Subject: Re: 360 degees
From: David Madore
Date: Fri, 17 Jan 1997 13:38:44 +0100
Angel Garcia wrote:
> The "HOURS" comes from latin "HORA" which in turn comes from
> ancient Egypt the God HORUS:
That is not true. "hora" comes from the proto-indo-european word
"y h1 e r h2" (where h1 stands for schwa1 which was pronounced, when
a vowel, something like o, and h2 stands for schwa2, which was
pronounced something like a). The initial "y" (yod) was transformed
into h in latin, and simply disappeared in classical greek, giving
"ora" (long o). In the germanic languages, the word in question
became "jahr" in german and "year" in english. This shows plainly
that the first sound was not like an h, and so the word could not
come from the name of the egyptian god (which I doubt was pronounced
even close to "horus", anyway; but even less like "yorus"). The
meaning of "y h1 e r h2" is unclear; it certainly had a relation to
a solar cycle of some kind, but whether the year or the hour, it is
difficult to tell.
David A. Madore
(david.madore@ens.fr,
http://www.eleves.ens.fr:8080/home/madore/index.html.en)
Subject: Re: Proof of infinite prime #'s
From: Jim Walters
Date: Fri, 17 Jan 1997 09:13:59 -0800
Dr. Richard L. Hall wrote:
>
> The standard proof is to assume that the number (say n) of primes is
> finite. Let the prime numbers be p1, p2, p3, ..., pn. Now form the new
> number p = p1 * p2 * p3 * ... * pn + 1. Now either (1) p is prime or
> (2) p is divisible by a prime. If (1) p is prime, repeat the process ad
> infinitum. If (2) p is not prime, it must be divisible by a prime. but
> clearly it is not divisible by any of the known primes p1, p2, p3, ...,
> pn. So there must be a new prime number p(n+1) that is a factor. Repeat
> this process ad infinitum.
Thanks.
Gregory Chaitin pointed out to me that his algorithmic information
theory
proof can be found at the end of section 3 on the following Web page:
http://www.research.ibm.com/people/c/chaitin/georgia.html
A couple of years ago I started a thread in this newsgroup speculating
on whether the technique of Chaitin's proof could be extended so as to
provide a information theoretic proof of the Prime Number Theorem. The
consensus in the newsgroup was that the bounds couldn't be made tight
enough.
Jim Walters
jwalters@brls.com
Subject: Clinton pardons Ohair on condition she converts to buddhism!
From: "John P. Boatwright"
Date: Fri, 17 Jan 1997 05:46:13 +0000
Your Name wrote:
> But I guess God must approve of incest, as only a short time later -
> Biblically speaking of course - He pares the population down to Noah,
> Noah's wife and their children. So from this small, inter-related
> coregroup, the world is repopulated.
Well again you're missing the whole point:
God makes everyone and everything. Do you demand Intel build
processors by your method? Nah, they do it the way they want to.
>
> Yep.
>
> Do the math. And use math to prove the existence of God.
OK, I have several calculators, what math?
>
> Then take the God the math's existence proves, and reconcile Him with
> the tenets you would have us live by.
Sorry to pop the thread bubble, but I don't use math to prove God,
God proves himself daily and in the bible prophecies coming true.
> Somehow, the equation just doesn't balance.
I have some old algebra books, what's the problem? It occurs to me
you're probably discouraged from getting non-substantial answers
to your math problems from the atheists and physicists frequenting
this newsgroup. Sure I could solve your problem, but then you'd
miss out on the knowledge you'd gain by doing it yourself.
Good luck anyway (double check your answers).
Subject: Re: How to count in hex Was - Re: What is binary?? (or: how do you explain it to an idiot?)
From: Ian Lynagh
Date: Fri, 17 Jan 1997 17:18:33 +0000
Once upon a time, in the land of sci.math, Dan Larsen eloquently
composed:
>I understand the fundamentals of binary, but how do you count in Hex?
>
>Ie, how do you translate 8F into base ten, and how do you translate 254
>into hex?
