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

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David1618
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What would be the all around "best" base to use?

Postby David1618 » Wed Apr 17, 2013 12:54 am UTC

Suppose we were able for some reason to change what base the world uses with no negative effects in the transition. Everything is converted automatically and everyone would instantly know the system as well as if they had been raised with it.
What would be the best choice?

Here are a few options I can think of.
Base 1: Very simple, but very impractical.
Base 2: I'd suppose this would be the second choice for simplicity. Much easier to work with than base 1 though.
Base 4/8/16/32/64: These bases seem natural in a way. Smaller ones Will require more space for expression. Larger ones require the memorization and use of more characters.
Base 10: Because we have 10 fingers and toes.
Base 12: Being highly divisible would give a limited advantage in expressing fractions.
Prime bases: A worse choice when the base is greater, as more and more numbers become difficult to express simply.
Base e/pi/phi: I don't know how that would work, but it seems like an interesting idea. Phi especially, if irrational bases are at all practical.
Base i: I have no idea, but I thought I'd throw it put there.

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

Postby Qaanol » Wed Apr 17, 2013 12:58 am UTC

-2
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Re: What would be the all around "best" base to use?

Postby David1618 » Wed Apr 17, 2013 1:17 am UTC

Qaanol wrote:-2

Any particular reason why? Would make counting interesting... I assume it is something like this?

0
1
110
111
100
101
11010
11011
1000
1001
11110
11111
11100
11101
10010
10011
10000
10001
1110010
1110011
10100
...

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

Postby Qaanol » Wed Apr 17, 2013 1:52 am UTC

David1618 wrote:
Qaanol wrote:-2

Any particular reason why? Would make counting interesting... I assume it is something like this?

Because unary negation operators are le weaksauce. This way negative numbers have an even number of digits, and positive numbers don't.

On the other hand, base 12 has some nice properties, such as making it really easy to check divisibility by most small primes:
2, 3, 11, and 13 are nearly trivial
5 and 29 are pretty simple as well
7 and 19 are not too tough
Only 17 and 23 are difficult, among the first ten primes.
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Re: What would be the all around "best" base to use?

Postby David1618 » Wed Apr 17, 2013 2:05 am UTC

Qaanol wrote:
David1618 wrote:
Qaanol wrote:-2

Any particular reason why? Would make counting interesting... I assume it is something like this?

Because unary negation operators are le weaksauce. This way negative numbers have an even number of digits, and positive numbers don't.

On the other hand, base 12 has some nice properties, such as making it really easy to check divisibility by most small primes:
2, 3, 11, and 13 are nearly trivial
5 and 29 are pretty simple as well
7 and 19 are not too tough
Only 17 and 23 are difficult, among the first ten primes.


Do you think it would work well for the average person doing things as simple as shopping, figuring distances and times, and... Counting? I could be wrong, but it seems that even being raised with base -2, a user would be overall less efficient than someone stuck with base 10.

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

Postby dudiobugtron » Wed Apr 17, 2013 2:08 am UTC

David1618 wrote:Suppose we were able for some reason to change what base the world uses with no negative effects in the transition. Everything is converted automatically and everyone would instantly know the system as well as if they had been raised with it.
What would be the best choice?


That depends on your definition of 'best'. I personally like the nihilistic aesthetics of base zero. Why do we need to represent non-zero numbers? Everything is zero in the end anyway. Multiplication, addition, subtraction, differentiation, etc etc... would all become trivial, and ultimately pointless.
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Re: What would be the all around "best" base to use?

Postby David1618 » Wed Apr 17, 2013 2:19 am UTC

That is why I said all around best haha.
How about this:

The base in which the combined purposes of all humanity in the present Time (individual purposes being weighted by their importance) take less time to be completed than in any other base (with the completion time modified by the accuracy of the results.)

Of course, importance is subjective and debatable, but I think that it is a minor rough influence that we can agree to a sufficient degree.

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

Postby dudiobugtron » Wed Apr 17, 2013 3:07 am UTC

Almost all calculations in base zero are effectively instantaneous, and as accurate as possible. As long as you don't mind the results being meaningless - but then perhaps that's the only true meaning anyway.
And of course, finding out that the sum total of the combined purposes of all humanity amounts to nothing is pretty poetic, if you ask me.
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Re: What would be the all around "best" base to use?

