COMPLEX NUMBERS AND QUADRATIC EQUATIONS

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1 COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x etc are not solvable n real number system Thus, there s a need to extend the real number system to a larger system so that we can have solutons of such equatons In fact, our man objectve s to solve the quadratc equaton ax + bx + c 0, where a, b, c R and the dscrmnant b 4 ac < 0, whch s not possble n real number system In ths chapter, we shall extend the real number system to a larger system called complex number system so that the solutons of quadratc equatons ax + bx + c 0, where a, b, c are real numbers are possble We shall also solve quadratc equatons wth complex coeffcents COMPLEX NUMBERS We know that the equaton x + 0 s not solvable n the real number system e t has no real roots Many mathematcans ndcated the square roots of negatve numbers, but Euler was the frst to ntroduce the symbol (read ota ) to represent, and he defned If follows that s a soluton of the equaton x + 0 Also ( ) Thus the equaton x + 0 has two solutons, x ±, where The number s called an magnary number In general, the square roots of all negatve real numbers are called magnary numbers Thus,, 9 4 etc are all magnary numbers Complex number A number of the form a + b, where a and b are real numbers, s called a complex number For example, +, +, +, 7 + are all complex numbers The system of numbers C { ; a + b ; a, b R} s called the set of complex numbers Standard form of a complex number If a complex number s expressed n the form a + b where a, b R and n the standard form, then t s sad to be For example, the complex numbers +, +, 7 are all n the standard form Real and magnary parts of a complex number If a + b (a, b R) s a complex number, then a s called the real part, denoted by Re () and b s called magnary part, denoted by Im ()

2 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 9 For example : () If +, then Re () and Im () () If +, then Re () and Im () () If 7, then 7 + 0, so that Re () 7 and Im () 0 (v) If, then 0 + ( ), so that Re () 0 and Im () Note that magnary part of a complex number s a real number In a + b (a, b R), f b 0 then a, whch s a real number If a 0 and b 0, then b, whch s called purely magnary number If b 0, then a + b s non-real complex number Snce every real number a can be wrtten as a + 0, we see that R C e the set of real numbers R s a proper subset of C, the set of complex numbers Note that, 0,, π are real numbers ; +, etc are non-real complex numbers ;, etc are purely magnary numbers Equalty of two complex numbers Two complex numbers a + b and c + d are called equal, wrtten as, f and only f a c and b d For example, f the complex numbers a + b and + are equal, then a and b Algebra of complex numbers In ths secton, we shall defne the usual mathematcal operatons addton, subtracton, multplcaton, dvson, square, power etc on complex numbers and wll develop the algebra of complex numbers Addton of two complex numbers Let a + b and c + d be any two complex numbers, then ther sum + s defned as + (a + c) + (b + d) For example, let + and + 4, then + ( + ( )) + ( + 4) + 7 Propertes of addton of complex numbers () Closure property The sum of two complex numbers s a complex number e f and are any two complex numbers, then + s always a complex number () Addton of complex numbers s commutatve If and are any two complex numbers, then + + () Addton of complex numbers s assocatve If, and are any three complex numbers, then ( + ) + + ( + ) (v) The exstence of addtve dentty Let x + y, x, y R, be any complex number, then (x + y) + (0 + 0) (x + 0) + (y + 0) x + y and (0 + 0) + (x + y) (0 + x) + (0 + y) x + y (x + y) + (0 + 0) x + y (0 + 0) + (x + y) Therefore, acts as the addtve dentty It s smply wrtten as 0 Thus, for all complex numbers

