# Complex Plane Explorer

Canonical: https://manyspace.aporia.systems/tools/complex-plane
Updated: 2026-10-13

A complex number is an arrow: how far (modulus) and which way (argument). Multiplication turns and stretches.

## Try this

- Turn with ×i. Put z anywhere, press ×i, and confirm the readout angle gained exactly 90° while the modulus held still. Press it three more times to come home.
- Mirror it. Take the conjugate and watch the arrow flip across the real axis. Same length, opposite angle — then multiply z by its conjugate and note the purely real result.
- Add the angles. Reveal the second number, set w = i, and drag z around. The dashed product stays one quarter-turn ahead of z forever — multiplication as choreography.

## How to read the plane

Write z = a + bi and plot (a, b). The modulus |z| is the arrow's length; the argument is its angle from the positive real axis. The conjugate mirrors across the real axis (same size, negated angle), and multiplying by i is a pure 90° turn — no arithmetic required, just rotation.

Multiplication is the payoff: magnitudes multiply, angles add. That one sentence explains i² = −1 (two quarter-turns), explains why squaring (1 + i) lands on the imaginary axis, and turns every complex product into geometry you can see before you compute it.

## Related tools

- [Derivative Visualizer](https://manyspace.aporia.systems/tools/derivative)
- [Integral Explorer](https://manyspace.aporia.systems/tools/integral)
- [Unit Circle Explorer](https://manyspace.aporia.systems/tools/unit-circle)
- [Probability Lab](https://manyspace.aporia.systems/tools/probability)
- [Bayes Lab](https://manyspace.aporia.systems/tools/bayes)
- [Eigen Explorer](https://manyspace.aporia.systems/tools/eigenvectors)
- [Fourier Series](https://manyspace.aporia.systems/tools/fourier-series)