a + bi, aεr, bεr i = z = a + bi a = Re(z), b = Im(z) give z = a + bi & w = c + di, a + bi = c + di a = c & b = d The complex cojugate of z = a + bi is z = a bi The sum of complex cojugates is real: z + z = a The product of complex cojugates is real: z z = a + b Properties of complex cojugates: Re(z) = (z + z ) Im(z) = (z z ) (z ± z ) = z ± z (z ) = z (z ± z ) = z ± z (z z ) = z z & ( z ) = z z z, z 0 (z ) = (z ) Presetatio of complex umber i Cartesia ad polar coordiate system Modulus or Absolute value: z = r = x + y Argumet: arg z = θ = arc ta y x be careful x = rcos θ, y = rsi θ z = x + yi Cartesia form = r(cos θ + i si θ) polar form modulus argumet form = re iθ Euler form = r cis θ Recall that the argumet should be measured i radias Arga plae is the complex plae Very useful for fast coversios from Cartesia ito Euler form it will give you visually positio of the poit, ad therefore quadrat for the agle z = i z = e iπ i = e iπ i = e iπ = e i0
Practicality of Euler s form z = x + iy = r (cos θ + i si θ ) & z = x + iy = r (cos θ + i si θ ) Product is: z z = (x + iy )(x + iy ) = r r (cos θ + i si θ )(cos θ + i si θ ) = r r (cos θ cos θ si θ si θ + i (cos θ si θ + si θ cos θ ) = r r [cos(θ + θ ) + i si(θ + θ )] z z = z z modulus of product is product of modulus & arg (z z ) = arg (z ) + arg (z ) z z = [ z e iθ ] [ z e iθ ] = z z e i(θ +θ ) Quotiet is: z = (x + iy ) z (x + iy ) = r cos θ + i si θ = r cos θ + i si θ cos θ i si θ r cos θ + i si θ r cos θ + i si θ cos θ i si θ argmet of product is sum of argumets = r r cos θ cos θ + si θ si θ + i (si θ cos θ cos θ si θ ) cos θ + si θ = r r [cos( θ θ ) + i si( θ θ )] z = z z z modulus of quotiet is quotiet of modulus & arg ( z z ) = arg (z ) arg (z ) argmet of quotiet is differece of argumets z = [ z e iθ] z [ z e iθ ] = z z ei(θ θ ) Coclusio: Euler form of complex umbers follows ordiary algebra: a a m = a +m It is easy to multiply two complex umber i ay form, but what if you have 0 factors or z 5 5 or z Properties of modulus ad argumet z = z & arg (z ) = arg z zz = z e iθ = e i(θ+kπ) kεz
De Moivre s Theorem z = [r(cosθ + i siθ)] = r (cosθ + i siθ) = [re iθ ] = r e iθ = r (cos θ + i si θ) ( z e iθ ) = z e iθ (cos θ + i si θ) = (cosθ + i siθ) Applicatio of DeMoivre s Theorem (cos θ + i si θ) = (cosθ + i siθ) (a + b) = ( r ) a r b r = a + ( ) a b + + ( r ) a r b r + + b r=0 ( ( )( ) ( r + ) ) = r r! =! r! ( r)! =! ( r)! r! = ( r ) ( 0 ) Certai trigoometric idetities ca be derived usig DeMoivre s theorem. We ca for istace express cos, si ad ta i terms of cos, si ad ta. Example: We ca fid a expressio for cos5 Re(cos5 i si 5 ) The: cos 5θ = Re (cosθ + i siθ) 5 Re cos isi (usig DeMoivre s theorem) = 5 = Re(cos 5 θ + 5i cos θ siθ + 0i cos θ si θ + 0i cos θ si θ + 5i cos θ si θ + i 5 si 5 θ) = Re(cos 5 θ + 5i cos θ siθ + 0i cos θ si θ + 0i cos θ si θ + 5i cos θ si θ + i 5 si 5 θ) = cos 5 θ 0 cos θ si θ + 5 cos θ si θ 5 cos5 cos 0cos si 5cos si. (*) If required, the right had side ca be expressed etirely i terms of cos. We get: 5 cos5 cos 0cos ( cos ) 5cos ( cos ) = 5 5 5 cos 0cos 0cos 5cos 0cos 5cos cos5 5 6cos 0cos 5cos
