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        riddles >> general problem-solving / chatting / whatever >> Base-12 Counting System
        
(Message started by: Christine on Jul 10^th, 2013, 9:38am)

Title: Base-12 Counting System
Post by Christine on Jul 10^th, 2013, 9:38am

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.

Title: Re: Base-12 Counting System
Post by towr on Jul 10^th, 2013, 10:35am

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.

Title: Re: Base-12 Counting System
Post by Christine on Jul 10^th, 2013, 12:19pm

on 07/10/13 at 10:35:01, 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.

Why base 8?

Title: Re: Base-12 Counting System
Post by towr on Jul 10^th, 2013, 1:02pm

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).

Title: Re: Base-12 Counting System
Post by 0.999... on Jul 11^th, 2013, 6:57am

"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?

Title: Re: Base-12 Counting System
Post by Grimbal on Jul 11^th, 2013, 10:16am

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?

Title: Re: Base-12 Counting System
Post by towr on Jul 11^th, 2013, 11:18pm

on 07/11/13 at 06:57:23, 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?

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 (http://mathworld.wolfram.com/BBP-TypeFormula.html)) 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.

Title: Re: Base-12 Counting System
Post by 0.999... on Jul 12^th, 2013, 3:53am

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 (https://en.wikipedia.org/wiki/P-adic_number) 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.

Title: Re: Base-12 Counting System
Post by towr on Jul 12^th, 2013, 5:47am

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 2^k, because there's a simple transformation where one digit in base 2^k corresponds to k digits in base 2.