HDTenEightyP/NetNews huggingface dataset:
From: hdev@dutiag.tudelft.nl (Hans de Vreught)
Newsgroups: sci.math
Subject: Re: Most important event in the history of mathematics
Date: 30 Sep 92 23:05:15 GMT

amnell@klaava.Helsinki.FI (Marko Amnell) writes:

>In article <hdev.717797409@dutiag> hdev@dutiag.tudelft.nl 
>(Hans de Vreught) writes:

>>wft@math.canterbury.ac.nz (Bill Taylor) writes:
>>
>>>Got to be the discovery of irrationals by a Pythagorean (whose name escapes me)
>>>Its supreme significance is affirmed by its unique (apocryphal?) status as 
>>>the only time a mathematician has been killed for making a shattering
>>>mathematical discovery ! 
>>
>>His name was Hippasos of Metapontum. The legend goes that his disclosure of
>>the discovery of irrational numbers resulted in the death penalty by drowning.
>>It certainly was his most important event in the history of mathematics...

Please note I said the disclosure not the discovery and furthermore I spoke of
the legend.

>The whole story is pure legend.  I challenge you to cite any reputable
>scholarly evidence that Hippasus discovered the irrational numbers.  As
>someone pointed out earlier in another thread in sci.physics, the
>discoverer was not executed, but perhaps did perish in a shipwreck.
>It is true that the Pythagoreans kept their discovery secret for some
>time because of unpleasant consequences in geometry...

OK, I found it in a Dutch book of an old professor of mine. He has worked on
the translation of several old papers and has worked in several teams on this
subject. To be short: a respectable scientist. The book gives some highlights
in the history of math using the original texts. The book is called:

	A.W. Grootendorst, "Grepen uit de Geschidenis van de Wiskunde",
	Delftse Uitgevers Maatschappij, 1988.

On page 49: "De ontdekking daarvan wordt toegeschreven aan Hippasos van
Metapontum, volgeling van Pythagoras en het verhaal gaat dat hij als straf
voor de bekendmaking daarvan de verdrinkingsdood heeft gevonden." Translated in
English: "The discovery of that (irrational numbers) is attributed to Hippasos
of Metapontum, a follower of Pythagoras and the story goes that for the
disclosure he was penalized by being drowned." Although Grootendorst used the
word "story", he meant "legend" (actually in the Dutch sentence it is used in
that context).

So there is my justification. I don't see why it is so much different from the
story of the shipwreck. I can easily imagine how the legend could emerge from
that story. But I did clearly state it was a legend. Maybe it's true but
probably it's not. Either way it is a nice story.

Personally I think you jumped the gun too soon. Well, in any case I end it
here.
--
Hans de Vreught                          | John von Neumann:
hdev@dutiaa.tudelft.nl                   | Young man, in mathematics
Delft University of Technology (TWI-ThI) | you don't understand things,
The Netherlands                          | you just get used to them.
Newsgroups: sci.math
Subject: Re: Fourier Trans. to Freq. domain
From: khill@vax1.umkc.edu
Date: Wed, 30 Sep 1992 14:07:15 GMT
Organization: University of Missouri - Kansas City


	Simply put, the Fourier Transform can be equated (albiet loosely) to
the dot product used in vector algebra.  Take A-comp-B and you get the shadow
of vector A on vector B.  Now take A' and B' as functions of time, with
the obligatory restrictions on the functions (ie, finitely discontinuous,
bounded, etc.) and you can imagine they are actual vectors of infinite
dimension.  Then you take SIN' and COS' as functions of time also (yes it is
convenient that these function names so closely correspond to two other
functions that we know so well), but these functions give us a frequency.
Now if we take the infinitely dimensioned A'-comp-SIN' we will get the "shadow" 
of vector A' on vector SIN'.  Now change SIN' to represent a set of frequency
functions and take A'-comp-SIN' for each frequency and you will get the
"shadow" of A' on each frequency.
	NO, NO, and NO!  This is not a sound explaination of how the FT
tranforms functions of time to functions of frequency, but it is a very nice
(and there is a LITTLE truth in this) and easy way to visualize what is
happening, and thus it makes it easier to remember what the FT does, how it
does it, and the integrals that do it!


