Due: Thursday Points: 20 Module 3 Application Activity: Mandelbrot Set It is often difficult to illustrate a real-world application of complex numbers for algebra courses. However, here's an attempt, a very simplified way, to give you an idea of how complex numbers can be used. (Also, see rubric below.) There is something called a complex plane, similar to the coordinate plane you've seen in previous courses. However, here, the vertical axis is called the imaginary axis while the horizontal axis is called the real axis. A point (a , b) on the complex plane corresponds to the complex number a+bi . 1. Plot the following complex numbers in a complex plane. Please label both axis and points appropriately. (up to 4 points) 12 i a. 5 b. i c. A famous application of complex numbers, under the topic of fractals, is the Mandelbrot Set. The Mandelbrot Set can be drawn on the complex plane. To do so, consider the sequence of numbers: x 2+ x 2 + x 2+ x , ... etc. x 2+ x 2 + x , 2 x , x +x , x=2 , then we have 2 + 2 +2 +2, ... etc. 22+2 2 +2, 2 2, 2 + 2, Notice the pattern. So if 2 2 2 This will eventually give us 2, 6,38, 1446,... etc. Similarly, if x=1+i , then we have 1+i2 +(1+ i) 2 +(1+i)2 +(1+ i) ,... etc. 1+i 2+(1+i)2 +(1+i), 1+i, (1+ i)2+(1+i ), Things become messy quickly, but when we simplify the above, we end up with 1+i, 1+3 i,7+7 i ,197 i, ... etc. Now, for some values of x , it is possible that the sequence can be bounded. That is, the absolute value of each number in the sequence is less than or equal to some fixed number, call that number M . So, for example if x=1 , then the sequence of numbers becomes, 1, 0,1, 0,. .. etc. The absolute value of each number in the sequence is less than or equal to 1 . In this case the Mandelbrot set is bounded by M =1, and we say the complex number x=1 is in the Mandelbrot Set. For x=2 , as we saw the sequence above, the Mandelbrot set is unbounded and as such, the complex number x=2 is not in the Mandelbrot Set. 2. A question still remains, what exactly is a Mandelbrot Set? Search the Internet or books, from a reputable sources, and briefly describe in a paragraph or two the meaning of a Mandelbrot Set. (up to 3 points) 3. Your instructor will provide you with two complex numbers. Determine whether they are in the Mandelbrot Set. If they are bounded, determine the value of M . (up to 6 points) 4. From part #3, if a complex number is in the Mandelbrot Set, please go to Wolframalpha to plot the set. In Wolframalpha, type in the complex number as well as a comma, and the word Mandelbrot Set. The first plot/graph you see corresponds to a Mandelbrot Set. Copy and paste here the image as well as the web address of your plot. (up to 2 points) 5. In a paragraph or two, describe what fractals are. Why are they important? (up to 3 points) Rubric In addition to the assigned points, your writing must follow proper rules of grammar and mechanics of writing. It must also comply with APA requirements especially referencing and citation. (up to 2 points) You must show your work and explanation in detail to receive full credit. Just an answer without any explanation is worth 0 points. To err on the side of caution here, if unsure, please touch base with your instructor. Question Total Points Possible 1 4 2 3 3 6 4 2 5 3 Proper mechanics of writing and APA compliance 2 Comments Total 20 (SLOs, 1a-d, 2e, 3b-d) NB: The above activity is a modification of another from College Algebra with Applications for Business and Life Sciences By Ron Larson, Anne V. Hodgkinse