The document outlines the fundamentals of complex numbers, introducing the j-operator as an imaginary unit defined by the property j² = -1. It explains various forms of complex numbers, including standard, polar, exponential, and trigonometric forms, along with the methods for transforming between these forms and performing arithmetic operations. Additionally, it defines the real and imaginary parts of a complex number and presents the theorem stating that if (x + jy) = 0, then both x and y must equal zero.

1. ADVANCE ENGINEERING MATHEMATICS
ENGR. MARIA LOREEN ROSE B. REGAÑON

2. COMPLEX NUMBERS
A Complex Number is a combination of a real
number and imaginary numbers.
The j-operator
It is a constant which when multiplied to a quantity
will rotate by 90° counterclockwise without changing
its magnitude. We now simply say that the j-operator
is the imaginary unit and define it by the property j2
=
-1. Using the imaginary unit, we build a general
complex number out of two real numbers. From the
definition j2
= -1, this can be generalized as:
z = a + jb
a. j4n+1
= j
b. j4n+2
= -1
c. j4n+3
= -j
d. j4n+4
= 1
The j-operator was originally denoted as i. The
symbol i is no longer used because the symbol
is also used to represent current. For the
purpose of consistency, the j-operator is now
utilized.
Terminology
The real number “a” in z = a + jb is called the
real part of z; the real number “b” is called the
imaginary part of z. For example, if z = 3 – j4, 3
is the real part of z and - 4 is the imaginary
part of z.

3. Theorem on Complex Numbers:
If (x +jy) = 0, then x = 0, and y = 0.
A. FORMS OF COMPLEX NUMBERS
1. Standard / Cartesian / Rectangular Form:
where: a = real part
jb = imaginary part
2. Steinmetz / Polar Form:
where: r = magnitude / absolute value/ hypotenuse of a right
triangle.
θ=argument / direction with respect to the real axis
θ= angle in degrees
θ= tan-1
b/a
z = a + jb
3. Exponential Form:
where: r = magnitude
θ = angle in radians
4. Trigonometric Form:
where: r = magnitude
θ = angle in degrees
z = rÐq
z = r ej θ
z = r (cosq + jsinq)

4. B. TRANSFORMATION
1. Rectangular form to Polar form
By Pythagorean Theorem:
And
r = √ a
2
+b
2
)
2. Polar form to Rectangular Form
𝑎=rcos ⁡(𝜃) 𝑏=rsin ⁡(𝜃)

5. C. OPERATIONS OF COMPLEX NUMBERS
Complex numbers can be added, subtracted,
multiplied, and divided. If z1 = a + jb and z2 = c + jd,
these operations are defined as follows:
C.1) Addition/Subtraction
Steps:
~ all complex numbers in rectangular form
~ combine all real parts
~ combine all imaginary parts
Add:
Subtract:
C.2) Multiplication
(a) Using rectangular form:
Note:
(b) Using polar form:
C.3) Conjugate of Complex Numbers

6. C.4) Division
(a) Using rectangular form:
(b) Using polar form: