Pressing Toward the Prize

Posts Tagged ‘complex number’

Capstone Ideas

Posted on: November 5, 2010

The time has come to declare the topic for my capstone project, and I still find myself somewhat undecided. The two areas that interest me most are voting paradoxes and working with complex numbers. I like the practical application of studying voting processes, the problems that can arise when trying to determine the will of the people, and the manipulations (inadvertent or otherwise) that can greatly affect the results. Fairness in voting is a concept that should be of interest to us all, since the outcomes dictate many aspects of our lives.

A few weeks ago, my math professor suggested considering for our capstone project topics that may have been introduced in past math classes, but not studied in depth. My interest in complex numbers is a result of this suggestion. My exposure to the complex number system has mainly consisted of performing algebraic manipulations, and I would like to know more. Analysis of the complex numbers is such a broad topic that, should I decide to go this route, I will need some direction in choosing a focus. An intriguing application that my professor mentioned is one in which complex numbers are used to solve integration problems that would be quite difficult in the real number system. Both of these are worthy topics, but our formal proposal is due next week, so I will need to hop off the fence soon!

Posted on: October 18, 2010

Part of preparing for our Capstone Senior Project is learning how to research information using mathematical resources, including articles, books, and textbooks. Although the database Mathscinet is a great place for gathering references on a particular topic of interest, we have discovered that most of the articles are a bit over our heads. In those research articles, mathematicians are writing to their peers, of which we are not (yet). An alternative for us is to consider using math journals geared more toward the level of undergraduates. This leads me to our current Capstone assignment. After a lesson on how to best read a math article, complete with an appropriate list of do’s and don’ts, each of us was assigned an article to “read” and then do a review in the fashion of a “2nd grade book report.” I found that as simple as this sounds, it was a pretty tall order!

My article was entitled “Gaussian Integers and Arctangent Identities for $\pi$” by Jack S. Calcut, taken from The Mathematical Association of America, June-July 2009 issue. Not knowing what Gaussian integers were, at least not by name, I was relieved to find that a Gaussian integer is simply a complex number whose real and “imaginary” parts are integers. But as I stated in an earlier blog, my exposure to complex numbers has been minimal, and my exposure to graphing complex numbers has been nonexistent. This made the article a bit of a challenge for me, but the author did a very good job of explaining and defining terms where he thought there might be confusion on the part of the reader; this gave the article the feel of an instructional textbook. Not every term was defined, however, and I was grateful for my current Abstract Algebra course when I had to look up the definition of a commutative ring. Since a commutative ring has to do with binary operations and groups, and we have recently been studying both, I was thrilled to find that I actually understood the definition!

The article began by addressing the connection of arctangent identities for $\pi$ to calculating the decimal digits of $\pi$. Then a lesson involving unique factorization, primes, and irreducibles in $\mathbb{Z}[i]$, concepts I understood only marginally, led to the main lemma about Gaussian integer $z \neq 0$ and the conditions under which $z^n$ is real. This was then used to show that a simple form of an arctangent identity for $\pi$ does not exist. The author also expanded the application of the lemma to show connections between rational vs. irrational values in the leg lengths and angles of right triangles, as well as various concepts involving angles created on geoboards. As it turns out, developments in this area have been ongoing since the 14th century, involving a multitude of mathematicians who have discovered, and at times rediscovered, the various concepts, identities, and applications presented. And with the advent of the computer, new calculations, identities, and avenues of study have been made possible. This was a very interesting article, even though there was much I did not fully understand. I am deeply grateful that I only needed to produce a “2nd grade book report” about it!