Complex Numbers Workshop

8/10/2019 Complex Numbers Workshop

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WORKSHOP #2The Complex Numbers

Abstract: In this workshop we explore the arithmeti! "eometr an$

al"ebra o% omplex numbers with a little tri"onometr thrown in& Wethen use the linear an$ on'u"ate linear %untions o% a omplex(ariable to in(esti"ate an$ lassi% the similarities an$ on"ruenes o%the )uli$ean plane& We start %rom the be"innin"* so! %or those alrea$%amiliar with the omplex numbers! the +rst %ew worksheets will simplbe a re(iew&

Format:We start with a $esription o% the workshop material as wepresente$ it %ormatte$ as a sin"le narrati(e& In an atual presentation!some material was simpl present (erball! some in the %orm o%pro'ete$ transparenies ,or pro'ete$ omputer sreen- an$ some!

partiularl the problems! was $istribute$ in han$outs& The maina$(anta"e o% ha(in" the material in Wor$ is that the presenter an$ei$e the best wa to $i(i$e up the material %or an "i(en au$iene!pro$uin" their own sli$es an$ worksheets& In most ases! theworksheets will be muh the same& The worksheets that we use$ areattahe$ to the en$ o% this +le&

Note:The +"ures in this workshop were all onstrute$ usin".eometer/s Skethpa$& I% ou woul$ like those ori"inal .SP +les! sen$ are0uest b email to 'e"ra(er1sr&e$u&


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Introduction&

Wh onsi$er omplex numbers The ori"inal moti(ation ame %romthe nee$ to sol(e 0ua$rati e0uations& In %at all o% our numbersstems were $e(elope$ to sol(e lar"er an$ lar"er lasses o% problems&

3or example! i% nan$ mare natural numbers! meanin" the are a

part o% the set 4 = 1, 2, 3K{ } an$ n5 m& Then the e0uation x+ =has

a uni0ue solution %ro x also a natural number& 6ut i% we $on/t want torestrit n> ! it is neessar to enlar"e our number sstem %rom 4 to7= 3,2,1, 0,1, 2, 3K{ } &! the set o% integers& Howe(er we must

enlar"e our number sstem e(en more i% we wish to sol(e a lineare0uation o% the %orm mx+= 0 where nan$ mare inte"ers& Thise0uation will ha(e a solution in i% an$ onl i% mis a %ator o% n&

So! to sol(e the "eneral linear e0uation! we one a"ain we enlar"e thenumber sstem! this time to the rational numbers!

8 = m

n: m,n ,n 0

& The properties o% the rational numbers ensure

that a "eneral linear e0uationpx += 0, ,8 has a solution within the

sstem o% rational numbers whene(er p 0 & The spei+ solution is

x=

&

The next step is to onsi$er 0ua$rati e0uations& 6ut the simple0ua$rati e0uation x2 2 = 0 $oes not ha(e a solution in 8 ! so we

enlar"e our sstem o% numbers a"ain to the sstem o% real numbers!9. 6ut! the e0uation x2 + 1 = 0 has no solution in the sstem o% realnumbers 9 & This is "ettin" tiresome! but it is important! so we enlar"eour number sstem a"ain to the complex numbers: & To arri(e atthe sstem o% omplex numbers! we intro$ue the ima"inar unit i whih has the propert that i2 = 1. The nie thin" is that this

enlar"ement proess en$s here; Sienti+


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In this workshop we explore the arithmeti! "eometr an$ al"ebra o%omplex numbers with a little tri"onometr thrown in& We start %romthe be"innin"* so! %or those alrea$ %amiliar with the omplexnumbers! the +rst %ew worksheets will simpl be a re(iew&


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The Complex Plane.

The complex numbersare the set o% numbers o% the %ormx+yiwherexan$yare real numbers an$ i$enotes the s0uare root o% A& We$enote this number sstem b &

Eust as we (isualiFe the real numbers as points on the real line! we will(isualiFe the omplex numbers as points on the omplex plane& Westart with the usual oor$inate plane& Thexaxis will be i$enti+e$ withthe real line& Theyaxis will represent the multiples o% ian$ is alle$the ima"inar axis& The omplex numberx+yi will be i$enti+e$ withthe point that has oor$inates(x,y-.