Postby David1618 » Wed Apr 17, 2013 3:27 am UTC

You don't get much sun in New Zealand, do you?

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

Postby snowyowl » Wed Apr 17, 2013 12:56 pm UTC

Why the assumption that a positional system is going to be the best? I can think of several technical situations where representing numbers in prime factorization form would be the best way to go.
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Re: What would be the all around "best" base to use?

Postby eSOANEM » Wed Apr 17, 2013 3:39 pm UTC

Do people not believe in looking at any of the other recent threads any more?

All of my comments about the best base for a positional number system can be found there.

As for non-positional systems, prime factorisation is definitely useful although can make addition a pain. That said, multiplication tends to crop a lot (which is why we use concatenation as a short-hand for it rather than for addition) so I'd be completely ok with this.

It's not entirely clear how non-integers would be represented.

Also, you need some system for writing your primes and their exponents so any factorisation-based system will have to be dependent on some other system (for instance a positional one).
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Re: What would be the all around "best" base to use?

Postby David1618 » Wed Apr 17, 2013 3:53 pm UTC

eSOANEM wrote:Do people not believe in looking at any of the other recent threads any more?

All of my comments about the best base for a positional number system can be found there.

As for non-positional systems, prime factorisation is definitely useful although can make addition a pain. That said, multiplication tends to crop a lot (which is why we use concatenation as a short-hand for it rather than for addition) so I'd be completely ok with this.

It's not entirely clear how non-integers would be represented.

Also, you need some system for writing your primes and their exponents so any factorisation-based system will have to be dependent on some other system (for instance a positional one).


I did read that thread. It was helpful, but concerned itself with a slightly different topic than the one here, which seems to have escaped notice so far. :(

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

Postby Voekoevaka » Wed Apr 17, 2013 4:15 pm UTC

As I said on the other thread, I think the best base is base 12. I often count in base 12 using the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, X and Ↄ. 12 have a lot of usual divisors, and I like this number for another reason (http://forums.xkcd.com/viewtopic.php?f=17&t=101445#p3330147).
Base 60 is also useful, so I use to combine the 12 brevious numbers with 5 new ones : N, U, D, T and Q (for Nihil, Unus, Duo, Tres and Quattuor).
So, it would make :

0→N0, 10→NX, 11→NↃ, 12→U0, 23→UↃ, 24→D0, 35→DↃ, 36→T0, 48→Q0, 59→QↃ, 60→N1N0, 3600→N1N0N0...
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Re: What would be the all around "best" base to use?

Postby arbiteroftruth » Wed Apr 17, 2013 4:57 pm UTC

Base 6.

The first two primes, 2 and 3, divide the base, so those fractions terminate and divisibility by them can be checked at a glance.

The next prime, 5, has the next best treatment. Being one less than the base, fifths will recur, but as a single repeating digit, in this case 1. Divisibility testing by 5 is also pretty easy if necessary.

The next prime, 7, has the third best treatment. Being one more than the base, sevenths will recur with a two-digit pattern, in this case 05. Divisibility testing by 7 is doable mentally.

Finger counting in base 6 is completely natural. Base 6 uses digits 0-5, and a standard human hand has 5 fingers, giving you an instant two-digit abacus. This means that technically you don't even need to learn your arithmetic tables if you're willing to take the time to count out the answer on your fingers.

Our existing system of time-keeping would instantly be full of nice round numbers. Because 6 divides the 60 seconds in a minute or minutes in an hour, the 24 hours in a day, and the 12 months in a year. As for the 7 days in a week, counting by 7s in base 6 is as easy as counting by 11s in base 10. Granted, timekeeping advantages could be subsumed under the "assume no cultural problems in the transition", but even if we wanted to reform the calendar to accomadate a given base, arguably even an optimized calendar would still have 12 months, 4-5 weeks per month, and 7 days per week, since 364=4*7*13.