3 0 MATHEMATICS XI (v) The exstence of addtve nverse For a complex number a + b, ts negatve s defned as ( a) + ( b) a b Note that + ( ) (a a) + (b b) Thus acts as addtve nverse of Subtracton of complex numbers Let a + b and c + d be any two complex numbers, then the subtracton of from s defned as + ( ) (a + b) + ( c d) (a c) + (b d) For example, let + and + 4, then ( + ) ( + 4) ( + ) + ( 4) ( + ) + ( 4) and ( + 4) ( + ) ( + 4) + ( ) ( ) + (4 ) + Multplcaton of two complex numbers Let a + b and c + d be any two complex numbers, then ther product s defned as (ac bd) + (ad + bc) Note that ntutvely, (a + b) (c + d) ac + bc + ad + bd ; now put, thus (a + b) (c + d) ac + (bc + ad) bd (ac bd) + (ad + bc) For example, let + 7 and +, then ( + 7) ( + ) ( ( ) 7 ) + ( + 7 ( )) 4 + Propertes of multplcaton of complex numbers () Closure property The product of two complex numbers s a complex number e f and are any two complex numbers, then s always a complex number () Multplcaton of complex numbers s commutatve If and are any two complex numbers, then () Multplcaton of complex numbers s assocatve If, and are any three complex numbers, then ( ) ( ) (v) The exstence of multplcatve dentty Let x + y, x, y R, be any complex number, then (x + y) ( + 0) (x y0) + ( x0 + y) x + y and ( + 0) (x + y) (x 0y) + (y + 0x) x + y (x + y) ( + 0) x + y ( + 0) (x + y) Therefore, + 0 acts as the multplcatve dentty It s smply wrtten as Thus for all complex numbers (v) Exstence of multplcatve nverse For every non-ero complex number a + b, we have the complex number

4 COMPLEX NUMBERS AND QUADRATIC EQUATIONS a a + b b a + b (denoted by or ) such that s called the multplcatve nverse of Note that ntutvely, a b a b a+ b a + b a b a + b a a + b b a + b (check t) (v) Multplcaton of complex numbers s dstrbutve over addton of complex numbers If, and are any three complex numbers, then ( + ) + and ( + ) + These results are known as dstrbutve laws Dvson of complex numbers Dvson of a complex number a + b by c + d 0 s defned as c d ac bd bc ad + ( a + b) + c + d c + d c + d c + d Note that ntutvely, a + b a + b c d ( ac + bd) + ( bc ad) c + d c + d c d c + d For example, f + 4 and 6, then ( ) + ( 6+ 4 ) Integral powers of a complex number If s any complex number, then postve ntegral powers of are defned as,,, 4 and so on If s any non-ero complex number, then negatve ntegral powers of are defned as :,, etc If 0, then 0 Powers of Integral power of are defned as : 0,,, ( ), 4 ( ) ( ), 4, 6 4 ( ), and so on

5 MATHEMATICS XI Remember that, 4 4, and so on 4 Note that 4 and 4 It follows that for any nteger k, 4k, 4k+, 4k+, 4k+ Also, we note that and ( ) Therefore, and are both square roots of However, by the symbol, we shall mean only e We observe that and are both the solutons of the equaton x + 0 Smlarly, and ( ) ( ) ( ) ( ) ( ), ( ) Therefore, and are both square roots of However, by the symbol, we shall mean only e In general, f a s any postve real number, then a a We already know that a b ab for all postve real numbers a and b Ths result s also true when ether a 0, b < 0 or a < 0, b 0 But what f a < 0, b < 0? Let us examne : we note that ( )( ) (by assumng a b ab for all real numbers) Thus, we get whch s contrary to the fact that Therefore, a b ab s not true when a and b are both negatve real numbers Further, f any of a and b s ero, then a b ab 0 Identtes If and are any two complex numbers, then the followng results hold : () ( + ) + + () ( ) + () ( + ) ( ) (v) ( + ) (v) ( ) + Proof () ( + ) ( + ) ( + ) ( + ) + ( + ) (Dstrbutve law) (Dstrbutve law) (Commutatve law) + + We leave the proofs of the other results for the reader

