Labs / Mathematics
Complex Plane
Every complex number is an arrow on this plane: how far it reaches is its modulus, the angle it points is its argument. Drag the tip of z₁ or z₂ and watch their product z₁z₂ — you'll see the product's angle is the two angles added, and its length is the two lengths multiplied. Hit “Animate powers” to watch that rule spiral outward.
z₁ 1.00 + 1.00iz₂ 1.00 + 1.00iz₁z₂ 0.00 + 2.00i∠ 45° + 45° = 90°
What to try
- Put both arrows exactly on the unit circle. What happens to the length of the product, no matter how you spin them?
- Set z₁ and z₂ both to i. Why does multiplying two “up” arrows land you on −1?
- Drag one number to sit on the negative real axis. How does an angle of 180° flip the other number?
- Animate the powers of a z just inside the unit circle, then just outside it. Which way does the spiral wind — inward or outward?
- Switch to polar form. Can you predict the product's r and θ in your head before you finish dragging?