Author Archives: tylermurphy

Iterated Linear Maps in the Plane

Iterated Linear Maps in the Plane.   That sounds complex, doesn’t it?   In fact, it’s something we are already very familiar with.  At the beginning of the semester we investigated repeated iteration of a single linear equation $f(x)= ax+b$.   This time we are investigating the repeated iteration of a matrix, which is a nice way to write a system of equations.

We can take the matrix $A$= $\left( \begin{array}{cc} a & b \\ c & d \\\end{array} \right)$

and initial values $x_0, y_0$ placed into a vector $x_n$ = $\left( \begin{array}{c} x_0 \\ y_0 \end{array} \right)$

We then investigate repeated iterations of $Ax_n = x_{n+1}$

Analysis of the convergence of these matrices  sees a connection to the direction of the matrix and a special number called an $eigenvalue (\lambda)$ and its corresponding $eigenvector$

 



					

Euclidean Algorithm for Complex Integers

We got into the Euclidean Algorithm at the beginning of this course and one of the questions I had at the end of the lab was how the algorithm applied to other numbers systems beyond the integers.  I was curious how it worked with polynomials and complex numbers.  Imagine my delight to discover this lab.

In this lab, we investigate the $Gaussian$ $Integers$, which take the form $a+bi$ and are named after Karl Friedrich Gauss who was the first person to discover the correct way to generalize the Euclidean Algorithm for these numbers.

Much like we did with the first Euclid lab, we will investigate using complex numbers in the division algorithm.  This lab also defines a new term.  In the complex plane, the distance a point is from the origin is called its $modulus$  and the $norm$ of a complex number $z$ is denoted by $N(z)$, which is the square of its modulus.

The Euclidean Algorithm is very fascinating and I’m excited to see how else it can be applied to other number systems.

Murphy and Fermat’s Last Theorem

My name is Tyler Murphy and I love puzzles.   Currently, my favorite puzzle is Fermat’s last Theorem.   I find it very fascinating that this one conjecture confounded the greatest mathematical minds in the world for over 300 years.  It’s even more fascinating that the math required to prove the theorem ($\nexists  a,b,c,n \in \mathbb{Z}/\{0\} \mid a^n+b^n=c^n, n>2$) didn’t exist for 250 years after Fermat made the claim that he had proven the conjecture.

I also love that most people think mathematics is a stale field with nothing more to discover.  Fermat’s Last Theorem is proof against that.  I love that this was proved in my lifetime.  To me it represents the continually changing and dynamic world of mathematics.  It’s like having special eyesight that allows me to see into a special and private world that permeates every aspect of our lives which most people will never glimpse or understand.

Today, most mathematicians believe that Fermat could not have had a viable proof.  Even once the conjecture was proven in 1995 by Andrew Wiles, the search for a proof didn’t end.   Today, the search is for a more concise proof, as Wiles’ proof was over 100 pages long.   The current search is trying to find a proof for a theorem about numbers that only talks about numbers.

Here’s an interesting article about the most recent advancement in Fermat’s Last Theorem as well as some historical context.