The one-digit arithmetic tables are nearly trivial. Then there are the mental shortcuts one often takes to multiply by small numbers. For example, in base 10 multiplication by 5 and division by 2 are related, multiplication by 9 is just a decimal shift and a subtraction, etc. In base 6, these types of shortcuts allow easy multiplication by numbers up to around 14.

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

Postby LaserGuy » Wed Apr 17, 2013 6:09 pm UTC

For everyday life, I think base 12 is probably the winner. Halves, thirds, and quarters are pretty natural divisors for a lot of practical purposes, and base 10 is pretty clunky for thirds and not as great for quarters. Dividing something into fifth are tenths is somewhat uncommon--even sixths are probably more common and base 12 gives us those--so trading those for thirds and quarters is a good bargain. There are enough applications in base 12 that we already have words to talk about things in twelves--dozens, gross. A lot of people prefer to work in feet and inches because base 12 is just more convenient (though they may not think about it in those terms). Even our number words already acknowledge the importance of 12s--we have unique words for all numbers between one and twelve, then we start using compound words for numbers larger twelve.

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

Postby eSOANEM » Wed Apr 17, 2013 6:13 pm UTC

Tenths are used quite a lot, but only when people are speaking roughly and I suspect this is for the same reasons or due to the fact that we use a base 10 system.
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Re: What would be the all around "best" base to use?

Postby Lenoxus » Wed Apr 17, 2013 8:34 pm UTC

arbiteroftruth wrote:Base 6.


Beat me to it. The repeating-decimal advantages are nice. Plus, kids would only have to memorize ten nontrivial one-digit multiplications (order doesn't matter, anything times one is itself, anything times the base just means adding a zero), as opposed to the thirty-six there are in base 10. [Edit: I calculated that by finding the sum of the numbers from 1 to 8, with the understanding that the 1s and 10s are trivial, as is reversing a memorized multiplication. The result being 36 is a coincidence. Using tricks, one can reduce the memorization of either even further, but of course six still beats ten.]

It's also very easy to convert into the spacially-efficient base 36, which is the highest posible base using digits and (English) letters. As they got older, people could transition to that base, though probably with its own set of symbols to avoid confusion (every English word is also a base-36 number).

There are also a couple good arguments for the oft-neglected base 3. One is that if you measure overall "efficiency" as the product of (a) the total number of digits the base has and (b) the number of digits in an arbitrary number, then 3 is the most efficient as the chosen number increases. I read that this is because 3 is the closest integer to e, and I half-understand how that works.

Someone brought up "-2". I wasn't quite sure how that worked (is it little-endian binary?), but there is a cool base called balanced ternary, where the digits (say, 0, 1, X) represent either 0, 1, or -1. Every integer has a unique balanced ternary representation, eg, seven is 1X1, meaning "nine minus three plus 1".

Wikipedia points out that this means that if your currency was in ordinary ("unbalanced") ternary denominations, and you and someone else had at least one of each bill, then you could transfer any amount of money from $1 to $(3^N), where N is the bills you each have. Just write the amount you owe in balanced ternary; for each 1 you give the corresponding bill, for each X you recieve the corresponding bill, and for each 0 you do nothing. The process can be reversed by reversing each digit (X to 1 and 1 to X), which is equivalent to making the number negative (there's no need for a minus sign in balanced ternary).

Of course, the disadvantages of using either balanced or unbalanced ternary likely outweight the advantages, so that's why I root for 6.

All that said, decimal isn't so terrible. Whereas "dozenal" has two smaller-than-the-base unit fractions that are written as messy multi-digit repeaters (one-fifth and one-seventh), decimal only has one: one-seventh. (Base six has none at all – it goes 1, 0.3, 0.2, 0.13, 0.1_, 0.1.) And smaller numbers (such as five) tend to come up more often in division anyway; it's more useful to accurately divide something into fifths than into thirteenths. Still, by that same logic, it's even more useful to have a non-repeating division of one-third, and this may well tip the balance in dozenal's favor.

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

Postby ahammel » Wed Apr 17, 2013 8:57 pm UTC

Lenoxus wrote:every English word is also a base-36 number
Lenoxus wrote:24980110 31893552737 1525081 676 494808 10 527198-114 1442151747
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Re: What would be the all around "best" base to use?