Worsheet !".The Complex Plane

Plot the

%ollowin"omplexnumbers;a G

b G Di

G Di

$ G Ji

e G Ji

% G Di

I$enti% theplotte$omplexnumbers;" G

h G Ji

' G DLi

k G Mi

r G Bi

s G Li

One (er use%ul wa to think o% a omplex numberzis as the (etor%rom the ori"in to the pointz& With that interpretation in min$! weintro$ue the lengthor absolute valueo% the omplex number its


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the len"th when thinkin" o% it as a (etor; i% z=+ ! then |z|= 2 +2 &

We use the absolute (alue notation sine this is the absolute (alue %orthe real number&

Compute the absolute (alue %or eah o% the omplex numbers on

Worksheet #A&

Complex Arithmetic # Addition

We a$$ two omplex numbers a+ an$ c+ to"ether b a$$in" thereal parts ,the a an$ the c in this ase- an$ the ima"inar parts ,bi an$di - separatel& So the sum will look like (a+ ) + (+) = (+) + (+) &

)xamples; (3 + 5) + (4 + ) = 7 + 6 *(3+ 8) + (1 5) = 2 + 3

I% we interpret the

omplex numberas the (etor in theomplex with initialpoint ,@!@- an$ terminalpoint ,a!b-! a$$ition o%omplex numbersbeomes (etora$$ition in the omplexplane& .eometriall! i%zan$ ware omplexnumbers thenz+w

ompletes aparallelo"ram with @!zan$ w&

I% we +x a omplexnumber! sa %orexamplezGDi! wema assoiate to F! the

translation z (w) that maps wontoz+w; that is the translation that

mo(es eah omplex number up +(e units an$ three units to the ri"ht&

Worsheet !$.


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0 * the i$entit translation eah point is +xe$&

.i(e the z %ormula %or eah o% the %ollowin" translations;

3in$ F wherez translates eah point $own M units* 4i3in$ F wherez translates eah point $own M units an$ D units to

the le%t*4

5i

3in$ F wherez mo(es eah point 3 2 units alon" the positi(e MDo

line; 3+ 3i3in$ F wherez mo(es eah points units alon" the positi(e MDo

line;

3+ 3i

2

Complex Arithmetic # %ultiplication

When we multipl two omplex numbers a+ an$ c+ we use thesame $istributi(e properties we woul$ when multiplin" polnomials&Howe(er in this ase we will also keep in min$ that i2 = 1. ?ultiplin"these two terms we "et

(a+ )(+) =

+() +() + ()() =

+ ()+ () (2 ) =

+ ()+ () =

( ) + (+ )

Worsheet !&.Complex


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34 40 = 1360

& | (5 + 3)(6 2) |

362 + 8 2 = 1360

J& | a+ ||+ |

(a

2

+

2

)(

2

+

2

) =

2

2

+

2

2

+

2

2

+

2

2

L& | (a+ )(+) |

(ac )2 + (+)2 =22 + 22 + 22 + 22

Interpretin" multipliation as a trans%ormation is rather ompliate$&So we will start with a %ew simple ases& 3irst onsi$er thetrans%ormation2 (z) = 2z ! that multiplies e(er omplex number b 2&

)(er (etor is $ouble$ in len"th an$ 2 is the dilationb a %ator o% 2

about the ori"in& 3or an omplex number wwe $e+ne w b w (z) = wz &The last two omputations on Worksheet # show the important %atthat wzGzw %or all omplex numbers& Hene w multiplies all

$istanes b w! the len"th o% w& When w is a positi(e real number wis the $ilation entere$ at the ori"in with ma"ni+ation m& The $ilationw atuall strethes ,or shrinks- all $istane b the %ator m&

Trans%ormations that alter all $istanes b the same %ator map"eometri +"ures onto similar +"ures an$ are alle$ similarities&

Qilations are 'ust one lass o% similarities& Trans%ormations thatpreser(e $istanes map "eometri +"ures onto on"ruent +"ures an$are alle$ congruences& Translations are on"ruenes&

Now onsi$er the trans%ormation 1 * this maps eah (etor onto itsne"ati(e an$ is easil reo"niFe$ as the hal%turn ,AJ@orotation- aboutthe ori"in& How $oes i map the omplex plane onto itsel% Sine i

GA! it shoul$ be a on"ruene& Chekin" a %ew (alues ,i (1) =i *i (i) =1 *i (1) =i *i (i) = 1-! we onlu$e that i is the L@$e"ree

ounterlokwise rotation about the ori"in& What about 2i We ma

think o% multiplin"zb 2ias +rst multiplin" b ian$ then b 2;2iz= 2(()) & Hene! 2i (z) =2 (i (z)) * in other wor$s! 2i rotates theplane L@ $e"ree ounterlokwise about the ori"in an$ $ilates b the%ator 2& In "eneral! w will split into a rotation part an$ a $ilation part&


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?ore to the point! the omplex number witsel% will split; w=| | (

| |) *

| w| is a positi(e real number while w

| w|is a omplex number o% len"th

one& To see exatl how a

omplex number o% len"th oneorrespon$s to a rotation! letz=+ be a omplex number o%len"th one* that is! where

a2 +2 = 1. Consi$erin" the

ri"ht trian"le with (erties @!zan$ the pro'etion o%zon thexaxis! we onlu$e that! %or somean"le between @ an$ B@!a= an$ b= & So a

omplex number o% len"th A hasthe %orm (cos+ ) & Hene;

Any complex number w 0 can be written in the formw=(+ ) , where r is a positive real number and(cos+ ) is the unit vector that makes a counterclockwise

angle of degrees with the positive real axis.