Note : We ca also get a idetity for si5 : si 5θ = Im (cosθ + i siθ) 5 = Im(cos 5 θ + 5i cos θ siθ + 0i cos θ si θ + 0i cos θ si θ + 5i cos θ si θ + i 5 si 5 θ) = Im(cos 5 θ + 5i cos θ siθ + 0i cos θ si θ + 0i cos θ si θ + 5i cos θ si θ + i 5 si 5 θ) = 5 cos θ siθ 0 cos θ si θ + si 5 θ si 5θ = 5 cos θ siθ 0 cos θ si θ + si 5 θ (**) If required, the right had side ca be expressed etirely i terms of si 5 si5 5( si ) si 0( si )si si 5 5 5 = 5si 0si 5si 0si 0si si si5 5 6si 0si 5si Note : We ca also get a expressio for ta5 by dividig equatio (*) by equatio (**): 5 si 5 5cos si 0 cos si si ta 5 cos 5 5 cos 0 cos si 5cos si Dividig every term o the top ad bottom by 5 cos gives: ta 5 5 5cos si 0 cos si si 5 5 5 cos cos cos 5 cos 0 cos si 5cos si 5 5 5 cos cos cos = 5 ta 0 ta ta 0 ta 5 ta
5 Questio: a) Fid a expressio for cos i terms of cos oly. b) Fid a expressio for si i terms of si oly. c) Show that t t ta, where t = taθ. 6t t Questios:. Fid a expressio for cos6 i terms of c cos ad s si. Fid a expressio for si 7 i terms of si oly.. Fid a expressio for ta 7 i terms of t ta. Fidig a geeral root of a complex umber Geeral problem: Fid the complex umbers z such that z =a + ib. Example: Fid the cube roots of 8 8i, i.e. fid z such that z = 8 8i. The th roots of the complex umber c are solutios of z = c. There are exactly th roots of c. let c = c e iθ = c e i(θ+kπ) The complete solutio of the z = c is give by z = c e i(θ+kπ) = c {cos ( θ+kπ Oly the values k = 0,,, - give differet values of z Geometrically, the th roots are the vertices of a regular polygo with sides i Arga plae. z = ) + i si ( θ+kπ ) } k = 0,,,, Example: Fid 5 th root of. Or show that if,,,, = e i(0+kπ) kεn 5 i e, the the 5 th roots of uity ca be expressed as 5 = e i(0+kπ) 5 k = 0,,,, So the 5 th roots of uity are 0 i i i i e, e, e, e, e 6 8 5 5 5 5 - -
6 Example: Fid the cube roots of 8 8i, i.e. fid z such that z = 8 8i. 8 8i = [ 8, ] c = c e iθ = c e i(θ+kπ) so [ 8, ] or [ 8, 7 ] or [ 8, 5 8 ] Let z = [r, θ] be a cube root of 8 8i. The [ 7 5 z r, ] = [ 8, ] or [ 8, ] or [ 8, ] Comparig the modulus ad argumets we get: r 8 i. e. r 8 / / 6 (8). 9097 7 5 or or so or 7 or 5 The cube roots of 8 8i are: 6 i 8(cos( ) si( )) 6 7 i 7 8(cos( ) si( )) 8(cos( ) si( )) The cube roots (to sf) are: 6 5 i 5.7-0.58i -0.58 +.7i -.59.59iNote: The cube roots of 8 8i ca be show o a Argad diagram: Notice that the cube roots form a equilateral triagle. - - 0 - -
7 REAL POLYNOMIALS are polyomials with real coefficiets. REMAINDER
8
9 P(x) is real ad + i is zero. i is zero [ax + 9x + ax 0] = [x ( + i)][x ( i)](ax + b) [ax + 9x + ax 0] = (x + 6x + 0)(ax + b) = ax + (6a + b)x + (0a + 6b)x + 0b 6a + b = 9 & 0b = 0 0a + 6b = a 0b = 0 b = 0a + 6b = a 9a + 6b = 0 a = or 6a + b = 9 a = liear factor (ax+b) = x so zeroes are: ± i ad Complex Numbers. Let z = x + yi. Fid the values of x ad y if ( i)z = i. []. (a) Evaluate ( + i), where i =. (b) Prove, by mathematical iductio, that ( + i) = ( ), where *. (c) Hece or otherwise, fid ( + i). [0]
