Complex Numbers as Rotations

Description

Shows the complex plane as a geometric framework where multiplication by i corresponds to a 90-degree rotation. Builds up to Euler's formula e^(iθ) as a general rotation, culminating in the famous identity e^(iπ) = -1.

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Phases

#	Phase Name	Duration	Description
1	Setup	5s	Title, draw complex plane with Re and Im axes
2	Complex Number as Vector	6s	Show z = 3 + 2i as a point/arrow, label components
3	Multiply by i	8s	Show z × i = -2 + 3i; rotate arrow 90°, annotate
4	Powers of i	8s	Show z × i², z × i³, z × i⁴ = z back; four positions trace a square
5	Euler's Formula	8s	Introduce e^(iθ) as point on unit circle, animate θ sweeping
6	e^(iπ) = -1	8s	Animate θ from 0 to π, arrive at -1; display full equation
7	Summary	7s	Show Euler's formula, unit circle, key identity

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Layout

+--------------------------------------------------+
|  Title: "Complex Numbers as Rotations"           |
+--------------------------------------------------+
|                                                  |
|   Complex plane (center)     | Equation panel   |
|   with unit circle           | (right)          |
|                              |                  |
|   z arrow (blue)             | z = 3 + 2i       |
|   Rotated arrows (colors)    |                  |
|   Unit circle (dashed)       | z·i = rotation   |
|                              |                  |
|   Animated θ sweep           | e^(iθ) = cosθ+i·sinθ |
|                              |                  |
+--------------------------------------------------+

Area Descriptions

• Center-left: Complex plane with arrows and unit circle
• Right 35%: Equation evolution — z×i, powers of i, Euler's formula
• Bottom: Geometric interpretation text

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Assets & Dependencies

• Fonts: LaTeX
• Manim version: ManimCE 0.19.1

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Notes

• Complex plane: x=Re, y=Im, with labeled axes
• Initial vector z = 1 + 0i on unit circle for clean rotation demo
• Then show arbitrary z = 3+2i for concreteness
• Use Arc/Circle for unit circle, ValueTracker for Euler sweep