6 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 4 Modulus of a complex number Modulus of a complex number a + b, denoted by mod() or, s defned as a + b, where a Re (), b Im () Sometmes, s called absolute value of Note that 0 For example : () If +, then ( ) + 4 () If 7, then + ( 7 ) Propertes of modulus of a complex number If, and are complex numbers, then () () 0 f and only f 0 () (v), provded 0 Proof () Let a + b, where a, b R, then a b ( a) + ( b) a + b () Let a + b, then a + b () Let Now 0 ff a + b 0 e ff a + b 0 e ff a 0 and b 0 e ff a 0 and b 0 e ff e ff 0 a + b, and c + d, then (ac bd) + (ad + bc) ( ac bd) + ( ad + bc) a c + b d abcd + a d + b c + abcd ( a + b )( c + d ) a + b c + d ( Q a + b 0, c + d 0) (v) Here 0 0 Let (usng part ()) Q REMARK From (), on replacng both and by, we get Smlary, e etc

7 4 MATHEMATICS XI Conjugate of a complex number Conjugate of a complex number a + b, denoted by, s defned as a b e a+ b a b For example : () +, + () 7 + 7, Propertes of conjugate of a complex number If, and are complex numbers, then () ( ) () + + () (v) (v), provded 0 (v) (v) (v) Proof () Let a + b, where a, b R, so that a b ( ) a b a + b () Let a + b and c + d, then + ( a + b) + ( c + d) ( a + c) + ( b + d) (a + c) (b + d) (a b) + (c d) + () Let a + b and c + d, then ( a+ b) ( c + d) ( a c) + ( b d) (a c) (b d) (a b) (c d), provded 0 (v) Let a + b and c + d, then ( a + b)( c + d) ( ac bd) + ( ad + bc) (ac bd) (ad + bc) Also (a b) (c d) (ac bd) (ad + bc) Hence (v) Here 0 0 Let (usng part (v)) Q (v) Let a + b, then a b a + ( b) a + b

8 COMPLEX NUMBERS AND QUADRATIC EQUATIONS (v) Let a + b, then a b (a + b) (a b) (aa b( b)) + (a( b) + ba) (Def of multplcaton) (a + b ) + 0 a + b ( a + b ) Remember that (a + b) (a b) a + b (v) Let a + b 0, then 0 (a + b) (a b) a + b Thus,, provded 0 REMARK From (v), on replacng both and by, we get e ( ) Smlarly, ( ) ( ) ( ) ( ) ( ) etc NOTE The order relatons greater than and less than are not defned for complex numbers e the nequaltes + +, 4, + < etc are meanngless ILLUSTRATIVE EXAMPLES Example A student says ( ) ( ) Thus Where s the fault? Soluton ( ) ( ) s true, but ( )( ) s wrong Because f both a, b are negatve real numbers, then a b ab s not true Example If 7 + 9, fnd Re(), Im (), and Soluton Gven Re () 7 and Im() ( 7) + ( 9) Example If 4x + (x y) 6 and x, y are real numbers, then fnd the values of x and y Soluton Gven 4x + (x y) 6 4x + (x y) + ( 6)

9 6 MATHEMATICS XI Equatng real and magnary parts on both sdes, we get 4x and x y 6 x 4 and y 6 4 x 4 and y Hence x 4 and y 4 Example 4 For what real values of x and y are the followng numbers equal () ( + ) y + (6 + ) and ( + ) x () x 7 x + 9 y and y + 0? Soluton () Gven ( + ) y + (6 + ) ( + ) x ( y + 6) + (y + ) x + x y + 6 x and y + x x and y 4 x and y ± Hence, the requred values of x and y are x, y ; x, y () Gven x 7 x + 9 y y + 0 (x 7 x) + (9 y) ( ) + ( y + 0) x 7 x and 9 y y + 0 x 7 x + 0 and y 9 y (x 4) (x ) 0 and (y ) (y 4) 0 x 4, and y, 4 Hence, the requred values of x and y are x 4, y ; x 4, y 4; x, y ; x, y 4 Example Express each of the followng n the standard form a + b : 7 4 () () (7 + 7) + (7 + 7) () ( + ) ( + Soluton () ) (v) 7 + () (7 + 7) + (7 + 7) ( + ) + (7 + 7 ) ( ) ( 7) + ( + 7) () ( + )( + ) ( + ) ( + ) (6 ) + ( 4 ) 0 7 ( + ) ( ) ( + ) ( )