0. Let z = 6 i, ad z = i. (a) π π Write z ad z i the form r(cos θ + i si θ), where r > 0 ad θ. (b) z Show that = cos + i si. z z (c) Fid the value of i the form a + bi, where a ad b are to be determied exactly i radical z (surd) form. Hece or otherwise fid the exact values of cos ad si. []. Let z = a cos i si ad z = b cos i si. Express z z i the form z = x + yi. 5. If z is a complex umber ad z + 6 = z + l, fid the value of z. 6. Fid the values of a ad b, where a ad b are real, give that (a + bi)( i) = 5 i. 7. Give that z = (b + i), where b is real ad positive, fid the exact value of b whe arg z = 60. 8. The complex umber z satisfies i(z + ) = z, where i. Write z i the form z = a + bi, where a ad b are real umbers. [] [] [] [] 9. The complex umber z satisfies the equatio z = + i. i Express z i the form x + iy where x, y. [] [5] 0. Cosider the equatio (p + iq) = q ip ( i), where p ad q are both real umbers. Fid p ad q.. Let the complex umber z be give by i z = +. i Express z i the form a +bi, givig the exact values of the real costats a, b.
. A complex umber z is such that z z i. (a) Show that the imagiary part of z is. (b) Let z ad z be the two possible values of z, such that z. (i) Sketch a diagram to show the poits which represet z ad z i the complex plae, where z is i the first quadrat. (ii) π Show that arg z =. 6 (iii) Fid arg z. (c) Give that arg z = π, fid a value of k. i [0]. Give that (a + i)( bi) = 7 i, fid the value of a ad of b, where a, b.. Give that z, solve the equatio z 8i = 0, givig your aswers i the form z = r (cos + i si). 5. Give that z = (b + i), where b is real ad positive, fid the exact value of b whe arg z = 60. 6. Give that z = 5, fid the complex umber z that satisfies the equatio 5 5 8i. z z * 7. a b The two complex umbers z = ad z = i i where a, b, are such that z + z =. Calculate the value of a ad of b. 8. The complex umbers z ad z are z = + i, z = + i. (a) Fid z z, givig your aswer i the form a + ib, a, b. (b) The polar form of z may be writte as 5,arcta. (i) Express the polar form of z, z z i a similar way. (ii) π Hece show that = arcta + arcta. π π 9. Let z = r cos isi ad z = + i. (a) Write z i modulus-argumet form. (b) Fid the value of r if z z =.
0. Let z ad z be complex umbers. Solve the simultaeous equatios z + z = 7, z + iz = + i Give your aswers i the form z = a + bi, where a, b.. The complex umber z is defied by z = π π cos isi cos π 6 π isi. 6 (a) Express z i the form re i, where r ad have exact values. (b) Fid the cube roots of z, expressig i the form re i, where r ad have exact values.. The polyomial P(z) = z + mz + z 8 is divisible by (z ++ i), where z ad m,. Fid the value of m ad of.. Let u =+ i ad v =+ i where i =. (a) (i) u Show that i. v (ii) u By expressig both u ad v i modulus-argumet form show that cos v. (iii) π Hece fid the exact value of ta i the form a b where a, b. (b) Use mathematical iductio to prove that for +, π π i cos isi. (c) Let z = v u. v u π isi π Show that Re z = 0. [8]. (a) Express the complex umber + i i the form ae, where a, b π i b +. i (b) Usig the result from (a), show that, where, has oly eight distict values. (c) Hece solve the equatio z 8 = 0. [9] 5. Fid, i its simplest form, the argumet of (si + i ( cos )) where is a acute agle. [7] 6. z Cosider w = where z = x + iy, y 0 ad z + 0. z
