Why the form exists, which is a question students rarely get answered.
Do the same product in Cartesian form on the board first.
Four terms and an i². Then show the figure: one multiplication of lengths and one addition of angles. That contrast is the entire justification for learning a second notation.
Turn z₂ and the product's arm follows exactly, which also makes the division rule obvious without deriving it separately.
A calculator left in degrees gives wrong arguments silently and the error propagates through every subsequent part. Check the mode at the start of every question in this sub-topic.
And give the argument in the range asked for, usually −π < θ ≤ π, which may mean adding or subtracting 2π after a multiplication.
| 1. |z₁z₂| | 2 × 3 = 6. |
| 2. arg(z₁z₂) | π/3 + π/6 = 1.571 rad. |
| 3. Real part of 2cis(π/3) | 2cos(π/3) = 1. |
1 markWorking in radians.
1 markModuli multiplied and arguments added, without converting to Cartesian.
1 markThe argument given in the required range.
| They give | What it means |
|---|---|
| 5 (Q1) | Added the moduli. |
| 0.524 (Q2) | Gave π/6 alone. |
| 90 (Q2) | Degrees where radians were asked for. |
| 1.732 (Q3) | Gave the imaginary part. |
| 0.5 (Q3) | Computed cos(π/3) and forgot to multiply by r. |
"Why bother with a third form?" Because multiplication and powers become trivial. Do the same product both ways once and the question answers itself.
"Is cis a real notation?" It is a shorthand the course uses for cos + i sin. Write it out if in doubt.
"What is e doing there?" Making the rotation rule look like the index laws, which is exactly what it is. A full answer is beyond the course; that sentence is not.
| What is happening | |
|---|---|
| 1 | The same product both ways, side by side. |
| 2 | The three forms and converting between them. |
| 3 | Multiplication and division in polar form. |
| 4 | Euler form and e^(i pi) + 1 = 0. |
| 5 | Argument ranges. |
Do not teach polar form before Cartesian multiplication has been felt as tedious. Without that, there is no reason for it to exist.
Do not let a calculator stay in degrees. Check it out loud at the start.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.