Visualizing how repeated multiplication by (1 + εi) approximates rotation as ε → 0
Final value: 1.00 + 0.05i
Magnitude: 1.001
Angle: 0.050 radians (2.86°)
Expected angle (n×ε): 0.050 radians
Error: 0.000 radians
This visualization demonstrates how repeated multiplication by (1 + εi) approximates a rotation in the complex plane.
(1 + εi)ⁿ ≈ e^{εni} = cos(nε) + i sin(nε) when ε is small
As ε approaches 0, multiplying by (1 + εi) repeatedly produces a result that is very close to a pure rotation by nε radians.
The blue point shows the result of the repeated multiplication, while the red point shows the ideal rotation for comparison.
Adjust the ε and n sliders to see how the approximation improves as ε becomes smaller and n increases.