What quantity did you change?
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.
Labs / Mathematics / Complex Plane
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.
Why it matters
Why explore this?
Mathematics helps us describe patterns precisely and predict what comes next. This model turns an abstract relationship into something you can change and observe.
Explore
Try the model
z₁ 1.00 + 1.00iz₂ 1.00 + 1.00iz₁z₂ 0.00 + 2.00i∠ 45° + 45° = 90°
Investigate
Questions to test
Predict first, try it in the model, then use what you observe to answer.
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?
Explain
Make sense of what happened
State the pattern in words, test it with another example, and then connect it to the mathematical rule or representation shown in the model.
Check
Before you leave
What pattern did you observe?
Does the pattern work for another example?
If you can answer all three using evidence from the model, you understand the main idea. If not, return to Explore and change one setting at a time.