10 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 7 (v) ( + ) ( ) ( + ) ( ) ( ) ( ) + + ((a + b) (a b) a + b ) ( ) Q 0 7 Example 6 Express the followng n the form a + b : () ( ) () 8 () 0 () 9 (v) ( ) (v) (v) Soluton () ( ) () 8 ( ) () ( Q 4k +, k I ) + 0 () 9 4 ( 0) + ( Q 4k +, k I) 0 + (v) ( ) ( ) ( ) ( Q 4k +, k I) ( ) 0 + (v) ( ) (v) ( 9) ( ) Example 7 Express each of the followng n the standard form a + b : () ( ) 4 () () ( ) + ( ) (v) (v) ( + ) 6 + ( ) Soluton () ( ) 4 (( ) ) ( + ) ( + ( ) ) ( ) 4 4 ( ) (NCERT Examplar Problems)

11 8 MATHEMATICS XI () ( ) ( ) ( ) () ( ) + ( ) ( + ) + ( ) (v) 8 + ( ) + ( ) ( ) ( ) Q ( ) + ( ) ( ) ( ) ( ) ( + ) [ ] [ ] + ( + ) (v) ( + ) 6 (( + ) ) ( + + ) ( + ) () 8 8 ( ) 8 and ( ) + ( ) + ( ) ( + ) 6 + ( ) 8 + ( ) 0 Example 8 Fnd the multplcatve nverse of + Soluton Let +, then and ( ) We know that the multplcatve nverse of s gven by Alternatvely + + ( ) ( ) 9 ( ) Example 9 Express the followng n the form a + b : () () + + () 4 4 (v) + + ( + )( + ) ( )( ) +

12 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 9 Soluton () + + ( ) ( ) + + () ( ) + 6 ( ) () ( 4) ( + ) ( + 9 ) ( 4 ) ( ) ( + ) ( 8) ( 0) ( ) ( ) (v) ( + )( + ) ( )( ) ( + )( + )( + ) ( )( )( ) + ( )( + ) ( + )( ) ( )( 8 6 ) 9 ( + 4 ) ( 4 ) Example 0 () If ( + ) x + y, then fnd the value of x + y (NCERT Examplar Problems) () If + x + y, then fnd (x, y) (NCERT Examplar Problems) + () If + 00 a + b, then fnd (a, b) (NCERT Examplar Problems) Soluton () x + y ( + ) ( ) + 4 ( ) x and y 4 4 x + y + 4 () We have, ( + ) ( ) + Q

13 40 MATHEMATICS XI Gven x + y + + ( ) + ( ) 0 x 0 and y Hence, the orders par (x, y) s (0, ) () Gven a + b + 00 ( ) 00 () a and b 0 Hence, the ordered par (a, b) s (, 0) ( ) 00 (see part ()) Example () If ( + ) ( ), then show that (NCERT Examplar Problems) () If and +, fnd Re Soluton () Gven ( + ) ( ) ( ) ( ) () Re ( )( ) ( ) ( ) 4 ( ) + + Example () Fnd the conjugate of ( ) ( + ) ( + ) ( ) () Fnd the modulus of + + () Fnd the modulus of ( ) + Soluton () Let ( ) ( + ) ( + ) ( ) ( ) ( ) Conjugate of 6 6 +

14 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 49 Example If α and β are dfferent complex numbers wth β, then fnd β α αβ Soluton We have β α ( β α) β (Note ths step) αβ ( αβ) β ( β α ) β β αββ ( β α ) β β α (Gven β β ββ ) Hence, ( β α ) β β α Q β α β β α ( Q ) β α β β α ( Q ) β ( β, gven) β α αβ Example If n, prove that n (NCERT Examplar Problems) n Soluton Gven n n,,, n n,,, n n n n ( Q ) n n EXERCISE Very short answer type questons ( to ) : Evaluate the followng : () 9 4 () ( 9) ( 4 ) () 6 (v) 6 (v) If, fnd Re (), Im(), and If, then s t true that ± ( )? 4 If + 4, then s t true that ± ( + )? If, then show that (x + + ) (x + ) x + x + 6 Fnd real values of x and y f () y + (x y) () (x ) + ( + y) () (y ) + (7 x) 0

