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December 12, 2017
x2=9
x2=196
x2=-1
x2=-25
Solve the following: ( for X )
* eat f f → ×=3 x= -3
§×= 1 4 ×= - /y
December 12, 2017
The imaginary number i
Purely Imaginary Numbers
fd
- -
e. =.→
-
any number in the form
qx
basewhere bis a real number and b¥O
11--2=1177--47 . k=ir
ii. i.i. VI. Fi= -1
December 12, 2017
The square root of a negative real number
Forany positive real number a
,
V=a=iva
Note : f
TVF )2=(iFyY=i2( F) 2=-1 -4=-4-
T
pts yT36ex . Simplify using to
. =iF if . Fs M if .Fs6
\Fz ✓ ya=i2' By=zg=i6VII.
.VI. rz
2iV3=2iB=6E
=tVI
December 12, 2017
Complex numbers-
a number in the formatbi
a and b are real numbers
0 = Otoi-
a → real partfcompenent of a complex #
bi → imaginary par Hcomponent of a complex #
EI write \F9 +6 in the form a tbi= VII +6 = if 59 +6=31 +6=6+31
December 12, 2017
Rewrite in a+bi form:
⇒T.FI +4 = if +4
=if6 +4 = if -56+4
p=VI. if , -6
=i -256+4
= to .i - g
= 4+2 if
= 4i - 6
= - 6t4i
December 12, 2017
Complex number plane
Horizontal axis
Vertical axis
- a coordinate plane where the horiz.
axis correspondsto the real number line
and the vertical axis corresponds to a number theof imaginary numbers
i - real axisimaginaryaxis
•A
1/3 real-
imaginary axisaxis
2+31 = A• q - 5- 8i=B
To plot a complex number on this planeuse the real & imaginary parts of the # as
coordinates
December 12, 2017
The absolute value of a complex number
ex.
aDc a2+b2= a
D
is its distance from the origin when
the # is plotted on the complexPlane ,
-4+3 i
in
e
bs.pe?eyeft4t3ilfat+ba=T :: 's is
a
latbitsyatbi v
atbi
pp2- 3i/2¥tbi=1a+bi1
leave answers in terms ofa simplified radical
a=2 bI3
12 -
3il.la#=i/22tT35=y-a
#1- 5+6il (25+36)
"
=yF5tn=yFE=F(X+2)2
December 12, 2017
Operations with complex numbers
Additive Inverses
Example:
Find the additive inverse of -2+5i
- 2 complex numbers are additive inverses
of each other if their sum is
Zero
↳ To find
⇒change signs ofa and b in a + Bi
0d
§+2i)+(7- Gi ) 2 -55=10-4's
December 12, 2017
Adding/subtracting complex numbers
Examples:
Combine real componentsand combine imaginary componentsseparately
•8-+31-1-4in 6 - i
0=0=
=3 + Bi
December 12, 2017
Multiplying imaginary numbers
Example
- multiply coefficients togethermultiply the is ( separately )
^= (5) C-4) i. i= -2012=-204 )=2O
December 12, 2017
Multiplying complex numbers
Example:
Apply properties ofmultiplying binomials
(2t3F)f3+5i)F 0 1 L
2. t 3) 2(5i ) 3it3 ) 3iC5i )
-6 + Ioi - 9i + l5i2
- 6-+ i-1-5-2€
( G-5i)(4-3i )F 0 1 L
641 6t3i ) . silyl . Sil .si )
24 + - 18in + -20in + Isis
24T - 38 's t IS Ct )
24+-38 it -15
Gt -38in
December 12, 2017
Solving quadratic equations with complex solutions
4×2+100=0TakingSquare roots
4×2=-100
€-255
XT±yI5=±iFs=±5@
December 12, 2017
Functions of the form where
c is a complex number generate fractal graphs, as the one above.
To test if z belongs on the graph, use 0 as the first input and use the output as the next input value. If the output values do not approach infinity, then z belongs on the graph.
b