Postby Lenoxus » Wed Apr 17, 2013 9:33 pm UTC

ahammel wrote:
Lenoxus wrote:every English word is also a base-36 number
Lenoxus wrote:24980110 31893552737 1525081 676 494808 10 527198-114 1442151747


53311120479340 3015093237090890 1976505365 847894339973 1446884362222

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

Postby dudiobugtron » Wed Apr 17, 2013 10:13 pm UTC

David1618 wrote:You don't get much sun in New Zealand, do you?

We get a lot, but it's nowhere near as hot as countries that are closer to the equator.

LaserGuy wrote:Dividing something into fifth are tenths is somewhat uncommon

Actually it's a lot more common that you claim - but mostly because we use the decimal system so it's easy. If we used base 12 instead then we wouldn't do it as much. But since we use it, dividing by 5 or 10 is very common as a result. In fact, I think I divide by ten more often than I divide by any other number. (Unless you count dividing by different powers of 10 separately - then probably 100 would be the most common one for me.)
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Re: What would be the all around "best" base to use?

Postby moiraemachy » Fri Apr 19, 2013 1:23 am UTC

Base 6 user reporting! Well kinda. Not there yet. Been kind of trying for maybe two months.

I even made some trigram-esque algarisms which are all easily discernible and fast to write, but I'm still getting used to them. It's really cool how it seems more natural to count things by lumping them in groups of three. Though needing three digits for numbers bigger than 35 is bad... I suspect that the cost to remember digits, to your brain, is not related to the number of available digits, so it's hard to remember more bits. Now I only need cool names for my numbers.

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

Postby snowyowl » Fri Apr 19, 2013 11:23 am UTC

eSOANEM wrote:As for non-positional systems, prime factorisation is definitely useful although can make addition a pain. That said, multiplication tends to crop a lot (which is why we use concatenation as a short-hand for it rather than for addition) so I'd be completely ok with this.

It's not entirely clear how non-integers would be represented.


Fractions!

Also, you need some system for writing your primes and their exponents so any factorisation-based system will have to be dependent on some other system (for instance a positional one).

True. It could be dependent on itself, of course...

Code: Select all

1
p_1
p_(p_1)
(p_1)^(p_1)
p_(p_(p_1)))
p_1*p_(p_1)
p((p_1)^(p_1))
(p_1)^(p_(p_1))
etc.


Anyway, it's p_1^-1 past p_(p_(p_(p_1)))) here, so I should head off...
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Re: What would be the all around "best" base to use?

Postby eSOANEM » Fri Apr 19, 2013 12:19 pm UTC

snowyowl wrote:
eSOANEM wrote:As for non-positional systems, prime factorisation is definitely useful although can make addition a pain. That said, multiplication tends to crop a lot (which is why we use concatenation as a short-hand for it rather than for addition) so I'd be completely ok with this.

It's not entirely clear how non-integers would be represented.


Fractions!


A lack of an equivalent to decimal notations means it's hard to easily compare the size of constants or values measured using instruments with finite precision.
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Re: What would be the all around "best" base to use?

Postby Xanthir » Fri Apr 19, 2013 9:35 pm UTC

moiraemachy wrote:Base 6 user reporting! Well kinda. Not there yet. Been kind of trying for maybe two months.

I even made some trigram-esque algarisms which are all easily discernible and fast to write, but I'm still getting used to them. It's really cool how it seems more natural to count things by lumping them in groups of three. Though needing three digits for numbers bigger than 35 is bad... I suspect that the cost to remember digits, to your brain, is not related to the number of available digits, so it's hard to remember more bits. Now I only need cool names for my numbers.

The two are related, but not in any obvious way.

If you pretend that the number of distinct digits and the length of numbers are weighted equally in our minds, then base 3 is the best, because it's closest to the global minimum of e.

Compute the "weight" of each base by multiplying the number of digits by the length of the number, for a large quantity of randomly-chosen numbers, and averaging. Base 2 is length-dominated, base 10 is digit-dominated.