>
Base 10 number Base 16 equivalent
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
10 A
11 B
12 C
13 D
14 E
15 F
16 10
To answer your questions,
8F hex in decimal is:
8*16^1 + F*16^0 = 8*16^1 + 15*16^0 = 128+15 = 143
In general it would be:
^n + ^n-1 ... + ^0
To translate 254 decimal into hex:
Find the highest power of 16 (you could use:
int(log(254)/log(16)) where log is of any base).
In this case 16^1 is the highest power of 16.
Keep subtracting 16^1 from 254 until the number if
less than 16^1.
ie. 254-16 = 238
238-16 = 222
.
.
.
30-16=14
14 < 16^1
I have subtracted 16^1 15 times, so the first digit
is F. Now do the same for 16^0. There are 14 of them,
so the next digit is E. When you reach 16^0 you have
all the digits, so the answer is FE.
If you don't understand anything, please feel free to
ask more questions.
See ya!
Ian
_ _____ _ _
(_)( _ )( ) ( ) Ian Lynagh - ian@lynagh.demon.co.uk
| || (_) || `\| | http://www.lynagh.demon.co.uk/
| || _ || , ` |
| || | | || |`\ | If a train station is where a train
(_)(_) (_)(_) (_) stops, what is a workstation?
Subject: Re: 360 degees
From: David Kastrup
Date: 17 Jan 1997 17:14:02 +0100
David Madore writes:
> That is not true. "hora" comes from the proto-indo-european word
> "y h1 e r h2" (where h1 stands for schwa1 which was pronounced, when
> a vowel, something like o, and h2 stands for schwa2, which was
> pronounced something like a). The initial "y" (yod) was transformed
> into h in latin, and simply disappeared in classical greek, giving
> "ora" (long o).
Minor nag: classical Greek uses breathings to represent an initial
"h" sound, so it is not really disappeared. The breathings became
necessary because the Ionic script being used for Attic had already
used up the H letter (which was used in some Western Greek scripts for
the Latin H sound) for Greek Eta. For some time, initial h was
still pronounced, but not transcribed. Finally, the spirit asper was
introduced for writing it.
Of course, today's Greek does not pronounce it any more, but has
written it up to a few decades ago...
--
David Kastrup Phone: +49-234-700-5570
Email: dak@neuroinformatik.ruhr-uni-bochum.de Fax: +49-234-709-4209
Institut f=FCr Neuroinformatik, Universit=E4tsstr. 150, 44780 Bochum, Germa=
ny
Subject: Re: Cpx. Anal. Q. about approx. with holomorphic functions
From: tleko@aol.com
Date: 17 Jan 1997 16:21:41 GMT
In article <5bm08i$cs9@gap.cco.caltech.edu>
ikastan@alumnae.caltech.edu (Ilias Kastanas) wrote:
:
:In article <19970116144401.JAA21708@ladder01.news.aol.com>,
: wrote:
:>In article <5beek3$lj$1@nuke.csu.net> ikastan@sol.uucp (ilias kastanas
:>08-14-90)
:>wrote:
:>:
:>:In article <5bdu97$sv7@engnews2.Eng.Sun.COM>,
:>:Jeffrey Rubin wrote:
:>:@
:>:@Lastly, I said, "all right, I can extend f to be continuous on all of
:>:@the unit circle and, using the Poisson integral, I can find a g which
is
:>:@harmonic in U and continuous on D which coincides with f on K."
:>However,
:>:@the problem asks for a holomorphic f, not a harmonic one.
:>:
:>: That works too, if need be. The Poisson kernel is after all the
:>: real part of 1+z/1-z , z in U.
:>
:> How does it work ?! Is it holomorphic ?
:
: Not in general; but you can obtain approximating polynomials. The
: Poisson kernel converges to 0 as r -> 1, uniformly for theta bounded
: away from 0.
Not in general. This is fine.
How about in particular when it is holomorphic ?
tleko@aol.com
Subject: Re: Geometry Problem
From: Jay Belanger
Date: 17 Jan 1997 13:41:30 -0600
"OD" writes:
> Alex Papazoglou wrote
> > Here's a little problem I came across. Can someone give
> > me its proof?
