STUDY PACKAGE. Subject : Mathematics Topic : COMPLEX NUMBER Available Online :

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2 Complex Numbers. The complex number system There s no real number x whch satsfes the polynomal equaton x + 0. To permt solutons of ths and smlar equatons, the set of complex numbers s ntroduced. We can consder a complex number as havng the form a + b where a and b are real number and, whch s called the magnary unt, has the property that. It s denoted by.e. a + b. a s called as real part of whch s denoted by (Re ) and b s called as magnary part of whch s denoted by (Im ). Any complex number s : () Purely real, f b 0 ; () Purely magnary, f a 0 () Imagnary, f b 0. NOTE : (a) The set R of real numbers s a proper subset of the Complex Numbers. Hence the complete number system s N W I Q R C. (b) Zero s purely real as well as purely magnary but not magnary. (c) s called the magnary unt. Also ² ; ; 4 etc. (d) a b a b only f atleast one of a or b s non - negatve. (e) s a + b, then a b s called complex conjugate of and wrtten as a b Self Practce Problems. Wrte the followng as complex number () () x, (x > 0) () b + 4ac, (a, c> 0) Ans. () 0 + () x + 0 () b + 4ac. Wrte the followng as complex number. () x (x < 0) () roots of x ( cos)x + 0 Algebrac Operatons: Fundamental operatons wth complex numbers In performng operatons wth complex numbers we can proceed as n the algebra of real numbers, replacng by when t occurs.. Addton (a + b) + (c + d) a + b + c + d (a + c) + (b + d).. Subtracton Multplcaton (a + b) c + d) a + b c d (a c) + (b d) (a + b) (c + d) ac + ad + bc + bd (ac bd) + (ad+ bc) 4. Dvson a b c d a b. c d c b c d ac ad bc bd c d ac bd (bc ad) ac bd bc ad + c d c d c d Inequaltes n complex numbers are not defned. There s no valdty f we say that complex number s postve or negatve. e.g. > 0, 4 + < + 4 are meanngless. In real numbers f a + b 0 then a 0 b however n complex numbers, + 0 does not mply 0. Example : Fnd multplcatve nverse of +. Soluton Let be the multplcatve nverse of +. then. ( + ) Ans. Self Practce Problem. Smplfy n+00 + n+50 + n+48 + n+4, n. Ans. 0. Equalty In Complex Number: Two complex numbers a + b & a + b are equal f and only f ther real and magnary parts are equal respectvely.e. Re( ) Re( ) and m ( ) m ( ). Teko Classes, Maths : Suhag R. Karya (S. R. K. Sr), Bhopal Phone : , page of 8

3 Fnd the value of x and y for whch ( + ) x ( ) y x y + 5 where x, y R. Soluton ( + )x ( )y x y + 5 x y x y x x 0 x 0, and x + y 5 Soluton. f x 0,y 5 and f x, y x 0, y 5 and x, y 5 are two solutons of the gven equaton whch can also be represented as 0, & (, ) 5 0,, (, ) Ans. Fnd the value of expresson x 4 4x + x x + when x + s a factor of expresson. x + x (x ) x x + 0 Now x 4 4x + x x + (x x + ) (x x ) 4x + 7 when x +.e. x x + 0 x 4 4x + x x ( + ) Ans. Solve for f + 0 Soluton. Let x + y (x + y) + x y 0 x y + x y 0 and xy 0 x 0 or y 0 when x 0 y + y 0 y 0,, 0,, when y 0 x + x 0 x 0 0 Ans. 0,, Fnd square root of Soluton. Let (x + y) x y 9...() and xy 0...() squng () and addng wth 4 tmes the square of () we get x 4 + y 4 x y + 4x y (x + y ) 8 x + y 4 from () + () we get...() x 5 x ± 5 and y y ± 4 from equaton () we can see that x & y are of same sgn x + y +(5 + 4) or (5 + 4) sq. roots of a + 40 ± (5 + 4) Self Practce Problem Ans. ± (5 + 4). Solve for : Ans. ±, 0, 4. Representaton Of A Complex Number: (a) Cartesan Form (Geometrc Representaton) : Every complex number x + y can be represented by a pont on the Cartesan plane known as complex plane (Argand dagram) by the ordered par (x, y). Length OP s called modulus of the complex number whch s denoted by & s called the argument or ampltude. Teko Classes, Maths : Suhag R. Karya (S. R. K. Sr), Bhopal Phone : , page of 8 x y & tan y x (angle made by OP wth postve xaxs)

