Complex roots of unity

1. 1
Complex Roots of Unity
We can find the roots of any complex number raised to a power that gives 1. In
symbols this means:
Some useful facts to know when working out problems:
Draw on an Argand diagram and write in terms polar form:
1
i -1 -i
How do we write 1 in polar form?
What other values for the argument give a complex number the value of 1?
Explain why.
This means:
Zn = cis 0, cis 2𝜋, cis 4 𝜋, cis 6 𝜋, …………
Using de Moivre’s Theorem make z the subject
Zn = 1

2. 2
Example 1
Find the cube roots of unity. In symbolic form this means solve
Z3 = 1
From before we know that
Z3 = cis 0, cis 2𝜋, cis 4 𝜋, cis 6 𝜋, …………
How many values are there? When do we stop?
Applying deMoivre’s Theorem find an expression for Z3
Do you notice some of the roots are the same?
How many roots are there that do not repeat?
Draw the roots on an Argand Diagram
How many solutions does zn=1 have?
The first one is always 1 what are the others?
What generalization can you make about the solutions to the equation: