DeMoivre's Theorem

Trigonometry · Axiom Academy

A powerful theorem for raising complex numbers to powers through rotation and scaling in the complex plane. 1. Complex Numbers in Polar Form Before we can understand De Moivre's Theorem, we need to represent complex numbers in polar form. Any complex number can be written as: where r is the magnitude (distance from origin) and θ is the angle (measured counterclockwise from the positive real axis). In other words: to raise a complex number to the nth power, raise the magnitude to the nth power and multiply the angle by n. This means that raising a complex number to a power involves two simple operations: scaling (raising r to the power) and rotating (multiplying the angle). 3. Visual Example: Squaring a Complex Number Notice how the magnitude grows from 2 to 4 (scaling), while the angle doubles from 30° to 60° (rotation). The result is 4(cos 60° + i sin 60°) . 4. Higher Powers: The Pattern Emerges Let's watch what happens as we raise the same complex number to increasing powers. Each time, we multiply the angle and raise the magnitude to that power. This theorem is particularly useful for finding roots of complex numbers, deriving trigonometric identities, and solving problems in physics and engineering involving periodic motion and waves.

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