What are imaginary and complex numbers?
Graph it
Solve for x: x2 + 1 = 0
2
x  1
2
x   1
?
What number when
multiplied by itself
gives us a negative one?
No such real number
parabola does
not intersect
x-axis NO REAL ROOTS
Imaginary Numbers
1  is not a real number,
then 1  is a non-real or
If
imaginary number.
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.
1 
ab  a  b
i
1
1)  5 25
25  25
25 (25
25
1 1
15i 5 1  5i b = 5
7
7 77 1
1 7i
7i ii 77
b 7
In general, for any real number b, where b > 0:
2
b  b
2
1  bi
1  i
 
1
If
i2
2
Powers of i
 i2 = –1

 1

2
 i2 = –1
i3
= – 1, then = ?
i3 = i2 • i = –1( 1 ) = –i
i4 = i2 • i2 = (–1)(–1) = 1
i5 = i4 • i = 1( 1) = i
i6 = i4 • i2 = (1)(–1) = –1
i7 = i6 • i = -1( 1 ) = –i
i8 = i6 • i2 = (–1)(–1) = 1
What is i82 in simplest form?
82 ÷ 4 = 20 remainder 2
i82 equivalent to i2 = –1
i0 = 1
i1 = i
i2 = –1
i3 = –i
i4 = 1
i5 = i
i6 = –1
i7 = –i
i8 = 1
i9 = i
i10 = –1
i11 = –i
i12 = 1
A little saying to help you
remember
Once
I
Lost one
Missing eye
1  i
Properties of i
16  9 
Addition:
16 1  9 1  4i + 3i = 7i
Subtraction: 25  16 
25 1  16 1  5i – 4i = i
Multiplication:
36 4 
36 1  4 1  (6i)(2i) = 12i2 = –12
note :
Division:
36 4  144
16  4 
16

4
16 1 4i

2
4 1 2i
Complex Numbers
Definition: 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.
a + bi
real numbers
pure imaginary
number
Any number can be expressed as a complex
number:
7 + 0i = 7
a + bi
0 + 2i = 2i
The Number System
5 76
-i
Complex Numbers
Real Numbers
i
i3
-i Irrational
Numbers
i9
i
2 + 3i
47
Rational
Numbers
i
Integers
i
i75
Whole Numbers
Counting
Numbers
1/2 – 12i
-6 – 3i
i
-i47
i
Model Problems
Express in terms of i and simplify:
100 = 10i  16 = 4/5i  1 300  5i 3
2
25
Write each given power of i in simplest terms:
i49 = i
i54 = -1 i300 = 1
i2001 = i
Add:
4 18  50  4i 9 2  i 25 2
 12i 2  5i 2  17i 2
Multiply: 4 5  80
 4i 5  4i 5  16i
2
 5
2
 4i 5  i 16 5
 16 5  80
Simplify: 72  32  3 8
 i 36 2  i 16 2  3i 4 2
 4i 2
 6i 2  4i 2  6i 2