Chapter 15: Square Root of i: Situation 9 From the MACMTL-CPTM Situations Project

Knowing that a Computer Algebra System (CAS) had commands such as cfactor and csolve to factor complex number expressions and solve complex number equations, a teacher was curious about what would happen if she entered $i$⁠. The result was $22+22i$⁠. She wondered why a CAS would give a result such as that.

When using a CAS, students and teachers can encounter situations that cause them to question why the CAS may give a particular result. Symbolic verification and manipulation can be used to confirm results given by a CAS. Focus 1 accounts for the reasoning behind the symbolic work by confirming that the result makes sense. To address the underlying mathematical logic relating to why $i=22+22i$⁠, Focus 2, Focus 3, and Focus 4 utilize representations of complex numbers on the complex plane. Focus 2 connects powers of i to points of the unit circle on the complex plane and their images under rotations, and Focus 3 uses Euler's formula to represent complex numbers in exponential and trigonometric form. Focus 4 considers the powers of i as elements of cyclic groups.