De Moivre's Theorem Visualizer | Complex Numbers Lab

De Moivre's Theorem

De Moivre's Theorem is a fundamental result in complex number theory that connects complex numbers with trigonometry. It provides a powerful way to compute powers and roots of complex numbers.

The Theorem

For any complex number in polar form and any integer n:

\[ [r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta) \]

Applications

• Computing Powers: The theorem simplifies raising complex numbers to integer powers by multiplying the modulus by itself n times and multiplying the argument by n.
• Finding Roots: The theorem can be extended to find the n-th roots of a complex number using the formula:

\[ \sqrt[n]{r(\cos\theta + i\sin\theta)} = \sqrt[n]{r} \left( \cos\frac{\theta + 2\pi k}{n} + i\sin\frac{\theta + 2\pi k}{n} \right) \]

for \( k = 0, 1, 2, ..., n-1 \) giving the \( n \) roots: \( \; w_0, \; w_1, ..., \; w_{n-1} \)

How to Use This Tool

• Enter the real and imaginary parts of your complex number
• Choose whether to compute powers or roots
• Set the exponent (for powers) or root degree (for roots)
• Click "Compute & Visualize" to see the results
• Use the zoom controls to adjust the view of the complex plane

The visualization shows the original complex number and its computed powers or roots on the complex plane, with angles measured counterclockwise from the positive real axis for positive angles and clockwise for negative angles.