> >
> > Assume a (finite) number of points on the same plane:
> > A1, A2, A3, ..., An (not necassarily in that order) with
> > the following property: On the line connecting two
> > of the n points at least one more of them can be placed.
> > Prove that they are all points of the same line.
> >
> > Alex
> >
>
> This is obviously true for n<=2. Now suppose it is true for n=k>=2, and
> consider a set of k+1 points. By hypothesis, the first k points lie on the
> same line L.
Why? It is not immediate that if k+1 points have the above property,
then some k of these points will also have the property. (This is
true in the Euclidean plane of course (since the points will be
collinear), just not immediate. A non-Euclidean counter-example is the
seven-point projective plane.)
If the last point were not on the line L, then the line L'
> joining the point A(k+1) to A1 would not contain other points of the set.
> Indeed L and L' are different because A(k+1) belongs to L but not to L',
> hence they have at most one point in common. Consequently A(k+1) belongs to
> L, and the proposition is proved.
>
Subject: Re: Vietmath War: If US had been parliamentary, no Vietnam war?
From: jaskew@yoyo.cc.monash.edu.au (Joseph Askew)
Date: 15 Jan 1997 23:29:10 GMT
Rich Rostrom (R-Rostrom@bgu.edu) wrote:
This has some bizzare follow-ups. If I could change them I would
but my poster will not let me. I apologise to the readers of all
the groups that got stuck on this list. How it happened I don't
know but I hope everyone will remove all but s.h.w-i if they can.
: > Interestingly enough, while the US was busy getting bogged down in
: > Vietnam, the UK was engaged in fighting in Borneo, in remarkably similar
: > political situations. The UK military position wasn't as good as that of
: > the US; the Borneo border was massively longer than that which the
: > Americans had to deal with, and the terrain very much harder.
: >
: > Nonetheless, the UK was successful.
As I point out to Richard Rostrum below you are confusing the
Emergency with the Confrontation. The fighting in Borneo was
against Indonesian "volunteers" aiming to create a Greater
Indonesia. Malays and Indonesians being more or less the same
people. The British fought the mostly Chinese Communist Party
of Malaya (ie excluding Borneo) in Malaya itself. The British
military position was excellent not only did the Communists have
no land border with friends on the other side, as Mr Rostrum
points out, but the Chinese Communists got little to no support
A
from the majority Malay community in Malaya. The British also
had their own administration which was tolerably corruption free
and had many local links. Above all the British kept the military
under civilian control whereas the greatest failing in Vietnam
was that the American military was not only a law unto itself
(whatever people say about fighting from Washington) but each and
every branch of the military did what it liked without reference
to and often in defiance of what everyone else was doing. Finally
the British "won" but only at the price of leaving Malaya to the
Malays. They defeated the Communists but they were forced out of
Malaysia. How is that a victory?
: The main difference was that there was no land border with a Communist
: state a few miles away. The Communist guerrillas in Malaya were forced
: to operate entirely on their own resources. The Viet Cong were supported
: by hundreds of thousands of North Vietnamese troops and lavish quantities
: of arms.
I think there is some confusion here between the Emergency fought
mostly in Malaya and Confrontation witht Indonesia fought in Borneo.
They are two separate struggles. Borneo does and did share a land
border with Indonesia which did supply the "rebels". The Communist
Party of Malaya only shared a land border with Thailand. As a sign
of how short the Communists were when it came to weapons the British
were able to keep files of *each* weapon they had and keep track of
where they were. Not all of them all the time but pretty much all
of them.
: The Borneo border was exposed; but the Indonesian side of it is far
: more remote from any civilized base area than was the inland border of
: South Vietnam. Sarawak was not the key part of Malaysia, and the
: infiltrators had relatively little local support. (My understanding
: is thyat he Dyaks took to collecting the heads of the infiltrators.)
This is not the Emergency (anti-Communist) fighting but the later
Confrontation with Indonesia. The Communists did not take part in
this. Notice that the Malayan Communists were mostly Chinese and so
had little support in the Malay community during the Emergency.
: Also, after 1966, the Indonesian government no longer supported
: insurrection in Malaysia, abandoning Suharto's "konfrontasi" and
: expelling or killing his Communist friends.
