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===== The square root of a complex number =========================

To find the two square roots of the

complex number a ib do t

e followi g:

(i) Let t

e sq are root be x iy ie

a ib x iy ! (ii) Sq are bot

sides, t

e apply t

e eq ality property to get two

sim

lta

eo

s eq

atio

s i

x a

d y, solve t

em.EXAMPLE: Fi d t

e two sq are roots of t

e complex mber 5-12i

SOLUION:

2 2

2 2

2 2

2 2

( i ) let 5-12i x iy

( ii ) Sq

are bot

e sides: 5-12i=x y 2 xyi

x y 5...........( 1)

2 xy 12......( 2 )

( 1 ) ( 2 )

x y 13........( 3 )

Solvi

g ( 1 )&( 3 ) : x 3 & y 25 12i ( 3 2 y ) A

s

!

@ !!

!

! s ! s@ ! s

Ot

er sol tio :

2 2 2 2

2 2

From eq

atio

s (1),(2)

e q

a

tities x y ;2 xy; x y form t

e t

ree sides of a

rig

t a

gled tria

gle i

w

ic

x y is t

e

ypoti

eo

s. So we ca

form a rt. a

gled tria

glew

ose legs are 5;12 a

d get t

2 2

2 2

e

ypoti

eo

s

eq

als 13, a

d write

x y 5..................( 1 )

2xy 12.............( 2 )x y 13...............( 3 )

Solvi

g( 1 ) &( 3 ) x 3 ; y 2 5 12i ( 3 2i ).

!

! !

! s ! ! s m

Ot

er sol tio :

2 2 2 2 2

1T

i

k abo t t

o

mbers

ose prod ct 6 ( t

e imagi

ary part )2

a

d t

e differe

ce of t

eir sq ares is 5 (t

e real part); t

e

mbers

ill

be 3 a d 2

5-12i

3 12i 2 i 9 12i 4i ( 3 2i )

5 12i ( 3 2i )

s.

@ ! !

@ !

Now, try t

ese problems:

3 4i ; 7 24i ; 8 15i

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EXAMPLE: Solve t

e followi

g eq atio

s i

C(i) !z z2 1 0 (ii) x ( i )x !22 5 6 0

SOLU ION

1 1 4 1 1 1 3i 1 3( i )z i

2 2 2 2

s v v s s

2

2 2 2 2

1 2

( 5 i ) ( 5 i ) 4 2 6 ( 5 i ) 24 10i ( ii ) x

4 424 10i l mi l m 24 ;l m 26

l 1 ; m 5

24 10i ( 1 5i )

( 5 i ) ( 1 5i ) 3 3i x x ; x 1 i

4 2

s v v s

s s

s s

EXAMPLE: Solve t!

e followi" g eq# atio" i" C2

( 1 i )z ( 1 3i )z 2( 2 3i ) 0 SOLU ION:

2

2

1 2

Divi de t$

e eq%

atio&

by (1+i) a&

d simplify

z ( 2 i )z ( 1 5i ) 0

( 2 i ) ( 2 i ) 4( 1 5i ) ( 2 i ) 7 24i z

2 2

7 24i ( 4 3i )

( 2 i ) ( 4 3i ) ( 2 i ) ( 4 3i )z 3 2i ; z 1 i

2 2

s s

s

@

EXAMPLE:

Form t!

e q#

adratic eq#

atio" wit!

real coefficie" ts, if o" e of its roots is 3+i

SOLU ION:

2

2

3 i is a root 3-i is t $

e ot$

er roots

%

m of roots = 3+i + 3-i=6

prod%

ct of roots=(3+i)(3-i)=10

t$

e eq%

atio&

is

x ( s%

m )x prod %

ct 0

i .e . x 6 x 10 0

@

@

Q

EXAMPLE:

1 2i 1 2i If x ; y ,then find 5x ' 3y .

1 i 1 i

! !

SOLU(

ION:1 2i 1 2i 1 i 1 3i x

1 i 1 i 1 i 21 2i 1 2i 1 i 1 3i

y1 i 1 i 1 i 2

2 3 25 x 3 y 4 3i ( i )

2 2

! ! v !

! ! v !

! ! s

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Exercise 2

2 2

4 2 2

2

2

1 If x ) then find the solution set of

a. x 25 0 b.x 4 x 13 0

c.x 7 x 12 0 d .x 6 x 9 2i 0

e.x ( 5 i )x ( 8 i ) 0

f .( 2 i )x ( 9 7 i )x 5( 3 2i ) 0

20

ind the square roots of the follo1

ing complex numbersa.z 5

! !

! !

!

!

!

3-2

1 1

2 2

2 2

12i b.z 3 4i

c.z 7 24i d .z 1 i

13 65i e.z i f .z

5 i

3 If x 3 4i , then find x

4 If x 2 21 3 20i, then find the value of x 29x .

5 If x and y are real values, find these values if

y iy 6

! ! !

! !

!

2 2 2

2

2 2

*

i 2 x ix

( x i ) ( 2 y i ) 4( 3 1 )i 2 y x

6 If l and m are the roots of the equation x ( 4 6i )x ( 10 20i ) 0 ,then find

then find the equation1

hose roots are l and m .

74

olve the simultaneously the follo1

ing equ

! !

