Why students lose marks in complex numbers and it is always the argument
Complex numbers is one of the most visually clear topics in IB AA HL — everything can be drawn on the Argand diagram — and yet marks disappear with unusual consistency. Almost always, the error is in the argument. A student converts a number to modulus-argument form and writes the reference angle directly as the argument. The reference angle is correct. But the point is in the third quadrant, so the argument requires adjustment. In 25 years of teaching, I have seen this error on more papers than I can count. The Argand diagram is not optional — it is the check. Sketch the point, identify the quadrant, then calculate the argument. In that order. These worksheets make that sequence automatic.
Recognition Training
Cartesian Form & Operations
Adding, subtracting, multiplying, and dividing complex numbers. Division requires multiplying by the complex conjugate. The most common error: writing the conjugate product as a squared minus b squared instead of a squared plus b squared.
Modulus-Argument Form & De Moivre's Theorem
Converting between forms, multiplying and dividing in polar form, and applying De Moivre's theorem. The argument must be placed in the principal range. Students who use the calculator result directly without checking the quadrant produce arguments in the wrong region.
Roots of Unity & Loci on the Argand Diagram
Finding all nth roots by dividing the argument equally, sketching roots as vertices of a regular polygon, and interpreting geometric loci. The error that costs the most marks: finding one root and stopping without generating all n roots.
The 4 Patterns Behind Every Lost Mark
Using the calculator result as the argument without checking the quadrant
The inverse tangent function gives a reference angle in a limited range. For points in the second or third quadrant this reference angle must be adjusted. Sketch the point on the Argand diagram first. The quadrant determines the adjustment.
Writing the conjugate product as a squared minus b squared
The product of a complex number and its conjugate always gives a real positive value: a squared plus b squared. Students who write minus b squared have forgotten that i squared equals negative one.
Finding only one nth root
The equation z to the n equals w has exactly n solutions. After finding the first root the remaining roots are obtained by adding 2 pi over n to the argument repeatedly. Students who stop after one root have answered one nth of the question.
Applying De Moivre's theorem with the modulus unchanged
Both the modulus and the argument are affected. The modulus is raised to the power n. Students who correctly multiply the argument but leave the modulus unchanged produce a complex number with the right direction and the wrong magnitude.
The Full Diagnostic Path
- 50 original exam-style questions across 5 sections
- Full worked solutions with M1/A1/R1 IB mark scheme
- Mistake analysis on every question
- Sections: Cartesian operations, modulus-argument form, De Moivre's theorem, roots of unity, loci
Need one-to-one help with complex numbers?
Book a session with Abdul Wadood and work through the techniques that are costing you marks.