Download The Polar Form of Complex Numbers

THE POL AR FORM
OF COMPLEX
NUMBERS
DR. SHILDNECK
THE COMPLEX NUMBER SYSTEM
Complex Numbers
Real Numbers
Rational Numbers
Integers
Whole Numbers
Natural Numbers
Irrational Numbers
Non-integers (fractions)
Negatives
Zero (0)
Imaginary Numbers
COMPLEX NUMBERS
Remember that complex numbers have two parts:
a real part and an imaginary part.
Thus, the complex number z takes the form
z = a + bi
for all real numbers a and b
and i = −1
PLOTTING COMPLEX NUMBERS
In the Cartesian coordinate plane, the horizontal axis represents our
real number part, while the vertical axis represents the imaginary part.
Plot: z = 3 + 4i
ABSOLUTE VALUE
The absolute value of any number is defined as its _______________.
For a real numbers, this amount is one dimensional.
However, for complex numbers, this “magnitude” is two dimensional.
GRAPHICALLY FINDING ABSOLUTE VALUE
Use the graph to determine the absolute value (distance from the
origin) of the complex number.
z = 3 + 4i
ABSOLUTE VALUE
Thus, the absolute value of a complex number can be defined as the nonnegative distance from the origin to the number on the complex plane.
|a + bi| =
2
𝑎
+
2
𝑏
THE POLAR FORM OF A COMPLEX NUMBER
Since a complex number can be thought of as a point in a Cartesian
coordinate plane, we can translate that point into polar form.
Find the magnitude and angle related
to the complex number z = -5 + 2i.
POLAR FORM OF A COMPLEX NUMBER
The distance the complex number is from the origin is called the modulus of
the number and is indicated by r (in polar form)
The angle for the complex number is called the argument of the number and
is indicated by Ѳ in polar form. (0 ≤ 𝜃 ≤ 2𝜋)
The complex number a + bi is written in polar form as
𝒂 + 𝒃𝒊 = 𝒓𝒄𝒐𝒔𝜽 + 𝒊𝒔𝒊𝒏𝜽 = 𝒓𝒄𝒊𝒔𝜽
Where 𝑟 =
𝑎2
+
𝑏2
and 𝑡𝑎𝑛𝜃 =
𝑦
(adjusted for the correct quadrant).
𝑥
EXAMPLES
[Example 1] Plot z = -2 – 2i on the complex plane and write in polar form.
EXAMPLES
[Example 2] Plot −1 + 𝑖 3 on the complex plane and write in polar form.
EXAMPLES
[Example 3] Write z = 2cis60o in rectangular form.
EXAMPLES
[Example 4] Write z = 4cis150o in rectangular form.
ASSIGNMENT
Will Be Posted on the Blog later today.