if you would do us all a service and take greater pride in completing worthwhile endeavors, however small they might be. Thank You.

While I don't really know enough about the subject to judge the validity of the complaints presented above, and while I certainly don't consider the tone quite appropriate, I do agree with the anonymous commentator that the article desperately needs clarification and some relevant citations. —Ilmari Karonen (talk) 13:23, 10 June 2007 (UTC)


 * I hope you will agree that the article has now been significantly improved. Gandalf61 (talk) 13:56, 26 August 2008 (UTC)


 * It certainly looks better, but I don't see how is relevant, and  is just a blog talking about the AmSci article. Melchoir (talk) 15:36, 26 August 2008 (UTC)


 * Radix economy explains why we might expect ternary trees to perform better than binary trees under certain assumptions; the Bentley and Sedgewick article describes an implementation that confirms that this performance advantage is realised in practice.
 * The blog is a minor reference that just confirms the calculation of the minimum of x/ln(x). This result is easy to verify - I just included the reference for completeness in case someone threw a fact tag at it. Gandalf61 (talk) 15:49, 26 August 2008 (UTC)


 * As far as I can tell, the Bentley and Sedgewick article confirms that ternary trees offer a compromise between two different types of performance while maximizing no particular quantity, and they don't mention the quantity in this article. Our own article Ternary search tree does the same. It may be possible to keep Bentley and Sedgewick around, but the article would have to approach the subject with a different perspective: more high-level discussion of trade-offs rather than percentage-point obsessing over the exact value of rw.
 * I think you can drop the blog, since the calculation is so simple. I'm much more concerned with the conclusions. Melchoir (talk) 16:07, 26 August 2008 (UTC)


 * Whatever. From time to time I see a deserving article at AfD and I spend a bit of time polishing it to see if I can save it from oblivion. So I've done my bit as a signed up member of the "improving is better than deletion" club and I've put my !vote on the AfD page and answered your questions as best I can, and I'll leave it at that. Gandalf61 (talk) 16:22, 26 August 2008 (UTC)


 * Fair enough, but someone's going to have to rewrite the article after the dust settles. It's probably going to be me, and I'll be inclined to cut everything I think is misleading, which is currently a lot. It's better to explain why now than later; fair warning and so forth. Melchoir (talk) 18:43, 26 August 2008 (UTC)


 * I strongly object to your charcterisation of my edits as "misleading". Nothing that I have added to the article is inaccurate, and the implication that I have editted the article in order to deliberately mislead readers is disgraceful.
 * In particular, Bentley and Sedgewick describe the investigation of their ternary tree implementation in detail. They conclude that "Ternary search trees are efficient and easy to implement. They offer substantial advantages over both binary search trees and digital search tries." They then give a list of five specific advantages of ternary trees. Radix economy provides a theoretical explanation for their findings. Which is just what the article says.
 * Anyway, if you feel the urge to chop up the article to justify your AfD nomination, then by all means go ahead and cut away. Just don't insult other editors while you are doing that - it's not big and it's not clever. I am done here. Gandalf61 (talk) 20:03, 26 August 2008 (UTC)


 * Whoa, calm down. What makes you think that minimizing the product of the nodes per level and the number of levels provides a theoretical explanation for Bentley and Sedgewick's findings? Melchoir (talk) 20:15, 26 August 2008 (UTC)


 * ...Well, I found the 1950 book I mentioned in the AfD, so I'm throwing the switch now. Melchoir (talk) 08:24, 27 August 2008 (UTC)

Restored material
I have restored the introductory definition, explanation and illustrations that Melchoir removed in the re-write. These sections are correct and sourced and I believe they provide a much-needed context for the new material that Melchoir has added. I have not restored the two three references that Melchoir objected to above and in the AfD. Gandalf61 (talk) 09:28, 27 August 2008 (UTC)


 * Yes, explanation of the log formulae is beneficial, but it is neither correct nor sourced to call any particular quantity the "radix economy". See Avoid neologisms. Melchoir (talk) 09:30, 27 August 2008 (UTC)


 * I don't give a hoot what we call the article or the concept. Important things for me are that the article is clear and understandable and survives AfD. Gandalf61 (talk) 09:48, 27 August 2008 (UTC)


 * Then you won't mind calling rn just rn and getting rid of all this categorical "base with the lowest average radix economy" nonsense. It's time to un-marry yourself from the original version of the article. Google for optimization of radix sort and behold authors writing about optimal radices as high as 16. There is no single theoretically anointed cost function.
 * You don't have to worry about AfD. It's important to me that this Wikipedia article is reflective of real-world research. Melchoir (talk) 10:26, 27 August 2008 (UTC)


 * You know what though, on second thought, it's not important enough. Y'all have my references, so happy hunting! I'm outta here. Melchoir (talk) 11:24, 27 August 2008 (UTC)