Indonesia never gave any support to the Malayan Communists. The
idea that Suharto had many Communist "friends" is a myth and the
Confrontation is a separate issue.
Now you would never make that mistake in this country as our War
Memorials list the two separately. We sent soldiers to both and
had a few minor casualties.
Joseph
--
"Blessed are the Peacemakers, for they shall inherit the Earth"
- President Bill Clinton
Subject: Re: Why do Black Holes Form at all?
From: odessey2@ix.netcom.com (Allen Meisner)
Date: 17 Jan 1997 18:52:42 GMT
In <5bktgs$1s2@nntp1.u.washington.edu> hillman@math.washington.edu
(Christopher Hillman) writes:
>
>In article <5bj9ia$562@dfw-ixnews7.ix.netcom.com>,
>odessey2@ix.netcom.com (Allen Meisner) writes:
>
>|> Ok Mr. Hillman, what I would like to ask now is whether the
>|> topology of the velocity space is responsible for the inertial
motion
>|> of a body in the same way that the topology of the gravitational
field
>|> is responsible for the acceleration of a body.
>
>I can't understand what you have in mind here. In the case of the
>acceleration of a body, it would be more correct to say that the
>GEOMETRY of space-time is modified by the presence of a
>massive body. The topology of space time can have global effects, but
>acceleration is a local phenomenom, and because all space-times are
locally
>homeomorphic to R^4 the TOPOLOGY must be irrelevant to any such
phenomena.
>Curvature and the presence of matter in some region of spacetime are
>both local phenomena.
>
>|> Why does a curvature in space time constrain an object to move?
>
>I think you still have a misunderstanding here. Nothing physical
MOVES in
>space-time; rather, the mental idea of a body being represented by a
>"moving point" is replaced by the idea of a body being represented
(throughout
>its existence) by a "world line", a (possibly non-straight) curve in
space-time.
>The "length" between two points (events associated with a time and a
spatial
>location) on such a curve is interpreted as the time interval between
these
>events as measured by a clock carried with the body during its motion.
>
Do you mean that the worldline of a body preexists in some sense
and that it is only because our cognition of reality is temporally
limited by our senses that the illusion of motion arises? If we could
be present throughout time, we would see a continous picture of the
body stretching from the infinite past to the infinite future? This is
very counterintuitive. I must admit that I do not like it. Here is
another idea. Time dilation or length contraction, whichever you
prefer, creates a gap in space which stresses or stretches space. The
body is contrained to move in order to reduce the gap and therefore the
stretching and stressing of space. Is this a plausible expalnation? I
admit that it is highly speculative, but it is simple and elegant.
Regards,
Edward Meisner
>Given these assumptions, it is reasonable to ask how gravitation can
be
>intrepreted as an effect of the curvature of the space-time. The
simplest
>answer is by analogy: on a sphere, the shortest distance (along any
curve
>constrained to lie on the surface of the sphere) between two points
lies
>along one arc of a great circle (a circle of maximal radius such as
the
>equator). For instance, all LONGITUDE lines on a globe are geodesics
>(shortest length curves) in this sense, but of the LATITUDE lines,
only
>the equator is a geodesic.
>
>Now consider two parallel geodesics (straight lines) in euclidean
space.
>As we all know, these lines remain parallel all along their length.
>Contrast two longitude lines on a globe. They start off being
parallel
>(imagine two travelers going North starting at the equator, for
instance) but
>thereafter they CONVERGE (eventually meeting at the North pole). This
>convergence of initially parallel geodesics is one effect of the
(positive)
>curvature of the globe. This is a LOCAL effect because you can detect
>convergence no matter how close the initially parallel geodesics are
to
>begin with.
>
>Going back to the computation in which two particles were falling
radially
>toward a massive object, the closer one pulls ahead as time increases.
>Remember that in the space time picture we replace the
>idea of two moving points with the idea of two world-lines which,
since
>no forces other than gravity are acting, we are assuming are
geodesics.
