Pre-Calculus Room A437, Mr. Schutt

NAME: ______DATE: ______

Pre-Calculus Room A437, Mr. Schutt

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Mandelbrot Set Point Test

A Project for Complex Numbers

Perform arithmetic operations with complex numbers.

N-CN.1.) Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real.

N-CN.2.) Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

N-CN.3.) (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

Represent complex numbers and their operations on the complex plane.

N-CN.4.) (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

N-CN.5.) (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + √3 i)3 = 8 because (-1 + √3 i) has modulus 2 and argument 120°.

N-CN.6.) (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

Essential Question: How can fractals be used to analyze real world situations that would normally be difficult or even impossible to model?

Goal: You will be able to calculate whether points are contained within or outside the fractal The Mandelbrot Set, using complex numbers to model each point by performing iterations of the fractal equation.

Role: You are a silviculturist, a scientist who studies the growth and cultivation of trees and entire forests.

Audience: You are with a group of scientists who are using fractals create an efficient model that describes how forests grow, so we can make accurate predictions that describe valuable effects of silviculture, such as the intake of carbon-dioxide.

Situation: Fractals are images that display “self-similarity,” for if you zoom into them repeatedly you observe the same image repeating itself over and over again. Even though fractals may look very complicated, they are actually very simple for just one simple set of instructions can be used to generate an entire fractal. Many living organisms use fractals to grow, such as the branches of a fern leaf or even the arteries and veins in your body. In order to properly understand how fractals work, you will be analyzing the arithmetic and geometry behind the fundamental fractal, The Mandelbrot Set.

The Mandelbrot Set is graphed on the complex plane, where the x-axis is the real component and the y-axis is the imaginary component of any complex number plotted—e.g. the complex number 3 + 4i would be located at the point (3, 4). The equation Zn2+C=Zn+1 is used to test whether a given complex number C is contained within The Mandelbrot Set—the black region of the graph. While some points deep inside the black zone are obviously inside the set, other points that are right on the edge are hard to tell whether they are in the set or not since the fractal image is so detailed. Hence we need a method for investigating them.

Product: You are to perform four calculations to test whether certain complex numbers are contained within The Mandelbrot Set or not. You are to then answer several short-answer questions to explain how the fractal is generated, and lastly you are to write a one-page report for how analyzing fractals can help model real-world situations that would normally be too difficult to comprehend without them.

PRACTICE TRIAL: Test the complex number . For the first row, let Z0 = 0 + 0i. Pay attention to the magnitude in each row.

Zn / Magnitude of Zn+1:
Z0 / 0 / 0i
0 / 0 / 0i
0i / 0i / 0i2
Z1 /
Z2 /
Z3 /

TRIAL 1: Test your first complex number C = ______. Does the magnitude ever exceed the number 2 in your work? YES or NO:

Zn / Magnitude of Zn+1:
Z0 / 0 / 0i
0 / 0 / 0i
0i / 0i / 0i2
Z1 /
Z2 /
Z3 /

TRIAL 2: Test your second complex number C = ______. Does the magnitude ever exceed the number 2 in your work? YES or NO:

Zn / Magnitude of Zn+1:
Z0 / 0 / 0i
0 / 0 / 0i
0i / 0i / 0i2
Z1 /
Z2 /
Z3 /

TRIAL 3: Test your third complex number C = ______. Does the magnitude ever exceed the number 2 in your work? YES or NO:

Zn / Magnitude of Zn+1:
Z0 / 0 / 0i
0 / 0 / 0i
0i / 0i / 0i2
Z1 /
Z2 /
Z3 /

QUESTIONS: Answer each question with complete sentences.

1.) For which trials did the magnitude exceed 2? For which trials did the magnitude remain less than 2? Refer to your math to explain how.

2.) If the magnitude of Z4 is less than 2, then the complex number a + bi that you tested is likely to be within the Mandelbrot Set, and hence the point (a, b) is a part of the black fractal image. Do the results of your math in Trials 1, 2, and 3 make sense when you compare with the image on the back page?

3.) A computer is normally used to do the calculations that test and plot points for the Mandelbrot Set fractal. In fact, for each individual trial for a given complex number, the computer will test up to iterations of Z50 or more. Why do some trials always remain less than 2, regardless of however long you test them?

4.) Even though you had to use imaginary numbers to solve the equations for this project, the math was still very similar to the real number math that you are used to doing. What difference(s) did you encounter with imaginary numbers that you had to adjust your math for during this activity?

REPORT: Write a one-page report for how analyzing fractals can help model real-world situations that would normally be too difficult to comprehend without them. You are to use additional resources to help answer this question, including the NOVA video: Fractals: Hunting the Hidden Dimension, dictionaries, encyclopedias, and the internet.

Exceeds Expectations / Meets Expectations / Almost Meets Expectations / Below Expectations
Complex Numbers
Understanding of what a complex number represents and how it may be mathematically manipulated / Successfully uses complex number arithmetic and shows all work to prove understanding of the Mandelbrot set. / Demonstrates successful use of complex number arithmetic in most trials for the Mandelbrot Set. / Demonstrates general understanding of complex arithmetic and the Mandelbrot set but lacks consistent success. / Little or no relation demonstrated between complex number math and the Mandelbrot set.
Mathematic Application
Consistent use of operations on complex numbers in recursive situations / Understanding of need for recursive complex arithmetic and successful application throughout all trials. / Successful application of complex number operations in all trials with only minor inconsistencies. / Demonstrated understanding of complex number arithmetic but without recursive success. / Unclear on necessary arithmetic to use or how to apply it beyond a single step.
Description and Presentation
Explanation of how complex number arithmetic represents the Mandelbrot set / Use of clear and relevant mathematical language based on the Standards and Mandelbrot set fractal description. / Uses vocabulary related to complex numbers in order to generally describe fractals and the relate to the Mandelbrot set. / Can describe the Mandelbrot set but lack mathematic vocabulary or clarity. / No inclusion of how complex numbers relate to the Mandelbrot set.
Enduring Understanding
Recognition of the relationship between the real and the complex plane / Firm grasp on complex numbers and how they can be illustrated visually to describe real concepts. / Understands complex numbers and fractals but does not completely connect the two. / Understanding of complex numbers and/or fractals but cannot demonstrate understanding of the connection. / Fails to develop an understanding of the relevance of complex numbers.

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