4.1 Complex Number
4.1.1 Definition
Any number of the form is called a complex number, where
and
are real numbers and
is the imaginary unit.
The real part of the complex number, , is
, and
is the imaginary part of
.
4.1.2 Arithmetic Operations
Addition
(1)
Subtraction
(2)
Multiplication
(3)
Division
(4)
4.1.3 Conjugate
If is a complex number, then the complex conjugate, or simply the conjugate, of
is
(5)
The arithmetic operations of a conjugate are as follows
(6)
4.1.4 Modulus or Absolute Value
The absolute value of is defined as
(7)
4.1.5 Geometric Interpretation
The following coordinate plane is known as the complex plane or simply the z-plane. The -axis is the real axis, and the
-axis is the imaginary axis. The absolute value of a complex number is the distance of that number to the origin.

4.2 Powers and Roots
4.2.1 Polar Form
The polar form of a complex number, , is
(8)

4.2.2 DeMoivre’s Formula
(9)
4.2.3 Integers Powers of 
The integer powers of a complex number can be found easily by applying the DeMoivre’s formula on the polar form of as follows
(10)
(11)
(12)
4.2.4 Roots of 
A number is the
root of a complex number
, if
.
Let’s assume that , then we can find the values of
and
as follows
(13)
(14)
So,
(15)
By summarizing the results, the
(16)
4.3 Problems
Reference
Dennis G. Zill. Advanced Engineering Mathematics, edition. Jones
Bartlett Learning. 2016.