Honors Precalculus Notes--Chapter 2, Polynomial and Rational Functions
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Course
Precalculus
Institution
Junior / 11th Grade
Chapter 2 notes for 11th grade Honors Precalculus from an A+ student. Includes definitions on course materials, graphs displaying concepts, numerous example problems including answers, equations, and step-by-step instructions designed to make learning a breeze! Chapter features 3 moduels, includin...
,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
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