-
((CCoommpplleexx NNuummbbeerrss)) x2 + 1 = 0 x2 = 1 (complex
number)
11.1 .1
(a, b) a b (a, b) (c, d) (1) (a, b) = (c, d) a = c b = d (2) (a,
b) + (c, d) = (a + c , b + d) (3) (a, b) (c, d) = (ac bd, ad + bc)
C 1 (4, 3) (7, 2) (4, 3) + (7, 2) = (4 + 7, 3 + 2) = (3, 5) (4,
3)(7, 2) = ( (4)(7) (3)(2), (4)(2) + (3)(7) ) = (28 6, 8+ 21) =
(34, 13) 2 (a, b) (a, b) + (2, 5) = (1, 2) (a, b) + (2, 5) = (1, 2)
(a 2, b + 5) = (1, 2) a 2 = 1 b + 5 = 2 a = 1 b = 7 (a, b) = (1,
7)
1
-
3 (x, y) (x, y) (3, 7) = (27, 53) (x, y) (3, 7) = (27, 53) (3x +
7y, 7x + 3y) = (27, 53) 3x + 7y = 27 ..(1) 7x + 3y = 53 ..(2) (1) 7
21x + 49y = 189 ..(3) (2) 3 21x + 9y = 159 ..(4) (3) + (4) 58y =
348
y = 58348 = 6
y 6 3x + 7(6) = 27 3x + 42 = 27 3x = 27 42 3x = 15
x = 315
= 5
(x, y) = (5, 6)
z = (a, b) a b a (real part) z Re(z) b (imaginary part) z
Im(z)
0 (purely imaginary number) (0, 1) 1 i i (imaginary unit) i = 1
i2 = ( 1 )2 = ( 1 )( 1 ) = 1 i3 = i2 i = (1)(i) = i i4 = i2 i2 =
(1)( 1) = 1 i5 = i4 i = (1)(i) = i
S M
T M
complex
S M T M
complex
2
-
n i4n = (i4)n = 1n = 1 i4n + 1 = (i4n)(i) = (1)(i) = i i4n + 2 =
(i4n)(i2) = (1)( 1) = 1 i4n + 3 = (i4n)(i3) = (1)( i) = i
im m 4 m
1. 4 m im = 1 2. 4 m 1 im = i 3. 4 m 2 im = 1 4. 4 m 3 im =
i
4 (1) i 35 = i32 + 3 = (i4)8 (i3) = (1)( i) = i ( 4 35 3) (2)
i125 = i124 + 1 = (i4)31 (i) = (1)(i) = i ( 4 125 1) (3) i258 =
i256 + 2 = (i4)64 (i2) = (1)( 1) = 1 ( 4 258 2) 5 (1) i504 + i505 +
i506 + i507 (2) i504 i505 i506 i507 (1) 4 504 i500 = 1 4 505 1 i501
= i 4 506 2 i502 = 1 4 507 3 i501 = i i504 + i505 + i506 + i507 = 1
+ i 1 i = 0 (2) i504 i505 i506 i507 = (1) (i) (1) (i) = 1 i504 i505
i506 i507 = i504 + 505 + 506 + 507 = i2022 = 1 ( 4 2022 2) a a = a
i
S M
T M
complex
S M T M
complex
3
-
6 (1) 4 = 4 i = 2i (4) 2 = 2 i (2) 9 = 9 i = 3i (5) 8 = 8 i = 2
2 i (3) 25 = 25 i = 5i (6) 27 = 27 i = 3 3 i 7 (1) 4 9 = ( 4 i) ( 9
i) = (2i)(3i) = 6i2 = 6 (2) )9)(4( = 36 = 6 7 4 9 )9)(4( a b a b
)b)(a( b bi = (b, 0)(0, 1) = (0, b) (0, b) b i (a, b) = (a, 0) +
(0, b) (a, b) = a + bi (a, b) a + bi a (real part) b (imaginary
part) (1) (4, 0) = 4 + 0i = 4 4 0 (2) (0, 3) = 0 3i = 3i 0 3 (3)
(2, 9) = 2 + 9i 2 9 (4) (5, 2 ) = 5 + 2 i 5 2 (5) (2 5 , 6 ) = 2 5
6 i 2 5 6 (a, b) = a + bi b = 0 a a = 0 b 0 bi (1) (4, 0) = 4 + 0i
= 4 (2) (0, 3) = 0 3i = 3i a + bi
(a + bi) + (c + di) = (a + c) + (bi + di) = (a + c) + (b + d)i
(a +bi)(c + di) = ac + bci + adi + bdi2 = (ac bd) + (ad + bc)i
S M
T M
complex
S M T M
complex
4
-
8 (1) (2 + 3i) + (5 6i) (2 + 3i) + (5 6i) = (2 + 5) + (3 6)i = 7
+ (3i) = 7 3i (2) (2 + 3i)(5 6i) (2 + 3i)(5 6i) = 10 + 15i 12i 18i2
= (10 + 18) + (15 12i) = 28 + 3i
