Subject: Re: Duodecimal nomenclature?
From: edgar@math.ohio-state.edu (G. A. Edgar)
Date: Wed, 13 Nov 1996 09:16:43 -0600
In article <56b9sf$p12@duke.squonk.net>, darshan@squonk.net (Kenneth
Lareau) wrote:
> This is something that has been on my mind a bit recently, and I thought
> that someone here might be able to answer it.
>
> I am wondering if anyone has ever bothered to create an actual nomenclature
> for the duodecimal system. I made a half-hearted attempt several years ago,
> but never came up with anything that sounded remotely coherent... then a-
> gain, I'm not a linguist. :)
As I recall from long ago, one way that was used had the two
extra digits: X, pronounced "dek", and a backward 3, pronounced "el".
Of course 10 is not pronounced "ten" but "dozen".
>
> I know there is, or at least used to be, a society specifically for the pro-
> motion of the duodecimal system, but recent searches on the web have brought
> up nothing, as well as searches through several mathematics-oriented sites.
> Any info anyone could give me on this subject would be greatly appreciated.
Probably quite obsolete.
Our library lists these three books:
AUTHOR Terry, George S. (George Skelton)
TITLE Duodecimal arithmetic [by] George S. Terry.
PUBLISH INFO London, New York [etc.] Longmans, Green and co., 1938.
DESCRIPTION 4 p.l., 407 p. incl. tables, 2 diagr. 30 x 25 cm.
AUTHOR Andrews, Frank Emerson, 1902-
TITLE New numbers; how acceptance of a duodecimal (12) base would
simplify mathematics, by F. Emerson Andrews.
PUBLISH INFO New York, Essential books [1944]
DESCRIPTION 168 p. illus. 20 cm.
NOTES "Complete multiplication table, base of 12" in pocket.
"Second edition."
"A list of references": p. 143-146.
AUTHOR Terry, George S. (George Skelton)
TITLE The dozen system, an easier method of arithmetic [by] George S.
Terry.
EDITION [1st ed.]
PUBLISH INFO London, New York, Longmans Green, 1941.
DESCRIPTION 53 (i.e. 63) p. illus., map., tables. 29 cm.
NOTES Reproduced from type-written copy.
Paged by the dozen system.
"Duodecimal protractor reading .002 circle" and table laid in.
--
Gerald A. Edgar edgar@math.ohio-state.edu
Subject: Concepts Was: Re: Why are we worse in math & science than foreign nations?
From: hrubin@b.stat.purdue.edu (Herman Rubin)
Date: 13 Nov 1996 09:46:40 -0500
In article <568rut$j8h@nntp1.best.com>,
Richard Alvarez wrote:
I am leaving in the entire article, but placing it at the end.
What is different about the teachers and administrators praised
and those who are not is not that they taught lots of details,
but that they presented the concepts. Mr. Alvarez then was able
to fill in at least many of the details. Those who merely told
him the details, like the kids who showed him how to do long
division, may or may not have understood the concepts involved.
And the principal who objected to children asking questions is
a good example of the far too numerous people in education who
are a major liability to ever doing anything about it.
Professor Langmuir did not teach that much more mathematics in
a short time, but he presented the CONCEPT of Fourier series.
One can build up on the concepts; doing the other way is the
hard one.
Mr. Alvarez also points out how his thesis advisor clarified things
by pointing out that it was a special case; the general situation,
not having so much special properties, is usually easier to grasp.
Quoting the last paragraph,
* In the teaching situations that I have stumbled into, I have tried
*to teach like those six great people. It is both satisfying and
*disappointing to have the students say "Why didn't somebody explain that
**long* ago!?!". It is satisfying because they are learning something.
*It is disappointing because they could have learned it much sooner.
This is a large part of the problem; we are not even teaching the
concepts as well as was done in the past. And Miss Welch's horrors
are legion at the college level; students are continually saying,
"Don't give me the THEORY; just tell me how to do the problems."
>Some fantastic math teachers...
>and a school principal.
> When I was a little kid in elementary school, the big kids spoke of
>division, and I wanted to know what it was. I couldn't get an
>explanation from the older kids, or from teachers; they just showed me
>examples of long-division, with no indication of what it was or what it
>was good for. Finally, the nice vice principal said something like "How
>many times does 2 go into 6?" I said "3". She said "OK, that's
>division.". Why couldn't somebody have explained that in the first
>place? Darned if I know!
