What would be the all around "best" base to use?

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DR6
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Re: What would be the all around "best" base to use?

Postby DR6 » Mon Jun 30, 2014 4:38 pm UTC

jaap wrote:I vaguely remember reading an argument somewhere that went something like this:

1) The two main factors determining the complexity of the number notation system are the number of symbols the system has (equals the base) and the average length of commonly used numbers when written in that system (average number of digits).
2) These two factors were combined in some way to give a complexity score to using any particular base.
3) Optimising for this score gave e as the best answer.
4) Therefore base 3 was the best base to use.

Looking at this now there is an obvious fudge factor in step 2, where you can weigh the importance of the two aspects differently to get whatever outcome you like. If you really hate using many kinds of symbols but don't mind writing a lot, use binary (or even 'unary'), if you want shorter numbers use a higher base.

At the time I read it, this fudge factor was not obvious, so it was probably implicit due to some assumption somewhere.

Does this ring a bell with anyone?


I'd say that number of symbols is not really that important: we already learn 27*2 symbols(plus punctuation) for writing and nobody complains. Maybe for numbers that would be too much, but 15 symbols are not much harder to learn that 5, specially if we are drawing from symbols we already know. Like arbitreroftruth said, it's easier to memorize short strings with many possible symbols than long strings with few possible symbols: just compare "∂e5@" and "1001101011101001".

And a factor your reasoning does not really include is the utility that comes from having easy multiplication tables and terminating fractions: things that depend on the prime factorization of the base. Small numbers like 3 are bound to be bad both at this and at number of digits(2 is different because multiplication becomes sums and shifts, but that's also the worst number possible when it comes to length).

In my opinion, the base that has risen naturally should be at least close to the optimal one, in terms of memory efficiency: so I'd go with 12, for that reason and the prime factors.

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Re: What would be the all around "best" base to use?

Postby moiraemachy » Mon Jun 30, 2014 4:41 pm UTC

ninja'd, but still:
Xanthir wrote:Compute the "weight" of each base by multiplying the number of digits by the length of the number.
I guess the idea is to minimize the number of cards in one of these:
Image
assuming you need all numbers up to some n. The paper I linked earlier basically discusses this fudge factor.
cyanyoshi wrote:One catch is that by using digits from -5 to 5, many more numbers (almost all of them?) would no longer have a unique representation, so it would be slightly harder to compare two arbitrary numbers.
My suggestion would be -5 to 4 in negatives and -4 to 5 in positives. Question: assuming this, do we have any 0.999... = 1 situation? I mean, I only found one "type" of ambiguous number:

Spoiler:
0.555... = 1.444... (underlined = negative)
Both are repeating, so less people losing sleep over it.

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Re: What would be the all around "best" base to use?

Postby cyanyoshi » Mon Jun 30, 2014 5:54 pm UTC

moiraemachy wrote:My suggestion would be -5 to 4 in negatives and -4 to 5 in positives. Question: assuming this, do we have any 0.999... = 1 situation? I mean, I only found one "type" of ambiguous number:

Spoiler:
0.555... = 1.444... (underlined = negative)
Both are repeating, so less people losing sleep over it.

In that system, the only positive numbers that can be written two different ways would be numbers with an infinite string of trailing "5"s or "4"s. For negative numbers, it would be "4"s or "5"s instead. Cue the people who would still claim that 1.444... is slightly larger than 0.555..., or that five ninths is somehow a special number to the structure of the universe!

Technically, terminating decimals could still be written many different ways (i.e. 1 = 1.0 = 1.00), but most people don't have a problem with that.

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Re: What would be the all around "best" base to use?

Postby chridd » Mon Jun 30, 2014 6:36 pm UTC

gmalivuk wrote:The universe loves pi and e for continuous things, but they're pretty terrible to work with as positional numbering systems, on account of integers not having terminating representations, and the fact that almost all numbers have multiple possible representations.
She said phi, though, not pi; base phi has unique (in standard form), terminating representations for all integers.
arbiteroftruth wrote:Or, for human brains, you might notice that memorizing a string of letters isn't really more difficult than memorizing a string of numbers, and the length of the string is far more important than the number of possible characters.
...so, base 26 then? Or base 24, to take advantage of the high number of factors (similar to base 12), using all but two letters as digits?
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Re: What would be the all around "best" base to use?

Postby Thesh » Mon Jun 30, 2014 7:51 pm UTC

You could create a numeral system based on simple rules, which could extend to large bases. Babylonians did this with base 60. Also, does base 12 really provide much of an advantage over base 6?
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Re: What would be the all around "best" base to use?

Postby Xanthir » Tue Jul 08, 2014 5:46 am UTC

While heximal has some nice advantages (namely a *trivial* multiplication table, which means that divisibility by numbers between 2 and N are trivial too), it gives you numbers about 50% larger than decimal. That's not killer, but it's annoying. Dozenal numbers are approximately the same size as decimal ones.

Dozenal's divisibility rules aren't *quite* as simple, but as illustrated previously, are still pretty easy - only 5 and 7 give any real trouble. Having two factors of 2 make a lot of larger divisibilities a lot easier, too - quite a lot of reasonably useful numbers can be determined with the last two digits, for example. (This is the same reason sexagesimal is actually much more useful than trigesimal.)

