Re: A Simplified Number System

From: Sean O'Leathlobhair (jwlawler_at_yahoo.com)
Date: 10/29/04 

Date: 29 Oct 2004 02:06:57 -0700

hrubin@odds.stat.purdue.edu (Herman Rubin) wrote in message news:<clri29$55l4@odds.stat.purdue.edu>...
> In article <d1835a57.0410280103.4ea8f30f@posting.google.com>,
> Sean O'Leathlobhair <jwlawler@yahoo.com> wrote:
> >hrubin@odds.stat.purdue.edu (Herman Rubin) wrote in message news:<clopm1$5q2m@odds.stat.purdue.edu>...
> >> In article <d1835a57.0410270050.f469166@posting.google.com>,
> >> Sean O'Leathlobhair <jwlawler@yahoo.com> wrote:
> >> >Arnold Victor <arvimide@earthlink.net> wrote in message news:<2u8b0hF22v56vU1@uni-berlin.de>...
> >> >> Sean O'Leathlobhair wrote:
>
> >> >> > Arnold Victor <arvimide@earthlink.net> wrote in message news:<EHhfd.7799$5i5.6829@newsread2.news.atl.earthlink.net>...
>
> >> >> >>Sean O'Leathlobhair wrote:
>
>
> ...................
>
> >> >The only advantage of hexadecimal is that it is easy to convert to and
> >> >from binary. This is of little use to humans. In the early days of
> >> >computing this was valuable but it is of little value today.
>
> >> There are many humans to which using binary is of considerable
> >> use. The compactness advantages of doing it in hexadecimal
> >> are considerable.
>
> >I am not saying that binary is of no theoretical use though I would
> >say that is has little day to day value for most people. With the
> >possible exception of the pi result mentioned by LSD, I don't think
> >that there is much theoretical interest in base 16. At most, it is a
> >short hand for what is really binary.
>
> I am a mathematical statistician, and there are many things
> in random number generation where base 2 is better.
> Information theory, which has numerous uses, was devised by
> people who probably still do not have proficiency in
> carrying out arithmetic in bases which are powers of 2, but
> nevertheless the unit is the bit, which was derived as
> "Binary digIT". The standardization of the byte is much
> later.
>
> >How many people outside research establishments (including computer
> >hardware and software development) use binary often? Even among those
> >that do, how many use hexadecimal for anything other than quickly
> >writing down binary numbers?
>
> They do have to work out binary designs, as the hardware is
> binary. Some of these use powers of 2; 4 is a common one.
> You are right in that hexadecimal is only used for writing
> down the numbers, but as I have remarked, this only puts a
> time penalty in doing the arithmetic by hand or on paper.
>
> >> I do not know the representation of Mayan, but the various
> >> representations of characters which need to be distinguished
> >> in a single "place" in the other systems of notation with
> >> which I have some familiarity is maximized in hexadecimal,
> >> with 16 distinct characters. Sumerian-Babylonian base 60
> >> had 15 characters, with 45 pairs occurring, and I have
> >> made up what I would use as a multiplication table, but
> >> it would take more multiplications and additions than using
> >> one character for each place.
>
> >> I do not know what Babylonians memorized, but I suspect
> >> it was the additions and multiplications corresponding
> >> to single characters.
>
> >> Greek used 27 characters, but only 9 could occur in a
> >> given place. They still did not have that much to memorize,
> >> as they were quite aware that the characters in the different
> >> places behaved similarly.
>
> >I think that the Greek system was decimal. The oddness was because
> >they did not use the place value notation. So instead of representing
> >10, 20, 30, . . . with the same symbols as 1, 2, 3, . . . and 0 as a
> >place marker, they used additional symbols.
>
> The Greek system, as that of most other people, seems to
> have been decimal. In fact, there seems to be a common
> origin of the representation, including vocabulary, in
> Indo-European languages up to 100. The Afro-Asiatic
> system seems also to have been decimal. I do not know
> if Sumerian was linguistically sexagesimal, but the
> numbering system evolved seems to have come from combining
> their one-hand representation with 12 alternatives with
> a basic decimal system. The Sumerian-Babylonian system
> is the only early one I know which uses place value, and
> it has been claimed that the Hindu use of place value was
> put in by those who knew the Babylonian.
>
> >Roman numerals had the same problem. In the absence of a place
> >marker, 10, 20, 30, . . . also required new characters. Again a
> >clumsy form of decimal.
>
> >Back to alternative bases which are represented similarly to modern
> >base 10, various simplicity arguments are opposed so it is hard to say
> >which base is easier.
>
> >With larger bases e.g. 12, 16 or 60, the representation is shorter and
> >arithmetic requires fewer steps. But more symbols are required and
> >the multiplication tables are larger.
>
> As I remarked, the sexagesimal system had alternating 10
> and 6, so only used 9 symbols for units, 5 for multiples of
> 10, and a 0 symbol, which was not always used.
>
> >With smaller bases e.g. 2, 5 and 8, fewer symbols are required and
> >multiplication tables are small. But representations are very long
> >and arithmetic requires more steps.
>
> >Bases which are powers of 2 allow easy conversion to and from binary
> >but few fractions terminate.
>
> >Bases with multiple distinct prime factors e.g. 12 and 30 have more
> >terminating fractions but cannot be easily converted to and from
> >binary.
>
> These are no worse to convert than base 10. In fact, 60
> might be easier, and 1000 another alternative.
>
> >I don't think that there is any objective answer to which base would
> >be best.
>
> >So let's stick with 10 since we are used to it.
>
> Communications with computers are becoming more important,
> including floating point. Coding theory seems to want to
> use powers of 2. The Fast Fourier Transform originally
> had to go by powers of 2, but now can go by small factors.
> In treating randomness, base 2 is quite convenient, with
> the useful operations starting with and, or, and not, and
> these do not translate at all to other bases. This is
> also the case in the other practical applications of set
> theory and Boolean algebras.

I don't think that we are disagreeing. I don't contest that binary
has many theoretical and technical uses. I am aware of most of the
things that you mention. Prior to entering computing, I taught pure
maths in a university.

My points were only:

Binary is of interest and use mainly to certain theoreticians and
technicians. I don't that an interest could be developed in the
general public.

Hexadecimal is of little interest at all except as a short hand for
writing binary. The odd pi in hexadecimal result mentioned by LSD
seems to be an exception.

Floating point can indeed pose difficulties. Corruption can occur if
it converted to decimal and back again. One of the early results in
Chaos theory was linked to this. A researcher was running a large
weather prediction program. In one run he saved the intermediate
results and later restored them and restarted the program. He was
surprised to find that the run started to diverge from a similar but
uninterrupted run. The explanation was that the save was in decimal
and hence the save / restore corrupted the data slightly. It is now
well known that these slight differences can have a significant effect
but at the time it was not.

My preferred solution to this is to not transfer data from computer to
computer via a human. If the human can be excluded, the computers can
talk in their native language or via a lossless translation (e.g. big
to little endian).

In my area of computing, we rarely use floating point. That is
because we are mostly dealing with money. So $0.01 means exactly that
neither more nor less. If it were stored in floating point then it
would be corrupted slightly. We usually use BCD.

Seán O'Leathlóbhair

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