Higher Level only. The guide links it to vectors at AHL 3.12, and the link is worth making: an Argand diagram is a plane of position vectors with a multiplication rule attached.
Step straight to the third view, −3 − 4i. The green arrow points into the bottom left and the red one, where arctan puts it, points into the top right. Two opposite corners of the plane from one calculation.
Then point at the dashed circle. All four numbers have modulus 5, so nothing about the size distinguishes them, and the argument is the only thing that does. That is why getting the argument wrong is expensive and why no size check will save you.
Views 1 and 4 are the control: in the first and fourth quadrants arctan is right. Having them in the same figure stops the class concluding that arctan is simply unreliable; it is reliable on exactly half the plane, and the half is predictable.
| Question | Answer |
|---|---|
| 1. |3 + 4i| | √(9 + 16) = 5. |
| 2. arg(3 + 4i) | 53.13°, or 0.927 radians. |
| 3. arg(−3 − 4i) | 53.13 − 180 = −126.87°. |
| 4. Real part of (3 + 4i)/(1 − 2i) | (−5 + 10i)/5 = −1 + 2i, so −1. |
| 5. Why arctan fails here | B. The two minus signs cancel in the fraction, so arctan cannot tell this number from 3 + 4i. |
Questions 2 and 3 are the same acute angle in two different quadrants, which is the point. A student who gets both right has the method; one who gets 2 right and 3 wrong has a calculator.
1 markThe modulus.
1 markThe acute angle from arctan.
1 markPlacing it in the right quadrant, with a sketch or a stated reason.
The third is effectively a mark for the sketch, and on Paper 1 there is no angle( function to go round it. A correct acute angle in the wrong quadrant scores one out of three, which is worth saying before they decide the sketch is optional.
| They wrote | What happened |
|---|---|
| 7 | On question 1, added the parts. The parts are at right angles, so it is Pythagoras. |
| 25 | Gave z z*, the modulus squared. Common and cheap to fix. |
| 36.87° | On question 2, arctan(3/4): the angle from the IMAGINARY axis. Worth naming, because it is 90 minus the right answer and looks like a quadrant error when it is not. |
| 0.927 | Radians. The mode, not the method, and on this course both are acceptable answers if the question does not specify. |
| 53.13° on question 3 | The quadrant error, straight from the calculator. It is exactly what the calculator returns, and the working looks perfect. |
| 126.87° | Second quadrant. They have used the second-quadrant rule, 180 − θ, where the third quadrant needs θ − 180. Usually a sketch with the point in the wrong corner. |
| 233.13° | The same direction, not the principal argument. Accept the geometry and insist on the convention: −180 < arg(z) ≤ 180. |
| 2 on question 4 | Gave the imaginary part. Read the question. |
| 3 on question 4 | Divided the parts separately. They have not met the conjugate trick, or have forgotten why it is needed. |
| 2.2 | Expanded with i² = +1, so 3 + 8 instead of 3 − 8. The single most common sign slip in the whole sub-topic. |
"Why not just always use the calculator's angle function?" Because Paper 1 has no calculator, and because the examiner is testing whether you know that the ratio b/a does not determine the point. Both machines get it right internally by looking at the two signs separately, which is exactly what the sketch does by hand.
"Is i a number?" Yes, in the same sense that −1 and √2 are: each was added to the system to make an equation solvable, and each looked suspicious when it arrived. The word "imaginary" is a historical insult that stuck. Worth two minutes, and it is the TOK link the guide itself suggests.
"What is the argument of 0?" Undefined, because there is no direction. The modulus is 0. It comes up when a student divides by a complex number that turns out to be zero, and the answer is that the question has no answer.
"Do I write the conjugate as z* or z-bar?" Either; the guide uses z*. Be consistent within a question. The thing that actually matters is that only the imaginary part changes sign, and students who write the conjugate of 3 + 4i as −3 − 4i have changed both.
| Stage | What to do |
|---|---|
| Demonstrate | Two lines on one screen: angle(-3-4i) giving −126.87, then tan⁻¹(-4/-3) giving 53.13. The machine disagrees with itself, and it is worth saying why: the first is handed two numbers and the second is handed one. |
| Where they stick | Finding i. On the Nspire it is the dedicated key under π, not the letter i, and a typed letter i becomes a variable that silently evaluates to nothing useful. On the Casio it is SHIFT 0, and Complex Mode must be set to a+bi in SET UP before any of it works. Both take a minute once. |
| The check | Multiply the answer back. If (3 + 4i)/(1 − 2i) is −1 + 2i, then (−1 + 2i)(1 − 2i) must be 3 + 4i. One line, and it catches every sign slip including the 2.2. |
Degrees or radians: set it deliberately per question. The guide accepts both for an argument, but a question that gives angles in radians expects them back.
| Step | What |
|---|---|
| 1 | i² = −1, and x² + 1 = 0 as the reason anyone bothered. |
| 2 | Real part, imaginary part, conjugate. Insist that Im(3 + 4i) is 4, not 4i. |
| 3 | Plot all four of ±3 ± 4i on an Argand diagram on the board. |
| 4 | Ask for the argument of each. Collect the answers before the figure. |
| 5 | Figure, view 3. The two opposite arrows. |
| 6 | The two-step method: sketch, then place the acute angle. |
| 7 | z z* = |z|², then division by the conjugate, with the multiply-back check. |
Do not say "the argument is tan⁻¹(b/a)". It is the sentence this page exists to prevent, and it is in a great many textbooks unqualified. If you want a formula on the board, write arg(z) = tan⁻¹(|b|/|a|) placed by quadrant, which is longer and is the method.
Do not say "imaginary numbers do not exist". Students repeat it and then cannot take the topic seriously. They are as real as any other number and they describe alternating current, quantum states and the stability of bridges. The name is the problem, not the object.