We ha(e illustrate$ this %orzin the +rst 0ua$rant* similar pitures anbe $rawn %orzin an other 0ua$rant&Now returnin" to multipliation! let w= (+ )an$z= (+ ) & ThenWe shoul$ reo"niFe the sine an$ osine %ormulas %or the sum o% twoan"les* hene

wz= [(+ ) + (+ )]&

So it is lear that multiplin"zb w! rotates it b the an"le an$strethes its len"th b the %ator r&

Worsheet !'.The Tri"onometri Representation o% ComplexNumbers


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Plot eah o%these omplexnumbers an$ompute itsabsolute (alue

an$ an"le o%rotation&A& z= 1 +

|z|= 2 an$

= 45o

2& z= 1 |z|= 2 an$

= 45o

& z= 2 2 3|z|= 4 an$

=240oI$enti% an$ ploteah o% theomplex withthese absolute(alues an$an"les&

M& r= 2 an$= 135o

z= 1 +

D& r= 2 an$ = 225 o

z= 2 2B& zG B an$ = 30o

z= 3 3 + 3

Complex Con(ugationThere is one more (er important trans%ormation o% the omplex plane!the reetion throu"h thexaxis alle$ conjugation; %or z= +, we$e+ne z ! the on'u"ate o%zb z = & The %ollowin" two propertieso% on'u"ation are eas to see "eometriall;

z = an$ w+=+& There are se(eral other properties whih ou will(eri% al"ebraiall on the next worksheet&

Worsheet !).The Properties o% Complex Con'u"ation


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eri% eah o% the %ollowin" e0ualities b $iret omputation&A& z =

2& z+= +

& (zw)= ()()

M& |z |=||

D& zz =||2

B& z

||2 = 1

& r(cos+ ) =(() + ())

J& I% z= +,a=

+

2 an$ b=

2.

Complex Arithmetic * +i,ision

What about $i(ision We know that we ant $i(i$e b @ %rom workin"with real numbers! so when writin" a omplex number as a %ration wemust alwas a$$ the on$ition that the $enominator is not Fero&


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The TIJM an$ TIJL alulators are able to $o omplex arithmeti usin"the same omman$s as %or real arithmeti& To aess this %eature oumust reset the %+/sreen& 6oth alulators "i(e ou the option o%the usual ,retan"ular- %ormat %or omplex numbers or the polar

%ormat& The latter %ormat uses rei

to represent r(cos+ ) & Pro(in"that this exponential notation is (ali$ is beon$ the sope o% thisworkshop&Usin" the retan"ular %ormat %or omplex numbers it mi"ht beinstruti(e to hek our han$ omputations on Worksheets ! D an$ B&The omman$s %or =on'u"ate> an$ =absolute (alue> an be %oun$ onthe CP0submenu o% the %AT1menu&

2uadratic /3uations

Qoes e(er omplex number ha(e a s0uare root I% a omplex numberhas a s0uare root! how man $oes it ha(e


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zp = (2

) + (

2

) ! %orpG@!A!2!!,nA-& One an also easil see

that; A an$ A are the roots o% 2 an$ that A! i! A an$ i are the Mthrootso% unit& On the ri"ht in the %ollowin" +"ure! we ha(e piture$ the r$root o% unit in blue&

Worsheet!7& The nth

Roots o% Unit

A& Usin" the(alues o% thetri" %untionson our

alulator!ompute theDthroots o%unit an$ plotthem&

2& Computethe Jthroots o%unit an$ plotthem&

Similarities

The similarities o% plane "eometr are those trans%ormations o% theplane that sale all $istanes b the same positi(e real numberalle$themagnication factor* the on"ruenes are the similarities withma"ni+ation %ator A& We an (isualiFe on"ruenes in the wa that)uli$ $i$; +"ureAis on"ruent to +"ure Bi%Aan be pike$ up an$plae$ on top o% B mathin" per%etl& Instea$ o% pikin" up +"ure . CONEU.


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Worsheet !"?. SI?IY

Documents

Complex Numbers Workshop