The explanation I like to use with pre-calculus students appeals to
intuition:
Consider matching the distance covered with the time required to cover it.
Call the total time 1 unit, that way you can match half the distance with
half the needed time, quarter the distance with quarter the needed time,
etc. Then using proof by arm-waving, you can show that the sum of all the
fractions is 1.
Calculus, of course, provides us with the concept of limits, which yields
a rigorous ("i.e.: satisfactory to mathematicians") proof.
John M. Flanigan <johnf@hawaii.edu> The equation is the final arbiter.
Math Resource Instructor --Werner Heisenberg
Kapi'olani Community College The scoreboard is the final arbiter.
Honolulu, Hawaii --Bill Walton
On Sun, 14 Apr 1996, Phil Mahler wrote:
> While the list is between threads - a question, a recommendation, and a
> comment.
>
> -------------------------------------------------------------------------
> Consider the following version of one of Zeno's Paradoxes. You are in the
> middle of a room. To leave the room you must first cover half the distance
> between you and the exit. Then you must cover half the remaining distance.
Then
> you must cover half the remaining distance again... Considering this infinite
> process, how do you manage to finally get out of the room?
>
> I have heard it said that this paradox, and perhaps others of Zeno, are
> "explained" in the calculus. Assuming you have heard this too, and agree,
> I'd be interested in how it is explained. Or a reference to an appropriate
> source.
>
> -------------------------------------------------------------------------
>
> Many years ago I saw the article I cite here:
> Brief Tabular History of Some Relevant Mathematical Notations, Allen C.
> Utterback, Cabrillo College, CA, in The MATYC Journal, Spring 1977, Volume 11,
> Number 2.
>
> (How many remember the MATYC Journal!)
>
> In the article Prof. Utterback cited another book,
> A History of Mathematical Notations, Florian Cajori, The Open Court Publishing
> Company, Chicago, vol I (1928), vol II (1929).
>
> I liked the idea of knowing, and using in my teaching and writing, such things
> as when pi was first used to mean the ratio of C/D, who first used negative
> exponents, etc. So, I used this article in my teaching and writing, and
> eventually drove 40 miles several times to a library that had Cajori's books.
I
> tried to order them, and was told they were out of print. Thus, I was very
> happily surprised when I came across them recently, in one paperback book for
> $19.95.
>
> I highly recommend this addition to your library if you have a penchant
> for history in mathematics. Here is some relevant info.
>
> Author: Cajori, Florian, 1859-1930.
> Title: A history of mathematical notations / by Florian Cajori
> Publication Info: New York : Dover Publications, 1993.
> Notes: Originally published: Chicago : Open Court
Pub.Co.,1928-1929.
> Notes: "Two volumes bound as one."
> Notes: Includes indexes.
> Notes: v. 1. Notations in elementary mathematics
> v. 2. Notations mainly in higher mathematics.
> ISBN: 0-486-67766-4 (pbk.)
>
> ---------------------------------------------------------------------
>
> One of our librarians recently suggested I might want to read the book Humble
> Pi, by Michael K. Smith - 1994, Prometheus Books. Since she handed it too me,
I
> couldn't say no. I'm not recommending it, but it's interesting because its
sole
> purpose is to make a case for requiring one year of HS math, with algebra,
> geometry etc. electives. Perhaps it was reviewed in the Mathematics Teacher or
> somewhere, but it's the first I've heard of it.
>
> The author feels that math is grossly overrated!
>
> I don't happen to feel that the book is convincing - the author really
> stretches some of his arguments -, but I do feel, as one writer on this list
> wrote very recently, that we need to reexamine every premise, and if the
answer
> still comes up algebra, so be it. If not, so be it. Also, forewarned is
> forearmed.
>
> ------------------------------------------------------------------------
>
> Philip Mahler
> Middlesex CC
> Bedford, MA 01730
>