1. Complex Numbers: Trigonometric Form
Section 8.3
Complex Numbers: Trigonometric Form

2. Objectives Graph complex numbers.
Given a complex number in standard form, find trigonometric, or polar, notation; and given a complex number in trigonometric form, find standard notation.
Use trigonometric notation to multiply and divide complex numbers.
Use DeMoivre’s theorem to raise a complex number to powers.
Find the nth roots of a complex number.
Just as real numbers can be graphed on a line, complex numbers can be graphed on a plane. We graph a complex number a + bi in the same way that we graph an ordered pair of real numbers (a, b). However, in place of an x-axis,we have a real axis, and in place of a y-axis, we have an imaginary axis. Horizontal distances correspond to the real part of a number. Vertical distances correspond to the imaginary part.

3. What are imaginary and complex numbers?
Do Now:
Graph it
Solve for x: x2 + 1 = 0 ?
What number when
multiplied by itself
gives us a negative one?
parabola does
not intersect
x-axis -
NO REAL ROOTS
No such real number

4. i Imaginary Numbers If is not a real number, then is a non-real or
Definition:
A pure imaginary number is
any number that can be expressed in the
form bi, where b is a real number such
that b ≠ 0, and i is the imaginary unit. i

5. i2 = i2 = Powers of i i –1 –1 i0 = 1 i1 = i i2 = –1 i3 = –i i4 = 1
If i2 = – 1, then i3 = ?
= i2 • i = –1( ) = –i i3
i4 =
i6 =
i8 = i5 i7
i2 • i2 = (–1)(–1) = 1
= i4 • i = 1( ) = i
i4 • i2 = (1)(–1) = –1
= i6 • i = -1( ) = –i
i6 • i2 = (–1)(–1) = 1
What is i82 in simplest form?
82 ÷ 4 = 20 remainder 2
equivalent to i2 = –1
i82

6. a + bi Complex Numbers A complex number is
any number that can be expressed in the
form a + bi, where a and b are real numbers
and i is the imaginary unit.
Definition:
a + bi
real numbers
pure imaginary
number
Any number can be expressed as a complex
number:
7 + 0i = 7
a + bi
0 + 2i = 2i

7. The Number System
Complex Numbers i -i i3 i9
Real Numbers
Rational Numbers i
i75
-i47
Irrational Numbers
Integers
Whole Numbers
Counting Numbers
2 + 3i
-6 – 3i
1/2 – 12i

8. Example
Graph each of the following complex numbers. a) 3 + 2i b) –4 – 5i c) –3i d) –1 + 3i e) 2
Solution:

9. Absolute Value
The absolute value of a complex number a + bi is

10. Example
Find the absolute value of each of the following.
Solution:

11. Trigonometric Notation
If we let  be an angle in standard position whose terminal side passes through the point (a, b), then

12. Example
Add/Subtract

13. Example
Multiply

14. Example
Divide