> The school principal had me on the carpet for asking questions about
>things like division, that I wasn't supposed to know yet. I wonder how
>many other kids she turned off. I hope that they were not as
>intimidated by her as I was.
> Later, in that same school, I tried to find out what algebra was.
>(And I suppose I landed on the principal's carpet for it.) Again, I got
>lots of examples, but no explanations. Here is how that question
>finally was answered:
> During my high school freshman year, Miss Welch substituted in our
>algebra class one day. Miss Welch called a girl to the board, and had
>the girl work a problem. Then Miss Welch said "OK, she did that right.
>Now *why* did she do so-and-so?". The class was horrified! "We don't
>ask *why*! We do it that way because that's how we're *told* to do it."
>Then it was Miss Welch's turn to be horrified. She explained "There are
>*reasons* for all of this. There's *logic* behind it!" During that one
>class period, math suddenly went from just an interesting subject, to a
>whole new world for me. I registered for Miss Welch's plane geometry
>class for my sophomore year.
> During my sophomore year, Miss Mathisen substituted for Miss Welch
>in our geometry class one day. Miss Mathisen was pacing around the
>front of the room, quite excited, waving her hands in the air as she
>talked. Suddenly she stopped in front of the board, and said "Now look
>at what we have here: three new theorems!". (My response: "My gosh, we
>*do*!") Miss Mathisen: "And look at what we can *do* with them!". (My
>response: "My gosh, we *can*!") I registered for Miss Mathisen's
>advanced algebra and trig classes for my junior year. Miss Mathisen
>showed us serious *uses* for our math. Of course that delighted my
>engineer's mind. Miss Mathisen was an engineer at heart. But few
>ladies went into engineering when Miss Mathisen was in college.
>Fortunately for us, Miss Mathisen went into teaching.
> Often I passed Mr. Barker's class room. The door always was open,
>and his students looked like they were *enjoying* his presentations.
>So, with fear and trembling, I went into his room and introduced myself
>to him. I soon saw that behind his stern-looking exterior, was a fine
>gentleman and a brilliant teacher. I registered for his classes during
>my senior year. He was truly outstanding.
> My big regret is that maybe I was too bashful to tell those three
>teachers how much I appreciated them. I hope that they realized how
>much they influenced the lives of us students, and that they understood
>that some of us were too immature and bashful to tell them that.
> During my junior year in college, in an applied math course,
>Professor Harold Wayland had a way of saying "This is what it's all
>about.". He could explain quickly and well, what most text books spend
>many pages beating around the bush and still not really *explaining*.
> During my senior year, in an electromagnetic theory course,
>Professor Robert Langmuir typically drew complicated hardware
>configurations on the board, then said "It is obvious that the solution
>is of the form...", and he wrote long product-solutions containing
>functions whose arguments were fractions with wild arguments consisting
>of constants and variables all over the place. He was such a good
>teacher that it *was* obvious! Then he would say "By inspection, we see
>that...", and he would insert the values of the constants and variables.
> During my junior year, through no fault of Professor Wayland, my
>class section missed out on Fourier series and transforms, which we
>needed for Professor Langmuir's course. Professor Langmuir said "You
>people don't understand Fourier series? Here, let me explain it.". In
>a few minutes, he taught us more math than most math courses could teach
>in two weeks. He was *good* !
> Later, often I went to my thesis advisor with particular questions
>about math or physics. Usually he said "This is just a special case of
>the following general situation.". I left his office feeling that he
>had opened a whole new section of the world to me.
> In the teaching situations that I have stumbled into, I have tried
>to teach like those six great people. It is both satisfying and
>disappointing to have the students say "Why didn't somebody explain that
>*long* ago!?!". It is satisfying because they are learning something.
>It is disappointing because they could have learned it much sooner.
> Dick Alvarez
> alvarez@best.com
--
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu Phone: (317)494-6054 FAX: (317)494-0558
Subject: Re: cross products in 4 dimensions
From: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Date: 13 Nov 1996 15:16:59 GMT
In article <56a9j2$l4h@sulawesi.lerc.nasa.gov>, Geoffrey A. Landis In article <55tclo$jsr@newsstand.cit.cornell.edu> Bryan W. Reed,
>breed@HARLIE.ee.cornell.edu writes:
>>In order to pick out a unique (to within minus signs) direction
>>that's perpendicular to all vectors in some linearly independent set
>>S, you need S to have n-1 elements (where n is the dimensionality).
