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7/12/2014 CBSE Class 11 Maths Notes : Complex Number
http://schools.aglasem.com/?p=45642 1/18
CBSE Class 11 Maths Notes : ComplexNumber
April27,2014byNeepur
ImaginaryQuantity
Thesquarerootofanegativerealnumberiscalledanimaginaryquantityorimaginarynumber.
e.g.,-3,-7/2
Thequantity-1isanimaginarynumber,denotedbyi,callediota.
IntegralPowersofIota(i)
i=-1,i =-1,i =-i,i =1
edurite.com
2 3 4
4n+1 4n+2 4n+3 4n+4 4n
7/12/2014 CBSE Class 11 Maths Notes : Complex Number
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So,i =i,i =-1,i =-i,i =i =1
Inotherwords,
i =(-1) ,ifnisaneveninteger
i =(-1) .i,ifisanoddinteger
ComplexNumber
Anumberoftheformz=x+iy,wherex,yR,iscalledacomplexnumber
Thenumbersxandyarecalledrespectivelyrealandimaginarypartsofcomplexnumberz.
i.e.,x=Re(z)andy=Im(z)
PurelyRealandPurelyImaginaryComplexNumber
Acomplexnumberzisapurelyrealifitsimaginarypartis0.
i.e.,Im(z)=0.Andpurelyimaginaryifitsrealpartis0i.e.,Re(z)=0.
EqualityofComplexNumbers
Twocomplexnumbersz =a +ib andz =a +ib areequal,ifa =a andb =b i.e.,Re(z )=
Re(z )andIm(z )=Im(z ).
AlgebraofComplexNumbers
1.AdditionofComplexNumbers
Letz =(x +iy )andz =(x +iy )beanytwocomplexnumbers,thentheirsumdefinedas
z +z =(x +iy )+(x +iy )=(x +x )+i(y +y )
PropertiesofAddition
4n+1 4n+2 4n+3 4n+4 4n
n n/2
n (n-1)/2
1 1 1 2 2 2 2 2 1 2 1
2 1 2
1 1 i 2 2 2
1 2 1 1 2 2 1 2 1 2
7/12/2014 CBSE Class 11 Maths Notes : Complex Number
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(i)Commutativez +z =z +z
(ii)Associative(z +z )+z =+(z +z )
(iii)AdditiveIdentityz+0=z=0+z
Here,0isadditiveidentity.
2.SubtractionofComplexNumbers
Letz =(x +iy )andz =(x +iy )beanytwocomplexnumbers,thentheirdifferenceisdefined
as
z z =(x +iy )(x +iy )
=(x x )+i(y y )
3.MultiplicationofComplexNumbers
Letz =(x +iy )andz =(x +iy )beanytwocomplexnumbers,thentheirmultiplicationis
definedas
z z =(x +iy )(x +iy )=(x x y y )+i(x y +x y )
PropertiesofMultiplication
(i)Commutativez z =z z
(ii)Associative(z z )z =z (z z )
(iii)MultiplicativeIdentityz1=z=1z
Here,1ismultiplicativeidentityofanelementz.
(iv)MultiplicativeInverseEverynon-zerocomplexnumberzthereexistsacomplexnumberz
suchthatz.z =1=z z
1 2 2 1
1 2 3 2 3
1 1 1 2 2 2
1 2 1 1 2 2
1 2 1 2
1 1 i 2 2 2
1 2 1 1 2 2 1 2 1 2 1 2 2 1
1 2 2 1
1 2 3 1 2 3
1
1 1
7/12/2014 CBSE Class 11 Maths Notes : Complex Number
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(v)DistributiveLaw
(a)z (z +z )=z z +z z (leftdistribution)
(b)(z +z )z =z z +z z (rightdistribution)
4.DivisionofComplexNumbers
Letz =x +iy andz =x +iy beanytwocomplexnumbers,thentheirdivisionisdefinedas
wherez #0.
ConjugateofaComplexNumber
Ifz=x+iyisacomplexnumber,thenconjugateofzisdenotedbyz
i.e.,z=xiy
PropertiesofConjugate
ModulusofaComplexNumber
Ifz=x+iy,,thenmodulusormagnitudeofzisdenotedby|z|andisgivenby
|z|=x +y .
Itrepresentsadistanceofzfromorigin.
InthesetofcomplexnumberC,theorderrelationisnotdefinedi.e.,z >z orz |z |or|z |0andy>0,thenarg(z)=0
(ii)Ifx0,thenarg(z)=-0
(iii)Ifx