>Then, the "pulling ahead" of the closer particle appears as DIVERGENCE
>of initially parallel geodesics, and this divergence corresponds to
the
>(negative) curvature measured by one component of the curvature
tensor.
>
>|> is time dilation the only explanation for the Lobachevsky
curvature?
>|> I must admit that I can
>|> not see how this is so. Wouldn't time dilation give you the flat
>|> spacetime of SR and the Lorentz metric?
>
>I addressed several separate issues in seperate postings, one of which
>concerned general relativity and the rest special relativity, in which
>only flat spacetime is considered. The "velocity spaces" I described
>in one of these postings are indeed surfaces of constant curvature,
but
>they are the exact analogs in Minkowsky space (flat t^2 - x^2 - y^2 -
z^2
>metric) of the sphere, which is the surface of constant distance from
>the origin in ordinary space (x^2 + y^2 + z^2 metric). In the case
>of Minkowsky space, it turns out that the analogous surface of
constant
>distance from the origin has three parts, two copies of hyperbolic
space
>(only one of which is the velocity space for ordinary particles)
>and the tachyon velocity space. If you have a book discussing
analytic geometry
>look at the figure depicting a hyperboloid of two sheets--- this
>is the three dimensional analog of the copies of hyperbolic space,
>and consists of two copies of the Lobachevsky plane-- and look at the
>figure depicting a hyperboloid of one sheet--- this is the three
dimensional
>analog of the tachyonic velocity space.
>
>Hope this clarifies things!
>
>Chris Hillman
Subject: Re: Evidence for God's Existence - TRY Math
From: tricer@news.HiWAAY.net (Richard Trice)
Date: 17 Jan 1997 15:21:22 GMT
: Well, for one thing, he doesn't have fingers (being "without body, parts
: or passions"). When it says Adam became a living soul as a result of
: God's imparting His own breath to him, it doesn't imply God has lungs,
: or breathes oxygen.
I beg to differ. The bible clearly states that God was a walking talking
human for a while. Actually, until the flood, if I am not mistaken. Not
that I believe it. But hey, if you think about it, it gives new credence
to the God was a space alien theory.
:
: > Then He forbade them from eating of the tree of knowledge. Eve, that
: > hussy, that temptress, talked Adam into it anyway.
: "The knowledge of good and evil," please. The tree and the forbidding
: could have been literal (or not), but obviously the acquisition of the
: "knowledge of good and evil" through ingestion of its fruit could not
: have been a matter of natural metabolism.
Not according to ABC... "Fish is good for your brain. It makes you smart"
Certainly the story doesn't
: suggest that the "knowledge of good and evil" immediately produced in
: Adam and Eve when they ate the fruit amounted to accurate and/or
: comprehensive ethical enlightenment. Their eyes were opened, as the
: serpent had promised, but what they saw was a need to cover themselves
: with fig leaves and hide from God. "Who told you that you were naked?"
Quick question. Why did God need to ask? Wasn't he all knowing back
then?
: demands God, clearly not ratifying their interpretation of the lesson of
: "good and evil" they had received. Confronted with his disobedience,
: Adam blames both Eve and God:
"a Fox condemns the trap, not himself."
"The woman whom Thou gavest to be with me,
: she gave me from the tree ..." Genesis doesn't imply the validity of
: either aspect of that excuse.
: > After eating of this fruit, Adam and Eve were cursed with the sin of
: > lust...yet from out of this sin comes procreation. Interesting paradox, no?
Which is why we are all born of sin, no? ....also why the baptists teach
everyone that sex is bad,... period. Of course, this makes no sense with
what god says later..... "be fruitful"
: Doesn't say that, at all. Of course, the alienation which potentially
: transforms Adam's exultant delight in finding a soulmate ("Here at last
: is bone of my bone and flesh of my flesh!") into a power relationship
: ("Yet your desire shall be for thy husband, and he shall rule over you")
: is an aspect of Adam's (and Eve's) fallenness.
I don't think I wanna be having sex with anyone who is at all related to
me..... "bone of my bone and flesh of my flesh"
: > The only people in the world. Yet Cain slays Abel, is exiled, goes away
: > and later comes back with a wife.