>>That's why you need three vectors in 4 dimensions to form the "cross
>>product." Or whatever the generalization of the cross product is called.
>Cross products really only work in 3 dimensions.
Cross products of the kind described by Bryan Reed work in any dimension
3 and up.
Some consider that cheating. "Real" cross products, the two vectors at a
time sort, work in dimensions 3 *and* 7. The familiar 3-dimensional one
can be thought of as the purely imaginary part of quaternion multiplication.
Similarly, the purely imaginary part of octonion multiplication can be read
as a 7-dimensional cross product. Most of the familiar identities hold in
both cases.
See the nice little paper:
W S Massey "Cross products of vectors in higher-dimensional Euclidean
spaces", AMER MATH MONTHLY, 90 (1983), #10, pp 697-701.
Massey's first theorem is that bilinear maps R^n x R^n -> R^n such that
the result is perpendicular to the factors and the norm of the result
is equal to the area of the parallelogram spanned by the factors is one
of these two products. His second theorem is if we instead assume the
product is continuous, keep perpendicularity as before, but only require
that the product of linearly independent nonzero vectors is nonzero, then
again, we have one of these two products.
--
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)
Subject: Re: BETROTHED numbers.
From: jlame@math.ohio-state.edu (John Lame)
Date: 13 Nov 1996 15:37:29 GMT
In article <56bkcq$foe@cantuc.canterbury.ac.nz>,
Bill Taylor wrote:
>Guy coined this nice terminology. It refers to numbers which are almost
>amicable, but each is the sum of the PROPER divisors of the other.
>Proper, hence betrothed. e.g.
>
>48 = 3 + 5 + 15 + 25 and 75 = 2 + 3 + 4 + 6 + 8 + 12 + 16 + 24.
>
>There are many other betrothed pairs, 140 & 195 is the next.
>
>My query is, are there any "affianced sequences", of length k > 2.
>
>i.e. n2 = f(n1), n3 = f(n2) ... n_k = f(n_(k-1)) & n1 = f(n_k);
>
>where f(n) = sigma(n)-n-1, (as above).
>
The only affianced sequences involving any natural numbers <= 10000 have
length k=2. These are:
{48,75} {140,195} {1050,1925} {1575,1648}
{2024, 2295} {5775,6128} {8892, 16587} {9504, 20735}
All other numbers <= 10000 eventually collapse to zero or lead
to one of the above 2-cycles.
This was obtained via Mathematica. (hopefully its correct as well)
Later,
John
Subject: Re: a plane twice bigger has a engine more powerful
From: Alexander Anderson
Date: Wed, 13 Nov 1996 15:32:54 +0000
In article <3288dfd9.0@news.cranfield.ac.uk>, Simon Read
writes
>Therefore: constant = L^3 / (V^2 L^2)
> * * *
> * * +---area
> * *
> * +---velocity^2
> *
> +---mass
>
>That gives us v^2 increases as length. So v scales as L^0.5
>
>Engine _thrust_ scales as
>the mass of the plane (length cubed) but engine _power_ is
>force x velocity, which is L^3.5
>
>
>That's your answer, I think. If a plane is twice as large, engine
>power must be multiplied by 2*2*2*root(2).
Wow! Yuh, it was just a heuristic guess. Finding out it's not
relevant is just as useful as if it is, I've always found.
A bit like the "3 dogends = 1 fag, there's 9 dogends, how many
fags?" question if you get the drift of that seeming non-sequitur.
Sandy
--
// Alexander Anderson Computer Systems Student //
// sandy@almide.demon.co.uk Middlesex University //
// Home Fone: +44 (0) 171-794-4543 Bounds Green //
// http://www.mdx.ac.uk/~alexander9 London U.K. //
Subject: Re: lim_(x -> 0) 0/x
From: David Kastrup
Date: 13 Nov 1996 17:43:04 +0100
nobody@nowhere (me) writes:
> Fredrik Sandstrom wrote:
>
> >I've got a simple(?) question. I came to think of it one day, and I
> >can't come to a conclusion. What is
>
> > lim 0/x ?