4 NOTE : () Argument of a complex number s a many valued functon. If s the argument of a complex number then n+ ; n I wll also be the argument of that complex number. Any two arguments of a complex number dffer by n () The unque value of such that < s called the prncpal value of the argument. Unless otherwse stated, amp mples prncpal value of the argument. () By specfyng the modulus & argument a complex number s defned completely. For the complex number the argument s not defned and ths s the only complex number whch s only gven by ts modulus. (b) Trgnometrc/Polar Representaton : r (cos + sn ) where r; arg ; r (cos sn ) NOTE : cos + sn s also wrtten as CS or e. e x e x Also cos x e x e x & sn x (c) Euler's Representaton : re ; r; arg ; re are known as Euler's denttes. (d) Vectoral Representaton : Every complex number can be consdered as f t s the poston vector of a pont. If the pont P represents the complex number then, OP & OP Express the complex number + n polar form. Soluton. + ( ) Arg tan tan (say) (cos + sn ) where tan Self Practce Problems. Fnd the prncpal argument and (9 ) Ans. 7 tan, 8 5. Fnd the and prncpal argument of the complex number (cos 0º sn 0 ) 5. Ans., 50 Modulus of a Complex Number : If a + b, then t's modulus s denoted and defned by a b. Infact s the dstance of from orgn. Hence s the dstance between the ponts represented by and. Propertes of modulus (). () () + + (v) (provded 0) (Equalty n () and (v) holds f and only f orgn, and are collnear wth and on the same sde of orgn). If 5 7 9, then fnd the greatest and least values of. Soluton. We have 9 (5 + 7) dstance between and Thus locus of s the crcle of radus 9 and centre at For such a (on the crcle), we have to fnd ts greatest and least dstance as from +, whch obvously 4 and 4. Fnd the mnmum value of + +. Soluton (trangle nequalty) + + mnmum value of ( + + ) Geometrcally whch represents sum of dstances of from and t can be seen easly that mnmu (PA + PB) AB Ans. / 4 e n 8 Teko Classes, Maths : Suhag R. Karya (S. R. K. Sr), Bhopal Phone : , page 4 of 8

5 Soluton. then fnd the maxmum and mnmum value of Let r Self Practce Problem r r + r r + r + r r R +...() and r r r r r (, )...() from () and () r (, ) Ans. r (, ). < and 4 > M then fnd the postve real value of M for whch these exst at least one complex number satsfy both the equaton. Ans. M (0, ). Agrument of a Complex Number : Argument of a non-ero complex number P() s denoted and defned by arg() angle whch OP makes wth the postve drecton of real axs. If OP r and arg(), then obvously r(cos + sn), called the polar form of. In what follows, 'argument of ' would mean prncpal argument of (.e. argument lyng n (, ] unless the context requres otherwse. Thus argument of a complex number a + b r(cos + sn) s the value of satsfyng rcos a and rsn b. Thus the argument of,, +,, tan or V th quadrant. b a, accordng as a + b les n,, Propertes of arguments () arg( ) arg( ) + arg( ) + m for some nteger m. () arg( / ) arg ( ) arg( ) + m for some nteger m. () arg ( ) arg() + m for some nteger m. (v) arg() 0 s real, for any complex number 0 (v) arg() ± / s purely magnary, for any complex number 0 (v) arg( ) angle of the lne segment PQ PQ, where P les on real axs, wth the real axs. Solve for, whch satsfy Arg ( ) and Arg ( 4). Soluton From the fgure, t s clear that there s no, whch satsfy both ray Sketch the regon gven by () Arg ( ) / () 5 & Arg ( ) >/ Teko Classes, Maths : Suhag R. Karya (S. R. K. Sr), Bhopal Phone : , page 5 of 8