	BTW A-comp-B = A*B/|B|
                       \-----/
			\   /
    			 \ /
			 See the similarities in the FT integrals???
-- 
===============================================================================
!        Ken Hill       ! This message is SHAREWARE.  You are authorized      !
!        --------       ! to use this message for up to 5 days.  If you       !
!  KHILL@VAX1.UMKC.EDU  ! like what you see please register it by sending     !
========================! a response.  This message may be freely distributed !
! Amateur Mathematician ! in its original and unmodified form.  Remember only !
! Professional Paperboy ! you can support SHAREWARE.  (c) 1992 Ken Hill  :>   !
===============================================================================
     This message has been brought to you by the letter Q and the number 7
Newsgroups: sci.math
From: hrubin@pop.stat.purdue.edu (Herman Rubin)
Subject: Re: Could have Archimedies invented Calclus?
Organization: Purdue University Statistics Department
Distribution: na
Date: Tue, 29 Sep 1992 13:25:26 GMT

In article <1992Sep27.204021.19995@galileo.physics.arizona.edu> seh@neutron.physics.arizona.edu (Steven E. Hoell) writes:
>I missed most of this discussion, but my freshman calculus instructor
>said that Archimides would have discovered calculus had he been
>familiar with the concept of limit.
>
>Steve Hoell		seh@neutron.physics.arizona.edu

Archimedes was certainly familiar with the concept of limit.  In particular,
he expressed 2*pi as the common limit of the perimeter of an inscribed
regular n-gon (clearly a lower bound) and the perimeter of a circumscribed
regular n-gon (not so clearly an upper bound; the argument for that is
attributed to him).  The "Riemann integral" as a limit of sums was at least
intuitively understood by that time as well.
-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@pop.stat.purdue.edu (Internet, bitnet)  
{purdue,pur-ee}!pop.stat!hrubin(UUCP)
Newsgroups: sci.math
From: gsmith@euterpe.uucp (Gene W. Smith)
Subject: Re: various number systems
Organization: IWR, University of Heidelberg, Germany
Date: Wed, 30 Sep 92 19:41:05 GMT

In article <BvCH4A.H7w@unix.amherst.edu> mkrogers@unix.amherst.edu (MICHAEL K ROGERS) writes:
>lloyd mitchell (lmitchel@nyx.cs.du.edu) wrote:

>: I am currently teaching a pre-calculus course at a local community college.
>: We have just covered the quadratic formula, which has served as an intro-
>: duction to complex numbers.  Now, I would like to expose my students to
>: other numbering systems.  My goal is not to teach them the ins and outs of
>: the systems, but to expose them to other ways of thinking about numbers.

>You could add to the real numbers a new invented number called
>epsilon with the following properties:
>	a) epsilon > 0;
>	b) for every real number  x > 0,  epsilon < x .
>You think of epsilon as an "infinitessimally small" number.
>You now take the closure under addition, multiplication, etc.
>You can use this new field of numbers to do calculus without
>limits (Nonstandard Analysis).  

This isn't a nonstandard real field unless "etc" is interpreted pretty
broadly. It sounds like what you have is R(epsilon) with ordering; in
other words an algebriac function field.  This does not mean it is a
bad example; in fact, it is a much better one for the purpose of a
community college class.

Other fields, easier than the p-adic numbers mentioned below, are
algebraic number fields.  You could have your students contemplate
ordered pairs of rational numbers (x, y) such that (x, y) + (w, z) =
(x+w, y+z), (x, y)*(w, z) = (xw+yz, xz+yw+yz).  If they could show this
allowed divison for all numbers, and satisfied the usual rules
(communitive, associative, distributive, etc.) they will have learned
something.  And you can be pretty sure they didn't find the answer in a
book, either!

