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 (-2) a. 5 b. (0) these are the labels of the points in the graph. Ignore (2) and (3) i c. (-1) 4 3 3 2 2 1 IMAGINARY AXIS -4 -2 0 0 0 -1 -1 -2 2 4 6 -2 -3 REAL AXIS 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: 2 2 2 x + x + x + x , ... etc. x 2+ x 2 + x , 2 x , x +x , Notice the pattern. So if x=2 , then we have 8 2 2 2 2 + 2 +2 +2, ... etc. 22+2 2 +2, 2 2, 2 + 2, This will eventually give us 2, 6,38, 1446,... etc. Similarly, if x=1+i , then we have 2 2 2 1+i +(1+ i) +(1+i) +(1+ i) ,... 2 2 etc. 1+i +(1+i) +(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) It's the set of all complex numbers z for which the sequence defined by the iteration z(0) = z, z(n+1) = z(n)*z(n) + z, n=0,1,2, ... (1) remains bounded. This means that there is a number B such that the absolute value of all iterates z(n) never gets larger than B. A bounded sequence may or not have a limit. For example, if z=0then z(n) = 0 for all n, so that the limit of the (1) is zero. On the other hand, if z=i ( i being the imaginary unit), then the sequence oscillates between i and i-1, so remains bounded but it does not converge to a limit. 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) Complex numbers: x=2+I and x=3 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) A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are selfsimilar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems - the pictures of Chaos