Newsgroup sci.math 157126

In article <32D2AB04.1983@daedal.net>, James Tuttle   wrote:
>Richard Mentock wrote:
>> Milo Gardner wrote:
>> > Roman numerals are base 10, stated in terms of the register of an
>> > abacus (for example).
>> > Multi-based? No base at all? Wow, what confusion.
>> > I hope this note clarifies something to somebody.
>> Well, usually "base n" implies positional notation, which Roman
>> Numerals clearly aren't.  X *does* equal ten, but V equals five,
>> so why wouldn't you claim they were base 5?
>Roman numerals are *highly* positional.  IX is 9 and XI is 11.
>Position matters a lot.
Most of the base 10 systems are positional.  This includes, for 
example, the Egyptian and the Greek.  From what I have seen of 
the Egyptian system, there was a symbol for each power of 10 up
to some point, and that symbol was used as many times as needed.
The Greek system had a symbol for j=10^k, j=1,...,9, k=0,1,2.
It used an overbar to multiply by 1000.  The Roman numerals are
definitely base 10, with symbols for 10^k and 5x10^k.
The oldest known system which uses the same symbols for multiples
of different powers of the base is the Babylonian base 60.  Some
have conjectured that the Hindus got the idea of using it for base
10 from this.
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu         Phone: (317)494-6054   FAX: (317)494-0558

Norman D. Megill (ndm@shore.net) wrote:
: Conlippert  wrote:
: >Bill Gates said in his book "The Road Ahead" He was talking about
: >internet security, and how an encrypted code could be broken.
: >That the web would be a secure place......
: >"Unless someone figures out how to factor large prime numbers."
: The security holes in MS products finally explained...
: I'm sure he wants to be the first to figure this out, so maybe he has a
: whole secret department working on it. [snip]
:
  Ah, but `you know he meant' "how to factor numbers composed of large
prime factors".  Perhaps he has a secret department working on an "I know
you meant" program that ISN'T designed to drive users stark, raving mad --
and I must say, that's a really well-kept secret!

ags@seaman.cc.purdue.edu (Dave Seaman) wrote:
>Although languages like Fortran and C do give 2^(1/2) = 1 (and so does
>BASIC, if I remember correctly),
I doubt it. Since you're not using exact FORTRAN, i.e. you're
using (^) instead of (**), I assume you're not mentioning other
details like (1./2.) is different from (1/2) because the former
is floating-point whereas the latter is integer. If you do not
mention this exact detail, you can't make blanket statements like
"2^(1/2)=1" .
Every BASIC I have ever used has given 2^(1/2) as the square root of
two to several decimal digits of accuracy. "Several" is a bit vague
because some BASICs let you put a "#" sign to signify extra precision
and not all BASICs store their default floating-point numbers to the
same precision.
I have never used a BASIC which even allows me to calculate 2^(1/2)
as an integer. I can calculate it as a floating-point number, and
explicitly truncate the result, but that's not the same as
calculating it as an integer to start with. One BASIC I know
allows integer division, but I don't know about explicitly
restraining powers to be integers.
C  I can't comment on.
"languages like" FORTRAN and C ...  that includes Pascal.
The original Pascal didn't even have a power statement/operator/function,
so you had to do 2^(1/2) by taking logarithms or writing your own
subroutine/function. This was clearly floating-point.
People soon got fed up with this and put a power facility in Pascal.
It sounds like languages like FORTRAN and C do indeed give floating-point
answers.