There are also tons of quaterion algebras over the rationals if you get
really ambitious. Finite fields are another good thing to look at.  For
instance,

(x, y) + (w, z) = (x+w (mod 2), y+z (mod 2)),
(x, y)*(w, z) = (xw+yz (mod 2), yw+xz+yz (mod 2))

The addition and multiplication are the same as before, but now x, y,
w, z are either 0 or 1.  These have finite times tables which could be
worked out, if we call (0, 0) = 0, (1, 0) = 1 and (0, 1) = a, for
instance.

-- 
     Gene Ward Smith/Brahms Gang/IWR/Ruprecht-Karls University 
               gsmith@kalliope.iwr.uni-heidelberg.de
Newsgroups: sci.math
From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: Last non-zero digit of 10^6!
Organization: Computer Science Department,  Stanford University.
Date: Sat, 26 Sep 1992 06:58:30 GMT

In article <1992Sep25.153409.2662@wkuvx1.bitnet> mccoyje@wkuvx1.bitnet writes:
>In article <19ufr0INNhpd@uniwa.uwa.edu.au>, doi@tartarus.uwa.edu.au (Daniel Kuan Li Oi) writes:
>> 	This is not a homework problem though it did appear in a math
>> competition which has finished.
>> 
>> Question
>> 
>> What is the last non-zero digit of one million factorial, 1000000!
>> 
>Just sitting here, I can tell you that the last non-zero digit is a 6.
>I think this is right, but I could be wrong.

Spoiler warning

Would you believe 4?

Here is a method for computing the least significant nonzero decimal
digit of n! for n >= 2.

Let a[0..19] denote the following array:

  i  =  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19
a[i] =  6  6  2  6  4  2  2  4  2  8  4  4  8  4  6  8  8  6  8  2

Then for n >= 2 the low order nonzero digit of n! is the product mod
10, for i = n, n/5, n/25, ..., of a[i mod 20].  (So now you can see how
to construct the table, if you start from i=2 and "wrap round." I'll
let you worry about why it has period 20 rather than 10. :-)

Applying this to n = 1000000 we obtain:

    i    a[i mod 20]
                      cumulative product
 1000000     6            6
  200000     6            6
   40000     6            6
    8000     6            6
    1600     6            6
     320     6            6
      64     4            4
      12     8		  2
       2     2		  4

So the answer is 4.  The sum of the left hand column gives exactly the number of
trailing 0's, here 1249998 (5n/4 - random fuzz).

No calculator or computer needed, it can be done in your head:
repeatedly double, shift right, look up, one-digit multiply.  In that
respect the method somewhat resembles the standard method for
determining the day of the week in convenience for mental arithmetic.
The differences are that the table has 20 elements instead of 12, the
result accumulates multiplicatively instead of additively, the
arithmetic is mod 10 instead of mod 7, and the problem is not useful
enough to bother memorizing the table.  (I memorized the
day-of-the-week table in 7th grade and by the end of the year I think
just about every kid in the school had asked me what day they were born
on, and a lot of them twice to see if I was faking it.  Got so I could
answer in about 3 seconds.  I still find it useful occasionally even
today.)

-- 
=================================================== In short, I am floundering
Vaughan Pratt   pratt@cs.Stanford.EDU  415-494-2545 around and don't take what
=================================================== I say seriously.
From: edgar@function.mps.ohio-state.edu (Gerald Edgar)
Newsgroups: sci.math
Subject: Re: Lebesgue measure in Fractal Dimension
Date: 6 Oct 1992 12:57:08 -0400
Organization: The Ohio State University, Dept. of Math.
Distribution: usa
NNTP-Posting-Host: function.mps.ohio-state.edu

