Higher Level only. It needs 1.12 and it feeds 1.14, where De Moivre's theorem is the multiplication rule applied n times.
Teach the product first and let them over-generalise. View 2 gives 6 cis 75° and the rule is clean: multiply the moduli, add the arguments. Then ask for the SUM and most of a class will offer 5 cis 75°, because they have just been handed a rule and no reason to doubt it.
View 3 is the correction, and the parallelogram is the reason. View 4 then draws 5 cis 75° beside the true sum with a dashed line between them, so the error is a distance on the screen rather than two numbers to compare.
The detail worth pointing at: 4.96 is less than 5, and it had to be. Two arrows pointing different ways give something shorter than their lengths added. If a class has done AHL 3.10 they have met this as the triangle inequality and will recognise it.
| Question | Answer |
|---|---|
| 1. Modulus of the product | 2 × 3 = 6. |
| 2. Its argument | 30 + 45 = 75°. |
| 3. Modulus of the sum | |3.853 + 3.121i| = 4.96. |
| 4. Argument of the sum | arctan(3.121/3.853) = 39.01°. |
| 5. Why the sum's modulus cannot be 5 | B. Equality in the triangle inequality needs equal arguments, and these differ by 15°. |
A useful sanity check to hand out with question 4: the argument of a sum always lies between the two original arguments, so it has to be between 30° and 45°. It leans towards the longer arrow, which is why 39.01 rather than the midpoint 37.5.
1 markThe right form chosen for the operation.
1 markThe conversion, where one is needed.
1 markThe arithmetic.
1 markThe answer in the form the question asked for.
The last one catches more candidates than the arithmetic does. "Give your answer in the form a + bi" means a polar answer scores nothing for that mark, however right the number is. It is worth a five-minute exercise where the only task is converting correct answers into the requested form.
| They wrote | What happened |
|---|---|
| 5 on question 1 | Added the moduli. The rule for a product multiplies them, and 5 is the bound for a SUM. Both facts are on the page and they have crossed. |
| 1350 | On question 2, 30 × 45. They have applied multiplication to both parts rather than to one. |
| −15° | Gave the quotient's argument. Subtracting rather than adding. |
| 5 on question 3 | The product rule applied to a sum. Combining r and θ as if addition followed the product rule. |
| 3.85 | Gave the real part of the sum rather than its modulus. They have done the hard part and stopped. |
| 24.59 | The sum of the squares, before the root. |
| 75° on question 4 | Same over-generalisation as the 5. Worth showing that a sum's argument must lie between 30 and 45, so 75 is impossible before any arithmetic. |
| 37.5° | The average of 30 and 45. A good wrong answer: they have reasoned rather than guessed, and the fix is that the longer arrow pulls harder. |
| 50.99° | arctan with the parts inverted. The argument is arctan(imaginary over real). |
| 0.681 | Radians. Acceptable if the question did not specify; wrong here because it asked for degrees. |
"Why does Euler form use radians?" Because the exponent of e is a number and a degree is not. More precisely, the series for ex and for sin x only line up when x is in radians, which is the same reason the derivative of sin x is cos x only in radians. Writing 2e30i is a different number: 30 radians is over four turns.
"Which form should I work in?" One question: what is the operation? Addition and subtraction go in Cartesian; multiplication, division and powers go in polar or Euler. For a mixed question, convert once, do the additions, convert back. Converting twice is still quicker than expanding a product in Cartesian form.
"Is cis a real notation?" It is the guide's notation and it is shorthand for cosθ + i sinθ. Examiners use it. It is worth writing out in full once a lesson for a while, because students who only ever see cis forget that the plus sign in the middle is a plus and start writing cos − i sin.
"What does multiplying by a complex number do?" Rotates and stretches. Multiplying by i is the cleanest case: a quarter turn and nothing else. Do (3 + 4i) × i on the board and watch the argument go from 53.13 to 143.13 with the modulus unchanged. It is also the single best preparation for 1.14.
| Stage | What to do |
|---|---|
| Demonstrate | On the Casio, 2∠30 x 3∠45 returns 6∠75 in one line, and then 2∠30 + 3∠45 returns 4.96∠39.01. Two entries, one operator apart, and the machine is doing the conversion for you both times. It is the fastest demonstration of the whole page. |
| Where they stick | The angle mode. In radians, 2∠30 means 30 radians and returns something unrecognisable with no error. Set Angle deliberately and check the status line. On the Nspire, the Real or Complex setting changes the display form rather than the arithmetic, which confuses students who think they have changed the answer. |
| The check | The sum's modulus against the sum of the moduli. It must be less, or equal only if the arguments match. Here 4.96 against 5. One comparison, and it catches the error this sub-topic is about. |
Analysis Paper 1 has no calculator and the questions there are built for exact work: arguments of π/6, π/4 and π/3 and moduli like 2 and √2. Set a batch with the machines away.
| Step | What |
|---|---|
| 1 | The three forms for one number, 2 cis 30°, written side by side. |
| 2 | Convert both ways, with the quadrant check from 1.12 still in force. |
| 3 | The product rule, derived from the exponent law in Euler form so it is not arbitrary. |
| 4 | Figure view 2. Then ask for the sum. |
| 5 | View 3: the parallelogram. View 4: what the over-generalisation costs. |
| 6 | Multiplying by i, as a rotation, on the board. |
| 7 | Euler's identity as a thirty-second closer and a sanity check on the form. |
Do not say "polar form is easier". It is easier for three operations and useless for two, and a student who believes it is simply easier will try to add in it. Say which form suits which operation, every time, until they ask the question themselves.
Do not say "cis is just notation". It is notation and it hides a plus sign that students lose. Expand it in full whenever the argument is negative, because 2 cis(−15°) written out is 2cos(−15°) + 2i sin(−15°) and the second term is negative while the first is not, which is exactly the thing they get wrong.