Doug McKean wrote:
> 
> What are they - base 5, base 10, base 50, ..., multibased?
What I am finding most interesting is the concensus 
appears to be "no base". It is convention that a 
number system is founded on successive powers of 
a specific specific number usually positional. 
The base 10 system is a "based/positional" number system. 
The Roman numeral system is a "non-based/positional" number system.
So, are there any "based/non-positional" number systems or 
any "non-based/non-positional" number systems out there? 
*******************************************************
-------------------------------------------------------
The comments and opinions stated herein are mine alone,
and do not reflect those of my employer.
-------------------------------------------------------
*******************************************************

 If the term "eternal" has been given a special sort of meaning, such as
  in theology, the phrase "eternal math laws" can be acceptible.  As is,
  it tends to carry the idea of "resistant to change over very long
  passages of time."
 That suggestion must be firmly resisted, I will urge.  When mathematical
 abstractions are treated as quasi-physicals endowed with special powers,
 todays gurus are only too happy to loop the label "Platonism" around the
 neck of the proposal and wind it tight.  The fate of phlogiston and the
 ether provide cautionary examples.
 I propose, rather, that permanence in mathematics is a sign of a state of
 existence independent from space-time.  The real puzzle is (as Plato
 realized) why we experience physical reality in fashions that make our
 awareness of mathematical ideas (of a sort) so inevitable.  I mean here
 very simple things, like counting from 1 to 10, comparison of shapes,
 ranking sizes, elementary logical if..then, and so on.
   Plato gave a solution to this puzzle that I along with others find
   brilliantly flawed.  The length of his shadow over several millenia
   of Western physical philosophy makes the needed revisions and fresh
   starts difficult indeed.
      Frank Bernhart,  Rochester, NY

Roger  Luther  wrote:
(snip)
>PS some of us think that the lottery is the most retrogressive tax since 
>the poll tax of the 1300's. Every other tax takes from those who have 
>most, to help the rest of society. The lottery takes, proportionally, 
>more from those who have least, and gives to those who have most eg the 
>Royal Opera House grant!
Correction:  The lottery takes from those who have the least........
intelligence!  A *smart* poor person does not play. 
/\/\att
- -
 ^
"Does anyone have a copy of my book, or have they all 
 been burnt by now?"
                                      - Howard Stern

In article <57721566490@einsteinium.universe>
Carl Friedrich Socrates Einsteinium  writes:
> The whole problem is you don't even understand where the problem is.
> Just what do you mean by "such numbers"? If you just mean ...000,
> ..001 and ...002, then there's nothing ridiculous about asking the
> question, and it's so easy to find the answer I think even David
> Madore could do it. But that is not the problem. You want to study
> a whole set of numbers ...000, ...001, ...002, and so on. But that
> won't work.
> 
> For, you see, all numbers are FUZZY
                                ^^^^^
 Your posts are readable up until you enter that above word, then they
are no longer interesting and in fact the next time you enter fuzzy I
will put your posts into my killfile.

From           dik@cwi.nl (Dik T. Winter)
Organization   CWI, Amsterdam
Date           Wed, 8 Jan 1997 02:23:21 GMT
Newsgroups     sci.math,sci.physics,sci.logic
Message-ID     
References     1 2 3
In article <5asb6a$ma6$1@dartvax.dartmouth.edu>
Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium) writes:
 > In article 
 > dik@cwi.nl (Dik T. Winter) writes:
 > 
 > >  > Let me recap here some facts :
 > >  >    p-adics form a field
 > > 
 > > No.  Obviously, if there *are* primes you do not have a field.
 > 
 > 3-adics form a field, 5-adics form a field
 > any p-adic where p is prime forms a field
Eh?  Did you read what I wrote after the sentence above?  If there *is*
a prime, there is no field.  In the 3-adics 3 is the only prime (%),
but
because there is a prime there is no field.  In a field every element
except 0 has a multiplicative inverse; there is no inverse of 3 in the
3-adics.  There is no inverse of p in the p-adics.  In the rationals
for
instance 3 is *not* prime.
--
% Prime in the sense of prime ideals.  Not in the sense of: there are
no a and b such that a.b = 3; because there are such a and b
(for instance: 2 * ...11111111120 = 10 = 3 in the 3-adics).  My
previous
proof was a bit wrong however.  If there is an element with a
multiplicative
inverse there is no prime in the traditional sense.  That was what the
original proved, but in that sense 3 is not even a prime in the
3-adics.
However, an element without multiplicative inverse can still generate a
prime ideal (as is the case with p in the p-adics), but in a field
there
are no such elements.
-- 
dik
This is a problem of *timing*. And a problem of timing is a big deal in
mathematics because mathematics is the science of precision.
 Yes I read your previous message.
  But you Dik assume that 3 is prime in the first place, do you not, in
order to construct the 3-adics. 
   Now, then, you construct the 3-adics and I say that the 3 is prime.
But you , Dik , accuse me of saying 3 is not a prime if 3-adics is a
field.
  I say to you Dik that we get the primeness of 3 from the special set
of Reals of this set { ....,  ..., ...002- , ....002+, ...003+,
....005+,....}
  I start with the Reals, and you Dik, I do not know where you started
from. Those special class of Reals, the Whole Reals are prime in Whole
Reals.
  So, if 2 or 3, 5, is not prime then Dik, how in the world can you
even start to construct the  2-adics or 3-adics or 5-adics.
  Personally I think this is a major problem of mathematics. Everyone
is saying these absolutist things that 3-adics is a field and yet they
do not want to recognize that 3 is prime in order to prove that it is a
field. And after they have proved that 3-adics is a field , they then
want to forget that 3 was prime in the first place in order to prove
that it is a field. All a bit hypocritical, I would say.
  So Dik, please tell when in the discussion you want to claim 3 is a
prime number in order to prove 3-adics is a field, and then, when do
you want to renounce that 3 was prime so that you can say that the
3-adics is not a field?

