Pressing Toward the Prize

Posts Tagged ‘integers

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!

How much do we do without thinking? We take for granted that the algebraic equation x+6=15 can be solved for x, but do we even consider the principles that make this possible? We know that x must equal 9 in order for this statement to be true, but how do we know this? In our Abstract Algebra class, we are looking at “abstracting” the essence of various mathematical structures in order to find commonalities among them. Part of that process has been identifying certain properties that can exist for a given structure, although they may not.

Consider the simple equation mentioned above. We can solve this because of certain properties that exist in the integers. Let’s take a look at what is actually involved in finding the solution:

\begin{aligned}x+6&=15\\ (x+6)+(-6) & = 15+(-6)\quad \textbf{(inverse)}\\ x+(6-6) &=(15-6)\quad\textbf{(associativity)}\\x+0&=9\qquad\textbf{(inverse)}\\x&=9\qquad\textbf{(identity)}\\ \end{aligned}
In reality, this simple equation can only be solved because the associative, inverse, and identity properties hold for addition on the integers. So the next time you are able to solve an algebraic equation, just say, “Thank you.”

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  • gramsonjanessa: I can't wait to listen to your capstone presentation in the spring! Your proposal was really interesting and I'm interested to see how the linear alge
  • dewittda: This is impressive! I thought I was good because I solved a rubik‚Äôs cube once in an hour. I served with a guy in the Air Force who could solve a r
  • ZeroSum Ruler: The Euclidean algorithm should me the mainstream way we teach students how to find the GCF. Why isn't it? A mystery.