Friday
Friday: Distance and Midpoint
"A cable runs from light A at 3 + 2i to light B at 1 + 4i," Comet says. "How long is it, straight across?"
Wren subtracts. "B minus A is -2 + 2i. That is the move from A to B. Its modulus is the length: 2√2."
"About 2.8 meters by the tape," Comet says. "And a new light goes exactly halfway. Where?"
Nova hovers between the two dots. "Would you like a hint? Halfway is an average."
"Add the two numbers and divide by 2," Wren says. "(3 + 2i + 1 + 4i) ÷ 2 = 2 + 3i."
"Distance is the modulus of the difference. Midpoint is the average," Comet says. "Those are the two rules."
"And both are just the moves from this week," Wren says. "A slide, then a modulus."
Distance is the modulus of the difference
To find the distance between two complex numbers, subtract them, then take the modulus.
From 3 + 2i to 1 + 4i: 1 + 4i - (3 + 2i) = -2 + 2i. The modulus of -2 + 2i is 2√2.
Subtracting the other way gives the opposite number, but the same modulus. Order does not matter for distance.
Midpoint is the average
To find the midpoint of the segment between two complex numbers, add them and divide by 2.
Halfway from 3 + 2i to 1 + 4i is (4 + 6i) ÷ 2 = 2 + 3i. Average the real parts, average the imaginary parts.
Check: the midpoint should be the same distance from each end. Subtract and take the modulus both ways.
Where the crew used the complex plane this week
| Question | Rule | Answer |
|---|---|---|
| The reflection of light A in the water | conjugate: flip the imaginary sign | 3 - 2i |
| How far light A is from the post | modulus: square root of a² + b² | √13 |
| Spot 2 + 2i as a distance and an angle | polar form | r = 2√2, θ = 45° |
| The cable from A to B | modulus of the difference | 2√2 |
| The new light halfway from A to B | average of the endpoints | 2 + 3i |
Which answer makes sense?
A cable length must be positive, so 2√2 is right and -2√2 is not. A modulus is never negative.
A midpoint must sit between its two ends. If your average lands outside the segment, check the signs.
- Read the question.
- Tap your answer.
The cable runs from light A at 3 + 2i to light B at 1 + 4i. How long is it?A new light goes halfway from 3 + 2i to 1 + 4i. Where is it?
Lights E at 2 - 3i and D at -4 - i get a second cable. How long is it?
What is the midpoint of the segment from -2 + 3i to 4 - i on the complex plane?- Read the question.
- Tap your answer.
A spot at -1 - i on the chalk grid. What are its modulus and angle?| Statement | True or false? |
|---|---|
| The distance between z and w is the modulus of z - w. | ? |
| The midpoint of z and w is z + w. | ? |
| The distance from z to w equals the distance from w to z. | ? |
| 3 + 1 = 4 | ? |
| A modulus can come out negative if you subtract in the wrong order. | ? |
A full week on the plane. Tomorrow is Dock Day: show your family a light and its reflection on graph paper.