From owner-freebsd-chat Sat Jan 27 14:37:20 2001 Delivered-To: freebsd-chat@freebsd.org Received: from relay4.inwind.it (relay4.inwind.it [212.141.53.75]) by hub.freebsd.org (Postfix) with ESMTP id 9E16837B401 for ; Sat, 27 Jan 2001 14:37:00 -0800 (PST) Received: from bartequi.ottodomain.org (62.98.171.35) by relay4.inwind.it (5.1.056) id 3A6DB81B000FF9B4; Sat, 27 Jan 2001 23:36:55 +0100 From: Salvo Bartolotta Date: Sat, 27 Jan 2001 22:39:42 GMT Message-ID: <20010127.22394200@bartequi.ottodomain.org> Subject: Re: OT again: Re: hexidecimal literacy To: Mike Meyer Cc: freebsd-chat@freebsd.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> X-Mailer: SuperCalifragilis X-Priority: 3 (Normal) MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Sender: owner-freebsd-chat@FreeBSD.ORG Precedence: bulk X-Loop: FreeBSD.org [ redirected to -chat before somebody flames both of us :-) ] >>>>>>>>>>>>>>>>>> Original Message <<<<<<<<<<<<<<<<<< On 1/27/01, 9:56:05 PM, Mike Meyer wrote regarding OT=20 again: Re: hexidecimal literacy: > Salvo Bartolotta types: > > >>>>>>>>>>>>>>>>>> Original Message <<<<<<<<<<<<<<<<<< > > On 1/27/01, 8:20:01 PM, Mike Meyer wrote regarding R= e: > > hexidecimal literacy: > > > Mark B. Withers 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 :-(. In the field (meant as a technical term of Algebra) of real numbers R,=20 "0" is the neutral element of the "+" (or "sum") operation. However, 0=20 has no inverse with respect to "*" (or "multiplication"). Also, 0=20 raised to a positive integer is defined (and equals zero), but 0=20 raised to 0 has, in a strictly algebraic sense, NO meaning. Therefore,=20 you canNOT generate integers; 0 cannot be a base. =20 =20 "1" -- a very important mumber since... Plato :-)) -- is the neutral=20 element for the "*" operation. Hence 1 raised to any (relative)=20 integer equals one, and canNOT be used to represent integers (ie N,=20 whence NxN=3DZ, whence Z*Z'=3DQ, whence its "adherence" -- in a=20 topological sense -- R, whence RxR=3DC).=20 Incidentally, 1 raised to any real number can be defined (cf the=20 construction of continuous homomorhisms mapping R to R+, viz the=20 exponential functions). =20 Oooooops. I was implicitly thinking in too an abstract mathematical=20 way when I wrote my first reply :-) > > > > 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 a **trascendental** irratio= nal > > number (!). I let you guess what kind of coefficients you have to us= e > > to generate integers. > > > 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=20 use positive and/or negative coefficients to express a given integer,=20 or transcendental coefficients. This recalls somehow to mind a vector=20 space of dimension 1. I was also thinking in Bourbakistic terms. Mathematics IS beauty as=20 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