6 FREE Download Study Package from webste: & Soluton () () Self Practce Problems. Sketch the regon gven by () Arg ( ) < /4 () Arg ( + ) /. Consder the regon 5 0. Fnd the pont n the regon whch has () max () mn () max arg () (v) mn arg () 7. Conjugate of a complex Number Conjugate of a complex number a + b s denoted and defned by a b. In a complex number f we replace by, we get conjugate of the complex number. s the mrror mage of about real axs on Argand's Plane. Propertes of conjugate () () () ( ) ( ) + ( ) (v) ( ) ( ) ( ) (v) ) ( (v) (v) + ( + ) ( ) () ( ( ) 0) (v) ( ) (x) If w f(), then w f( ) (x) arg() + arg( ) 0 If s purely magnary, then prove that Soluton. Re Hence proved Self Practce Problem. If s unmodulus and s not unmodulus then fnd. Ans. 8. Rotaton theorem () () 0 If P( ) and Q( ) are two complex numbers such that, then e where POQ If P( ), Q( ) and R( ) are three complex numbers and PQR, then Teko Classes, Maths : Suhag R. Karya (S. R. K. Sr), Bhopal Phone : , page of 8 e

7 () If P( ), Q( ), R( ) and S( 4 ) are four complex numbers and STQ, then FREE Download Study Package from webste: & If arg then nterrupter the locus. Soluton arg Soluton. Self Practce Problems 4 e arg Here arg represents the angle between lnes jonng and and +. As ths angle s constant, the locus of wll be a of a crcle segment. (angle n a segment s count). It can be seen that locus s not the complete sde as n the major are arg wll be equal to. Now try to geometrcally fnd out radus and centre of ths crcle. centre 0, Radus Ans. If A( + ) and B( + 4) are two vertces of a square ABCD (take n antclock wse order) then fnd C and D. Let affx of C and D are + 4 respectvely Consderng DAB 90º + AD AB ( ) we get 4 ( 4 ) ( ) e AD AB 4 ( + ) ( + ) Z ( 4) ( ) ( 4) and e CB AB + 4 ( + ) ( ) ,,, 4 are the vertces of a square taken n antclockwse order then prove that ( + ) + ( ) Ans. ( + ) + ( ). Check that and 4 are parallel or, not where, Ans. Hence, and 4 are not parallel.. P s a pont on the argand dagram on the crcle wth OP as dameter two pont Q and R are taken such that POQ QOR If O s the orgn and P, Q, R are represented by complex,, respectvely then show that cos cos Ans. cos 9. Demovre s Theorem: Case Statement : If n s any nteger then () (cos + sn ) n cos n + sn n () (cos + sn ) (cos ) + sn ) (cos + sn ) (cos + sn )...(cos n + sn n ) cos ( n ) + sn ( n ) Case Statement : If p, q Z and q 0 then k p (cos + sn ) p/q cos k p + sn q q where k 0,,,,..., q Teko Classes, Maths : Suhag R. Karya (S. R. K. Sr), Bhopal Phone : , page 7 of 8