In article <Oct.6.12.39.28.1992.1882@no.rutgers.edu> bumby@no.rutgers.edu (Richard Bumby) writes:
>
>On the original question of finding the actual measure of certain sets
>in the critical dimension, there is some work by H. Wegmann in Math.
>Ann. 193 (1971) that deals with the sets defined by forbidding a digit
>in the base-g expansion (generalizing the idea that the classical
>Cantor set is obtained by forbidding the digit 1 in the base-3
>expansion).  He obtained some results which I found incredibly precise
>when I first saw them.  The thing that makes this work so impressive
>is that the results are obtained for the measure based on coverings by
>all intervals.  The best coverings may not be the "cylinders" of your
>favorite Cantor set construction, but there will be come construction
>for which the coverings by intervals occurring in the construction
>suffice for computing the Hausdorff measure.

For the original Cantor set (omit the middle third and repeat) the
exact Hausdorff measure (which is 1 if you define Hausdorff measure without
normalizing constant) is computed in Hausdorff's original 1918 paper.



-- 
  Gerald A. Edgar                Internet:  edgar@mps.ohio-state.edu
  Department of Mathematics      Bitnet:    EDGAR@OHSTPY
  The Ohio State University      telephone: 614-292-0395 (Office)
  Columbus, OH 43210             -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)
From: hdev@dutiag.tudelft.nl (Hans de Vreught)
Newsgroups: sci.math
Subject: Re: Most important event in the history of mathematics
Date: 29 Sep 92 20:10:09 GMT

wft@math.canterbury.ac.nz (Bill Taylor) writes:

>Got to be the discovery of irrationals by a Pythagorean (whose name escapes me)
>Its supreme significance is affirmed by its unique (apocryphal?) status as 
>the only time a mathematician has been killed for making a shattering
>mathematical discovery ! 

His name was Hippasos of Metapontum. The legend goes that his disclosure of
the discovery of irrational numbers resulted in the death penalty by drowning.
It certainly was his most important event in the history of mathematics...
--
Hans de Vreught                          | John von Neumann:
hdev@dutiaa.tudelft.nl                   | Young man, in mathematics
Delft University of Technology (TWI-ThI) | you don't understand things,
The Netherlands                          | you just get used to them.
Newsgroups: sci.math
From: barnes@bert.eecs.uic.edu (Craig Barnes)
Subject: Help with Solving Frensel Integrals
Organization: University of Illinois, Chicago
Date: Tue, 6 Oct 1992 18:12:32 GMT

Hello,

   I am trying to write a program that computes and plots the values
of two integrals:

      v /            2
       |       Pi * v
       |  sin (------ ) dv
      0/          2


      v /            2
       |       Pi * v
       |  cos (------ ) dv
      0/          2

What I would like is something that can be put into a computer (i.e. a
nice for loop). However my calculus is very rusty and I can't seem to get
the right form of the integral. I tried to use mathmatica! but all it gave 
me was the function FrenselS and FrenselC not the definition of them. 

I am sorry for wasting bandwidth for something like this but I haven't been 
able to find any other source for the answer.


Thank you for your time.
     

C Barnes.
barnes@uicbert.eecs.uic.edu
From: hrubin@pop.stat.purdue.edu (Herman Rubin)
Newsgroups: sci.math
Subject: Re: A more virus resistant architecture
Keywords: computer,virus,Babbage,analytic engine
Date: 6 Oct 92 13:43:32 GMT
Organization: Purdue University Statistics Department

In article <2024@coyote.UUCP> aspnes@molokini.almaden.ibm.com (J. Aspnes) writes:

>In article <BvMJAo.24F@world.std.com>, rjk@world.std.com (robert j
>kolker) writes:
>|> The question is this: Is a computer whose program space is totally
>|> disjoint from the the data space less likely to be buggered by viruses
>|> than a  Von Neuman Machine, where the data and the program are kept in the
>|> same store.

>Yes.  Unfortunately, it's also less likely to be infected by compilers.

