1. What are imaginary and complex numbers?
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

2. 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
b = 5
In general, for any real number b, where b > 0:

3. i2 = i2 = i Powers of 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

4. A little saying to help you remember
Once I Lost one Missing eye

5. i
Properties of i
Addition:
4i + 3i = 7i
Subtraction:
5i – 4i = i
Multiplication:
(6i)(2i) = 12i2 = –12
Division:

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. Model Problems
Express in terms of i and simplify:
= 10i
= 4/5i
Write each given power of i in simplest terms:
i49
= i
i54
= -1
i300
= 1
i2001
= i
Add:
Multiply:
Simplify: