MATH 17100 28446
09/14/2009
If we insist on getting a solution to the quadratic equation ax
2  bx
 
0 regardless of the sign of the discriminant, i.e. even for the case D
 b
2  system complex numbers : z is called a complex number if z
 a
4 ac

0 , we will have to admit to our number
 bi where a and b are real and i is the imaginary unit defined by i
2  
1 , hence it is also called the principal square root of 
1
. a is called the real part of the complex number z, and b is called the imaginary part of z . Note that both the real part as well as the imaginary part of a complex number are real numbers.
Equality of two complex numbers: a
 bi = c
 di , , , , real  a

,
 d
Sum of two complex numbers: ( a
 bi ) + ( c
 di
Product of two complex numbers: ( a
 bi )( c
 di
) = ( a
 c
) = ( ac

) + ( bd b

) + ( d ) bc
 i ad
Quotient of two complex numbers: a c

 bi di

( a
( c

 bi )( c
 di )( c
 di ) di )

( ac
 bd ) c
2


)
( bc d
2 i
 ad ) i
The complex number z
 a
 bi can also be represented by the ordered-pair ( a , b ), and graphically by the point ( a , b ) in the Argand plane (similar to the Cartesian plane except for the name), also called the complex plane , where the horizontal axis is called the real axis , and the vertical axis is called the imaginary axis .
Complex Conjugates z
 a
 bi is called the complex conjugate of z . In the complex plane, the complex conjugate z is the reflection of z about the real axis.
Complex conjugates have the following properties: z
 w
 z
 w zw
 z w z n  z n
Note that z z
 a
2  b
2 is real, a result taken advantage of in rationalizing the denominator .
The modulus , or absolute value , of the complex number z is z
 zz
 a
2  b
2 and represents its distance from the origin in the complex plane.

Polar Form
Recalling the relation between the Cartesian coordinates and polar coordinates of a given point in the plane, we extend the expression of the complex number: z x iy
 r cos
  ir sin

 r (cos
  i sin

)
 re i

by
Euler’s Formula
Here r
 z is the modulus of z and  is called the argument of z and sometimes designated
  .
While addition and subtraction is best done with the real and imaginary parts, multiplication and division is less tedious when using the polar form: z
1
 r e
1 i

1 , z
2
 r e
2 i

2
 z z
2
 i (
2
) and z
1 z
2
 r
1 e i ( r
2
2
)
De Moivre’s Theorem z
 re i

 r (cos
  i sin

) n  n in z r e

 r n
(cos n
  i sin n

)
Roots of a complex number z
 re i
  r (cos
  i sin

)
 z
1 / n  1 / n i r e
(
 
2 k

) / n  r
1 / n
(cos
 
2 k

 i sin
 
2 k

), k

0,1, 2,  , k

1 , n a natural number n n are the n roots of z . In the complex plane, these n roots are evenly distributed around the circle centered at the origin, radius r
1 / n , adjacent roots subtending an angle of
2
 at the origin, the root n with k

0 being the principal root.
HW: App G # 1, 3, 5, 7, 15, 21, 25, 29, 35, 39, 43
R.Tam