Honors Precalculus Notes--Chapter 2, Polynomial and Rational Functions

,2. I Complex Functions
Complex Numbers and Imaginary numbers


The
imaginary unit i is defined as




i - FI where i2= -

I
,




he set of all numbers in the form



at bi



numbers and b and i, the is called the set of
with real a
, imaginary unit
, complex numbers .




The standard form of a complex number is



at bi




Orientation of complex Numbers


form of bi like the binomial atbx To add , subtract , and methods that we
The a complex number at is .



multiply complex numbers
,
we use the same




use for binomials .




Adding and Subtracting Complex Numbers

Perform the indicated standard form
'



operations , writing the result is .




A . (S -
Zi ) + ( 3t3i ) b .
(2+6 ; ) ( 12 - -
i )

( 5- 3) it (-2+3) ; (2-12)+(6+1) i
'
-
-
-




=
Sti =
lot > i
-




Multiplying Complex Numbers


Find the product :


a .
7i( 2 Gi ) -
b .
( 5-14; )(6 -
Ti )



141 6312 516) -151 Ti ) -14 :( 6) Hit > i )
'
-
= +
= -




141 -163C ) =
30 35 ; -12%-2812
'


= - I -




= 14 i -

63
=
30 -
Ili -
28C -

t)




= 63+14 ; = 30 -

Ili -128



=
58 -
hi

, Conjugate of a Complex Number


For the at bi , define its be bi
conjugate to at
complex number we complex .




The product of complex number and its conjugate is a real number .




( at billa bi ) -




=
Ala ) tac bi ) -
-1 bila) +
bit bi ) -




' '
=
a -
abi +
abi -
b' i


b 't 1)
'
=
a
-
-




Complex Number Division


The of number division obtain the denominator denominator of
goal complex is to a real number in . We
multiply the numerator and a complex number




denominator
quotient by of the to obtain this
the
conjugate real number .




Using Complex conjugates to Divide complex Numbers


Divide and express the result in standard form :




-
5t4i
4- i
St4i Uti
- -



=
4 -
i ai
2
20+5 's -1161 -14;
'




-

= '
16+4 9; i
'

- -
,



20+21 i -11-4 )
=



16+21
'




if Ii
.


- '
-




In form :
-
-
17 standard the result is
,




Principal Square Root of a Negative Number


For positive b the of defined by
any real number principal square root the
negative number b is
-




,




Fb -

-
irb




Operations Involving Square Roots of Negative Numbers


Perform the indicated operations and write the result in standard form .




A .
FL> +
Ft b .
C- 2. tf ) '


=
3. if -141-53 = 1-2-1 if )t2- if )

=
7;D = 4- 2ir3 -

2iBt3i2

= 4- Hit -13ft )

=
I -
4if3