Complex Numbers Adventure

z = a + bi | polar: r ∠ θ

a real part b imaginary part r modulus θ argument

A complex number has the form z = a + bi, where a is the real part and b is the imaginary part. On the complex plane, a goes along the horizontal axis and b along the vertical. In polar form, r ∠ θ gives the distance from the origin and the angle from the positive real axis. Multiplication stretches and rotates; addition shifts the point on the plane.

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Your Goal: Enter two complex numbers and choose an operation to see the result on the complex plane. Explore how multiplication rotates and stretches, how addition shifts, and render stunning fractals. Click Generate to test yourself! Real-World: Complex numbers drive AC circuit analysis, signal processing, quantum mechanics, and the fractal geometry found in nature.

Complex Number Operations

Enter two complex numbers and pick an operation. Watch how addition shifts, multiplication rotates and stretches, and division does the reverse on the complex plane.

z₁:

+ i

Polar: 5 ∠ 53.13°

z₂:

+ i

Polar: 2.24 ∠ -63.43°

Cartesian: 4 + 2i

Polar: 4.47 ∠ 26.57°

|z|: 4.47

arg(z): 26.57°

Fractal Explorer

Fractals are infinitely detailed patterns built by repeating a simple formula on complex numbers. The Mandelbrot set tests each point c to see if z = z² + c stays bounded. Julia sets fix c and vary the starting point. Increase iterations for finer detail and zoom in to discover hidden structures.

Type:

Julia c:

+ i

Iterations: 80

Zoom: 1.0x

Challenges

Complex Plane

The Argand diagram plots each complex number as a point: the horizontal axis is the real part, the vertical axis is the imaginary part. Vectors show how the two inputs combine under your chosen operation.

Grid Unit Circle Vectors Animate

z₁ z₂ Result

Fractal Visualization

Hover over the canvas to see the complex coordinate and how many iterations it takes for that point to escape. Bright colours escape quickly; dark regions stay bounded — the boundary between them is where the stunning detail lives.

Point: Hover over fractal Iter: -

Progress

0Completed

0%Success

0Fractals