Complex plane / field notes 01
Turn, scale,
repeat.
Choose a starting point z and a multiplier w. See how each multiplication changes the point’s distance and direction, then follow the complete sequence.
Calculate the current inputs.
Choose Calculate products to see the single product and every indexed step for the inputs shown.
One multiplication
From z to zw
Measured direction change, z → zw:
Dashed segments join discrete multiplication steps; they do not describe a continuous motion or winding. The gold vector marks w separately.
Every indexed point
Shown to at most 8 significant digits. The JSON keeps full floating-point values; decimal arithmetic and polar values may be approximate. Zero has no argument. Directions use degrees in [−180°, 180°), modulo a full turn.
Scroll sideways to inspect every coordinate and angle.
| Step | Real | Imaginary coefficient | Modulus | Argument |
|---|
Keep learning
Connect the picture
with the algebra.
Twelve original questions link coordinates, distance and direction, complex multiplication, and repeated products. Each answer includes a worked explanation and a prompt to apply the idea.
The lesson file opens in the existing RecallWeave learner. Nothing is uploaded.
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Read the worked guide for the definitions, examples and calculation limits. If this explorer was copied alone, use the learner and guide from the same RecallWeave project.
Mathematical conventions and original content
For z = a + bi, Re(z) = a and Im(z) = b. Nonzero multipliers scale distance by their modulus and add their argument modulo 360°. The zero multiplier collapses a point to the origin; zero itself has no direction. P₀ is assigned directly, so a zero-step sequence does not require 0⁰.
The original course and diagrams were authored for RecallWeave. Mathematical background was checked against OpenStax Precalculus 2e, Complex Numbers and Polar Form of Complex Numbers. The lesson does not claim measured learning improvement. This explorer performs finite products; it does not model signals or elapsed time.