Education Technology

Imaginary Explosions

Activity Overview

What happens when you square an imaginary number?  Does it get bigger or smaller? How can you tell if the number will get bigger or smaller? This seems straight forward enough, but a really surprising result occurs if you add something after you square, then repeat, square and add, square and add. While the activity involves only adding, squaring and graphing complex numbers, the results are truly amazing. 

Objectives

  • Number systems and recursion
  • Representation of complex numbers
  • Arithmetic of complex numbers in Cartesian form
  • Algorithms for computation in a variety of contexts.

Vocabulary

  • Imaginary
  • Real component
  • Imaginary component
  • Argand Plane

About the Lesson

Using only addition and multiplication of complex numbers, students conduct an exploration into the patterns (Julia Sets) generated by complex numbers. The activity works through checking if numbers get bigger or smaller under repeated multiplication. The second and third stage has students adding a constant to the 'mix'. The dynamic nature of TI-Nspire makes this activity particularly interesting as the graphs update very quickly and the animations can be somewhat unexpected.