Plus, dozens are already a well-known unit. People do plenty of work in dozens, and occasionally grosses, on a regular basis. It's not foreign, like heximal would be.

On the other hand, heximal works well with our hands - two 5-finger hands can be a heximal digit each, letting us count to 35 on our hands real easily. Of course, if you're doing finger counting binary works even better, and lets you count to 1k.
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Re: What would be the all around "best" base to use?

Postby Thesh » Tue Jul 08, 2014 5:50 am UTC

Xanthir wrote:it gives you numbers about 50% larger than decimal. That's not killer, but it's annoying.


Why's that a problem?

EDIT: What I mean is, there's nothing really special about base 10, other than that's what we are used to. I don't see why it would make it annoying.
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Re: What would be the all around "best" base to use?

Postby arbiteroftruth » Tue Jul 08, 2014 2:46 pm UTC

Xanthir wrote:While heximal has some nice advantages (namely a *trivial* multiplication table, which means that divisibility by numbers between 2 and N are trivial too), it gives you numbers about 50% larger than decimal. That's not killer, but it's annoying. Dozenal numbers are approximately the same size as decimal ones.


It's a lot closer to 30%. log610~=1.28. And besides, the need for more digits to reach a certain level of precision is balanced out by the increased freedom in deciding what level of precision to use.

Xanthir wrote:Plus, dozens are already a well-known unit. People do plenty of work in dozens, and occasionally grosses, on a regular basis. It's not foreign, like heximal would be.


If you're someone who holds out any hope of civilization actually changing bases, dozenal is the best candidate for exactly that reason. But the premise of this thread was to ignore issues of social momentum.

Xanthir wrote:On the other hand, heximal works well with our hands - two 5-finger hands can be a heximal digit each, letting us count to 35 on our hands real easily. Of course, if you're doing finger counting binary works even better, and lets you count to 1k.


Well, unless you're concerned about rude gestures.

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Re: What would be the all around "best" base to use?

Postby Xanthir » Thu Jul 10, 2014 4:09 am UTC

Thesh wrote:
Xanthir wrote:it gives you numbers about 50% larger than decimal. That's not killer, but it's annoying.


Why's that a problem?

EDIT: What I mean is, there's nothing really special about base 10, other than that's what we are used to. I don't see why it would make it annoying.

Longer numbers are longer. That's a problem. We don't use binary, because it produces numbers ~3 times longer than decimal, and it's a lot harder to remember those. You have to optimize number of digits vs length of numbers based on human limitations.
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Re: What would be the all around "best" base to use?

Postby cyanyoshi » Thu Jul 10, 2014 6:14 pm UTC

Base 6 is nice because it represents common fractions particularly well. Most importantly, one fifth and one seventh are much nicer in base 6 than in dozenal. The multiplication table is way simpler (10 nontrivial multiplications versus 55 in base 12), but whole numbers in base 6 are on average 39% longer than in base 12. Whether or not this trade-off is worth it might depend on the person or application, but I think the shorter length of numbers in base 12 suits my own purposes better.

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Re: What would be the all around "best" base to use?

Postby ThirdParty » Tue Jul 29, 2014 8:07 am UTC

I'd like to nominate Base 36 for consideration.

It's a multiple of 2, 3, 4, 6, 9, and 12, so most divisions are easy and most fractions have short expansions. 36-1 is a multiple of 5 and 7, so for the few divisions that aren't easy you can at least easily compute the remainder by summing digits, and those fractions with infinite expansions tend to at least have simple ones such as "0.777..." or "0.3LLL...".

Having a large base makes memorizing addition and multiplication tables much harder, but once you've learned them you'll be able to do a huge number of everyday calculations by rote. Numbers are short, making them easy to remember and to store in working memory. (If you look at our current system, you see that base 10 is too small for many applications: we divide dollars into 100 cents; we divide days into 24 hours; we divide hours and minutes into 60 minutes and 60 seconds respectively; an address such as "204 First Street, room 311" means "the 11th room on the 3rd floor of the 4th building on the 2nd block of First Street"; license plates and bar codes are frequently in base 36; phone numbers are unwieldy; etc. 36 is large enough and divisible enough that this hodgepodge could be eliminated and everything regularized.)

To count on your fingers in base 36, use binary to represent 0 thru V, and pinch together your thumb and a finger to represent W, X, Y, and Z.

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Re: What would be the all around "best" base to use?

Postby Girl-With-A-Math-Fetish » Wed Jul 30, 2014 1:17 am UTC

ThirdParty wrote:I'd like to nominate Base 36 for consideration.


Didn't base 39 once get used? People counted off all their body parts, including their dong. I wonder how women counted. This was long before the age of female oppression. xD
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Re: What would be the all around "best" base to use?

Postby Elmach » Mon Oct 13, 2014 12:52 am UTC

Kind of resuscitating this topic, but seems better than to make a new one.

I'd also prefer a "partially-balanced" numeral system (Wikipedia calls it a signed-base representation, but I have no idea if that is the actual term). Base 6 is interesting because all the reciprocals of one through ten are nice, but is too small to be useful; I think a relatively large number would be good. Base 36 or 12...

Anyways, I was going to suggest a -2...9 positional system (base a dozen), perhaps with numerals #,-,0,1,2,3,4,5,6,7,8,9, and representing negation with -:n, where : is the multiplication operator. (so we also have 1 + -:1 = 0, or 1 + - = 0)


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