z = a + bi n zn (x + y)2 = x2 + 2xy + y2 ( x y)2 = x2 2xy + y2
(x + y)3 = x3 + 3x2y + 3xy2 + y3 (x y)3 = x3 3x2y + 3xy2 y3
9 z = 2 + 3i z2, z3, z4 z5 (1) z2 = (2 + 3i)2 = 22 + 2(2)(3i) +
(3i)2 = 4 + 12i 9 = (4 9) + 12i = 5 + 12i
(2) z3 = (2 + 3i)3 = 23 + 3(2)2(3i) + 3(2)(3i)2 + (3i)3 = 8 +
36i 54 27i = (8 54) + (36 27)i = 46 + 9i
5
S M
T M
complex
S M T M
complex
-
(3) z4 = (z2)2 = (5 + 12i)2 ( z2 = (5 + 12i (1)) = (5)2 +
2(5)(12i) + (12i)2 = 25 120i 144 = 119 120i (4) z5 = z4 z = (119
120i)(2 + 3i) = 238 + 360 357i 240i = 122 597i
() (a, b) + (0, 0) = (a + 0, b + 0) (0, 0) + (a, b) = (0 + a, 0
+ b) = (a, b) = (a, b) (0, 0) 0 + 0i (a, b) + (a, b) = (a a, b b)
(a, b) + (a, b) = (a + a, b + b) = (0, 0) = (0, 0) () (a, b) (a, b)
() a + bi a bi
10 (3, 4) (3, 4) 2 + 5i 2 5i 5 + 2 i 5 2 i 3 2 i 3 + 2 i
S M
T M
complex
S M T M
complex
6
-
() (a, b)(1, 0) = (a 1 b 0, a 0 + b 1) (1, 0)(a, b) = (1 a 0 b,
0 a + 1 b) = (a 0, 0 + b) = (a 0, 0 + b) = (a, b) = (a, b)
(1, 0) 1 + 0i (a, b) (0, 0) () (a, b) (a, b) (1, 0) (x, y) ()
(a, b) (a, b)(x, y) = (1, 0) (a, b)(x, y) = (ax by, bx + ay) (ax
by, bx + ay) = (1, 0) ax by = 1 bx + ay = 0
(x, y) =
2222 ba
b,
ba
a
() (a, b)
2222 ba
b,
ba
a
() a + bi iba
b
ba
a2222
z = (a, b) z (0, 0) () z z-1
z-1 =
2222 ba
b,
ba
a
11 (2, 5)
2222 52
5,
52
2 =
29
5,
29
2
(3, 4)
2222 )4(3
4,
)4(3
3 = 254
253 ,
3 + 5i i5)3(
5
5)3(
32222
= i34
5
34
3
S M
T M
complex
S M T M
complex
7
-
(a, b) (c, d) = (a, b) + (c, d)
(a, b) (c, d) = (a c, b d) (a + bi) (c + di) = (a c) + (b d)i 12
(1) (6, 15) (2, 4) = (6, 15) + (2, 4) = (6 + 2, 15 4) = (8, 11) (2)
(5 4i) (3 + 2i) = (5 4i) + (3 2i) = (5 3) + (4 2)i = 2 6i (0,
0)
(a, b) (c, d) = (a, b)
2222 dc
d,
dc
c (c, d) (0, 0)
(a, b) (c, d) = )d,c(
)b,a(
=
2222 dc
adbc,
dc
bdac
(a + bi) (c + di) = dic
bia
= (a + bi)
i
dc
d
dc
c2222
= 22 dc
bdac
+ idc
adbc22
S M
T M
complex
S M T M
complex
8
-
13 i45
i23
i45
i23
= (3 2i)
i
45
4
45
52222
= (3 2i)
i
41
4
41
5
= 41
815 +
41
1210 i
= 41
7 41
22 i
= 417 41
22 i 14 (2 2 i) (3 2 i)
(2 2 i) (3 2 i) = (2 2 i)
i
)2(3
2
)2(3
32222
= (2 2 i)
i
11
2
11
3
= 11
6 + i11
22 i11
23 + 11
2
= 11
8 i11
2
(conjugate)
z = a + bi a bi (conjugate) z z z = bia = a bi
(1) (5, 4) (5, 4) (3) 3 + 2i 3 2i (2) (9, 2) (9, 2) (4) 8 7i 8 +
7i
(a + bi)(a bi) = a2 + b2 abi + abi = a2 + b2 S M
T M
complex
S M T M
complex
9
-
15 i45
i23
i45
i23
= i45
i23
i45
i45
= 1625
i12i10815
= 41
i227
= 41
7 41
22 i
16 i43
i2314
+ i73
i527
i43
i2314
+ i73
i527
= )i43)(i43(
)i43)(i2314(
+ )i73)(i73(
)i73)(i527(
= 169
i)6956()9242(
+ 499
i)15189()3581(
= 25
i12550 + 58
i174116
= 25
)i52(25 + 58
)i32(58
= (2 + 5i) + (2 + 3i) = 4 + 8i
17 z (3 + 2i)z = 2 i (3 + 2i)z = 2 i
z = i23