!

ations, 1 here x and y are real

2 5i and x y i

x y ! !

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Geometric Represe5 tatio5 Of Complex N6

mbers

Arga5

d Fig6 res

7

8

e Fre5 c8

mat8

ematicia5 Arga5 d establis8

ed t8

e geometric represe5 tatio5

of t8

e complex5 6 mber x + y i as a5 ordered pair (x , y) R2,8

e called t8

e x-axis as

t8

e real axis a5

d t8

e y-axis as t8

e imagi5

ary axis. So, t8

e Cartesia5

pla5

e is5

amed

as Arga

5

d pla

5

e a

5

d t

8

e fig6

res t

8

at represe

5

t t

8

e complex

5

6

mbers or a

5

yoperatio5 performed o5 t8

em as Arga5 d fig6 res.

For t8

is, t8

e complex5 6 mbers will be represe5 ted i5 Arga5 d pla5 e by a poi5 t

(x , y), ( sometimes t8

ey call it vector), as yo6

see i5 t8

e fig6

re:

9

@

e complexAB

mberz1 = 3 + 4i is represeA ted by poiA t A(3,4), z2 = -1 + 2i is

represeA ted by t@

e poiA t B(-1,2) aA dz3 = -2 3i is represeA ted by t

@

e poiA t C(-2,-3),

aA d so oA .

RepreseA tatioA of SB

m

Ifz1=(x1,y1) aA dz2=(x2,y2) t@

eA

z1 + z2 = (x1 + x2 , y1 + y2). From t@

e

figB re we fiA d t@

at t@

e poiA ts (x1,y1) ;

(x2,y2); aA d (x1+x2 ,y1+y2) are t@

ree vertices

of t@

e parallelogram OACB, t@

at is t@

e s B m of

two complexAB

mbers is t@

e foB

rt@

vertex C. A(x1,y1)

B(x2,y2)

C(x1+x2,y1+y2)

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TheC

D mbers izaC

d iz

RepreseC t theC

Dmberz= 1+2i, the

C

represeC t the two C D mbers izaC d iz. What do

yoD

C otice.

1

2

1

1

2

z1 2i is t

he poi

E

t A (1,2)iz=-2+i is t he poi

E

t A (-2,1)

-iz=2-i is t he poiE

t A (2,-1)

UsiE

g the slope, we fiE

d that OA OA

aE

d OA OA . This meaE

s that we rotate OA

aE

ticlock 90 . OE

the otherhaE

d OA OA

OA

!

B

!

r B

2OA ,This mea

F

s that we rotate OA

clockwise90

!

r

ModG lG s- ArgG meH t-TrigoH ometric Form

of a ComplexHG

mber

We kH ew that the complexH

G mberz = x + iy caH be represeH ted iH the arga

H d

plaH

e by the poiH

t A(x , y). This poiH

t caH

also fG lly determiH

ed as sooH

as we kH

ow

the distaH ce OA aH d the polar a

H gle betweeH OA aH d the (+)ve x-axis (Ur), meas G red

aH ticlockwise.

The distaH ce OA is called the modG lG s ofz, a

H d is deH oted by r

2 2r x y@ ! The aH gle U is called the ArgG meH t ofz, aH d is deH oted by Arg(z) where

ytaH

x!U

SiH

ce if we kH

ow taH

U, theH

U willhave maH

y valG es ( 2FU Ts ), it is agreed

that [0 ,2 [ aH

d is called the priH

cipal ArgG meH

t ofzU T .

the priH

cipal ArgG meH

t [0,2 [ U T@

From the figG re we fiH

d that

xcos x r cos

ry

siH y r si H

r

U U

U U

! !

! !

z r(cos i si H )U U!

A(1,2)

A1(-2,1)

(2,-1)A2

U

A

x

y

O

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The previo I s form is called the TrigoP

ometric Form of the complexP

I mber.

YoI

mI

stP otice that the previoI

s form is the priP cipal form, which mI

st be, wheP

weP

eed to fiP

d its modI lI s aP

d ArgI meP

t, aP

d yo I o I ght to modify aP

y other form

before yoI

determiP e the modI

lI

s aP d the Arg.

SpecialTrigoP

ometric forms

z 1 z cos 0 i si Q

0

z i z cos i si Q

2 2z 1 z cos i si

Q

3 3z i z cos i si

Q

2 2

T T

T T

T T

! R ! r r

! R !

! !

! !

EXAMPLE:

FiS

d the modT lT s aS

d the priS

cipal Arg. of the followiS

g complexS

T mbers

theS write the trigoS ometric form of eachST

mber.

( i )2 2i ; 3 3i ; 1 3i ; 3 i

( ii )5 ; 2i ; 4 ; 4i

SOLUTION:

2 2

2 2

2 2

( i ) z 2 2i r 2 2 2 2

2ta

U

1 ( x 0 , y 0 )2 4

z 2 2(cos i si U

)4 4

z 3 3i r 3 3 3 2

3 3ta

U

1 ( x 0, y 0 )

3 43 3z 3 2(cos i si

U

)4 4

z 1 3i r ( 1 ) ( 3 ) 2

3 4ta

U

3 ( x 0, y 0 )1 3

4 4z 2(cos i si

U

)3 3

z 3

TU U

T T

TU U

T