> >x -> 0
>
> >One possible answer would perhaps be that lim_(x->0+) 0/x =3D 1 and
> >lim_(x->0-) 0/x =3D -1. Other possibilites are +/- infinity, or perhaps
> >0. What do you think?
>
> I think that you should read the MANY past comments on this topic
> (already beaten to death) and reach your own conclusion.
Nope, this particular tpoic has not been beaten to death as there is
no real question involved.
lim x->0 0/x is 0, and no mistake. 0/x stays 0 if x gets arbitrary
close to 0, and the value at 0 itself (where it happens to be
undefined) does not interest when you are taking the limit.
--
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: automorphisms
From: denny@GFZ-Potsdam.DE (Patrick Denny)
Date: 13 Nov 1996 16:47:56 GMT
In article <55r44m$io0@epx.cis.umn.edu>,
Laura M Walbrink wrote:
>Can anyone clarify homomorphisms, isomorphisms, and any other related
>-isms for me? Thanks.
A homomorphism is a function mapping one algebraic structure into another
such that if a,b are elements of the algebraic structure X and f:X->Y is a
homomorphism, then f(a.b) = f(a).f(b) where the "." in the left side of the
equation is the algebraic operation of the first algebraic structure and
the "." on the right hand side is the operation of the second algebraic
structure.
An isomorphism is a bijective homorphism.
Related stuff ? Well, do you want to know about homEomorphisms, bijections,
inverse functions, injections, surjections, modules, rings, fields, etc..
There is a LOT of related stuff !!
Best Regards,
Patrick Denny
Subject: Re: Mission Impossible: Can probability=0 events occur?
From: David Ullrich
Date: Wed, 13 Nov 1996 10:56:05 -0600
Robert E Sawyer wrote:
>
> Concerning an infinite sequence of iid 0/1 random variables
> X_1, X_2,... , with fixed pr(X_i=1), the following comments
> have appeared in this thread.
>
> Alan Douglas wrote:
>
> >... one can easily imagine an infinity
> > of sequences whose relative frequencies never converge.
>
> G. A. Edgar wrote:
>
> >Sure. Such a sequence is no less likely (or more likely)
> >than any given sequence whose frequencies do converge.
>
> I haven't thought about this in a long time, but something
> is wrong here, I think. *All* binary sequences have
> *convergent* relative frequencies, so convergence is
> not only "almost sure", but "sure".
> (The questions concern not *whether* there is convergence,
> but rather the *values* to which there is convergence.)
>
> Denote by R_n the (random) relative frequency of "1" among
> X_1, X_2, ..., X_n: R_n = (X_1+...+X_n)/n; n=1,2,...
> and let non-random values be denoted by corresponding
> lower-case symbols.
>
> Claim:
> *Every* binary sequence has a convergent relative frequency.
>
> (It suffices to look at the non-random case, just noting
> r_n = r_(n-1) + (x_n - r_(n-1))/n, hence
> |r_n - r_(n-1)| <= 1/n -> 0.)
Supposing for a second that you added the fractions
correctly, you seem to be asserting that if
|r_n - r_(n-1)| -> 0 then the sequence (r_n) converges. This
is nonsense (maybe you're misremembering the definition of
"Cauchy sequence"?)
It would be very easy to give a counterexample, like
maybe r_n = log(n). But no example is needed: If this were true
then it would in fact imply that any sequence of 0's and 1's
had a convergent relative frequency, and that's obviously false:
Start with 0. Now the relative frequency of 1's so far is 0.
It's clear that if you add enough 1's you make the relative
frequency of 1's greater than 2/3. Then if you add enough
0's you make the frequency < 1/3. Repeat.
> It follows trivially that for infinite *random* sequences,
> pr("R_n converges")=pr({(x_1, x_2, ...): r_n converges})=1,
> since *every* (x_1, x_2, ...) is such that r_n converges.
>
> Lastly, as a consequence of the Law of Large Numbers,
> pr("R_n converges to t")=1 if and only if t=pr(X_i=1).
>
> Robert E Sawyer (soen@pacbell.net)
> _____________________________
--
David Ullrich
?his ?s ?avid ?llrich's ?ig ?ile
(Someone undeleted it for me...)