8 NOTE : Contnued product of the roots of a complex quantty should be determned usng theory of equatons. 0. Cube Root Of Unty : () The cube roots of unty are,,. () If s one of the magnary cube roots of unty then + + ² 0. In general + r + r 0; where r I but s not the multple of. () In polar form the cube roots of unty are : cos 0 + sn 0; cos sn, cos + sn (v) The three cube roots of unty when plotted on the argand plane consttute the vertes of an equlateral trangle. (v) The followng factorsaton should be remembered : (a, b, c R & s the cube root of unty) a b (a b) (a b) (a ²b) ; x + x + (x ) (x ) ; a + b (a + b) (a + b) (a + b) ; a + ab + b (a bw) (a bw ) a + b + c abc (a + b + c) (a + b + ²c) (a + ²b + c) Fnd the value of Soluton Ans. If,, are cube roots of unty prove () ( + ) ( + ) 4 () ( + ) 5 + ( + ) 5 () ( ) ( ) ( 4 ) ( 8 ) 9 (v) ( + ) ( + 4 ) ( )... to n factors n Soluton. () ( + ) ( + ) ( ) ( ) 4 Self Practce Problem 0 r. Fnd ( r0 Ans.. n th Roots of Unty : r ) If,,,.... n are the n, n th root of unty then : () They are n G.P. wth common rato e (/n) & () p + p + p n p 0 f p s not an ntegral multple of n n f p s an ntegral multple of n () ( ) ( )... ( n ) n & ( + ) ( + )... ( + n ) 0 f n s even and f n s odd. (v) n or accordng as n s odd or even. Fnd the roots of the equaton where real part s postve. Soluton. 4. e + (n + ) x (n ) e 5 e, e, e, e roots wth +ve real part are e e Ans. 7 e + e, e, e Teko Classes, Maths : Suhag R. Karya (S. R. K. Sr), Bhopal Phone : , page 8 of 8

9 k k Fnd the value sn cos 7 Soluton. k k 7 k k sn 7 cos 7 k k k sn 7 cos 7 + k0 Self Practce Problems k0 k Ans. k0 (Sum of magnary part of seven seventh roots of unty) (Sum of real part of seven seventh roots of unty) +. Resolve 7 nto lnear and quadratc factor wth real coeffcent. Ans. ( ) 4 cos. cos. cos Fnd the value of cos + cos + cos Ans.. The Sum Of The Followng Seres Should Be Remembered : sn n / () cos + cos + cos cos n sn / sn n / () sn + sn + sn sn n sn / sn NOTE : If (/n) then the sum of the above seres vanshes.. Logarthm Of A Complex Quantty : () Log e (+ ) Log e (² + ²) + () n represents a set of postve real numbers gven by e Fnd the value of cos n n tan where n. n, n. () log ( + ) Ans. log + (n + ) () log( ) Ans. () Ans. cos(ln) + sn(ln) e (ln) (4n). (v) Ans. e (8n). (v) ( + ) Ans. e 4 (v) arg (( + ) ) Ans. n(). n Soluton. () log ( + ) log e log + n () e n cos (n ) cos (n ) + sn (n ) ] Teko Classes, Maths : Suhag R. Karya (S. R. K. Sr), Bhopal Phone : , page 9 of 8