Now wait, to be fair. Eve lived to be very old, and menopause didn't
exist back then and men could have kids at 400 and there wasn't this whole
don't have sex with your cousin routine..... or its all a bunch of bull.
: > Hmmm. But it was clearly stated these were the only people. Now, an
: > unknown has somehow entered the equation. Guess Cain's wife is an
: > X-factor.
She was the cousin to the sister of the son's niece's brother of the
nephew's uncle's father.......yakko,wakko,and dot.
: Cain could have had a wife who went into exile with him. But the
: problems of who she was, and whether their marriage was incestuous (to
: the extent they are problems) are obviously unaffected by that. The
: whole book (I mean, the whole Book) is concerned only with Adam, who
: became a living soul when God (Who doesn't breathe and hasn't got lungs
: or a diaphragm or lips) breathed into his nostrils, and Eve, who was
: taken from Adam's side, and their descendants. The words for "man" and
: "woman" used in the creation narrative(s) in Genesis _are_ "Adam" and
: "Eve," and in that respect our English translations might be
: characterized as somewhat misleading (or not, depending on one's point
: of view). Adam was the only Adam, and Eve was the only Eve, but they may
: or may not have been the only reproductive units. This is the
: possibility circumspectly hinted at by C.S. Lewis when he refers
: poetically to Cain's wife as an "unsmiling hominid."
Ah... is that an unsmiling hominid in your pocket or are you just glad to
see me routine.
: > Okay, fine. From two surviving men, and three surviving women (Adam,
: > Cain, Eve, Cain's wife and Adam and Eve's daughter) EVERYONE else who
: > eventually populates the planet springs.
:
: > How many generations does this equation have to be drawn before the
: > factor of "incest" is removed?
Well none. According to the church, their genes were pure, and the
incestual variations that occured are from what springs all of our
maladies today!
: > But I guess God must approve of incest, as only a short time later -
: > Biblically speaking of course - He pares the population down to Noah,
: > Noah's wife and their children. So from this small, inter-related
: > coregroup, the world is repopulated.
Well... Noah was pretty pure.
: Even without taking the "unsmiling hominid" speculation into account,
: Christian (and Jewish and Muslim) intellectuals have never had a problem
: conceding that any laws against marriage of siblings, cousins, etc.
: would have given way, for as long as needed be, to the higher necessity
: of launching the Adamic (and Noachic) race. In Old Testament terms,
: intrinsic right and wrong are one thing, and specific "laws" (actually,
: "commandments/blessings") are something quite different. There weren't
: any divine "laws" against incest or anything else (at least, the OT
: doesn't report any) until Moses, and the legislation he transmitted was
: only for a particular people who chose to receive those laws.
: And now, having sufficiently defended the illustrious cause of Gay
: Christianity against the wily slurs of the wicked atheists, I pose once
: again my oft-repeated question: WHAT THE HECK HAS IT GOT TO DO WITH
: ALT.POLITICS.HOMOSEXUALITY ONE WAY OR THE OTHER?!?!
Don't know! but thanks for your comments!
Subject: Re: 360 degees
From: "Ben Z. Tels"
Date: 17 Jan 1997 19:57:38 GMT
As far as I know it is because this number is easily divisible by several
other numbers, including 1,2,3,4,5,6,15, etc., which gives you nice numbers
in lots of basic divisions.
--
Ben Z. Tels
B.Z.Tels@stud.tue.nl
optimusb@stack.nl
"The Earth is the cradle of the mind, but one cannot stay in the cradle
forever."
--Tsiolkovsky
Loren Hardy wrote in article
<32DD9B3F.77D6@pantheon.yale.edu>...
> Does anyone know why there are 360 degrees in a circle?
