Pressing Toward the Prize

Posts Tagged ‘algebra

Most of us have probably heard the saying, “You can’t teach an old dog new tricks.” And I would venture to say that many of us know someone who would qualify as the “old dog” that is mentioned. It is easy to settle into a routine over time, and the older one gets, the more concrete that routine can become. There are different reasons why folks get set in their ways. For some, they truly believe their way of doing or believing is the only right way, and they aren’t willing to entertain any conflicting viewpoints. For others, they have just become so comfortable in their ways, that any deviation is unthinkable. But then there are those who continue to do what they have always done for a much deeper reason. My dad is an example. My mom passed away at the beginning of this past summer after 59 1/2 years of marriage to my dad. In fact, this coming Monday would have been my parents’ 60th wedding anniversary. Some of the ways my dad does things don’t always make sense to me, but he will not deviate from them because it was the way he and my mom always did things. I think it is his way of keeping her alive, and I would not take that away from him. I miss my mom, and I love my dad, so we just keep moving forward.

By now you are probably saying, “Nice sentiment, but what does this have to do with math?”

I had a conversation yesterday with my math professor about some proofs I did on our recent abstract algebra exam. I don’t remember his exact words, but the essence (as I remember it) was that my proofs are detailed, but reasonably concise. This was a HUGE compliment for me, because I have struggled greatly with trying to figure out this whole “mathematical proofs” business. I told him I was doing well because I had a great teacher – he was my proofs professor – and in his typically humble fashion he said, “Yeah, well… maybe.” But it’s true!! I took proofs with him last spring, and I remember trying to learn how much to include in a formal proof – not too wordy, but not wanting to leave out critical information, either. It was an interesting balancing act for a while, and my professor was quite patient with me, so I guess our perseverance is starting to pay off.

To explain how much of an improvement this is, I must tell you about my first “proofs experience.” Nearly two years ago, I took a linear algebra course at another institution. It was a heavily proofs-based course, but with one hitch: none of us had ever had any kind of formal proofs training! My instructor knew this and was trying to teach us as we went along, but he was forever writing on my assignments, “You could probably condense this down” or “A little wordy; see what you can do.” You wouldn’t believe the stuff that poor man had to wade through; it was astronomical! The problem was that I didn’t understand a formal proof should not contain all one’s thought processes, only what is essential to prove one’s point. And because I am a very detailed person by nature, trying to figure out what to trim has been a challenge. So with a sense of accomplishment I can say that I was thrilled to hear that I am finally making some progress in this arena, and there may be hope for me yet. So I guess sometimes you can teach an old dog new tricks!!

p.s. I love you, Dad!

In a recent conversation with my math professor regarding a possible topic for my Capstone, I commented that I really enjoyed studying linear algebra. I told him I might be interested in doing some type of real-world application involving linear algebra for my project, and he began to tell me about Donald Saari. Mr. Saari is a mathematician who studies what he calls “the paradoxes and problems of voting procedures,” and analyzes voting methods using linear algebra. I am currently looking at two books he wrote: “Chaotic Elections!” and “Basic Geometry of Voting.”

In “Basic Geometry” Mr. Saari makes it clear that when there are only two choices up for a vote, there is generally no difficulty in determining the winner. But when there are more than two choices, things can get rather interesting. In order to demonstrate some of the challenges that can arise,  Mr. Saari opens the book with the story of a fictional academic department chair who finds himself in hot water as a result of a departmental vote. The problem is that the same vote can be interpreted different ways, depending on which voting method is used. Even if a method ranking preferences is used, various outcomes can result depending on which way one counts the rankings. In fact, each of the options can be deemed the winner, depending on the method used.

Now things can become even more complicated if a group of voters attempts to manipulate the process. For example, assume Al, Bob, and Chuck are candidates for the new Chair position. Of the 15 people voting, 7 are for Al, 7 are for Bob, and the only one pulling for Chuck is, well, Chuck. A ranking system is used that gives 2 points to one’s first choice, 1 point for one’s second choice, and 0 points for the third. Al receives 14 points from his group, 7 points from Bob’s, and 1 from Chuck. Bob receives 14 points from his group, 7 points from Al’s, and 0 from Chuck. So Al wins, 22-21-2, with Chuck’s vote determining the winner.

Let’s assume now that Bob’s supporters predict this outcome and decide to ensure Bob’s success. They each vote for Bob as their first choice, Chuck for their second, and Al for their third. Now Bob wins 21-15-9. Al’s supporters see this coming, so they decide to be “strategic” and vote for Al as their first choice, Chuck as their second, and Bob as their third. With this turn of events, that is, both groups voting for Chuck as their second choice, even though he was the first choice for only 1 out of the 15 voters, Chuck wins 16-15-14.

With this simple story, Mr. Saari demonstrates that it is not difficult for a candidate to win an election, even though that person was not the first choice of the majority of voters. As it turns out, this is not an anomaly. The outcome of a vote does not necessarily reflect the will of the people. As Mr. Saari states, his intent is simply to share what can go wrong in elections and why, in the hopes that voting errors can be prevented in the future. All in all, I found this to be a very intriguing topic, and with a little more research, it could turn into my Capstone project.

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’s 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.

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