Mathematical Investigations IV
Name:
Mathematical Investigations IV
Complex Concepts-Making “i” contact
Finding Roots
We can use DeMoivre’s Theorem to find roots.
Let’s find the three cube roots of 125i. To do this, we solve the equation z3 = 125i, which is
equivalent to finding the roots of the equation z3 – 125i = 0.
To begin, we need to write 125i in cis form but with a slight twist. We need to determine ALL
values of  for which 125i  125  cis 

125i  125cis 90  360k 
 
(since i  cis 90 )
Now, let’s go back to the equation
z3 = 125i. Substitute r cis  for z.
____________________________________
Substitute the above cis form for 125i
in the right side of the equation
____________________________________
Using DeMoivre’s Theorem to rewrite
the left side of the equation in cis form.
____________________________________
Now, for the complex numbers on each side of the above equation to be equal, their values of r
and  must be equal. Set these values of r and  equal to each other and then solve.
The three cube roots of 125i are:
If k = 0, then z =
If k = 1, then z =
If k = 2, then z =
Plot these roots in an Argand diagram.
What is z for k = 3? k = 4? Do these values of k give new roots? Why or why not?
Complex 3.1
Rev F06
Mathematical Investigations IV
Name:
Try some other values of k, and then explain the role of the integer k in finding roots of complex
numbers. Specifically, in solving zn= a + bi, which values of k are needed to account for all
possible solutions?
ADDITIONAL PROBLEMS:
1.
Find the eight eighth roots of 1 in both cis form and rectangular form. Compare these
roots to the ones you found on page 1 of packet 1.
2.
Describe the graph of the 9 solutions of z9 = 1.
Complex 3.2
Rev F06
Mathematical Investigations IV
Name:
3.
Find the five fifth roots of –243 in cis form.
4.
Find the three cube roots of –8i in cis form.
What is the sum of these cube roots?
What is the product of the cube roots?
Does this make sense with what we know about the coefficients of polynomials and the
sum and product of their roots? Explain.
5.
 1
3 
i  in cis form and rectangular form
Find the four fourth roots of 625   
 2 2 
(rounding final values to the nearest hundreth).
Complex 3.3
Rev F06
Mathematical Investigations IV
Name:
6.
Find the four fourth roots of –81 in both cis and rectangular form.
Find the four fourth roots of +81 in both cis and rectangular form.
Plot both sets of fourth roots in the complex plane (Argand diagram):
Roots of –81
7.
Roots of +81
How are the roots of z4 = a4 and z4 = –a4 related?
Bonus: How are the roots of zn = an and zn = –an related?
Complex 3.4
Rev F06