>
Subject: here's a program to calculate pi, version 3.0
From: jasonp@Glue.umd.edu (Jason Stratos Papadopoulos)
Date: 17 Jan 1997 18:55:47 GMT
Hello again. Here's the next version of my QuickBasic program to
crunch out pi. Give it to anyone you want, I don't care.
jasonp
-------------CUT HERE INCLUDING THIS LINE--------------------------
DECLARE SUB PrintOut (words%)
DECLARE SUB Multiply (term%(), mult&)
DECLARE SUB Divide (term%(), denom&)
DECLARE SUB Add (sign%)
DECLARE SUB FastDivide (denom&)
'Program to calculate pi, version 3.0
'I've changed the algorithm to Machin's formula using Euler's atan series
'and cleaned up the subroutine calls. All in all, this version is about 25%
'faster than version 2.0; when compiled it requires 65 seconds for 5000
'digits on a 486 66MHz computer. Also, since Machin's formula converges much
'faster than Euler's, the program can find many more digits of pi before
'hitting overflow. In fact, I think the number of digits is limited only by
'the size of the 64K segment QuickBasic assigns for arrays (tops out around
'120,000 digits)
'
'Sadists who want even more digits from this program can run QuickBasic
'with the /AH option for huge arrays, and change the program to work with
'3 digits at a time instead of 4. Just change any instance of 10000 to 1000,
'change \4 to \3 in the variable "words" below, and fiddle with the PrintOut
'subroutine.
'
'This program has come a long way from version 1.0; thanks are due to
'Christian Goldbach, Randall Williams, and Bob Farrington for good ideas.
'One final note for speed freaks: this program will run about 4 times faster
'if written in C using an optimizing comiler. I also plan to code the Divide
'and Multiply SUBs in 386 assembly sometime to get monster speed.
DEFINT A-Z
CLS
INPUT "how many digits"; digits&
words = digits& \ 4 + 4
DIM SHARED sum(words), term(words)
'--------------------16*atan(1/5)
PRINT TIME$: denom& = 3: firstword = 1
CALL FastDivide(26)
CALL Multiply(term(), 80)
CALL Add(1)
DO UNTIL firstword = words
CALL Divide(term(), denom&)
CALL Multiply(term(), denom& - 1)
CALL Divide(term(), 26)
CALL Add(1)
denom& = denom& + 2
IF term(firstword) = 0 THEN firstword = firstword + 1
LOOP
'-------------4*atan(1/239)
denom& = 3: firstword = 2
CALL FastDivide(57122)
CALL Multiply(term(), 956)
CALL Add(-1)
DO UNTIL firstword = words
CALL Divide(term(), denom&)
CALL Multiply(term(), denom& - 1)
CALL Divide(term(), 57122)
CALL Add(-1)
denom& = denom& + 2
IF term(firstword) = 0 THEN firstword = firstword + 1
LOOP
CALL PrintOut(words)
END
'--------------------------------------------------------------------
SUB Add (sign)
SHARED words, firstword
IF sign = 1 THEN
'add it on
FOR x = words TO firstword STEP -1
sum(x) = sum(x) + term(x)
IF sum(x) >= 10000 THEN
sum(x - 1) = sum(x - 1) + 1
sum(x) = sum(x) - 10000
END IF
NEXT x
ELSE
'subtract it off
FOR x = words TO firstword STEP -1
sum(x) = sum(x) - term(x)
IF sum(x) < 0 THEN
sum(x - 1) = sum(x - 1) - 1
sum(x) = sum(x) + 10000
END IF
NEXT x
END IF
END SUB
'-------------------------------------------------------------------
SUB Divide (term(), denom&)
SHARED words, firstword
FOR x = firstword TO words
dividend& = remainder& * 10000 + term(x)
quotient = dividend& \ denom&
term(x) = quotient
remainder& = dividend& - quotient * denom&
NEXT x
END SUB
'---------------------------------------------------------------------
SUB FastDivide (denom&)
'not really a fast divide, but there are fewer operations
'since dividend& below doesn't have term(x) added on (always 0)
SHARED words
remainder& = 1
FOR x = 2 TO words
dividend& = remainder& * 10000
quotient = dividend& \ denom&
term(x) = quotient
remainder& = dividend& - quotient * denom&
NEXT x
END SUB
'---------------------------------------------------------------------
SUB Multiply (term(), mult&)
SHARED words, firstword
FOR x = words TO firstword STEP -1
product& = mult& * term(x) + carry
carry = product& \ 10000
term(x) = product& - carry * 10000&
NEXT x
END SUB
'------------------------------------------------------------------
SUB PrintOut (words)
PRINT : PRINT "pi=3."
