# Pressing Toward the Prize

## Archive for the ‘General Information’ Category

### The Euclidean Algorithm

Posted on: October 27, 2010

I recently did a review of an article involving Gaussian integers (complex numbers whose real and “imaginary” parts are integers) in which the Euclidean algorithm was used to find the greatest common divisor (gcd) of two complex numbers. I had never heard of the Euclidean algorithm before, and the process described in the article involved complex numbers and several Greek letters, so I had no idea what was going on. Then, as luck would have it, only a few days later we discussed in math class how to use the Euclidean algorithm to find the gcd of two integers, that is, the largest number that divides both of them without leaving a remainder. It is based on the principle that the gcd does not change if the smaller number is subtracted from the larger number. If this process is repeated, the numbers keep getting smaller until one of them is zero. At this point, the gcd is the larger, non-zero number.

Here is a simple video that describes the process used to find the gcd of 123 and 36:

So what happens if the numbers are a bit larger?

Let’s find the gcd of 5,624 and 3,959:

\begin{aligned} 5,624 &= 3,959 \cdot 1 - 1,665\\ 3,959 &= 1,665 \cdot 2 - 629\\ 1,665 &= 629 \cdot 2 - 407\\ 629 &= 407 \cdot 1 - 222\\ 407 &= 222 \cdot 1 - 185\\ 222 &= 185 \cdot 1 - 37\\ 185 &= 37 \cdot 5 - 0 \end{aligned}

So the gcd(5,624, 3,959) = 37.

No matter how large the numbers, the process can be completed in a finite number of steps. It has been found that the maximum number of steps required is five times the number of digits in the smaller integer. If the numbers are sufficiently large, this still could take a very long time. The beauty of this algorithm is that a computer can be programmed to complete the steps in a matter of seconds, rather than the untold hours if calculated by hand. Although I still don’t fully understand this algorithm as applied to Gaussian integers, I have found it to be a wonderful tool for calculations on real integers.

### What are you thinking?

Posted on: October 6, 2010

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.”

### Hello world!

Posted on: September 11, 2010

Hi! My name is Linda Nusser, and I am currently pursuing a BS in Mathematics at Pacific Lutheran University for the purpose of one day teaching math at the community college level. Although understanding math can be challenging at times, I thoroughly enjoy the feeling of accomplishment I get when a problem has been solved successfully. I have been very blessed to have teachers throughout my education who were very passionate about math, and they lit the spark of passion within me.

For the most part, math makes perfect sense to me, and I always feel badly for people who think math is too hard. The desire to help people discover that understanding math is well within their reach is what inspired me to become a teacher. I am a math tutor at Pierce College, and seeing a student’s eyes light up as they grasp a math concept for the first time is very gratifying. When being a student feels overwhelming, I remind myself that I am working toward fulfilling my lifelong dream of becoming a teacher. This is one of the prizes I am pressing toward.

I was asked to discuss the most interesting class that I have taken at PLU, and I found this to be a very tall order. Every course I have taken has been enlightening, and the professors are wonderful. To choose the “most” interesting is very difficult, but after much thought I have decided it was the Intro to Computer Science class. Computer programming was new territory for me, and learning the language and how to use it effectively was very time-consuming. But I liked the mix of learning a “foreign language” and then using that language to perform procedures and solve problems that included math. I was amazed at how much is involved in creating and running even the simplest programs, and I felt euphoric every time a program worked the way it was supposed to. As a result of my own experience, I found myself in awe of the amazing things that programmers do.

Last, but certainly not least, is the Capstone project. At this point, I really don’t have a clue what I would like to do.  I have taken three different classes on statistics and probability, and I find these topics intriguing and relevant to our world. The application of statistics concepts can be found in nearly every aspect of life. Perhaps performing a hypothesis test on a claim, or using probabilities in some way, would be interesting. I don’t know what I want to do with this, though, and I will definitely need some ideas and direction.