## Base 12 kicks all your asses

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zenten
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### Base 12 kicks all your asses

It's got both three and 4 as factors, not to mention 6. Sure it doesn't have 5, but who needs five when you have three others?

It also fits well with our time keeping systems.

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### Re: Base 12 kicks all your asses

So then we have ROT11? Hmm...
Wait. With a SPOON?!

btilly
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### Re: Base 12 kicks all your asses

zenten wrote:It's got both three and 4 as factors, not to mention 6. Sure it doesn't have 5, but who needs five when you have three others?

It also fits well with our time keeping systems.

The main reason why the Imperial system is able to hang on in some places is that it has lots of 12s in convenient places. Making it easy to divide things into halves, thirds and quarters.

If we only counted in base 12 instead of base 10, then we could have all of the notational convenience of metric with all of the practical convenience of being able to easily take our most commonly needed fractions.
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### Re: Base 12 kicks all your asses

if the base is not a power of 2 its useless.
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Yakk
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### Re: Base 12 kicks all your asses

9699690 (base 10) kicks any power of base2's ass!
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

Cheese
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### Re: Base 12 kicks all your asses

Okay, base 60, if you're just going for factors. 1,2,3,4,5,6,10,12,15,20,30,60.
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### Re: Base 12 kicks all your asses

Hex forever and always. Period. There's nothing better.
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### Re: Base 12 kicks all your asses

Cheese wrote:Okay, base 60, if you're just going for factors. 1,2,3,4,5,6,10,12,15,20,30,60.

Yakk
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### Re: Base 12 kicks all your asses

Repeated factors are a waste.

I mean, base 2 can do 1/8 just as well as base 8 can.

1/810 in base 2 is 0.001

Base 60 is just 2*2*3*5. Base 30 is much more efficient, it doesn't duplicate the 2!
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

Ari
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### Re: Base 12 kicks all your asses

segmentation fault wrote:if the base is not a power of 2 its useless.

For machines or for humans?

Humans don't really care what a base is a power of, they care what it factors to and whether it's relatively small.

Cheese wrote:Okay, base 60, if you're just going for factors. 1,2,3,4,5,6,10,12,15,20,30,60.

Essentially, base 12 is base 60 lite- in terms of factors under 12, all you lose is 5 and 10, and it comes in a much more practical size for human counting. Also, it's countable on one hand still- you just have to be able to hide, crook, and fully extend your thumb.

On another note: Bidecimal systems remove the most annoying repeating number of all, .333..., and replaces it very simply with .4. They simplify a quarter to .3, and an eighth to .15. They can represent modular (am/pm) time in three digits, the month of the year in one digit, essentially clarifying immediately what format a date is written in, and the ease of factoring by three, four, and six outweighs the increased difficulty of factoring by five.

Also, a transition to a new base in human digits would allow us to use digits that use geometric tricks to allow for visual addition and subtraction.
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### Re: Base 12 kicks all your asses

9,699,690A is relatively small and has all of those properties.

1/1710 becomes {0}.{570570}9,699,690, a one digit decimal number.

I mean, how useful is that?

...

But practically, you could instead have a non-constant base.

That gives the nice property that all rational numbers are terminating decimals.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

zenten
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### Re: Base 12 kicks all your asses

photosinensis wrote:Hex forever and always. Period. There's nothing better.

Um, no. There's no advantage of 16 over 2, except that it looks a bit more normal to people who are used to base 10.

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### Re: Base 12 kicks all your asses

segmentation fault wrote:if the base is not a power of 2 its useless.
I tend to agree with this. I've always suspected that if we were trained in base 16 right from the start, rather than base 10, that all arithmetic would be much cleaner and significantly easier to rapidly see patterns in our heads. Everything would "just line up", and the patterns would make rapid memorization and arithmetic much more efficient for our brains. It's all about patterns and logical ordering.

Of course, the only way to test this, is to teach some kids nothing but base 16 right from the start, and see if they end up math geniuses. Anyone want to volunteer their kids?

Yakk
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### Re: Base 12 kicks all your asses

You do know that a computer that runs in base 10 is perfectly buildable?

And that there is some theoretical indication that a base 3 computer might be more efficient than a base 2 computer.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

HappySmileMan
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### Re: Base 12 kicks all your asses

zenten wrote:
photosinensis wrote:Hex forever and always. Period. There's nothing better.

