Tuesday

"How far is light A from the gauge post?" Comet asks. "Straight across the water, not along the dock."

Wren draws the right triangle. "3 across, 2 up. The hypotenuse is the square root of 13."

"That distance has a name," Nova says. "The modulus of 3 + 2i. It is written with bars, like absolute value."

"Here is a second way," Wren says. "Multiply 3 + 2i by its conjugate 3 - 2i. You get 13, with no i at all."

"So the modulus squared is the number times its conjugate," Comet says. "What do you notice about division?"

Nova glows. "Would you like a hint? A bottom with no i is a bottom you can divide by."

"Multiply top and bottom by the conjugate of the bottom," Wren says. "Then the bottom is just a real number."

The modulus of a + bi is the square root of a² + b². It is the distance from 0 to the point, by the Pythagorean theorem.

For 3 + 2i: 3² + 2² = 13, so the modulus is √13. Written |3 + 2i| = √13.

For 3 + 4i: 9 + 16 = 25, so |3 + 4i| = 5. A modulus is never negative.

Multiply a number by its conjugate: (a + bi)(a - bi) = a² - (bi)² = a² + b². The i vanishes.

So (a + bi)(a - bi) is the modulus squared. For 3 + 2i: (3 + 2i)(3 - 2i) = 13, and the square root is √13.

Both ways give the same number. The triangle shows why it is a distance. The conjugate shows why it has no i.

Some moduli are whole, some are square roots. A square root is an exact answer. Only round when a tape measure asks you to.

Clean work. Tomorrow is Boathouse Lab: chalk, string, a tape measure and a protractor on the dock.