This is only the case if there is too much simplification.  If a computer
can go backward in its code, then any computer can be emulated on any
other by using a single fixed program depending only on the machine on
which the emulation is done, given enough memory.  It might not be very
efficient, but it only corresponds to a linear factor.
-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@pop.stat.purdue.edu (Internet, bitnet)  
{purdue,pur-ee}!pop.stat!hrubin(UUCP)
Newsgroups: sci.math
Subject: Re: Most important event in the history of mathematics
From: wft@math.canterbury.ac.nz (Bill Taylor)
Date: Tue, 29 Sep 1992 11:04:40 GMT
 <1992Sep28.174237.26784@Cookie.secapl.com> <1992Sep28.223806.27669@u.washington.edu> <BvBED3.Ly2@mentor.cc.purdue.edu>
Organization: Department of Mathematics, University of Canterbury

Got to be the discovery of irrationals by a Pythagorean (whose name escapes me).

Its supreme significance is affirmed by its unique (apocryphal?) status as 
the only time a mathematician has been killed for making a shattering mathematical discovery ! 
Xref: sparky sci.math:12548 rec.puzzles:6549
From: kubo@zariski.harvard.edu (Tal Kubo)
Newsgroups: sci.math,rec.puzzles
Subject: Conway's Recursive Sequence
Date: 1 Oct 92 02:08:51 GMT
Article-I.D.: husc3.1992Sep30.220851.15985
Distribution: world
Organization: Dept. of Math, Harvard Univ.


I am looking for information about a recursive sequence
defined by John Conway in 1988:

   a(n)= a(a(n-1)) + a(n-a(n-1))        for n>2
   
   a(1) = 1, a(2)=1

In spite of the bizarre defining recurrence, this
sequence displays an amazingly high degree of
combinatorial structure (involving powers of 2,
Pascal's triange, and Catalan numbers).

The only published work on this sequence appears to
be an article by C. Mallows in the January 1991 issue
of American Mathematical Monthly.  Also, problem E3274,
solved in the Dec. 1991 issue of the Monthly, involves 
some related sequences. 

I have been doing some work (in collaboration with
another graduate student here) on this sequence and would like
to know if any progress has been made since Mallows'
1991 article. Pointers to relevant literature, or
any information whatsoever, would be most welcome.

-Tal Kubo   kubo@math.harvard.edu
Newsgroups: sci.math
From: ara@zurich.ai.mit.edu (Allan Adler)
Subject: Albanese
Organization: M.I.T. Artificial Intelligence Lab.
Distribution: sci
Date: Sat, 26 Sep 1992 07:44:10 GMT


Can someone tell me where I might find a biography and a list of all of the 
publications of the mathematician Albanese? In case there is more than
one by that name, I mean the one for whom the Albanese variety is named.

Allan Adler
ara@altdorf.ai.mit.edu
From: petry@pythagoras.math.washington.edu (David Petry)
Newsgroups: sci.math
Subject: Re: Z <-> Q formula
Summary: Corrected version of a previous post
Date: 29 Sep 92 01:01:20 GMT
Organization: University of Washington, Mathematics, Seattle

In article <1992Sep27.085952.7534@Csli.Stanford.EDU>
 hiraga@Csli.Stanford.EDU (Yuzuru Hiraga) writes:

>The set of intergers Z and the set of rationals Q have the same cardinality,
>i.e. there exist a bijection between them.  So we all know that.
>But then, is there a "simple" formula that expresses any such mapping?
>You may use any device you please, but note that the subjective little
>"simple" is the criterion.

Here's a simple way to do it.   Given the rational number A/B, let {p_i} 
be the set of prime factors of B.   Let f(A/B) =  A^2*B^2/product(p_i) .
Then f is a bijection from the positive rationals to the positive integers.

