Complex numbers & the complex plane

COMPLEX NUMBER SIMULATION

▸ Building the complex plane (real & imaginary axes)…

▸ Placing z = a + bi as a point & vector

▸ Computing modulus |z| = √(a²+b²) & argument θ = atan2(b,a)

▸ Complex multiply = rotate (add angles) + scale (multiply moduli)

▸ Starting the unit circle e^iθ = cos θ + i sin θ…

▸ Ready — Online. ✅

Complex z & polar coordinates

ℂ Complex plane

a = Re · b = Im · |z| · θ

Real part a = Re(z)—

Imaginary part b = Im(z)—

Modulus |z|—

Argument θ (degrees)—

Argument θ (radians)—

Result (product/power)—

Tip

Each complex number z = a + bi is a point (a, b) on the plane — and a vector from the origin. The vector length = |z|; the tilt angle = θ. Multiplying two complex numbers = add angles & multiply lengths.

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Drag a, b to move point z · change the Scenario (multiply = rotate+scale · e^iθ · n-th roots of 1 · conjugate · powers)

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Re(z) · Im(z) · |z| over time ReIm|z| (÷2)