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Christine
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Base-12 Counting System
« on: Jul 10th, 2013, 9:38am » |
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Should we switch to a base-12 counting system?
http://io9.com/5977095/why-we-should-switch-to-a-base+12-counting-system
Why do some people propose that we learn to count in twelves in addition to counting by tens?
Groups like the Dozenal Society of America advocate converting to numeral systems based on divisors of 60 because of their comparative ease with fractional computations.
Share you thoughts, please.
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towr
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Re: Base-12 Counting System
« Reply #1 on: Jul 10th, 2013, 10:35am » |
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I think the costs of switching outweighs the benefits. And if we were to switch to something else, I think I'd prefer base 8 or 16.
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Christine
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Re: Base-12 Counting System
« Reply #2 on: Jul 10th, 2013, 12:19pm » |
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on Jul 10th, 2013, 10:35am, towr wrote:| I think the costs of switching outweighs the benefits. And if we were to switch to something else, I think I'd prefer base 8 or 16. |
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Why base 8?
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towr
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Re: Base-12 Counting System
« Reply #3 on: Jul 10th, 2013, 1:02pm » |
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Fewer digits; in fact fewer than there are on two hands. And it gives you more to work with than binary; so although your numbers will be larger in octal than in decimal or hexadecimal, it'll still be a lot shorter than in binary.
If you consider teachability, octal is a lot better than hexadecimal or dozenal, precisely for that first reason: fewer digits for children to remember, and they can count on their fingers. And like hexadecimal it makes the step to thinking in binary a lot easier than decimal or dozenal (as well as having all the benefits of a base that's divisible by only one prime, which certainly isn't only a disadvantage as the dozenazis  might have you think).
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0.999...
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Re: Base-12 Counting System
« Reply #4 on: Jul 11th, 2013, 6:57am » |
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"as well as having all the benefits of a base that's divisible by only one prime"
What are some of these, if you don't mind?
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Grimbal
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Re: Base-12 Counting System
« Reply #5 on: Jul 11th, 2013, 10:16am » |
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Anyway, the correct term is "duodecimal". Creating a new word and repeating it over and over sounds like what a sect would do. It has nothing to do with improving arithmetics. It is about defining themselves as a select group.
I believe duodecimal would have made life a bit easier. But base 10 is there to stay. Maybe in the context of the French revolution such a radical change could have been pushed through. (And then the British would cling to their 10-base numbers). But if the French couldn't even maintain the use their reformed calendar, what hope is there to reform numbers?
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| « Last Edit: Jul 11th, 2013, 10:31am by Grimbal » |
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towr
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Re: Base-12 Counting System
« Reply #6 on: Jul 11th, 2013, 11:18pm » |
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on Jul 11th, 2013, 6:57am, 0.999... wrote:"as well as having all the benefits of a base that's divisible by only one prime"
What are some of these, if you don't mind? |
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I'm afraid I'm going to have to disappoint you and admit I was bluffing.  I've been trying to think of some, but it doesn't generally make a lot of difference. Though of course hexadecimal has that rather nice option to calculate arbitrary digits of pi; and there's similar (BBP-type) formulas for other constants which often (but not always) use a power-of-prime base. To be honest I had expected some more advantages in the area of modular arithmetic, but I'm struggling to find them.
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0.999...
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Re: Base-12 Counting System
« Reply #7 on: Jul 12th, 2013, 3:53am » |
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Initially, I had thought some less than obscure properties of base p would carry over to base pn for arbitrary n.
However, if we cannot even establish a rule for exactly the number of trailing zeros of a product of numbers based on the corresponding numbers of the factors, then as you noted modular arithmetic seemingly becomes equally difficult in base pn as it does in an arbitrary base regardless of the modulus.
This is exactly the same reason that the n-adic numbers--using composite n--are not studied as thoroughly as the p-adic numbers are. No matter how clever one is, he or she will never be able establish an n-adic norm on the rationals, by the way, because there is a result stating that such a norm would have to be a power of one of the p-adic norms.
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| « Last Edit: Jul 12th, 2013, 3:54am by 0.999... » |
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towr
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Re: Base-12 Counting System
« Reply #8 on: Jul 12th, 2013, 5:47am » |
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Well, you can easily transform a base pn number to base p and vice versa. (Although that's true of any bn <-> b) So you can easily use any advantages a prime-based number gives.
For example, since we can calculate the nth hexadecimal digit of pi without calculating the preceding ones, we can also do this in any other base 2k, because there's a simple transformation where one digit in base 2k corresponds to k digits in base 2.
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