Um, no. There's no advantage of 16 over 2, except that it looks a bit more normal to people who are used to base 10.

Much more easy for mental arithmetic, Since in base16 you add two digits for each 8 in binary, also saves space and time when writing numbers down.

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### Re: Base 12 kicks all your asses

zenten wrote:
photosinensis wrote:Hex forever and always. Period. There's nothing better.

Um, no. There's no advantage of 16 over 2, except that it looks a bit more normal to people who are used to base 10.

And it takes up less space on the page...but that's about it.

btilly
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### Re: Base 12 kicks all your asses

HappySmileMan wrote:
zenten wrote:
photosinensis wrote:Hex forever and always. Period. There's nothing better.

Um, no. There's no advantage of 16 over 2, except that it looks a bit more normal to people who are used to base 10.

Much more easy for mental arithmetic, Since in base16 you add two digits for each 8 in binary, also saves space and time when writing numbers down.

Uh, no. 2 digits for each 4 in binary.

Yakk wrote:You do know that a computer that runs in base 10 is perfectly buildable?

And that there is some theoretical indication that a base 3 computer might be more efficient than a base 2 computer.

Do not laugh. Some mainframes were built to run in base 10 because it was more convenient for machines that were going to spend all of their time handling currency.
Yakk wrote:But practically, you could instead have a non-constant base.

That gives the nice property that all rational numbers are terminating decimals.

And makes all math difficult to do.

If you want to make the representation of all rational numbers terminate, use continued fractions instead. Conceptually far simpler, and it terminates much, much faster.
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### Re: Base 12 kicks all your asses

If a number representation system doesn't have a closed form equation for the nth digit of pi, it ain't for me.
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### Re: Base 12 kicks all your asses

btilly wrote:
HappySmileMan wrote:Much more easy for mental arithmetic, Since in base16 you add two digits for each 8 in binary, also saves space and time when writing numbers down.

Uh, no. 2 digits for each 4 in binary.

What?

bin 1111 = dec 15 = hex f
bin 11111111 = dec 255 = hex ff

Two hex digits for each 8 binary digits.

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### Re: Base 12 kicks all your asses

Be oldskool and use base 60. It worked for the Babylonians and we still use it today for timekeeping and angular measurements!

Except for those deviant mathematicians and physicists that use radians. May they burn in hell!

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### Re: Base 12 kicks all your asses

Ari wrote:For machines or for humans?

for when humans become machines
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### Re: Base 12 kicks all your asses

The "use powers of 2 as a base" people are crazy, I tell you. There's no advantage in factoring those numbers as you get increasingly large. Using powers of 2 means every non power of 2 is going to be coprime to it and that means you get ugly decimal expansions. 1/3? 1/5? 1/6? 1/7? 1/9? Any non power of 2 gives you a decimal expansion. Base 10 does a little better, in that every number that can't be factored into 2s and 5s is given an infinitely long decimal expansion. Base 12 does no better, because you simply replace the 5 by a 3.

It's a matter of economy, that's all. Ten is convenient for us right now thanks to all the history behind it. Base twelve (and base six) aren't much harder to use and would provide us with different fractions, 1/5 would be harder but 1/3 would be easier.

Why not write everything in base 2*3*5*7*11*13*17*19 as another poster suggested? Well, for one thing, it's an awful lot of symbols to learn. We'd probably end up chunking everything in such a way that we'd split the symbols up, perhaps dividing each into two symbols, maybe even dividing each of those into three symbols, then dividing each of those... oh wait, we're just turning it into base 19. You can start at the other end, and you end up with base 2.

Economy. Even the Chinese don't expect everyone to know all the symbols of their language as base units, how could we expect people to be able to write something like nine million different numerals and remember what each is distinctly? In order to represent base 9699690 we'd need square symbols over 3000 pixels wide by 3000 pixels tall. Seriously? You expect people to remember the difference between two arbitrary symbols when they differ by only 1 black or white pixel?

The reason base 12 doesn't kick my ass all over the place is you only trade one factor for another. Teh lame. 2*3*5 = 30 combines the advantages of the two, from the alphabet that we have we know it's easy to get people to remember approximately thirty unique symbols, hopefully they'd remember the order a bit better, and we'd actually gain something instead of losing. I see no compelling reason to switch though.
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btilly
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### Re: Base 12 kicks all your asses

Notch wrote:
btilly wrote:
HappySmileMan wrote:Much more easy for mental arithmetic, Since in base16 you add two digits for each 8 in binary, also saves space and time when writing numbers down.

