Square Roots of a Complex Number

Author:
James R Olsen
Topic:
Angles, Function Graph, Root, Square
This worksheet graphs a number N and finds the 2 square roots of N. The roots are are and .
Experiment: Manipulate N (click and drag) and see the relationships involving the 2 square roots. Uncheck the three boxes showing the values for the numbers. Questions: Q1. Move N so that N = 2 + 0i (which is the real number 2). What are the values for the two square roots:
  • angle,
  • modulus (distance from the origin), and
  • a + bi.
(You found the two square roots of 2.) Q2. Move N so that N = 0 + 1i (which is the number i). What are the values for the two square roots:
  • angle,
  • modulus (distance from the origin), and
  • a + bi.
(You found the two square roots of i.) Q3. Move N so that N = -1 + 0i (which is the real number -1). What are the values for the two square roots:
  • angle,
  • modulus (distance from the origin), and
  • a + bi.
(You found the two square roots of -1.) Q4. (challenge Move N so that N = 2 + 2i. What are the values for the two square roots:
  • angle (exact),
  • modulus (estimate) (distance from the origin), and
  • a + bi (estimate).
(You found the two square roots of 2 + 2i.) [You can check your answers by checking the three checkboxes to reveal the values.]