Argand diagram inquiry
Mathematical inquiry processes: Explore; generate examples; test cases; generalise and prove. Conceptual field of inquiry: Argand diagram; multiplying and dividing complex numbers; conjugate; modulus and argument.
The prompt originated in a student's question after a year 12 class had worked on the task below. The task requires students to identify the reciprocal of F on the Argand diagram. The class approached the task by suggesting that F represented the complex number 2 + 2i.
Through multiplying the numerator and denominator by the conjugate, the class quickly realised that the reciprocal is represented by C. The solution raised more questions:
Are a complex number and its reciprocal always perpendicular?
If they are not perpendicular, is there a connection between their arguments?
Is the modulus of the reciprocal always a fraction of the modulus of the complex number?
If the complex number is outside the unit circle, is the reciprocal always inside? Is it true the other way round?
The prompt does not require prior knowledge of Argand diagrams. Indeed, students have created the diagram with just the suggestion given in the prompt that one exists to represent complex numbers.
In the first phase of the inquiry, students' questions have included whether the meaning of 'reciprocal' is the same for numbers with an imaginary part, whether the reciprocal is itself complex, and how it might be possible to work out the angle between a complex number and its reciprocal.
Reasoning by analogy with reciprocals of real numbers (such as a half and two), students in one class conjectured that the length (or modulus) of the reciprocal would be a fraction of the length of the complex number.
One pair also speculated that the complex number and reciprocal could not be perpendicular; rather, they would be parallel just as y = 0.5 and y = 2 are parallel. However, many others disputed whether the analogy would hold for complex numbers because the two parts (real and imaginary) would make matters more complicated.
The prompt is true when a = b or a = -b in the complex number a + bi, where a is a real number. If a and b are positive integers, for example, the argument of the complex number is π/4 and the argument of its reciprocal is -π/4.
More generally, for any a or b, the angles between the real axis and the complex number and the real axis and the reciprocal are equal.
There is also a connection between the moduli. The modulus of the reciprocal is the reciprocal of the modulus of the complex number. Therefore, If the complex number is outside the unit circle, the reciprocal is inside and vice versa.
Lines of inquiry
1. Explore and generalise
After the initial phase, two main lines of inquiry arise in the choice of regulatory cards. Either students opt to test cases by generating their own examples (perhaps using the teacher's suggestions as a starting point) or they work with the general case immediately.
Of those who wish to explore, they aim to record accurate results (see the table below) and spot patterns inductively.
Generating examples has proved to be an effective way to develop fluency in multiplying and dividing complex numbers as well as finding moduli and arguments.
2. Reason and prove
The other line of inquiry involves considering the general case. If the initial phase of questions and conjectures has included examples, students might feel confident to reason deductively straightaway.
The picture below shows one year 12 student's proof. At the end of the inquiry lesson, the student presented his reasoning to the class and answered his peer's requests for clarification.