Polyhedra! -Jordan

This lab is all about polyhedrons and investigating their properties. This is the only chapter that does not use the computer to view data. This instead requires you to use models to help you visualize and and think about the observations that are made.

Some of the questions that are explored include finding the diameter of each of the regular solids if the edge length is one and, “what is the diameter of each of the regular solids if the faces each have area 1?.”

A reason why this topic is of interest to me because it does involve geometry which is a subject that I enjoy. Also the fact that this lab does not use the computer means to me that it will be somewhat hands on which interests me a well.

Iterations of Quadratic Functions

Iteration of linear functions was the first lab we covered.  Towards the end of the lab, I felt like this was a precursor to fractal geometry.  After all, the Mandelbrot set is based on an iteration of a quadratic function with the use of complex numbers.  Chapter 14 deals with iteration of quadratic functions.  This is one step closer to fractal geometry…..and a precursor to chaos theory, both of which are very interesting to me.

Early definitions in the chapter are:

  • fixed point:  given a function $f(x)$,  a point $u$ is a fixed point of  $f$ if $f(u)=u$.
  •  attractor:  a fixed point is an attractor when all nearby points move towards it under iteration.
  • repeller:  a fixed point is a repeller when all nearby points move away from it under iteration

Here’s a type of question from the chapter:

Given a function $f(x)=ax(x-1)$, how do various values of $a$ affect fixed points, attractors, repellers, and zeroes.  What about changing initial values?

Marc and Prime Numbers

A lab that I found interesting in the textbook was the Prime Numbers lab. Prime numbers are numbers that cannot be divided by any number other than one and itself. Because any number can be written as a product of prime numbers, prime numbers are basically the building blocks of arithmetic. Even though one cannot be divided by any number other than itself, it is not considered a prime number due to mere convenience because every number would have one as one of its prime multipliers which would be tedious. Yay, simplicity!

One question in this lab involves finding a proof that there are infinitely many prime numbers. This has always been a complicated question in the realm of mathematics!

Because prime numbers are such a fundamental and important group of numbers, I find it interesting to learn more about them for it always leads to learning more about the natural numbers in general. I can’t wait to see what labs others are interested in on Wednesday!

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.

Arthur, Interdependent Complexity

My name is Arthur and my favorite aspect of mathematics is not a particular field of theory, but rather how the fundamentals are used to describe systematically more complex applications of those core concepts. Two of the founding principles in mathematics come in the form of operations, one being the Symmetric Operation, and the other being the Inverse Operation. Continue reading

Square root of 2 is irrational?

My name is Farighon. I have a deep interest in the field of Combinatorics and Number Theory although I do like to take time to study Complex Analysis when I do have some free time. Since the inception of my interest in mathematics, I have always been interested and fascinated by proofs for theorems, lemmas, and propositions. But I would be lying if I said that I understood each proof that I have come across. However, there is one proof that has made perfect sense. It is none other than the proof for the square root of 2.

The proof for the square root of 2 being irrational has been one of the primary interests of ancient mathematicians starting with the Babylonian’s. Then the ancient Indians. Although later on it was proven by a simple yet elegant proof that square root of 2 is irrational, the hunt for determining the square root of 2 to as many decimal places as possible is an ongoing task for mathematicians teamed up with Computer Scientists. After all, who can expect a mathematician to be ever satisfied when perfection is what shapes and disciplines them?

The article for the proof of $\sqrt(2)$ can be found at the following article for the curious reader: http://en.wikipedia.org/wiki/Square_root_of_2.

 

Marc on why Calculus is the Bomb

My name is Marc and I am a math student at Boise State University in Boise, Idaho which is best known for its potatoes as well as recreational opportunities. One area of mathematics that I enjoy learning and studying about is calculus. Calculus is the study of limits, derivatives, and integrals which are useful in studying continuous functions. Limits play a massive role in a variety of fields ranging from economics to engineering, and limits help to define derivatives. Derivatives are functions that describe the slope of a continuous function. The opposite of a derivative, an integral, can be used to find areas under curves in two dimensions as well as volumes in three dimensions. All of these tools can be used for optimization purposes which are significant in practically any field.

For more information about calculus, you can visit the Wikipedia page here.

Because the derivative represents the slope of a function,  the derivative of a function is thus calculated with this idea of change in distance in mind. The equation associated with the derivative is:

$m=\lim_{h\to 0 }\frac{f(a+h)-f(a)}{h}$

Notice that this equation represents a change in distance on a graph. Slopes are meant to represent changes of functions and functions represent a variety of trends! For example, the derivative of a position function in physics would produce a velocity graph which represents speed, or how the distance is changing over time. The derivative of a velocity function would give an acceleration function which reveals how the speed is changing over time. Pretty cool, isn’t it!?

Jordan: Euclidean Geometry

My name is Jordan and a sub-topic of mathematics that I enjoy would be Euclidean Geometry. I like this topic because  with just 5 axioms and 5 common notions, many theorems can be proved. When looking through Euclid’s first book of elements there are many postulates with proofs from these axioms and notions. While the first 4 axioms seem to be very simple and almost common sense, the 5th axiom is more complicated and has had many people try to disprove it. Continue reading