10 Self Practce Problem FREE Download Study Package from webste: & Fnd the real part of cos ( + ) Ans. e e 4. Geometrcal Propertes : Dstance formula : If and are affxes of the two ponts P and Q respectvely then dstance between P + Q s gven by. Secton formula If and are affxes of the two ponts P and Q respectvely and pont C devdes the lne jonng P and Q nternally n the rato m : n then affx of C s gven by m n m n If C devdes PQ n the rato m : n externally then m n m n (b) If a, b, c are three real numbers such that a + b + c 0 ; where a + b + c 0 and a,b,c are not all smultaneously ero, then the complex numbers, & are collnear. () If the vertces A, B, C of a represent the complex nos.,, respectvely and a, b, c are the length of sdes then, () Centrod of the ABC : () Orthocentre of the ABC asec A b sec B csecc tan A tanb tan C or aseca bsec B csecc tana tanb tanc () Incentre of the ABC (a + b + c ) (a + b + c). (v) Crcumcentre of the ABC : (Z sn A + Z sn B + Z sn C) (sn A + sn B + sn C). () amp() s a ray emanatng from the orgn nclned at an angle to the x axs. () a b s the perpendcular bsector of the lne jonng a to b. (4) The equaton of a lne jonng & s gven by, + t ( ) where t s a real parameter. (5) ( + t) where t s a real parameter s a lne through the pont & perpendcular to the lne jonng to the orgn. () The equaton of a lne passng through & can be expressed n the determnant form as 0. Ths s also the condton for three complex numbers to be collnear. The above equaton on manpulatng, takes the form r 0 where r s real and s a non ero complex constant. NOTE : If we replace by e and by e then we get equaton of a straght lne whch. Passes through the foot of the perpendcular from orgn to gven straght lne and makes an angle wth the gven straghtl lne. (7) The equaton of crcle havng centre 0 & radus s : 0 or ² 0 whch s of the form + k 0, k s real. Centre s & radus k. Crcle wll be real f k 0.. (8) The equaton of the crcle descrbed on the lne segment jonng & as dameter s arg ± or ( ) ( ) + ( ) ( ) 0. (9) Condton for four gven ponts,, & 4 to be concyclc s the number 4. should be real. Hence the equaton of a crcle through non collnear 4 Teko Classes, Maths : Suhag R. Karya (S. R. K. Sr), Bhopal Phone : , page 0 of 8

11 ponts, & can be taken as s real. (0) Arg represent () a lne segment f () Par of ray f 0 () a part of crcle, f 0 < < () Area of trangle formed by the ponts, & s () Perpendcular dstance of a pont 0 from the lne r 0 s () () Complex slope of a lne r 0 s () Complex slope of a lne jonng by the ponts & s () Complex slope of a lne makng angle wth real axs e (4) & are the compelx slopes of two lnes. () If lnes are parallel then () If lnes are perpendcular then + 0 (5) If + K > then locus of s an ellpse whose foc are & () If 0 (7) If r lne r 0 r then locus of s parabola whose focus s 0 and drectrx s the k, 0, then locus of s crcle. (8) If K < then locus of s a hyperbola, whose foc are &. Match the followng columns : Column - Column - () If + + 0, () crcle then locus of represents... () If arg, 4 () Straght lne then locus of represents... () f () Ellpse then locus of represents... (v) 4 If arg 5 5, (v) Hyperbola then locus of represents... (v) If (v) Major Arc then locus of represents... (v) + + (v) Mnor arc then locus of represents... (v) 5 (v) Perpendcular bsector of a lne segment 5 (v) arg (v) Lne segment Ans. () () () (v) (v) (v) (v) (v) (v) (v) (v) (v) () (v) () () Teko Classes, Maths : Suhag R. Karya (S. R. K. Sr), Bhopal Phone : , page of 8

