De Moivre’s
Theorem
Powers of Complex Numbers
De Moivre’s Theorem
•
We use this theorem to:
•
I.
•
•
Simplify complex numbers raised to a power.
II.
Solve certain types of equations, or find the
nth roots of a complex number.
I.
POWERS
De Moivre’s Theorem:
You need to know:
A. Convert rectangular form into trigonometric form
B. Simplify fractions
C. Simplify radicals
Problem 1
Then, find the angle:
Your turn…
Practice:
Use De Moivre’s theorem to find (-1 + i√3 )12
a. Convert the complex number to trig form:
b. Then use De Moivre’s Theoem to find the value
Answer: 4096
Grab your book and calculator
De Moivre’s
Theorem
Powers of Complex Numbers
Who was De Moivre
A brilliant French mathematician who was
persecuted in France because of his
religious beliefs. De Moivre moved to
England where he tutored mathematics
privately and became friends with Sir Isaac
Newton.
De Moivre made a breakthrough in the fields of probability
(writing the Doctrine of Chance), but more importantly for IB HL
students he moved trigonometry into the field of analysis
through complex numbers with De Moivre’s theorem.
Warm - up
1. (3 -
5
2i)
2. (√5 -
3
4i)
= - 597- 122i
= - 43√5 + 4i