From bloke_without_a_favourite_colour@yahoo.co.uk Sat Nov 10 08:06:15 2001 Return-Path: X-Sender: bloke_without_a_favourite_colour@yahoo.co.uk X-Apparently-To: lojban@yahoogroups.com Received: (EGP: mail-8_0_0_1); 10 Nov 2001 16:06:15 -0000 Received: (qmail 11025 invoked from network); 10 Nov 2001 16:06:15 -0000 Received: from unknown (216.115.97.172) by m2.grp.snv.yahoo.com with QMQP; 10 Nov 2001 16:06:15 -0000 Received: from unknown (HELO n25.groups.yahoo.com) (216.115.96.75) by mta2.grp.snv.yahoo.com with SMTP; 10 Nov 2001 16:06:15 -0000 X-eGroups-Return: bloke_without_a_favourite_colour@yahoo.co.uk Received: from [10.1.10.127] by n25.groups.yahoo.com with NNFMP; 10 Nov 2001 16:05:56 -0000 Date: Sat, 10 Nov 2001 16:06:15 -0000 To: lojban@yahoogroups.com Subject: Re: possible A-F... Message-ID: <9sjj9n+k3ce@eGroups.com> In-Reply-To: User-Agent: eGroups-EW/0.82 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Length: 3412 X-Mailer: eGroups Message Poster X-Originating-IP: 62.64.151.46 From: bloke_without_a_favourite_colour@yahoo.co.uk X-Yahoo-Profile: bloke_without_a_favourite_colour X-Yahoo-Message-Num: 12007 --- In lojban@y..., And Rosta 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