High school · Complex numbers
Rotate with a complex exponential
Follow real and imaginary components as multiplication by eⁱᶿ turns a point around the origin.
Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.
Watch the relationship
PausedHD animation studio
From experiment to screen.
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Understand what you are seeing
The idea behind the motion.
A complex number can be viewed as a point or vector in the plane. Multiplication by cos θ + i sin θ rotates it through angle θ without changing its distance from the origin. This joins algebra, trigonometry, and geometry in one operation.
A relationship to keep
z = r eⁱᶿ = r cos θ + i r sin θ
Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.
- STEP 1
Start on the real axis
The initial number is the positive real number r. Its imaginary component is zero, and its distance from the origin is r.
- STEP 2
Follow the angle
Playback increases the angle from zero to your chosen value. Positive angles turn counterclockwise; negative angles turn clockwise. The dashed projections give the current components.
- STEP 3
Keep the modulus
The point stays on the same circle because cos² θ + sin² θ = 1. Its real and imaginary parts change, but the modulus remains r.
Your turn to explain
Make a prediction. Test your reasoning.
Starting at z = 2, what number results from a 90° counterclockwise rotation?
Use the values specified in the question. Reset restores the initial values for this investigation.
Compare your explanation
2i. Multiplication by i sends the real vector (2, 0) to (0, 2), preserving its length.
Work through a full lesson
Connect the animation to a worked example and practice questions.