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

Simplify:

@= t.EF.ru . k=i2r

as =L if

VFT = VII = if

<

,

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

Sets of numbers

man

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

Plot

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@

→Solve by taking the

square root.

-5×2-150=0

- 5×2 = 150

Is Is

#=-30TX=±F3o×=±iiFo

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

December 12, 2017

Example:

Find the first 3 output values for:Use 2=0 as the 1st input€ 1st input is 0Ootgpotfco)=02ti=j 2nd input is i - Iti

f ( i )=i2+e= - |+i3rd input 's - Hi - g

ffitil .it#itti=@2i#i= ( I - i -iti9+i=1-2i+i2+i@tiktti )

-

= -2iti=. ;