Kurt Foster wrote:
> and millennia.  There is no prefix I've heard of for 10^4 or 10^(-4), so
> one is probably free to make them up.  I've seen "eon" used for 10^9
In the first versions of the metric system, the prefix myria- was used
for 10^4 (from the greek myrioi=10000). I don't think it was ever used
outside France, though. But perhaps a myriadium is a nice word for a
10000 year cycle.
     David A. Madore
    (david.madore@ens.fr,
     http://www.eleves.ens.fr:8080/home/madore/index.html.en)

In article <58pieq$4vm@news.ycc.yale.edu> tdhui@pantheon.yale.edu () writes:
> Can someone give me a reference for the distance between two
> matrices in SL(n,R)?
The geodesic between two such matrices A and B is c(t) = exp(t*log(B/A))*A
(using the invariant metric  = tr(X*Y) at the identity); therefore,
the Riemannian distance between A and B is given by the Frobenious norm
of log(B/A).  In MATLAB code,
        d = norm(log(eig(A,B)))
Steven Smith

Herman Rubin wrote:
> theorem is weaker than that.  The existence of the algebraic
> closure of a finite field follows from the principle of dependent
> choices from finite sets, which is known to follow from the
> Axiom of Determinacy, which is not compatible with the Axiom
> of Choice.  One can even go a little farther here.
Strangely, you say that just at the point when Ilias Kastanas had
me convinced that AC was not at all necessary to construct an
algebraic closure of F_p.
Now what is wrong with the following:
We have to define F_(p^r). To do that, factor out the (p^r-1)-th
cyclotomic polynomial over F_p. We know that one of the irreducible
factors must be of degree exactly equal to r. Consider the one those
such factors which is least in the lexicographical ordering of F_p[X],
with F_p ordered in the following way: 0<1<...F_(p^s) for
r dividing s. We do this by induction on s, letting the maps
F_(p^r)->F_(p^s) be the least possible which are compatible with
all the previously defined ones (least in the sense of the
lexicographical order on the set of finite sequences of functions
F_p->F_(p^s),F_(p^d1)->F_(p^s),...,F_(p^dk)->F_(p^s), where
d1...dk are the proper divisors of s).
We finally let (F_p)~ be the inductive limit of the F_(p^r) under
these maps.
Have I made a mistake (most probable)? Or, if this is true, is there
a simpler way to do it without AC?
     David A. Madore
    (david.madore@ens.fr,
     http://www.eleves.ens.fr:8080/home/madore/index.html.en)