i = 2
DO UNTIL i = words - 1
PRINT " " + RIGHT$("000" + LTRIM$(STR$(sum(i))), 4);
IF (i - 1) MOD 15 = 0 THEN PRINT
i = i + 1
LOOP
PRINT : PRINT : PRINT TIME$
END SUB
Subject: Re: paradox
From: shepard@tcg.anl.gov (Ron Shepard)
Date: Fri, 17 Jan 1997 14:24:15 -0600
In article <5bnv9r$2iv@amenti.rutgers.edu>, owl@rci.rutgers.edu (Michael
Huemer) wrote:
[...]
>Here's a start at that. Here are some intuitively plausible
>principles that ought to govern the 'confirmation' relation:
>
>1. The observation of an A that is B confirms "All A's are B."
>2. The observation of an A that is non-B disconfirms "All A's are B."
>3. The observation of a non-A is irrelevant to (neither confirms nor
>disconfirms) "All A's are B."
>4. If P is logically equivalent to Q, then whatever confirms P
>confirms Q.
>
>(The first three principles are collectively called "Nicod's
>criterion".)
>
>The Ravens Paradox results because we see that these 4 principles,
>which at least appear obviously true, are inconsistent. For consider
>the observation of a white shoe. This object is a non-black
>non-raven. Therefore, by (1), it confirms "All non-black things are
>non-ravens." But "All non-black things are non-ravens" is logically
>equivalent to "All ravens are black." Therefore, by (4), the
>observation of a white shoe confirms "All ravens are black." However,
>by (3), the observation of a white shoe is irrelevant to whether all
>ravens are black.
>
>Thus, one of these principles has to go. Which one?
It seems that there should be a difference between proving/confirming
relations between members of finite sets and those of infinite sets. If
there are a finite number of objects, then confirmation of "All non-black
things are non-ravens" does indeed make progress toward proving that all
ravens are black; in principle, if the process were continued until all
non-black objects were tested, we would have the complete proof. But if
there are an infinite number of such non-black objects, then little, or
perhaps even no progress at all has been made.
$.02 -Ron Shepard
Subject: Re: Fibonacci
From: "Ben Z. Tels"
Date: 17 Jan 1997 20:32:36 GMT
I'd say the following is wrong, in your assumption and in your proof:
You've defined:
f(n) = f(n-1) + f(n - 2) (n >= 1, f(1) = 1, f(2) = 1)
g(n) = g(n-1) + g(n-2) + 1 (n >= 1, g(1) = 1, g(2) =1)
giving you (in your assumption):
g(n) = f(n) + 1
Now, here's where you make a small but strange mistake; you've come this
far, and now you go off trying to prove that
g(n) = f(n) - 1
which cannot be. Starnge. Maybe a typo?
But there is another mistake, a more fundamental one:
Look at the Fibonacci row and the one produced by g:
f g
1 1
1 1
2 3
3 5
5 9
8 15
... ...
You see what you are doing? You are adding 1 to the Fibonacci row, but
those 1's don't just disappear in the next step!
--
Ben Z. Tels
B.Z.Tels@stud.tue.nl
optimusb@stack.nl
"The Earth is the cradle of the mind, but one cannot stay in the cradle
forever."
--Tsiolkovsky
Tri Tram wrote in article
...
> I am wondering what is wrong with this proof:
> Given the recurrence relation for Fibonacci numbers F(n)=F(n-1)+F(n-2).
We
> considered the function G(n)=G(n-1)+G(n-2)+1. It seems obvious that
G(n)>F9n)
> We can prove that G(n)=F(n)-1 by induction.
> We assume that G(k)=F(k)-1 for all k such athat 1<=k<=n and we consider
> G(n+1):
> G(n+1)=G(n)+G(n-1)+1 =F(n)-1+F(n-1)-1+1 = F(n+1)-1.
>
> --
> -----------------------------------------------------------------
> Tri Tram, Computer Science and Engineering at UCLA
> http://www.seas.ucla.edu/~tram
>
>