i2
= i23
i2
i23
i23
= 22
2
23
i2i4i36
= 49
2i4i36
= 13
i74
= 13
4 13
i7
S M
T M
complex
S M T M
complex
10
-
z z-1 z-1 = z1 z 0
18 z-1 z = i43
i181
z-1 = z
1
= i181i43
= )i181)(i181(
)i181)(i43(
= 3241
i)454()723(
= 325
75 325
50 i
= 13
3 13
2 i
z, z1 z2
1) Re(z) = 2
1 (z + z ) Im(z) = i2
1 (z z )
2) 21 zz = 1z + 2z 3) 21 zz = 1z 2z 4) 21 zz = 1z 2z
5)
21
zz
= 21
zz
z2 0
6) ( 1z ) = ( z )-1 7) z = z
1 1) z = a + bi z = a bi
2
1 (z + z ) = 2
1 [( a + bi) + (a bi)] = 2
1 (2a) = a = Re(z)
i2
1 (z z ) = 2
1 [( a + bi) (a bi)] = i2
1 (2bi) = b = Im(z)
Re(z) = 2
1 (z + z ) Im(z) = i2
1 (z z )
S M
T M
complex
S M T M
complex
11
-
2) z1 = a + bi z2 = c + di 21 zz = )dic()bia( = dicbia =
i)db()ca( = (a c) (b d)i = a c bi + di = (a bi) (c di) = 1z 2z 21
zz = 1z 2z 3) z1 = a + bi z2 = c + di 21 zz = )dic)(bia( =
i)bcad()bdac( = i)bcad()bidiac( ( bd = bidi) = (ac + bidi) (ad +
bc)i = ac + bidi adi bci = (ac bci) (adi bidi) = (a bi)c (a bi)di =
(a bi)(c di) = 1z 2z 21 zz = 1z 2z 7) z = a + bi z = bia = bia = a
+ bi = z z = z
S M
T M
complex
S M T M
complex
12
-
11..11
1. Re(z) Im(z) (1) 5 4i
..................................................
................................................... (2) 3 + 9i
..................................................
................................................... (3) 9
..................................................
................................................... (4) 7i
..................................................
................................................... (5) (1 i)2 (1
i)2 = .................................................
................................................... (6) (2 + 5 i)2
(2 + 5 i)2 =... .................................................
...................................................
2. a + bi a, b R (1) (3, 4) + (2, 6) (3, 4) + (2, 6) = .... (2)
(5, 2) + (1, 3) (5, 2) + (1, 3)
=.............................................................................................
(3) ( 2 , 2) + (3 2 , 7) ( 2 ,2) + (3 2 , 7) = (4) (4 3i) + (5 +
6i) (4 3i) + (5 + 6i) =.. (5) (5 + 2i) + (1 4i) + (3 + i) (5 + 2i)
+ (1 4i) + (3 + i) =
13
-
3. a + bi a, b R (1) (3, 4)( 2, 6) (3, 4)( 2, 6) = (2) (1, 7)( ,
) (1, 7)( , ) =.. (3) ( 2 ,2)( 3 2 , 7) ( 2 ,2)( 3 2 , 7) =. (5) (4
3i)( 5 + 6i) (4 3i)( 5 + 6i) =. (6) (5 + 2i)(1 4i)( 3 + i) (5 +
2i)(1 4i)( 3 + i) =... (7) (1 + 7i)(2 + 2i)(4 5i) (1 + 7i)(2 +
2i)(4 5i) =. (8) (6i)(7 + 4i)( 8 i) (6i)(7 + 4i)( 8 i) = 4. (1) i97
i97 =.... (2) i130 i130 = (3) i500 i500 = (4) i635 i635 =. (5) i5 +
i6 + i7 + i8 + i9 + i10 i5 + i6 + i7 + i8 + i9 + i10 =. (6) i10 i11
i12 i13 i14 i15 i10 i11 i12 i13 i14 i15 = .