Why assume that the cost of each digit is proportional to b?
This seems like a completely baseless (no pun intended) assumption. It seems more sensible to assume the cost to be proportional to the entropy, as measured in bits or nits. &mdash; Sebastian 06:53, 4 April 2013 (UTC)


 * In many situations, you may be right. I think other ways of calculating the "cost" would be a great addition to this article. I suspect that different situations will have very different cost functions, giving very different results as to the "best" radix for those situations.
 * We (finally) have two references that give two different situations that (coincidentally?) give this same particular cost function.
 * (a) In 1950, computers were very expensive, and most of the cost was in the tubes. To compete with mechanical calculators, they needed to handle numbers up to 10^10 (see 36-bit for details). To minimize the cost of the computer, they needed to minimize the number of tubes needed to store a number in the range 0 to 10^10. The number of tubes required to store a digit was equal to b, the number of possible digits at that place in the numeral. For example, many early machines used mixed-base bi-quinary coded decimal because it required 2+5=7 tubes, rather than the 10 tubes that would be required to represent a decimal digit directly in a ring counter, or the 8 tubes that would be required to represent a decimal digit as binary-coded decimal.
 * (b) In 2001, people are annoyed at phone menus. Each "level" of the menu is analogous to the next digit in the number. The cost of listening to all the options at that level is roughly proportional to how many options there are.
 * How could we improve this article to make it more clear that there are practical situations where the cost of each digit is proportional to b?
 * How could we improve this article to point out other situations that have a different cost? --DavidCary (talk) 01:36, 9 July 2013 (UTC)


 * While their are other valid measures of cost per digit for various circumstances, the key concept here is that their is a cost per possible value each digit can have. Weather analysing mechanical, analogue, digital, software, or organic systems, the cost of processing and storing numbers is roughly proportional to the number of digits (width) times the number of potential values for each digit (depth).


 * Entropy and radix economy are related, but are measures of different things: the processing cost versus the information content. Bcharles (talk) 21:37, 7 August 2013 (UTC)


 * Thanks, DavidCary, for your explanation. (And thanks for making me aware of our article bi-quinary coded decimal.) Why not simply write it along the lines of what you wrote: "Cost functions can differ. The following considerations assume that cost is proportional to b, as in the case of (a) an abacus [where btw the bi-quinary coded decimal was applied already 2000 years ago, presumably for the same reason], (b) a 1950s computer with a tube for each number or (c) a phone menu ..." &mdash; Sebastian 14:23, 9 August 2013 (UTC)

About cognitive function and learning
If this is supposed to mean that the bases with the lowest radix economy are the best for humans to use, I think this is complete uncited BS. Firstly, it's not cited. Secondly, just try performing calculations in a base that's 4 or less. I can guarantee you that unless you septuple-check everything, you will mess up some digits. And you'll need such long numbers! There's a need to quickly distinguish numbers like 64 and 32: suppose you're driving quickly past a speed limit. Will you be able to quickly count the number of zeroes in "1000000" vs "100000"? I think not. I do not know what the optimal range of bases for humans to use is, but it self-evidently is not centered around e. 10 is definitely in it (citation definitely not needed), probably close to the centre. My gut feeling is that it would be in the range between 6 (senary) and 16 (hexadecimal) inclusive, with 6, 10, and 12 as outstandingly good choices, and 8 and 16 somewhat further behind, the rest not even being in the running. Given that this statement isn't cited, and doesn't make sense, I have removed it. (Of course I have not replaced it with what I just wrote, as that is uncited too!) Double sharp (talk) 17:35, 28 March 2015 (UTC)

Binary has the lowest radix economy for humans
A recent YouTube video, "the best way to count", observes that if leading zeroes are prohibited, as is typical for human use, then the leading digit contains less information, causing the calculations to be different. As a result, binary has the lowest radix economy for human use.

I am just putting this information in the talk page for now until I (or someone else) can draft a revision to the article to add a section for how applying radix economy to human (rather than computer) use causes the calculations to change. taulover (talk) 03:19, 15 January 2024 (UTC)

ðe equation is wrong.
ðe equation multiplies by b, when it should be multiplied by log(b). also, because ðe leading digit cant be zero, ðe most significant digit carries less information. as such, instead of a minimum at e, ðe function is strictly decreasing. Jan Eten (talk) 04:49, 6 June 2024 (UTC)

Page move and revision
As I said in the recent deletion debate, the book from 1950 uses phrases like "The economy to be gained by choice of radix", but not radix economy specifically. The sources that come up for that term in the literature are more recent than the creation of this article; they seem to start around 2012, when the article had been around for six years already. I have moved and lightly edited the page to break the citogenesis cycle. XOR&#39;easter (talk) 19:36, 23 June 2024 (UTC)