Hello,
My name is Joseph Galante and I would like to formally request that my
paper, intitled "Generalized Cantor Expansions" be published in the
RHIT Mathematics Journal. The article is attached to this email
(zipped) in MS Word format. The work for the paper was completed during
sophomore year at the University of Rochester, between the dates 11/02
and 5/03 and was put into its final form this summer (2003). The
mathematician who is sponsoring me is Ken McMurdy at the University of
Rochester.
Abstract:
There are many ways to represent a number, commonly known as base
expansions. The most frequently used base is ten, which is the basis
for our decimal number system. However a more uncommon way to represent
a number is the so called cantor expansion of the number. This system
uses factorials rather than numbers to powers as the basis for the
system, and it can be shown that this produces a unique expansion for
every natural number. However, if you view factorials as products,
then it becomes natural to ask what happens if you use other types of
products as bases.
This paper explores that question and shows there are an uncountably
infinite number of bases which can be used to represent the natural,
and real numbers uniquely. By using these new and interesting types of
bases, it becomes possible to formulate bases in which all rational
numbers have a terminating expansion.
About the Author:
My name is Joseph Galante and am junior mathematics major at the
University of Rochester. I initially learned of the Cantor Base
Expansion from Kenneth Rosen's Discrete Mathematics textbook. During
sophomore year, I mentioned to Prof. Ken McMurdy that there are
probably more types of strange expansions and he encouraged me to
explore this further. It ultimately resulted in the paper, completed
during my free time sophomore year. At college, I am the VP of the
Society for Undergraduate Math Majors (SUMS) and I am the webmaster for
my Tae Kwon Do club. In the future I plan to pursue mathematics with
hopes of graduate school.
Thank you for your time in reading this email and processing my
request.
Sincerely,
Joseph Galante
Dear Editors,
I would like to recommend a paper by Joe Galante entitled
"Generalized Cantor Expansions". Joe was a student of mine in honors
calculus as a freshman in the Fall of 2001. The work was done in the
Fall of 2002 and Spring of 2003, though, while Joe was a sophomore.
It's actually a very nice story how the work came about. Joe had a
class right after my Discrete Math class and in the same room. So we
would sometimes chat for a few minutes about what I had been covering.
One day I explained what the Cantor Expansion of a natural number was
and casually wondered out loud if there were actually many such
expansions with their own individual merits. Joe picked up on the
suggestion and worked on the problem unprompted by me. A couple of
weeks later he showed me his initial result, an uncountable collection
expansions for which the Cantor Expansion and the usual base expansions
were examples. I was very excited and immediately suggested that he
should write a paper and submit it to an undergraduate journal for
publication.
A semester has passed since then, and Joe has now proven some
results which further put his generalized Cantor expansions in
perspective. For example, he has shown that fractions and even real
numbers can also be represented in this way. I must warn that I have
not carefully checked every detail of the paper and do recommend that
it be carefully refereed (as I'm sure it will). I have a great deal of
confidence in Joe from his course work, though. I will be surprised if
there are more than a few minor mistakes.
Overall I think the paper is very appropriate for the RHIT
journal. The subject matter (digit-related number theory) should be
accessible to many undergraduates, and yet the results are laid out and
proven in a very sophisticated way, with carefully stated (and well
motivated) definitions and theorems. I will look forward to hearing how
it stands up to the refereeing process, but I honestly feel very good
about the paper.
Sincerely,
Ken McMurdy
Assistant Professor
University of Rochester