The Third Week

III. The Third Week - The Riemann Sphere

This meeting, we branched into a new way of thinking about the complex number system geometrically. In complex analysis, one learns that the complex numbers are graphed on a type of three-dimensional graph called a Riemann sphere (named after Bernhard Riemann). Fig. 3-1 below depicts the type of graph visually. 

[Fig 3-1 - The Riemann sphere, with labeled terms]. 

The illustration above attempts to show how a complex number is graphed onto the Riemann sphere - the complex number (e.g. 3 + 4i) is graphed with a coordinate of the form (x, y, z). 

The First and Second Weeks

I. The First Week - Special Fractals

The first weekly session mainly consisted of going over the definition of a complex number, its significance in mathematics, and one of the most curious, new mathematical concepts - fractals. 

[Fig. 1-1 - Mandelbrot set, surrounded by different Julia sets]

Fractals are the graphed projections of figures that, when examined at any given point, seemingly expand, like the examination of an atom under a microscope. An example of a fractal is the Mandelbrot set (see Fig. 1-1). These microscopic-like projections keep seemingly expand to infinity and project all sorts of differently-shaped figures, like seen in Fig. 1-1, called Julia sets. 

II. The Second Week - Proper Introduction to Complex Geometry

Our second meeting went back to basics in terms of discussions of our course material. This week's material consisted of reviewing how to add, subtract, multiply, divide, and find nth roots of complex numbers. The review session also touched on definitions, such as amplitude (or absolute value) and modulus (or principle value). 

MATH 199 - Independent Study

Hello, and welcome! I'm Darcy Ireland, a junior at Drake University (in Des Moines, Iowa). I'm a pure mathematics major at Drake U and have had a passion for the mathematics world since I was in elementary school. I have yet to declare a specialty in the broad field of mathematics, but I have an interest in algebra and number theory.

This blog is meant to serve as a sort of journal about my trek through my independent study course I'm currently taking this Spring 2009 semester. As of 2 February 2009 (the day I began this blog), I've already completed two sit-in sessions, have taken a few sheets of notes, and have completed two sets of homework assignments for the course. Each weekly post will have pictures that will aid the material covered in each post, which is meant to review material that I will have learned in Math 199.

The course is intended as a sort of prerequisite to the complex analysis course taught at most universities. In order to understand the material that will be posted in this blog in later time, the reader must have a firm grasp of trigonometry, modern algebra, and the concept of the imaginary number and the imaginary number system. This course begins with a review of the complex (or imaginary) number system, denoted by I, unlike the real number system, denoted by R. Later through the course, I'm supposed to learn about Moebius sets and some other concepts I won't talk about right away. The course concludes with a presentation that is to be given to a board of mathematicians, and an approximately ten-page paper that relates to the course material. On the side, the course is a great resume-booster for my graduate school applications (My top choices are the University of Oregon, Boston University, and the University of Vermont).

Dr. Daniel Alexander, my professor (and one of the greatest ones I've ever had at that) for this course, has a B.A. from Colby College, and an M.S. and Ph.D. in mathematical history from Boston University. He's been one of the greatest people I've met at Drake U, and is arguably the best aid I've recently had in mathematics.

In terms of posting for this particular blog, this post is the first one. The second post will be posted by the end of the day 2 February and the third post will arrive the afternoon of 4 February. After 4 February, unless noted later onwards otherwise, new posts will be posted every Wednesday afternoon.

I surely hope that you will 'subscribe' to my blog, so to speak. So far, the course has been entertaining and fascinating for me, and I hope that you will take interest in the course in the same fashion.

If the reader has any questions about the author, the blog, any course material, or anything in relation to those three subjects, please don't hesitate to make contact at darcy.ireland@drake.edu or D05000856@nwfstatecollege.edu (D - zero - 5 - zero - zero - zero - 8 - 5 - 6).