Uh, no. 2 digits for each 4 in binary.

What?

bin 1111 = dec 15 = hex f
bin 11111111 = dec 255 = hex ff

Two hex digits for each 8 binary digits.

Oops, brain fart on my part.
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### Re: Base 12 kicks all your asses

Anpheus wrote:It's a matter of economy, that's all.

According to one of last year's ARML power questions, ternary notation is the most efficient (#digits used*#available for each digit), and balanced ternary is a rather elegant notation.
17510 = 201113 = 1n0111balanced 3, if n = -1. So base 3 is nice that way.
As things stand for me, (base) 30 > 12 > 60 > 3 > 10 > 2 > higher prime numbers.
Also, stay away from base 20. The Mayans used that, and look what happened to them .
Oh, and yes, we do handle our time in a combination of base 60 and base 24. The numbers are still represented in base 10 though.
The base 12 society is still around, it seems at the Dozenal Society of America. But if they get their way, we have to modify the metric system while we Americans have fun because there are 10 inches to a foot, and a yard is .4 feet.
Yes, miles would be messy (3*5*11*25 isn't going to look nice in any base)
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### Re: Base 12 kicks all your asses

Economy of writing is pretty important too, and that's why base 16 is favored over base 2 when writing out numbers in binary, though most people can't be assed to remember Hex to binary conversions (F -> 1111, E -> 1110, D -> 1101...), it's a defacto standard.
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Yakk
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### Re: Base 12 kicks all your asses

0 = 0
1 = 1
2 = 1,0
3 = 1,1
4 = {2},0
5 = {2},1
6 = {3},0
7 = {3},1
8 = {4},0
9 = {4},1
10 = {5},0
11 = {5},1
12 = 1,0,0
13 = 1,0,1
...
359 = {29},{5},1
360 = 1,0,0,0
361 = 1,0,0,1
...
75599 = {209},{29},{5},1
75600 = 1,0,0,0,0
75601 = 1,0,0,0,1
...
174635999 = {2309},{209},{29},{5},1
174636000 = 1,0,0,0,0,0
174636001 = 1,0,0,0,0,1
...

I mean, how simple is that?

Addition is easy: just add place-wise, and do carrying based on the base of each place.

And you can turn any and all fractions into terminating decimal representations.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

Anpheus
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### Re: Base 12 kicks all your asses

Yeah, people would love to have to write out the commas and mark all the { } bits.

You know what simple is? Not changing a de facto standard that works.
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### Re: Base 12 kicks all your asses

LikwidCirkel wrote:
segmentation fault wrote:if the base is not a power of 2 its useless.
I tend to agree with this. I've always suspected that if we were trained in base 16 right from the start, rather than base 10, that all arithmetic would be much cleaner and significantly easier to rapidly see patterns in our heads. Everything would "just line up", and the patterns would make rapid memorization and arithmetic much more efficient for our brains. It's all about patterns and logical ordering.

Of course, the only way to test this, is to teach some kids nothing but base 16 right from the start, and see if they end up math geniuses. Anyone want to volunteer their kids?

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### Re: Base 12 kicks all your asses

base phi ftw

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### Re: Base 12 kicks all your asses

You can't really have a non-integer base, can you? because natural numbers (base e) would rawk.
Anyway, duodecimal ("dozenal") is good because:
-you gain so many factors
-only 2 extra numerals (ten/T and eleven/E)(T is a 2 rotated 1800, E is a backwards 3)(A & B are too hex-associated to work for dozenal, and don't look good as digits)
-check out these multiples (until the last digit repeats):
DECIMAL:
3: 3,6,9,12,15,18,21,24,27,30,33...
4: 4,8,12,16,20,24...
5: 5,10,15...
6: 6,12,18,24,30,36...
7: 7,14,21,28,35,42,49,56,63,70,77...
8: 8,16,24,32,40,48...
9: 9,18,27,36,45,54,63,72,81,90,99...

DOZENAL:
3: 3,6,9,10,13...
4: 4,8,10,14...
5: 5,T,13,18,21,26,2E,34,39,42,47,50,55...
6: 6,10,16...
7: 7,12,19,24,2E,36,41,48,53,5T,65,70,77...
8: 8,14,20,28...
9: 9,16,23,30,39...
T: T,18,26,34,42,50,5T...
E: E,1T,29,38,47,56,65,74,83,92,T1,E0,EE...