The "compute a bunch and average" thing is only necessary because lengths are rounded to integers. You can do it much easier by letting the "length" of a number just be the logn of it. This gives you identical results, and makes the equation easier to work out - you just have to find the 0 of the derivative of n*log_n(x). When you do this, the answer pops out as e, which is pretty natural since it relates to logs.

Of course, we don't actually weight digits and lengths equally. In reality, I think the two are largely uncorrelated, and their individual weightings are non-linear. We can remember a decent number of distinct digits - more than 10, but probably less than 26 - and our ability to remember length is pretty insensitive up to about 7-10, at which point it hits a ceiling.

This suggests that we should be trying to find a base that has close to the maximum number of digits we can remember, so we can fit more numbers into the length we can easily remember. With this, dozenal probably has some nice advantages, being slightly larger than 10 and having a lot of divisors.

Still, though, six does have some great qualities. Numbers in base 6 are only about 30% larger than ones in base 10, so the 7-10 length limit is still pretty comfy - 6^9 (1 billion6) is approximately equal to 10^7 (ten million10), so you barely lose anything at all. Phone numbers would be a bit harder to memorize, is all.
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Re: What would be the all around "best" base to use?

Postby arbiteroftruth » Fri Apr 19, 2013 11:08 pm UTC

Examples like phone numbers aren't particularly relevant. The pure need for a great variety of memorizable combinations only requires an arbitrary set of symbols that people know. There's no reason not to use some letters as well if we want to increase the number of possible symbols for that purpose.

The length issue only really matters when we're actually doing math and are thus limited to the digits of the base. And in that context, the need for more digits to reach a certain level of precision is balanced out by having greater control over how much precision to use. For purposes of scientific notation, you can only indicate your precision to the nearest order of magnitude of the base, so a smaller base gives you a better ability to accurately indicate your level of precision.

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

Postby moiraemachy » Mon Jun 02, 2014 2:12 pm UTC

So, doing homework, found this. About number of digits and working memory:
http://www.musanim.com/miller1956/

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

Postby phillip1882 » Mon Jun 02, 2014 4:27 pm UTC

i personally really like balanced ternary.
i think of all the numbering systems its the most efficient in terms of being able to add, subtract, multiply and divide quickly. though division requires a bit of extra thought compared to binary.
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Re: What would be the all around "best" base to use?

Postby Cauchy » Fri Jun 06, 2014 4:49 am UTC

No one's interested in quarter-imaginary, i.e. base 2i?

http://en.wikipedia.org/wiki/Quater-imaginary_base
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Re: What would be the all around "best" base to use?

Postby cyanyoshi » Fri Jun 06, 2014 5:58 am UTC

Cauchy wrote:No one's interested in quarter-imaginary, i.e. base 2i?

http://en.wikipedia.org/wiki/Quater-imaginary_base

That's one way to do it, but there are some useful properties of complex numbers that are most apparent when they are treated as two-dimensional. That being said, we also can cover all the complex numbers with the digits 0 and 1 in base √(2)i if we stop caring about practicality. [Edit: I just found out about base (±1±i) and the fact that the truncation error is a cool fractal that should look vaguely familiar if you've read Jurassic Park. Neat.]

I really like balanced ternary because it lets the negative numbers flow seamlessly into the positive numbers, and it lets us do away with that pesky negative sign. It also doesn't hurt that 3 is close to e, so estimating the natural log of a number from just counting digits is not too shabby. Base 12 is an improvement over base 10 in my opinion, but it looks like we're more or less stuck with base 10 because of the metric system. If the world would just adopt the US customary system of units, switching to a better base would be so much easier! :twisted:
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Re: What would be the all around "best" base to use?

Postby Thesh » Fri Jun 06, 2014 6:01 am UTC

Hexadecimal, for the sole purpose that it simplifies working with computers. Then we should abandoned metric for a new system based on powers of 10. We should also move to a new clock and calendar based on 20 hours in a day, 40 minutes in an hour, 40 seconds in a minute, with 10, 16-17 day months in a year. I would then change to an eight-day week, with a three-day weekend, because fuck yeah, three day weekends. I would have the first day of the first month be approximately winter solstice in the norther hemisphere.

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

Postby andrewxc » Mon Jun 23, 2014 2:34 pm UTC

Everything is base 10... :twisted:
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Re: What would be the all around "best" base to use?

