What would be the all around "best" base to use?
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What would be the all around "best" base to use?
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.
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.
Re: What would be the all around "best" base to use?
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
...
Re: What would be the all around "best" base to use?
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.
wee free kings
Re: What would be the all around "best" base to use?
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.
 dudiobugtron
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Re: What would be the all around "best" base to use?
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 nonzero numbers? Everything is zero in the end anyway. Multiplication, addition, subtraction, differentiation, etc etc... would all become trivial, and ultimately pointless.
Re: What would be the all around "best" base to use?
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.
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.
 dudiobugtron
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Re: What would be the all around "best" base to use?
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.
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.
Re: What would be the all around "best" base to use?
You don't get much sun in New Zealand, do you?
Re: What would be the all around "best" base to use?
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.
The preceding comment is an automated response.
Re: What would be the all around "best" base to use?
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 nonpositional 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 shorthand for it rather than for addition) so I'd be completely ok with this.
It's not entirely clear how nonintegers would be represented.
Also, you need some system for writing your primes and their exponents so any factorisationbased system will have to be dependent on some other system (for instance a positional one).
All of my comments about the best base for a positional number system can be found there.
As for nonpositional 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 shorthand for it rather than for addition) so I'd be completely ok with this.
It's not entirely clear how nonintegers would be represented.
Also, you need some system for writing your primes and their exponents so any factorisationbased system will have to be dependent on some other system (for instance a positional one).
my pronouns are they
Magnanimous wrote:(fuck the macrons)
Re: What would be the all around "best" base to use?
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 nonpositional 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 shorthand for it rather than for addition) so I'd be completely ok with this.
It's not entirely clear how nonintegers would be represented.
Also, you need some system for writing your primes and their exponents so any factorisationbased 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.
 Voekoevaka
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Re: What would be the all around "best" base to use?
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...
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...
I'm a dozenalist and a believer in Tau !

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Re: What would be the all around "best" base to use?
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 twodigit pattern, in this case 05. Divisibility testing by 7 is doable mentally.
Finger counting in base 6 is completely natural. Base 6 uses digits 05, and a standard human hand has 5 fingers, giving you an instant twodigit 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 timekeeping 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, 45 weeks per month, and 7 days per week, since 364=4*7*13.
The onedigit 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.
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 twodigit pattern, in this case 05. Divisibility testing by 7 is doable mentally.
Finger counting in base 6 is completely natural. Base 6 uses digits 05, and a standard human hand has 5 fingers, giving you an instant twodigit 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 timekeeping 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, 45 weeks per month, and 7 days per week, since 364=4*7*13.
The onedigit 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.
Re: What would be the all around "best" base to use?
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 uncommoneven sixths are probably more common and base 12 gives us thoseso 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 twelvesdozens, 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 12swe have unique words for all numbers between one and twelve, then we start using compound words for numbers larger twelve.
Re: What would be the all around "best" base to use?
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?
arbiteroftruth wrote:Base 6.
Beat me to it. The repeatingdecimal advantages are nice. Plus, kids would only have to memorize ten nontrivial onedigit multiplications (order doesn't matter, anything times one is itself, anything times the base just means adding a zero), as opposed to the thirtysix 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 spaciallyefficient 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 base36 number).
There are also a couple good arguments for the oftneglected 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 halfunderstand how that works.
Someone brought up "2". I wasn't quite sure how that worked (is it littleendian 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 smallerthanthebase unit fractions that are written as messy multidigit repeaters (onefifth and oneseventh), decimal only has one: oneseventh. (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 nonrepeating division of onethird, 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?
Lenoxus wrote:every English word is also a base36 number
Lenoxus wrote:24980110 31893552737 1525081 676 494808 10 527198114 1442151747
He/Him/His/Alex
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 dudiobugtron
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Re: What would be the all around "best" base to use?
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?
Base 6 user reporting! Well kinda. Not there yet. Been kind of trying for maybe two months.
I even made some trigramesque 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.
I even made some trigramesque 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.
Re: What would be the all around "best" base to use?
eSOANEM wrote:As for nonpositional 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 shorthand for it rather than for addition) so I'd be completely ok with this.
It's not entirely clear how nonintegers would be represented.
Fractions!
Also, you need some system for writing your primes and their exponents so any factorisationbased 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...
The preceding comment is an automated response.
Re: What would be the all around "best" base to use?
snowyowl wrote:eSOANEM wrote:As for nonpositional 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 shorthand for it rather than for addition) so I'd be completely ok with this.
It's not entirely clear how nonintegers 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?
moiraemachy wrote:Base 6 user reporting! Well kinda. Not there yet. Been kind of trying for maybe two months.
I even made some trigramesque 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 randomlychosen numbers, and averaging. Base 2 is lengthdominated, base 10 is digitdominated.
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 log_{n} 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 nonlinear. 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 710, 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 710 length limit is still pretty comfy  6^9 (1 billion_{6}) is approximately equal to 10^7 (ten million_{10}), 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?
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.
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?
So, doing homework, found this. About number of digits and working memory:
http://www.musanim.com/miller1956/
http://www.musanim.com/miller1956/
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Re: What would be the all around "best" base to use?
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.
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.
good luck have fun
Re: What would be the all around "best" base to use?
No one's interested in quarterimaginary, i.e. base 2i?
http://en.wikipedia.org/wiki/Quaterimaginary_base
http://en.wikipedia.org/wiki/Quaterimaginary_base
(∫p^{2})(∫q^{2}) ≥ (∫pq)^{2}
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Re: What would be the all around "best" base to use?
Cauchy wrote:No one's interested in quarterimaginary, i.e. base 2i?
http://en.wikipedia.org/wiki/Quaterimaginary_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 twodimensional. 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!
Last edited by cyanyoshi on Fri Jun 06, 2014 6:21 am UTC, edited 2 times in total.
Re: What would be the all around "best" base to use?
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, 1617 day months in a year. I would then change to an eightday week, with a threeday 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.
We will need to come up with 6 new symbols. We should probably choose them to be representable on a Sevensegment display.
We will need to come up with 6 new symbols. We should probably choose them to be representable on a Sevensegment display.
Summum ius, summa iniuria.
Re: What would be the all around "best" base to use?
Everything is base 10...
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Re: What would be the all around "best" base to use?
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 ncycles before the last digit repeats itself.
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 ncycles before the last digit repeats itself.
"We never do anything well unless we love doing it for its own sake."
Avatar: I made a "plastic carrier" for Towel Day à la So Long and Thanks for All the Fish.
Avatar: I made a "plastic carrier" for Towel Day à la So Long and Thanks for All the Fish.
Re: What would be the all around "best" base to use?
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.
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.
Re: What would be the all around "best" base to use?
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?
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?
I like this, but I think I'd rather have more negative digits. 5 to 5 anyone?
Re: What would be the all around "best" base to use?
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?
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 nonterminating 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?
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.
Re: What would be the all around "best" base to use?
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?
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?
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.
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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