Complex numbers and Euler's formula
One rotation, read as complex multiplication, a point on the unit circle, and the cosine and sine of the same angle.
Complex numbers
9 pieces about this concept, across works, topic guides, and exam visualizations.
Topics
Works
Phasor diagram
Two rotating phasors join head to tail, and their sum traces how harmonic waves superpose as amplitude, frequency ratio, and phase difference change.
Geometric significance of complex arithmetic
On the complex plane, addition combines vectors, and multiplication scales by the moduli and rotates by the sum of the arguments.
Polar form of a complex number
Any nonzero complex number is written as z = re^(iθ), with modulus r and argument θ, linking rectangular and polar coordinates.
Euler's formula, rotating
Euler's formula e^(iθ) = cos θ + i sin θ rotates a vector on the complex plane and traces the sine wave of its imaginary part.
Julia set
Iterating z to z squared plus c on the complex plane draws a fractal boundary and its self-similar structure.
De Moivre's formula and nth roots
Drag a complex number z and watch a power multiply the argument by n, or roots divide the circle into n equal parts.
