7/12/2014 CBSE Class 11 Maths Notes : Complex Number

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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

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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