Fractals and complex numbers:

Sample questions:

1. Find the next Julia function iterations, using the function f(z) = z² + .5 - .3i  starting with the point  z₀=0

Step 1: plug in z₀=0:

f(0) = 0*0 + .5 - 3i = .5 - 3i
z₁ =.5 - 3i

Step 2: plug in z₁ =.5 - 3i

f(.5-3i) = (.5 - .3i)(.5 - .3i) + .5 - .3i = .25 - .15i + -.15i + .09i² + .5 - 3i = .25 - .3i - .09 + .5 - .3i = .66 - .6i
z₂ = .66 - .6i

Step 3: plug in z₂ = .66 - .6i

f(.5-3i) = (.66 - .6i)(.66 - .6i) + .5 - .3i = .4356 - .396i + -.396i + .36i² + .5 - 3i = .4356 - .792i - .36 + .5 - .3i = .5756 - 1.092i
z₃ = .5756 - 1.092i

2. Find the next Julia function iterations, using the function f(z) = z² + .2 - .6i  starting with the point  z₀=0

Step 1: plug in z₀=0:

f(0) = 0*0 + .2 - 6i = .2 - 6i
z₁ =.2 - 6i

Step 2: plug in z₁ =.2 - 6i

f(.2-6i) = (.2 - .6i)(.2 - .6i) + .2 - 6i = .04 - .12i + -.12i + .36i² + .2 - 6i = .04 - .24i - .36 + .2 - 6i = -.12 - .84i
z₂ = -.12 - .84i

Step 3: plug in z₂ = -.12 - .84i

f(-.12 - .84i) = (-.12 - .84i)(-.12 - .84i) + .2 - 6i = .0144 + .1008i + .1008i + .7056i² + .2 - 6i = .0144 - .2016i - .7056 + .2 - 6i = -.4912 - .3984i
z₃ = -.4912 - .3984i

3. Find the next Julia function iterations, using the function f(z) = z² + -.7 + .5i  starting with the point  z₀=0
Step 1: plug in z₀=0:

f(0) = 0*0 + -.7 + 5i = -.7 + 5i
z₁ =-.7 + 5i

Step 2: plug in z₁ =-.7 + 5i

f(-.7 + 5i) = (-.7 + .5i)(-.7 + .5i) + -.7 + 5i = .49 - .35i + -.35i + .25i² + -.7 + 5i = .49 - .7i - .25 + -.7 + 5i  = -.46 - .2i
z₂ = -.46 - .2i

Step 3: plug in z₂ = -.46 - .2i

f(-.46 - .2i) = (-.46 - .2i)(-.462 - .2i) + -.7 + 5i = .2116 + .092i + .092i + .04i² + -.7 + 5i = .2116 - .184i - .04 + -.7 + 5i = -.5284 + .684i
z₃ = -.5284 + .684i

4. If these are some Julia iterations, what can you say about the Julia set?
A. z₀ = 0,   z₁ = -.2 +.3i,   z₂ = -.25 + .18i,    z₃ = -.17 + .21i,    z₄= -.21 + .23i   , z₅ = -.21 + .20i
These are all staying small--this Julia set is connected

B. z₀ = 0,   z₁ = .5 +.1i,   z₂ = .745 + .2i,    z₃ = 1.01 + .40i,    z₄= 1.36 + .903i   , z₅ = 1.54 + 2.54
These ones are getting larger and larger--this Julia set is not connected (hint: once the numbers get bigger than 1, you're pretty much doomed so far as staying small goes)

A. z₀ = 0,   z₁ = .2 +.6i,   z₂ = -.12 + .848i,    z₃ = -.49 + .401i,    z₄= .28 + .21i   , z₅ = .24 + .72i
These ones look like they might be staying small, but it's hard to tell because the last one has numbers that are significantly larger than the one before it.