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Week4ŒComplexNumbers
RichardEarl
MathematicalInstitute,Oxford,OX12LB,
November2003
Abstract
Cartesianandpolarformofacomplexnumber.Th
eArganddiagram.Rootsofunity.Therelation-
shipbetweenexponentialandtrigonometricfu
nctions.ThegeometryoftheArganddiagram.
1TheNeedForComplexNumbers
Allofyouwillknowthatthetworootsofthequadraticequation
2++=0are=±242(1)andsolvingquadraticequationsissomethingtha
tmathematicianshavebeenabletodosincethetime
oftheBabylonians.When
24
0thenthesetworootsarerealanddistinct;graphicallytheyare
wherethecurve
=2++cutsthe
-axis.When
24=0thenwehaveonerealrootand
thecurvejusttouchesthe
-axishere.Butwhathappenswhen
24
0?Thentherearenoreal
solutionstotheequationasnorealsquarestogivethenegative
24
Fromthegraphicalpointof
viewthecurve
=2++liesentirelyaboveorbelowthe
-axis.-1123-1123Distinctrealroots
-11231234Repeatedrealroot
-11230.511.522.533.54Complexroots
Itisonlycomparativelyrecentlythatmathematicianshavebeencomfortablewiththeserootswhen
24
0DuringtheRenaissancethequadraticwouldhavebeenconsideredunsolvableoritsroots
wouldhavebeencalled
imaginary.
(Theterm‚imaginary™was
ÞrstusedbytheFrenchMathematician
RenéDescartes(1596-1650).Whilstheisknownmoreas
aphilosopher,Descartesmademanyimportant
contributionstomathematicsandhelpedfoundco-ordinategeometryŠhencethenamingofCartesian
co-ordinates.)Ifweimagine
1toexist,andthatitbehaves(addsandmultiplies)muchthesameas
othernumbersthenthetworootsofthequadraticcanbewrittenintheform
=±1(2)where=2and=422arerealnumbers.
ThesehandoutsareproducedbyRichardEarl,whoistheSchoolsLiaisonandAccessO
cerformathematics,statistics
andcomputerscienceatOxfordUniversity.Anycomments,suggestionsorrequestsforothermaterialarewelcomeat
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