In article <5av3n0$9c9@mercury.iusb.edu>, Simone P Powell
 wrote:
:Can someone recommend geometry software for our school.
both Geometer's Sketchpad and the Cabri II geometry program are a lot of
fun and work really well.
i downloaded demo versions of these from the Swarthmore web site.
-- rob

Lotto simulation programs should be easy to write.
The key argument opposing that approach is how
closely your RNG matches the lotto "ball machine"
or whatever.
Although you can get an answer via simulation, I
still think that there would be a statistcal
method that would give good results.
Uenal Mutlu wrote:
> It would be useful if we had a simulation software which for example
> looks something like the following:
> 
> LOTSIM v k b nruns fFixedTickets fFixedDraw ...
> 
> v             = total numbers (ie. 49)
> k             = nbrs per ticket (ie. 6)
> b             = nbr of random tickets (>= 1)
> nruns         = nbr of random drawings (simulation) (>= 1)
> fFixedTickets = randomly fill tickets once OR refill each time (0/1)
> fFixedDraw    = randomly draw once and keep OR redraw each time (0/1)
> ...
> 
> Output:
> 
>  m  ep en  rn rp  dn dp ...
> ---------------------------
>  k  .. ..  .. ..  .. .. .
>  .  .. ..  .. ..  .. .. .
>  .  .. ..  .. ..  .. .. .
>  0  .. ..  .. ..  .. .. .
> ---------------------------
> Sum: .....
> 
> m  = matching nbrs (0..k)
> ep = expected theoretic probability
> en = expected theoretic frequency
> rn = simulated real frequency
> rp = simulated real probability
> dn = +/- diff frequency
> dp = +/- diff probability
> ...
> 
> If there is already a similar publicly available program I would
> like to hear about it. I think of programming such a thing too.
> Further comments/options/ideas to include in the program are welcome.
> IMHO such a program would be helpful in 'practically' answering some
> still outstanding problems like in the case of the above 54 tickets
> in question.

John R. Black (blackj@toadflax.cs.ucdavis.edu) wrote:
: What is your favorite "cute" proof?  The irrationality of sqrt(2)?  The
: fact that there are an infinite number of primes?  The proof that all
: numbers are interesting? (This one's more of a joke of course)
Here's another:
Theorem:
1^3 + 2^3 + 3^3 + 4^3 + ... + n^3 = (1 + 2 + 3 + ... + n)^2
Proof:
Two colors: '+' --> for odd areas
            '-' --> for even areas
            '$' --> where two even areas overlap twice
     1  2    3      4        5
     + - - + + + - - - - + + + + +
     - $ - + + + - - - - + + + + +
     - -   + + + - - - - + + + + +
     + + + + + + - - - - + + + + +
     + + + + + + - - - - + + + + +  <--- The dimension of square is 15 X 15
     + + + + + + - - - - + + + + +
     - - - - - - $ $ - - + + + + +
     - - - - - - $ $ - - + + + + +
     - - - - - - - -     + + + + +
     - - - - - - - -     + + + + +
     + + + + + + + + + + + + + + +
     + + + + + + + + + + + + + + +
     + + + + + + + + + + + + + + +
     + + + + + + + + + + + + + + +
     + + + + + + + + + + + + + + +
-Godfrey Degamo,
 gotd@jimmy.harvard.edu

In article <5ar10s$321$4@gruvel.une.edu.au>, ibokor@metz.une.edu.au writes:
> electronic monk (donniet@sqruhs.ruhs.uwm.edu) wrote:
> : John Briggs, VAX system manager, x4411 wrote:
> :  
> : > The limit of 1/x as x --> 0 does not exist.  At least not in the reals.
> : 
> : that's right, it is an infinite number.
> 
> Which infinite number?
Go to the real number line.  It's the last on the right.  You can't miss it.
[sorry, couldn't resist]
---------------------------------------------------------------------------
Tim Hollebeek         | Disclaimer :=> Everything above is a true statement,
Electron Psychologist |                for sufficiently false values of true.
Princeton University  | email: tim@wfn-shop.princeton.edu
----------------------| http://wfn-shop.princeton.edu/~tim (NEW! IMPROVED!)