15 0 MATHEMATICS XI 7 If x, y R and ( y ) + ( x y) 7, fnd the values of x and y 8 If x, y are reals and ( y + ) + (x + y) 0, fnd the values of x and y 9 If x, y are reals and ( ) x + ( + ) y, fnd the values of x and y 0 For any two complex numbers and, prove that Re ( ) Re ( ) Re( ) Im ( ) Im ( ) For any complex number, prove that Re() + and Im() If s a complex number, show that s real Express the followng ( to 0) complex numbers n the standard form a + b : () ( ) 8 () 4 () ( ) ( + 6 ) () ( ) + () ( + ) + + ( ) () (7 + ) (7 ) 6 () ( + ) ( ) () ( + ) 7 () () ( ) 8 () + () ( + 7) ( 7) 9 () 99 () 0 () ( 4 ) () Fnd the value of ( + ) + ( ) If n s any nteger, then fnd the value of () ( ) 4n + () 4n+ 4n (NCERT Examplar Problems) Fnd the multplcatve nverse of 4 Express the followng numbers n the form a + b, a, b R : () () + + If (a + b) (c + d) A + B, then show that (a + b ) (c + d ) A + B 6 Fnd the modulus of the followng complex numbers : () ( 4 ) ( + ) () Fnd the modulus of the followng : () ( ) 4 + () ( 7 ) 8 () If 7, then fnd () If x + y, x, y R, then fnd 9 Fnd the conjugate of 7 0 Wrte the conjugate of ( + ) ( ) n the form a + b, a, b R Solve for x : + x Express the followng ( and ) complex numbers n the standard form a + b : () ()

16 MATHEMATICS XI 44 If a + b ( x + ) x +, then prove that a + b ( x + ) ( x + ) 4 If +, then fnd the value of Show that f, then s a real number + 47 () If x + y and, show that (x + y ) 0x + 0 (NCERT Examplar Problems) () If x + y and + 4, show that x + y x y 7 n + 48 Fnd the least postve ntegral value of n for whch s a real number 49 Fnd the real value of θ such that + cos θ s a real number cos θ 0 If s a complex number such that, prove that ( ) s purely magnary + What s the excepton? (NCERT Examplar Problems) If, and are complex numbers such that + +, then fnd the value of + + (NCERT Examplar Problems) ARGAND PLANE We know that correspondng to every real number there exsts a unque pont on the number lne (called real axs) and conversely correspondng to every pont on the lne there exsts a unque real number e there s a one-one correspondence between the set R of real numbers and the ponts on the real axs In a smlar way, correspondng to every ordered par (x, y) of real numbers there exsts a unque pont P n the coordnate plane wth x as abscssa and y as ordnate of the pont P and conversely correspondng to every pont P n the plane there exsts a unque ordered par of real numbers Thus, there s a one-one correspondence between the set of ordered pars {(x, y) ; x, y R} and the ponts n the coordnate plane The pont P wth co-ordnates (x, y) s sad to represent the complex number x + y It follows that the complex number x + y can be unquely represented by the pont P(x, y) n the co-ordnate plane and conversely correspondng to the pont P(x, y) n the plane there exsts a unque complex number x + y The co-ordnate plane that represents the complex numbers s called the complex plane or Argand plane The complex numbers,,,, +,, + and whch correspond to the ordered pars (, 0), (, 0), (0, ), (0, ), (, ), (, ), (, ) and (, ) respectvely have been represented geometrcally n the coordnate plane by the ponts A, B, C, D, E, F, G and H respectvely n fg X X O < + O B (, 0) x Fg G (, ) < C (0, ) y D (0, ) H (, ) Fg x + y P(x, y) + E (, ) A X (, 0) F (, ) X