Give that Im w = 0, show that z =. [7] 7. (z + i) is a factor of z z + 8z. Fid the other two factors. [] 8. Let P(z) = z + az + bz + c, where a, b, ad c. Two of the roots of P(z) = 0 are ad ( + i). Fid the value of a, of b ad of c. De Moivre s Theorem. Let x ad y be real umbers, ad be oe of the complex solutios of the equatio z =. Evaluate: (a) + + ; (b) ( x + y)( x + y).. (a) Express z 5 as a product of two factors, oe of which is liear. (b) Fid the zeros of z 5, givig your aswers i the form r(cos θ + i si θ) where r > 0 ad π < θ π. (c) Express z + z + z + z + as a product of two real quadratic factors. [0]. (a) Express the complex umber 8i i polar form. (b) The cube root of 8i which lies i the first quadrat is deoted by z. Express z (i) i polar form; (ii) i cartesia form. π π π π cos i si cos i si. Cosider the complex umber z =. π π cos i si (a) (i) Fid the modulus of z. (ii) Fid the argumet of z, givig your aswer i radias. (b) Usig De Moivre s theorem, show that z is a cube root of oe, ie z =. (c) Simplify (l + z)( + z ), expressig your aswer i the form a + bi, where a ad b are exact real umbers. [] 5. (a) Prove, usig mathematical iductio, that for a positive iteger, (cos + i si) = cos + i si where i =. (b) The complex umber z is defied by z = cos + i si.
(i) Show that z = cos ( ) + i si ( ). (ii) Deduce that z + z = cos θ. (c) (i) Fid the biomial expasio of (z + z l ) 5. (ii) Hece show that cos 5 = 6 (a cos 5 + b cos + c cos ), where a, b, c are positive itegers to be foud. [5] 6. (a) Use mathematical iductio to prove De Moivre s theorem (cos + i si) = cos () + i si (), +. (b) Cosider z 5 = 0. π π (i) Show that z = cos i si is oe of the complex roots of this equatio. 5 5 (ii) (iii) Fid z, z, z, z 5, givig your aswer i the modulus argumet form. Plot the poits that represet z, z, z, z ad z 5, i the complex plae. (iv) The poit z is mapped to z + by a compositio of two liear trasformatios, where =,,,. Give a full geometric descriptio of the two trasformatios. [6] 7. Give that z, solve the equatio z 8i = 0, givig your aswers i the form z = r (cos + i si). 8. Cosider the complex umber z = cos + i si. (a) Usig De Moivre s theorem show that z + = cos. (b) By expadig z show that z (c) Let g (a) = cos d. (i) z a 0 Fid g (a). cos = (cos + cos + ). 8 (ii) Solve g (a) = []
5 9. π π Let z = cos + i si, for. (a) (i) Fid z usig the biomial theorem. (ii) Use de Moivre s theorem to show that cos = cos cos ad si = si si. (b) si θ siθ Hece prove that cosθ cosθ = ta. (c) Give that si =, fid the exact value of ta. [] 0. Let y = cos + i si. (a) dy Show that dθ = iy. [You may assume that for the purposes of differetiatio ad itegratio, i may be treated i the same way as a real costat.] (b) Hece show, usig itegratio, that y = e i. (c) Use this result to deduce de Moivre s theorem. (d) (i) si 6θ Give that si θ = a cos 5 + b cos + c cos, where si 0, use de Moivre s theorem with = 6 to fid the values of the costats a, b ad c. (ii) si 6θ Hece deduce the value of lim. 0 si θ [0]. Prove by iductio that + (5 ) is a multiple of 7 for +. [0]. Prove that i i is real, where +. a. Express i the form where a, b. [5] i b. Let w = cos isi. 5 5 (a) Show that w is a root of the equatio z 5 = 0. (b) Show that (w ) (w + w + w + w + ) = w 5 ad deduce that w + w + w + w + = 0. (c) Hece show that cos cos. 5 5 [] m 5. z = i ad z = i.
6 (a) Fid the modulus ad argumet of z ad z i terms of m ad, respectively. (b) Hece, fid the smallest positive itegers m ad such that z = z. []