David Petry
Newsgroups: sci.math
From: niall@nmsu.edu (Niall Graham)
Subject: Re: Rename the Calculus sequence?
Organization: Computing Research Lab
Date: Sun, 27 Sep 1992 21:51:51 GMT

How about Indifferential Calculus, for those students who couldn't
care less about mathematics.  And speaking of "couldn't care less"
isn't that idiom a perfect illustration of the concept behind
a delta-epsilon proof, as in: For all epsilon greater than zero,
the amount that you care about this class, delta, is less than
epsilon.
--
Niall Graham          "Patriotism is the virture of the vicious"
Los Alamos                                            Oscar Wilde
               
                                                        
Newsgroups: sci.math,sci.math.stat
From: R1STORY@VM1.CC.UAKRON.EDU
Subject: Re: Stat problem...
Organization: The University of Akron
Date: Tue, 6 Oct 1992 19:13:21 GMT

In article <92280.011410ROCHC@CUNYVM.BITNET>
<ROCHC@CUNYVM.BITNET> writes:
 
>     I'm studying for the GRE Subject Test in Mathematics and I'm having
>trouble with the following 2 statistics questions that I encountered in a
>practice exam:
>
>      1) In a game, two players take turns tossing a fair coin; the winner
>is the first one to toss a head.  What is the probability that the player who
>makes the first toss is going to win?
>
>            Answer: 2/3
>
>      2) Alternately a coin is tossed and a die is thrown, beginning with
>the coin.  What is the probability that the coin will register a "head"
>before the die registers a "5" or a "6"?
>
>            Answer: 3/4
>
>     Any information on how the answers above were derived will be greatly
>appreciated.
>
>     Thank you very much.
>
>                                          -- Ron --
===========================================================================
Set up a system of equations as follows (for problem #2)
 
let p1 = probability player A wins game (when player A goes first)
    p2 = probability play B wins game (when player B goes first)
 
    p1 = (1/2) + (1/2)(1 - p2)
    p2 = (1/3) + (2/3)(1 - p1)
 
Solving, we get p1 = 3/4.
Newsgroups: sci.math
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Infinity
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Thu, 1 Oct 92 20:45:15 GMT

In article <1992Sep30.214743.18350@cs.rose-hulman.edu> goddard@NeXTwork.Rose-Hulman.Edu (Bart Goddard) writes:
>
>In article <1367.2ac9ae4e@atlas.nafb.trw.com> dsimon@atlas.nafb.trw.com  
>writes:
>> Does infinity actually exist, or is it simply a mathematical tool?  
>> Does an infinity of something actually exist?  Is it *possible*
>> for there to be an infinity of something?

>Does "one" actually exist, or is it simply a mathematical tool, 
>(an abstraction) with which we are so familiar that it seems 
>concrete?

Hear, hear!  I get rather tired of the constant fussing over infinity as
if it were somehow drastically more mysterious than other mathematical 
abstractions.  The notion of "i" is an abstraction that cost people much
serious head-scratching when it was invented - hence the term
"imaginary".  The notion of "sqrt(2)" is an abstraction - hence the term
"irrational".  The notion of "0" is a far subtler abstraction - that
having no apples at all still counts as having a certain number of
apples, namely "0" apples.   Ditto for "-1".  And yes, the whole notion
of integers is a tremendous abstraction from the seething flux of
reality.  If I had to propose a "most important event in the history of
mathematics" it would be the invention of counting.  Too bad we may
never know how this happened, or how many times it happened (they didn't
keep count).
From: ara@zurich.ai.mit.edu (Allan Adler)
Newsgroups: sci.math
Subject: Re: Opinions Sought (Why major in math?)
Date: 2 Oct 92 21:29:10 GMT
Organization: M.I.T. Artificial Intelligence Lab.

I took and passed the first actuarial exam during the 60's. It didn't
require any more math than was required to do well on the MAA exams
that I had played with in high school.

Allan Adler
ara@altdorf.ai.mit.edu
From: dsimon@atlas.nafb.trw.com
Newsgroups: sci.math
Subject: Infinity
Date: 30 Sep 92 21:40:30 GMT

Does infinity actually exist, or is it simply a mathematical tool?  
Does an infinity of something actually exist?  Is it *possible*
for there to be an infinity of something?
----------
Dan Simon