Date: Sat, 27 Jan 2001 22:39:42 GMT From: Salvo Bartolotta <bartequi@inwind.it> To: Mike Meyer <mwm@mired.org> Cc: freebsd-chat@freebsd.org Subject: Re: OT again: Re: hexidecimal literacy Message-ID: <20010127.22394200@bartequi.ottodomain.org> In-Reply-To: <14963.13797.116165.382738@guru.mired.org> References: <14963.8033.752142.149320@guru.mired.org> <20010127.20140200@bartequi.ottodomain.org> <14963.13797.116165.382738@guru.mired.org>
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[ redirected to -chat before somebody flames both of us :-) ] >>>>>>>>>>>>>>>>>> Original Message <<<<<<<<<<<<<<<<<< On 1/27/01, 9:56:05 PM, Mike Meyer <mwm@mired.org> wrote regarding OT again: Re: hexidecimal literacy: > Salvo Bartolotta <bartequi@inwind.it> types: > > >>>>>>>>>>>>>>>>>> Original Message <<<<<<<<<<<<<<<<<< > > On 1/27/01, 8:20:01 PM, Mike Meyer <mwm@mired.org> wrote regarding Re: > > hexidecimal literacy: > > > Mark B. Withers <mwithers@one.net> types: > > > > Oh gosh! > > > > I thought I understood it before, but looking at it like this > > > > simplifies it dramaticly!! > > > Just remember that this applies to interesting bases like 0, 1, Pi and > > > negative numbers :-). > > Hmm, I am afraid you are exaggerating a bit :-) > Actually, I'm not. I'm pretty sure this was from Knuth, back when I > was an undergrad. Unfortunately, my books are all in storage, so I > can't check on it :-(. <yet more nit-picking :-)> In the field (meant as a technical term of Algebra) of real numbers R, "0" is the neutral element of the "+" (or "sum") operation. However, 0 has no inverse with respect to "*" (or "multiplication"). Also, 0 raised to a positive integer is defined (and equals zero), but 0 raised to 0 has, in a strictly algebraic sense, NO meaning. Therefore, you canNOT generate integers; 0 cannot be a base. <aside> In Mathematical Analysis, or Calculus, "0 raised to 0" (ie something tending to 0 raised to something else tending to 0) is an indeterminate form (just express it in terms of exp and you see why), with a well-defined meaning. </aside> "1" -- a very important mumber since... Plato :-)) -- is the neutral element for the "*" operation. Hence 1 raised to any (relative) integer equals one, and canNOT be used to represent integers (ie N, whence NxN=Z, whence Z*Z'=Q, whence its "adherence" -- in a topological sense -- R, whence RxR=C). Incidentally, 1 raised to any real number can be defined (cf the construction of continuous homomorhisms mapping R to R+, viz the exponential functions). Oooooops. I was implicitly thinking in too an abstract mathematical way when I wrote my first reply :-) > > <nit-picking mode> > > 1) a positional notation making use of negative bases looks very > > awkward/impractical (you would have to utilize negative coefficients); > > OTOH, when working with such a positional notation, you are supposed > > to be working on N (ie the set of the "natural" numbers, or positive > > integers); and to add to all this, there are a number of approaches to > > the "construction" of N itself (Peano's, Cipolla's, the set theory > > with all its subtle problems... cf Bourbaki[sm]) > > > > 2) Pi, as "e" ~ 2.71828182..., is <gasp> a **trascendental** irrational > > number (!). I let you guess what kind of coefficients you have to use > > to generate integers. > > </nit-picking mode> > You seem to be focused on generating N. If that's the goal, then using > negative (or any base less than 2) or irrational bases is indeed > problematical. However, that doesn't change the fact that a string in > some base has a single, fixed value even for negative and > transcendental bases. As such, they can be safely used to represent > numbers, and make a perfectly valid base. Yup (except for 0 and 1). Possible but ackward/impractical. Either you use positive and/or negative coefficients to express a given integer, or transcendental coefficients. This recalls somehow to mind a vector space of dimension 1. I was also thinking in Bourbakistic terms. Mathematics IS beauty as well as elegance... :-) Best regards, Salvo (ok, back to compiling code now :-) To Unsubscribe: send mail to majordomo@FreeBSD.org with "unsubscribe freebsd-chat" in the body of the message
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