Week 01 · The Complex Plane
Dock Day
Dock Day: the family grid
// Dock lights on a grid
⏱ about 20 min
Dock Day: The Family Grid
Saturday the graph paper comes to the kitchen table with a ruler, a protractor and a pencil.
"Pick a spot for a light," Comet tells the family. "Say it as a + bi. Then find its reflection."
"Someone picked 3 - 4i," Wren says. "Its conjugate is 3 + 4i. Its modulus is 5."
Nova dims to a hint. "Would you like a hint? Measure the angle too and give the point its second name."
"Now two lights," Comet says. "Stretch a string between them. How long is it, and where is halfway?"
"Subtract and take the modulus. Then average," comes the answer, and the ruler agrees.
"That is the whole week on one sheet," Wren says. "Points, reflections, distances and halfway marks."
- A complex number a + bi is the point (a, b) on the complex plane. Its conjugate a - bi is its reflection in the real axis.
- The modulus is the square root of a² + b², the distance from 0. A number times its conjugate is the modulus squared.
- To divide, multiply top and bottom by the conjugate of the bottom.
- Polar form names a point by modulus and angle. Multiplying multiplies moduli and adds angles.
- Distance between two numbers is the modulus of the difference. The midpoint is the average.
◇ FAMILY LIGHTS ON A GRID
Draw the two axes on graph paper. Each family member marks one "light" and names it as a + bi.
For each light, someone finds the conjugate and someone else finds the modulus. Plot the conjugate in a second color.
Pick two lights. Measure the string between them with a ruler, then compute the modulus of the difference. Compare.
Find the midpoint by averaging and mark it. Check with the ruler that it sits halfway. Trade roles and repeat.
For each light, someone finds the conjugate and someone else finds the modulus. Plot the conjugate in a second color.
Pick two lights. Measure the string between them with a ruler, then compute the modulus of the difference. Compare.
Find the midpoint by averaging and mark it. Check with the ruler that it sits halfway. Trade roles and repeat.
For a grown-up
This is a pencil-and-paper activity. The string is for measuring on the table, not for play near water.
A wrong answer is a good moment: plot the point and the answer, and let the picture show the slip.
The lights and distances are the family's own made-up grid, not a plan for a real dock.
A wrong answer is a good moment: plot the point and the answer, and let the picture show the slip.
The lights and distances are the family's own made-up grid, not a plan for a real dock.
DOCK DAY CHECK
- Read the question.
- Tap your answer.
A family member picks a light at 3 - 4i. What is its reflection, the conjugate?
What is the modulus of 3 - 4i, its distance from 0?
Two family lights sit at 3 - 4i and -1 + 2i. Where is the midpoint?Two family lights sit at 1 + i and 4 + 5i. The string between them is how long? Type the number.
Subtract: 4 + 5i minus 1 + i. Then square the two parts, add and take the square root.
WHY THIS EXERCISEDistance is the modulus of the difference, and this one comes out whole.
Draw the family grid with every light, every conjugate and one midpoint marked. Label each with its a + bi name.