Complex Plane Explorer
A complex number is an arrow: how far (modulus) and which way (argument). Multiplication turns and stretches.
Tip: press ×i twice and you're facing backwards — two quarter-turns make a negation.
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.
Complex numbers click once you've turned one by hand
This explorer is one small piece of how ManySpace teaches mathematics — every idea as something you can drag, rotate, and multiply into sense.