tag:blogger.com,1999:blog-78190269457695307062014-10-02T21:17:18.811-07:00"The Geometry of Complex Numbers"Darcyhttp://www.blogger.com/profile/12373803658770605253noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-7819026945769530706.post-53270822963161013312009-02-04T23:11:00.001-08:002009-02-04T23:17:31.787-08:00The Third WeekIII. <span class="Apple-style-span" style="font-weight: bold;">The Third Week - <span class="Apple-style-span" style="font-style: italic;">The Riemann Sphere</span></span><div><span class="Apple-style-span" style="font-style: italic; font-weight: bold;"><br /></span></div><div>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). <span class="Apple-style-span" style="font-weight: bold;">Fig. 3-1</span> below depicts the type of graph visually. </div><div><br /></div><div style="text-align: center;"><img src="http://media-2.web.britannica.com/eb-media/23/70823-004-2AD75C17.jpg" /><br /></div><div><br /></div><div><span class="Apple-tab-span" style="white-space:pre"> </span>[<span class="Apple-style-span" style="font-weight: bold;">Fig 3-1 </span>- <span class="Apple-style-span" style="font-style: italic;">The Riemann sphere, with labeled terms]. </span><br /></div><div><span class="Apple-style-span" style="font-style: italic;"><br /></span></div><div>The illustration above attempts to show how a complex number is graphed onto the Riemann sphere - the complex number (e.g. 3 + 4<span class="Apple-style-span" style="font-style: italic;">i</span>) is graphed with a coordinate of the form (x, y, z). </div>Darcyhttp://www.blogger.com/profile/12373803658770605253noreply@blogger.com0tag:blogger.com,1999:blog-7819026945769530706.post-84611512510693361092009-02-04T22:56:00.000-08:002009-02-04T23:10:57.546-08:00The First and Second Weeks<div style="text-align: left;">I. <span class="Apple-style-span" style="font-weight: bold; ">The First Week - <span class="Apple-style-span" style="font-style: italic; ">Special Fractals</span></span><br /></div><div><div><br /></div><div>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. </div><div><br /></div></div><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://www.somlev.net/chaos/diagrams/man.2.00.gif"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 340px; height: 335px;" src="http://www.somlev.net/chaos/diagrams/man.2.00.gif" border="0" alt="" /></a><div><span class="Apple-tab-span" style="white-space:pre"> </span>[<span class="Apple-style-span" style="font-weight: bold;">Fig. 1-1</span> - <span class="Apple-style-span" style="font-style: italic;">Mandelbrot set, surrounded by different Julia sets</span>]<br /></div><div><br /></div><div>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 <span class="Apple-style-span" style="font-weight: bold;">Fig. 1-1</span>). These microscopic-like projections keep seemingly expand to infinity and project all sorts of differently-shaped figures, like seen in <span class="Apple-style-span" style="font-weight: bold;">Fig. 1-1</span>, called Julia sets. </div><div><br /></div><div>II. <span class="Apple-style-span" style="font-weight: bold;">The Second Week -</span> <span class="Apple-style-span" style="font-weight: bold;"><span class="Apple-style-span" style="font-style: italic;">Proper Introduction to Complex Geometry</span></span></div><div><br /></div><div><br /></div><div><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://cnx.org/content/m0081/latest/complex.png"><img src="http://cnx.org/content/m0081/latest/complex.png" border="0" alt="" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 273px; height: 275px; " /></a></div><div><br /></div><div>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). </div>Darcyhttp://www.blogger.com/profile/12373803658770605253noreply@blogger.com0tag:blogger.com,1999:blog-7819026945769530706.post-33873651824270780992009-02-01T22:54:00.000-08:002009-02-02T01:04:29.935-08:00MATH 199 - Independent Study<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_YsQH0WXhCS8/SYa0yobPpiI/AAAAAAAAAAM/zziu_wTj8mY/s1600-h/d_ireland.jpg"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 150px; height: 200px;" src="http://2.bp.blogspot.com/_YsQH0WXhCS8/SYa0yobPpiI/AAAAAAAAAAM/zziu_wTj8mY/s200/d_ireland.jpg" alt="" id="BLOGGER_PHOTO_ID_5298120793586116130" border="0" /></a><br /><br /><br />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.<br /><br />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.<br /><br />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 <span style="font-weight: bold;">I</span>, unlike the real number system, denoted by <span style="font-weight: bold;">R</span>. 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).<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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).Darcyhttp://www.blogger.com/profile/12373803658770605253noreply@blogger.com0