17 COMPLEX NUMBERS AND QUADRATIC EQUATIONS Note that every real number x x + 0 s represented by pont (x, 0) lyng on x-axs, and every purely magnary number y s represented by pont (0, y) lyng on y-axs Consequently, x-axs s called the real axs and y-axs s called the magnary axs If the pont P(x, y) represents the complex number x + y, then the dstance between the ponts P and the orgn O (0, 0) x + y Thus, the modulus of e s the dstance between ponts P and O (see fg ) X O < x + y P(x, y) < y x Fg X Geometrc representaton of, and If x + y, x, y R s represented by the pont P(x, y) n the complex plane, then the complex numbers,, are represented by the ponts P ( x, y), Q(x, y) and Q ( x, y) respectvely n the complex plane (see fg 4) X Q ( x, y) O P(x, y) X Geometrcally, the pont Q (x, y) s the mrror mage of the pont P (x, y) n the real axs Thus, conjugate of e s the mrror mage of n the x-axs P ( x, y) Fg 4 Q(x, y) POLAR REPRESENTATION OF COMPLEX NUMBERS Let the pont P (x, y) represent the non-ero complex number x + y n the Argand plane Let the drected lne segment OP be of length r ( 0) and θ be the radan measure of the angle whch OP makes wth the postve drecton of x-axs (shown n fg ) Then x + y P(x, y) r x + y and s called modulus of ; and θ s called ampltude or argument of and s wrtten as amp() or arg() From fgure, we see that x r cos θ and y r sn θ x + y r cos θ + r sn θ r (cos θ + sn θ) r θ O < x M Fg Thus, r (cos θ + sn θ ) Ths form of s called polar form of the complex number X < y < X P P r r X O < θ X X < O θ X () ()

18 4 MATHEMATICS XI X r < O θ X X O < θ r X P P () (v) Fg 6 For any non-ero complex number, there corresponds only one value of θ n π < θ π (see fg 6) The unque value of θ such that π < θ π s called prncpal value of ampltude or argument Thus every (non-ero) complex number x + y can be unquely expressed as r (cos θ + sn θ ) where r 0 and π < θ π and conversely, for every r 0 and θ such that π < θ π, we get a unque (non-ero) complex number r (cos θ + sn θ ) x + y Note that the complex number ero cannot be put nto the form r (cos θ + sn θ) and so, the argument of ero complex number does not exst REMARK If we take orgn as the pole and the postve drecton of the x-axs as the ntal lne, then the pont P s unquely determned by the ordered par of real numbers (r, θ), called the polar co-ordnates of the pont P (see fg 6) ILLUSTRATIVE EXAMPLES Example Convert the followng complex numbers n the polar form and represent them n Argand plane : () + () + () (v) (v) (v) Soluton () Let + r (cos θ + sn θ) Then r cos θ and r sn θ On squarng and addng, we get r (cos θ + sn θ) ( ) + r 4 r cos θ and sn θ The value of θ such that π < θ π and satsfyng both the above equatons s gven by θ π 6 X O < π/6 + P X π π Hence, cos + sn, whch s the requred polar 6 6 form The complex number + s represented n fg 7 Fg 7

19 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 7 ANSWERS EXERCISE () 6 () 6 () 0 (v) 60 (v) 0 ; ; + ; 0 es 4 es 6 () x 6, y () x, y () x 7, y 7 x, y 8 x, y 9 x, y () () () 7 () 0 () () () ( 6 + ) + ( + ) () () 8 () () () () 0 () () () () () 4 () + () 0 6 () 6 () 7 () () 64 8 () 4 () x + y () () 0 + () () () () (v) () () 4 47 () 4 + (v) 6 9 ; () 6 4 () + (v) ; () + ; ; () + ; ; (v) ; 9 46 ; 97 7 () 0 + () (v) n 8 () 9 () x, y 4 () () x 6, y () x () 0, y 8 40 () s the only soluton () all purely magnary numbers (n + ) π, n I 0 Excepton s EXERCISE () True () True () True (v) True (v) True 0 θ 4 () + () () 0 (v) + +