12 5. (a) Reflecton ponts for a straght lne : Two gven ponts P & Q are the reflecton ponts for a gven straght lne f the gven lne s the rght bsector of the segment PQ. Note that the two ponts denoted by the complex numbers & wll be the reflecton ponts for the straght lne r 0 f and only f; r, where r s real and s non ero complex constant. 0 (b) Inverse ponts w.r.t. a crcle : Two ponts P & Q are sad to be nverse w.r.t. a crcle wth centre 'O' and radus, f: () the pont O, P, Q are collnear and P, Q are on the same sde of O. () OP. OQ. Note : that the two ponts & wll be the nverse ponts w.r.t. the crcle r 0 f and only f r. 0. Ptolemy s Theorem: It states that the product of the lengths of the dagonals of a convex quadrlateral nscrbed n a crcle s equal to the sum of the products of lengths of the two pars of ts opposte sdes..e If cos + cos + cos 0 and also sn + sn + sn 0, then prove that () cos + cos + cos sn + sn + sn 0 () sn + sn + sn sn () () cos + cos + cos cos () Soluton. Let cos + sn, cos + sn, cos + sn + + (cos + cos + cos ) + (sn + sn + sn ) () () Also (cos + sn ) cos sn cos sn, cos sn + (cos + cos + cos ) (sn + sn + sn ) () Now + + ( + + ) ( + + ) , usng () and () or (cos + sn ) + (cos + sn ) + (cos + sn ) 0 or cos + sn ) + cos + sn + cos + sn Equaton real and magnary parts on both sdes, cos + cos + cos 0 and sn + sn + sn 0 () + + ( + ) ( + ) + ( ) ( ) +, usng () (cos + sn ) + (cos + sn ) + (cos + sn ) (cos + sn ) (cos + sn ) (cos + sn ) or cos + sn + cos + sn + cos + sn {cos( + + ) + sn ( + + ) Equaton magnary parts on both sdes, sn + sn + sn sn ( + + ) Alternatve method Let C cos + cos + cos 0 S sn + sn + sn 0 C + S e + e + e 0 () C S e + e + e 0 () From () (e ) + (e ) + (e ) (e ) (e ) + (e ) (e ) + (e ) (e ) e + e + e e e e (e + e + e ) e () + e + e 0 (from ) Comparng the real and magnary parts we cos + cos + cos sn + sn + sn 0 Also from () (e ) + (e ) + (e ) e e e e + e + e e () Comparng the real and magnary parts we obtan the results. Teko Classes, Maths : Suhag R. Karya (S. R. K. Sr), Bhopal Phone : , page of 8 If and are two complex numbers and c > 0, then prove that

13 + (I + C) + (I +C ) Soluton. We have to prove : + ( + c) + ( + c ).e ( + c) + ( +c ) or + c + c or c + c 0 (usng Re ( ) ) or c c 0 whch s always true. If, [/, /],,,, 4, 5, and 4 cos + cos + cos. + cos 4 + cos 5, Soluton. Soluton. then show that > 4 Gven that cos. 4 + cos. + cos. + cos 4. + cos 5 or cos. 4 + cos. + cos. + cos 4. + cos 5 cos. 4 + cos. + cos. + cos 4. + cos 5 [/, /] cos < < e < 4 > > 4 + Two dfferent non parallel lnes cut the crcle r n pont a, b, c, d respectvely. Prove that a b c d these lnes meet n the pont gven by a b c d Snce pont P, A, B are collnear a a b b Smlarlym, snce ponts P, C, D are collnear b 0 b a (a b) + b ab a 0 () a (c d) c d (a b) c d cd (a b) b ab k k k r k (say) a, a b, b c etc. c From equaton () we get a (c d) () k k k k ck kd ak bk (c d) (a b) (a b) (c d) a b c d d c b a a b c d a b c d Teko Classes, Maths : Suhag R. Karya (S. R. K. Sr), Bhopal Phone : , page of 8