OLD: 3,7,9 are inconvenient to learn/remember, 5 is very easy.
NEW: 5,7,E are inconvenient, 3,4,6,8,9 are very easy, T is hard-ish, but okay once you know your 5s.

The main advantage is the 0.4 = 1/3 deal and all those pretty factors. It isn't a trade from 5 to 3, it's from 5 to 3*2.
Hex is lousy for fractions and has too many digits for times tables, etc.
Octal is even worse for fractions.

The one problem with dozenal: seven-line digital clock and calculator displays can't handle my favorite T symbol (the rotated 2).
solutions:
-backwards 7 looks like T
-extra line to make T not look identical to 2

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### Re: Base 12 kicks all your asses

It is a trade from 5 to 3, because 10 has as prime factors 5 and 2. You lose the 5, and gain a 2 in Dozenal.
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Xbehave
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### Re: Base 12 kicks all your asses

just taking number of (prime) factors is pointless, surely the best way to measure is in factors per size

base 60 has 4 prime factors 0.06'
base 12 has 3 prime factors 0.25
base 10 has 2 prime factors 0.2 (not much worse than 12)
base 6 has 2 prime factors 0.3'
base 2 has 1 prime factors 0.5

so in summary 60 is rubbish, 6 and 2 are too small to be practical, and 12 aint worth changing to.

i agree with using base 2 being useful as the stuff is binary (it is or isnt) but as they only have 2 as a prime factor theyre not practical for everyday useful.

p.s how can computers work in base 3? in a clocked quantum computer there are three states but clocking a quantum computer is abit pointless.
maybe base 8 could be used unclocked but that breaks down 2 a really fast way of sending base 2.
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### Re: Base 12 kicks all your asses

Some people say the three base states are: current flowing one direction, current flowing the other direction, no flow. I guess some circuits can be more efficient that way.

Maybe in our lifetimes we'll see bio computers build on base four, using DNA to store data.

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### Re: Base 12 kicks all your asses

could somebody explain the difference between bases 4,8,16 and just specail ways of showing base 2, i know the same can be said for any base converting between base 2,4,8,16 is just a case of replacing every 100 with 10 or every 10000 with 10 etc ...
its only base 3 that confused me with computers but now i get it, btw for base 8 i figure in a quantum computer youd use 2 electrons and that leaves 8 states uu, ud, du , dd, 0u, 0d, u0. d0 . but then again im basing it on what my friend explain and he said they dont use all 3 spins on a single electron which means every electron caried 3bits and which is can be used to represent 1 to 8 anyway. although ... figures out how to work a stupid base using the same concept as before
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### Re: Base 12 kicks all your asses

Base 4 allows you to use 2 and 3, Base 8 (Octal) allows the additional digits 4, 5, 6, 7, and Base 16 (Hexadecimal) allows the additional digits 8, 9, 10, A, B, C, D, E, F.

It's a matter of compactness. It's much easier to express a 32-bit number as only 4 hex digits, rather than write out 32 bits.
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### Re: Base 12 kicks all your asses

Would anyone consider themselves to be "proficient" in a base other than 10? For example, I quickly know that 180392 is about 3 times 60000, and I can imagine how big it is pretty easily. If you gave me 2C0A816, I couldn't tell you anything without converting to decimal.
spjork.

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### Re: Base 12 kicks all your asses

2C0A816?

That's about 3 times 1000016. That's really easy?

Every hex digit is 4 binary digits -- so that means the above number is about 1810 to 1910 binary digits.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

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### Re: Base 12 kicks all your asses

I'm pretty fast in dozenal, but long division is easier to begin with. You just need to take a few hours to learn the multiplication table and get used to the new digits.
Interestingly, the year in dozenal is 11E4 (Oneteen eleventy-four) which is not only fun to say, but only 8 years from the 13th century!
Clearly, we need to convert the world to base12 before then or we'll be the only ones celebrating the new century.

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### Re: Base 12 kicks all your asses

Actaeus wrote:You can't really have a non-integer base, can you? because natural numbers (base e) would rawk.

I would think it would come out to something along the lines of 221 in base 2.5 would be 2(2.5)^2 + 2^(2.5) + 1 which would be 8.5 in decimal. I think the same would follow for numbers like e and pi

Now what I'd like to see is base i

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base pi?