Section 3.1 The Complex Numbers
Today:
3.1 Complex Numbers
Announcements
 Homework 3.1 due Mon
 quiz Monday on alrithmetic of complex
numbers
 Midterm Review will be passed out.
Recall:
linear: f ( x)  mx  b
non-linear: f ( x)  ax 2  bx  c
Motivation: Linear Equations always have a solution to f(x)=c in R but not necessarily
non-linear equations.
Example: f ( x)  22 x  99
f ( x)  10 : 10  22x  99  109  22x  x 
109
22
Example: f ( x)  x 2
f ( x)  1 :  1  x 2  x   1
Geometric explanation:
linear equations hit all values of y, but a function with a curve can ‘turn away’ from
values
Example: y  x 2  1
Doesn’t hit any y-values less than 1.
So 0  x 2  1 has no solution.
Different interpretation:
Consider the image of the graph rotated 180 degrees… then all values of points are hit
either by the real part of the graph or the image of the graph.
If we consider the image, what values of x would yield y  0 ?
0  x 2  1  x=   1
We say the zeros are imaginary and call them  i
The imaginary number i:
Properties of i:
 i  1
 i 2  1  1  1
 i  i  2i
 i i  0
 i  4  4  i is called a complex number since
it has both a real and an imaginary
component.
 i  7  7i
i  1
Worksheet: Computing with complex numbers.
In general a complex number z is expressed as z  a  bi .
The conjugate of a complex number z  a  bi is z  a  bi and is useful for getting rid of
imaginary numbers in the denominator of a fraction.
Example
1  4i 1  4i
1  4i 1 1  4i 1  4i  1  4i   1  4i 

1 
 


1  4i 1  4i
1  4i 1 1  4i 1  4i  1  4i   1  4i 
1  4i   1  4i   1  4i  4i  16i 2
1  4i   1  4i  1  4i  4i  16i 2

1  8i  16(1) 1  16  8i  15  8i
15 8


  i
1  16(1)
1  16
17
17 17