howels@erols.com  wrote on 4th Jan
 > I need to know what the frequency spectrum is of a function divided by 
> it's envelope.
> 
> The function, amplitude vs. time, will be any waveform that is 
> band-limited - there are no frequency components below a or above b hz, 
> where b>a.
> 
> The envelope would be as calculated using the Hilbert transform, or 
> SQRT(I^2 + Q^2).
I can't point to any specific texts with certainty but this is a problem 
that was in vogue in 1960's and 1970's communications.     Look up 
"Bandpass limiting".   Try Davenport and Root "Intro. to communications 
theory".  There should be papers from that time in IEEE Trans. Acoustics 
Speech and Signal Processing since there were applications in limiting of 
single sideband modulation which was common at the time.
JEH

Michael A. Stueben wrote:
> 
> I was trying to list the different kinds of proof for my H.S.
> precalculus students. So I gave direct, indirect, math
> induction and proof by contraposition. Fine. But later I
> thought what about this: proof by example (offer am
> illustration of a situation or give directions for a
> construction)? Or proof by verification
> (substitute and complete a calculation). Then proof by cases
> could be a proof made up of a mixture of all types of proof.
You missed proof by intimidation.
--
email: stevens@math.chalmers.se
Matematiska Institutionen
Chalmers Tekniska H"ogskola
SE 412 96 G"oteborg
Sweden

Simone P Powell wrote:
> 
> Can someone recommend geometry software for our school.  We have a teacher
> who wants to take her class down to a computer lab to teach Geometry.  She
> wants the students to learn about geometric properties by maninpulating
> drawings on the screen.  Thank you in advance.
> 
> --
> 
> Sam Powell
> Northridge High School
> Middlebury Indiana
An extensive resource list may be found at:
http://forum.swarthmore.edu/math.topics.html
Regards,
phil kenny

Can anyone help.
I would like to convert some VRML scripts into a raytraciung fromat
via Q-basic.  
In VRML rotations can be defined about an arbitary vector
eg   1 1 0   Theta
where 1 1 0 is the vector defining the axis of rotation and theta is
the angle to rotate by.
I need to convert this into rotations about one or more of the normal
x, y, z axis
As Q-basic does not handle matrices I would prefer a formular not
involving these.  (They had not been invented when I was at school and
I have never been happy working with them)
I would also prefer an e-mail reply but that is not so important.
Thanks
john@jacy.demon.co.uk
:
We are born naked, wet and hungry. Then it gets worse.

Can anyone help.
I would like to convert some VRML scripts into a raytraciung fromat
via Q-basic.  
In VRML rotations can be defined about an arbitary vector
eg   1 1 0   Theta
where 1 1 0 is the vector defining the axis of rotation and theta is
the angle to rotate by.
I need to convert this into rotations about one or more of the normal
x, y, z axis
As Q-basic does not handle matrices I would prefer a formular not
involving these.  (They had not been invented when I was at school and
I have never been happy working with them)
I would also prefer an e-mail reply but that is not so important.
Thanks
john@jacy.demon.co.uk
:
We are born naked, wet and hungry. Then it gets worse.

Truth is truth and the person in your scenario told a lie.
Truth brings peace and happiness, but that does not mean _immediate_ peace
and happiness.
   Similarly, God can be all powerful and all loving if you understand
that suffering can lead to greater eventual happiness.  You can't know one
without feeling the other.

In article , Michael Weiss  wrote:
>Anyway, whatever the truth about the U.S., I'd be interested in
>hearing impressions from other countries.  If you, kind reader, were
>educated outside the U.S. and still reside there, what is your feeling
>about the average level of math and science instruction in your
>country?  Has it declined over the years?  How would you assess it
>today?
I live in Italy. The math education level is very low through primary and 
secondary school. Only if you go to university, and get to study chemistry, 
engineering, physics or, of  course, mathematics you are going to have 
significant education. I am currently in business school and I wish I had been 
taught more maths...