14 Short Revson. DEFINITION : Complex numbers are defnted as expressons of the form a + b where a, b R &. It s denoted by.e. a + b. a s called as real part of (Re ) and b s called as magnary part of (Im ). EVERY COMPLEX NUMBER CAN BE REGARDED AS Note : (a) (b) (c) Purely real Purely magnary Imagnary f b 0 f a 0 f b 0 The set R of real numbers s a proper subset of the Complex Numbers. Hence the Complete Number system s N W I Q R C. Zero s both purely real as well as purely magnary but not magnary. s called the magnary unt. Also ² l ; ; 4 etc. (d) a b a b only f atleast one of ether a or b s non-negatve.. CONJUGATE COMPLEX : If a + b then ts conjugate complex s obtaned by changng the sgn of ts magnary part & s denoted by..e. a b. Note that : () + Re() () Im() () a² + b² whch s real (v) If les n the st quadrant then les n the 4 th quadrant and les n the nd quadrant.. ALGEBRAIC OPERATIONS : The algebrac operatons on complex numbers are smlar to those on real numbers treatng as a polynomal. Inequaltes n complex numbers are not defned. There s no valdty f we say that complex number s postve or negatve. e.g. > 0, 4 + < + 4 are meanngless. However n real numbers f a + b 0 then a 0 b but n complex numbers, + 0 does not mply EQUALITY IN COMPLEX NUMBER : Two complex numbers a + b & a + b are equal f and only f ther real & magnary parts concde. 5. REPRESENTATION OF A COMPLEX NUMBER IN VARIOUS FORMS : (a) Cartesan Form (Geometrc Representaton) : Every complex number x + y can be represented by a pont on the cartesan plane known as complex plane (Argand dagram) by the ordered par (x, y). length OP s called modulus of the complex number denoted by & s called the argument or ampltude. eg. x y & tan y x (angle made by OP wth postve xaxs) f 0 NOTE :() s always non negatve. Unlke real numbers s not correct f 0 () Argument of a complex number s a many valued functon. If s the argument of a complex number then n+ ; n I wll also be the argument of that complex number. Any two arguments of a complex number dffer by n. () The unque value of such that < s called the prncpal value of the argument. (v) Unless otherwse stated, amp mples prncpal value of the argument. (v) By specfyng the modulus & argument a complex number s defned completely. For the complex number the argument s not defned and ths s the only complex number whch s gven by ts modulus. (v) There exsts a one-one correspondence between the ponts of the plane and the members of the set of complex numbers. Teko Classes, Maths : Suhag R. Karya (S. R. K. Sr), Bhopal Phone : , page 4 of 8

15 (b) Trgnometrc / Polar Representaton : r (cos + sn ) where r ; arg ; r (cos sn ) Note: cos + sn s also wrtten as CS. x x x x e e e e Also cos x & sn x are known as Euler's denttes. (c) Exponental Representaton : re ; r ; arg ; re. IMPORTANT PROPERTIES OF CONJUGATE / MODULI / AMPLITUDE : If,, C then ; (a) + Re () ; Im () ; ( ) ; + ; ;. ; 0 (b) 0 ; Re () ; Im () ; ;. ;, 0, n n ; + + ] [ ; + + [ TRIANGLE INEQUALITY ] (c) () amp (. ) amp + amp + k. k I () amp amp amp + k; k I () amp( n ) n amp() + k. where proper value of k must be chosen so that RHS les n (, ]. (7) VECTORIAL REPRESENTATION OF A COMPLEX : Every complex number can be consdered as f t s the poston vector of that pont. If the pont P represents the complex number then, OP & OP NOTE : () If OP r e then OQ r e ( + ). e. If OP and OQ are () () of unequal magntude then OQ OP e If A, B, C & D are four ponts representng the complex numbers,, & 4 then 4 4 AB CD f s purely real ; AB CD f s purely magnary ] If,, are the vertces of an equlateral trangle where 0 s ts crcumcentre then (a) (b) DEMOIVRE S THEOREM : Statement : cos n + sn n s the value or one of the values of (cos + sn ) n n Q. The theorem s very useful n determnng the roots of any complex quantty Note : Contnued product of the roots of a complex quantty should be determned usng theory of equatons. 9. CUBE ROOT OF UNITY :() The cube roots of unty are,,. () If w s one of the magnary cube roots of unty then + w + w² 0. In general + w r + w r 0 ; where r I but s not the multple of. () In polar form the cube roots of unty are : 4 4 cos 0 + sn 0 ; cos + sn, cos + sn (v) The three cube roots of unty when plotted on the argand plane consttute the vertes of an equlateral trangle. (v) The followng factorsaton should be remembered : (a, b, c R & s the cube root of unty) Teko Classes, Maths : Suhag R. Karya (S. R. K. Sr), Bhopal Phone : , page 5 of 8