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As the world turns Name:________________________
Important Stuff:
1. Put i21+ on thecomplex plane on theright.
2. Multiply i21+ by i , andput what you get on theplane
also.
3. Multiply what you got in question 2 by i , and put that on
the planeas well.
4. Multiply again! Put it on the plane! Again!
5. Multiply and put on plane! Again!
6. What do you notice about multiplication by i ?
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As the world turns Name:________________________
7. Start with )21( i+ again, but this time,multiply it by )1( i+
.
8. Then do it again.
9. Then do it again.
10. Then do it again.
11. What do you notice?
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As the world turns Name:________________________
12. Put i25+ on thecomplex plane on theright.
13. Divide i25 + by i , andput what you get on theplane
also.
14. Divide what you got in question 2 by i , and put that on the
planeas well.
15. Divide again! Put it on the plane! Again!
16. Divide and put on plane! Again!
17. What do you notice about division by i ?
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As the world turns Name:________________________
18. Start with )25( i+ again, but this time,divide it by )1( i+
.
19. Then do itagain.
20. Then do it again.
21. Then do it again.
22. What do you notice?
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As the world turns Name:________________________
Interesting Stuff:
Let cxxM += 2)( , where c is a number that will change
throughout the
problem, but will always be carefully specified.
A. If we let c = 1, calculate the following:
a. =)0(M d. =))))0(((( MMMMb. =))0((MM e. =)))))0((((( MMMMM
c. =)))0((( MMM
B. Lets say that you were to calculate )))...)0((.....( MMM ,
where you
were calculating the composition of M(x), say, 100 times.
Wouldthe result be a larger or smaller number than composing M(x)
just99 times?
C. Now were going to change c. Now, let c = -1. Calculate
thefollowing:
a. =)0(M d. =))))0(((( MMMM
b. =))0((MM e. =)))))0((((( MMMMM
c. =)))0((( MMM
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As the world turns Name:________________________
D. In one brief sentence, describe the difference between the
wayM(x) behaves when c = 1 and when c = -1.
A complex number c is in the Mandelbrot Set if it behaves more
like
c = -1 than c = 1. That is, c doesnt cause )))...)0((.....((((
MMMMMM
to spin off to infinity. If the value of c causes the repeated
compositionsof M to eventually cycle back on itself, then that
point is in theMandelbrot Set.
E. Is c = i in the Mandelbrot Set? Justify your answer by
calculating:a. M(0) =
b. M(M(0)) =c. M(M(M(0))) =d. M(M(M(M(0)))) =e. M(M(M(M(M(0)))))
=f. M(M(M(M(M(M(0)))))) =
F. The Mandelbrot Set is on the wall by the windows of our
classroom. The points that are shaded in are in the
Mandelbrotset. Can you find another complex number thats in
theMandelbrot Set?
