Inquiry prompt

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.

Sometimes true

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.

October 2023