00lhramer@bsu-ucs.uucp (Leslie Ramer) (01/17/91)
College level mathematics such as Calculus have many theorems and technical
proofs. In my efforts to become more mathematically oriented, I've started
keeping a sort of mathematical journal.
In my journal, I often study things independently, sometimes material that
is not covered in my classes. I find it quite interesting to ponder
mathematical methods common to numerical analysis and calculus.
i.e. -> I've derived Simpson's rule from an expansion of the trapezoidal
rule and its error bound. I'm positive that there must exist some
sort of function to relate each of the coefficients in numerical
integration.
From a geometric representation, I've come across an interesting
integration formula that allows integration of the inverse function
rather than the function itself. I haven't fully examined it, but
it has worked with many of the functions that I've tried it on.
In high school, I devised a method of computing approximations of Pi
A = 0
0 n _________
_________ P = 2 * \/ 2 - A
A = \/ 2 + A n n-1
n+1 n
What I found odd?
__________
Lim A = 2 Lim \/ 2 - A = 0
x n-1
x->inf x->inf
but...
Lim P = Pi
x
x->inf
My notebooks tend to be mathematical diaries, in which I put some form of
mathematical wisdom. (What's wisdom? What's commonsense? Who really knows?
It's more or less up to the individual.)
I'm quite interested in promoting my own mathematical maturity. In the past,
I've always been quite creative and artistic. I tend to like flavor in both
my mathematical and computer programming experiences.
My questions seem to stem from methods of proof. What kinds methods of proof
are there? In particular, I'm interested in what I might call an undirected
proof that ties mathematical concepts together. Those similar to the entries
that I have made in my notebook(s).
I believe Newton once said, "If I have seen farther than the average man, it
was by standing on the shoulders of giants."...and..."I seem to have been
a boy walking on a beach. Occasionally finding a pretty shell, or a beautiful
pebble while the ocean of truth lay undiscovered in front of me."
==== "No one runs so fast as he that is chased."
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