Rotations and powers of complex numbers

Lehman - Pesonen 2007

1. How the successive products a (eiC)n = a einC rotate a

Other interpretation: iterate a ⟼ a eiC.
Play first freely with the dynamic figure (called sketch). Push the buttons, move a in the plane and C on the real argument (radian angle) line.
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Problems 1

Reset by pushing the computer keyboard key 'R' and clear traces by clicking the red cross in the lower right corner (appears only when tracing has been used).
1a) Give values of C between 0 and 2π such that the point a ei2C is on top of a.
Answer 1a1) C1 =       
Answer 1a2) C2 =       
1b) Give values of C between 0 and 2π such that the point a ei3C is on top of a.
Answer 1b1) C1 =       
Answer 1b2) C2 =       
Answer 1b3) C3 =       
1c) Give values of C between 0 and 2π such that the point a ei4C is on top of a.
Answer 1c1) C1 =       
Answer 1c2) C2 =       
Answer 1c3) C3 =       
Answer 1c4) C4 =       

2. Test your geometric ability ...

can you imagine a square?
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Problems 2

Reset by pushing the key 'R' and clear traces (the red cross, if seen).

You can mark a position in the plane by dragging the point a to that position and clicking the Trace and then the Stop trace buttons. Then you can move a to some other position.

First step. Choose a position and mark it (see above).

Second step. There is a square centered at 0 having the marked position as a vertex. Guess where the other three vertices are; mark these positions.

Third step. Check the accuracy of your intuition using the Go to -buttons and a suitable value of C (or C/π).


3. Test your geometric ability ...

can you imagine an equilateral triangle?
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Problems 3

Reset by pushing the key 'R' and clear traces (the red cross, if seen).

First step. Choose a position and mark it (see above).

Second step. There is an equilateral triangle centered at 0 having the marked position as a vertex. Guess where the other vertices are; mark these positions.

Third step. Check the accuracy of your intuition using the Go to -buttons and a suitable value of C (or C/π).


4. Complete circle or not ?

Parametric curve: t ⟼ a eitC
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Problems 4

4a) Reset and clear. Put C/π = 1.8. Click Go to a*e^i4C. Click Fix, click Trace.
Will you trace a complete circle if you click on Go to a*e^i3C ?
Answer 4a)       
Try it.
4b) Clear the traces but do not reset. Will you trace a complete circle if you click on Go to a*e^iC ?
Answer 4b)       
Try it.