Postby andrewxc » Mon Jun 23, 2014 2:45 pm UTC

I like the dozenal system, as explained in Here's Looking at Euclid. Numbers are: zero, one, two, three, four, five, six, seven, eight, nine, dec, el, do, written: 0,1,2,3,4,5,6,7,8,9, X, E, 10. 'El' is usually a '3' that is rotated 180º (looks a bit like an epsilon), but HTML code has it as a capital 'E'.

The reason it works so nicely is that there are not a lot of numbers that are relatively prime (only 5 & 7 & 11), so the multiplication table in dozenal has a lot of repeating patterns and shorter n-cycles before the last digit repeats itself.
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Re: What would be the all around "best" base to use?

Postby cyanyoshi » Mon Jun 23, 2014 8:58 pm UTC

A simple modification to the standard decimal system is to have a -1 digit instead of the 9 digit. That gets rid of that pesky negative sign once and for all! It's not even that weird if you are used to Roman numerals, like how IX is really "ten minus 1". Negative numbers take some getting used to, though, but I like to think of that more as a weakness of the standard base 10 system rather than the one with a digit for -1. It's not as pretty as balanced ternary, but it works.

I've also been messing around with base -1+i for a little bit, and it is very cool. Every complex number can be written in that base with the digits 0 and 1, and every Gaussian integer can be written without a decimal point! It is certainly not be the best system to use, but it's neat in its own way.

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

Postby korona » Sat Jun 28, 2014 10:10 am UTC

cyanyoshi wrote:A simple modification to the standard decimal system is to have a -1 digit instead of the 9 digit. That gets rid of that pesky negative sign once and for all!

? How would you represent -2?

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

Postby moiraemachy » Sat Jun 28, 2014 10:28 am UTC

I guess N8 (-1*10 + 8). 9 would be 1N.

I like this, but I think I'd rather have more negative digits. -5 to 5 anyone?

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

Postby cyanyoshi » Sat Jun 28, 2014 9:29 pm UTC

moiraemachy wrote:I like this, but I think I'd rather have more negative digits. -5 to 5 anyone?

Sure. 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. Letting an underline represent the negation of that digit, tau could be written as 14.3232253... or as 14.3232153... equally well. Some authors prefer using digits from -5 to 4 to avoid this problem in base ten, but balanced numeral systems tend to work best with odd bases anyway.

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

Postby Girl-With-A-Math-Fetish » Mon Jun 30, 2014 1:06 pm UTC

I'd say base phi, followed by base e. I'd not know how that would work, but if the universe seems to love these numbers so much, there must be a reason to it. Base 12 is the most practical, I'd think. Fewer non-terminating decimals (1/3 is ugly in base 10, but is much nicer in base 12).
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Re: What would be the all around "best" base to use?

Postby gmalivuk » Mon Jun 30, 2014 1:37 pm UTC

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

Postby jaap » Mon Jun 30, 2014 2:06 pm UTC

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?

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

Postby arbiteroftruth » Mon Jun 30, 2014 4:34 pm UTC

The argument there is that the complexity of writing a number is just the sum of the complexities of writing the individual digits, and the complexity of writing an individual digit is just the number of possible choices. So the complexity of writing a number is (# of digits)*(# of digit options). The number of digits in a number n in base b is ln(n)/ln(b), and the number of digit options is b. So the complexity is ln(n)*b/ln(b). The only effect the base has is on the coefficient b/ln(b), which is minimized when b=e.

But neither of those starting assumptions is necessarily accurate when it comes to computers or human brains. For a computer, the hardware complexity of directly selecting one of b possible voltages to represent one of b possible digits grows more like b^2, in terms of how many transistors it would take. In that case, you want to minimize b^2/ln(b), which happens at sqrt(e) or about 1.65, making binary optimal.

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. To keep things simple, we might say that the complexity of a single digit is still linear with the number of possibilities, but the complexity of the string grows with the square of the length. In that case, you want to minimize b/ln(b)^2, which happens at e^2 or about 7.4, making 7 hypothetically optimal, although you'd probably go with 6 or 8 due to other considerations.


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