Subject: Re: possible A-F...

From: [email-masked]
Date: Sat, 10 Nov 2001 16:06:15 -0000

--- In lojban@y..., And Rosta <arosta@u...> wrote:
> Why is it [16] being a power of 2 a reason? Why is it a good thing 
that it is a power of two.

One startling advantage of this can be shown by taking hexadecimal as 
the example: the base of hexadecimal is 16, and representing all 
numbers in hexadecimal, if you halve 16, you get 8; if you halve 8, 
you get 4; if you halve 4, you get 2; if you halve 2, you get 1 -- 
bingo! Then, if you halve 1, you get 0.8; if you halve 0.8, you get 
0.4; if you halve 0.4, you get 0.2; if you halve 0.2, you get 0.1 -- 
bingo again! If you halve 0.1, you get 0.08, and so on. It's nice and 
clean and tidy and regular and simple. This works for any number 
system whose base is a positive integral power of 2.

What happens with duodecimal? The base of duodecimal is 12, and 
representing all numbers in duodecimal, if you halve 12, you get 6; 
if you halve 6, you get 3; if you halve 3, you get 1.6; if you halve 
1.6, you get, 0.9; if you halve 0.9, you get 0.46; if you halve 0.46, 
you get 0.23, and so on. Yuck! It's nasty and dirty and messy and 
irregular and complicated.

Why should this matter anyway? Consider the whole basis upon which 
different bases are being suggested in the first place. It's because 
of the fractions -- because certain fractions can be represented 
nicely in that particular base. *NO* base can represent all fractions 
nicely (this can be mathematically proven with a precise mathematical 
definition of what is meant by "nice", and I think it's pretty 
obvious anyway), so which fractions should we consider? Well, you 
could contrive to just pick any arbitrarily, but the simplest, 
smoothest, most natural, obvious, logical choice is the fraction 1/2, 
hence the above. A number system can represent nicely all numbers 
arrived at by successive halving if and only if it has a base that is 
a positive integral power of 2.

Also, consider that the prime representation (the prime 
representation of a number is positive integral powers -- any or all 
of the powers could be 1 and hence ignored -- of prime numbers 
multiplied together to give that number, so for example for 45 it is 
(3^2) * 5) of any number that is a positive integral power of 2 will 
clearly just be 2 to the power of something, the simplest form of 
prime representation possible. Whereas prime representations of any 
other numbers, such as 12 and 60 ((2^2) * 3 and (2^2) * 3 * 5 
respectively) are more complicated and are messy and irregular and 
not a natural, obvious choice.

> The benefits of 12 and 60 have been pointed out by Mark in several
> messages -- they make it easy to divide by 2, 3, 4, 5, 6, (10, 12, 
15, 20, 30).

So what though? Why does this outweigh the above? Dividing by 2 is 
the important thing. Why pick those other numbers? What's so special 
about them?

> I guess one can use traditional nonmetric measures to gauge which 
bases are most useful, at least for dividing things up. 12 and 60 are 
familiar of
> course. Old money had 12 pence to the shilling and 20 shillings to 
the
> pound and 21 to the guinea. There are 14 pounds to the stone 
(why?!?)
> and -- lo! -- 16 ounces to the pound (tho we in Britain talk about 
> half a pound and quarter of a pound when in grocery contexts).

Only the old-fashioned people! ;) ...Anyway, is 21 a good, useful 
choice? Or 20 or 14?

Sincerely,
Robert