14
-
5. z (1) z = (5, 2) (5, 2)
..........................................................................................
(2) z = 4 3i 4 3i
........................................................................................................
(3) z = 3 + 2 i 3 + 2 i
................................................................................................
6. z (1) z = (5, 2) (5, 2)
..........................................................................................
(2) z = 4 3i 4 3i
........................................................................................................
(3) z = 3 + 2 i 3 + 2 i
................................................................................................
7. a + bi a, b R
(1) i2
1
.....
(2) i43
5
............
(3) i24
i32
................
15
-
5. z = 4 3i a + bi a, b R (1) z z =... . (2) z z z z =. . (3) z
+ z z + z =..
.........................................................................................
(4) z(z + z ) z(z + z ) =.... . (5) z z z z =.... . (6) (z z )i (z
z )i =.....
.........................................................................................
(7) 1z 1z =.... .
(8) i
z
i
z =..
.
15 SM.TM 3
16
-
1. (1) 241i + 242i + 243i + + 644i 241i + 242i + 243i + + 644i =
..... (2) ni + 1ni + 2ni + 3ni n I+ ni + 1ni + 2ni + 3ni =. . (3)
505i 506i 507i 508i 505i 506i 507i 508i = ..... (4) ni 1ni 2ni 3ni
n I+ ni 1ni 2ni 3ni =... .....
2. a + bi a, b R (1) (2 5i) + (8 + 6i) (2 5i) + (8 + 6i) = ....
. (2) (8 + 4i) (2 3i) (8 + 4i) (2 3i) = . (3) 2(3 + 4i) + 3(2 i)
5(4 + 2i) 2(3 + 4i) + 3(2 i) 5(4 + 2i) =. . (4) (3 + 4i)(2 + 5i) (2
+ 3i)(3 7i) (3 + 4i)(2 + 5i) (2 + 3i)(3 7i) =. .
11.1.1
17
-
3. z1 = 2 + 3i, z2 = 3 + 4i z3 = 4 + 7i (1) z1(z2 + z3) . . .
(2) z1z2 + z1z3 . . . 4. x y (1) 3 + 2i + x + yi = 5 7i . . . (2)
(3 + 5i)( x yi) = 3 + 29i . . . . (3) x + y + (x y + 3)i = 1 + 7i .
. . . (4) x2 + y2 + 2xyi 1 + i = 0 . . . .
18
-
5. z (1) z = (7, 4) . .... (2) z = (1, 3 ) .......... (3) z = 3
2 i ...... (4) z = (2 3i) + (4 + 7i) ...... (5) z = (2 + 3i)(3 4i)
...... 6. z (1) z = (7, 4) ...... (2) z = 3 + 2i ........... (3) z
= (2 3i) + (4 + 7i) ...............
19
-
7. z-1 z
(1) z = i543
....................
(2) z = 5i2
....................
(3) z = i23
i2
..........................
(4) z = i45i2i31
))((
........................ .
20
-
8. 1z = 3 i 2z = 2 + 3i (1) 1z . (2) 2z ..... (3) 1z 2z . (4)
21zz . (5) 1z 2z . (6) 1z + 2z . (7) 21 zz . (8) 1z + 2z . 9. a +
bi a, b R
(1) i21
i24
................
(2) i34
1
+
i2
i35
..... ...... . .
21
-
(3) i34
i71
i21
i52
...... ...... .
(4) i2i5
i3i2
..... ...... ......
(5) 3 + i )i1(
)i2( 2
...... ...... ......
(6)
i1
i1
i1
i1
..................... .
22
-
10. z z = x + yi (1) (2 + i)z = 4 2i . . . (2) 4(1 3i)z = 2 + i
. . . (3) (3 2i)z + i = 7 . . . 11. z (1) iz = i z . . (2) Im(iz) =
Re(z) . . (3) Re(iz) = Im(z) . .
(4) Re(z) = 2
zz
(5) Im(z) = i2
zz
. .
23
-
11..2 2 ((SSqquuaarree RRoooottss ooff CCoommpplleexx
NNuummbbeerrss)) z z w w2 = z w z w z z = x + yi x y w = a + bi z z
= x + yi = w2 = (a + bi)2 = (a2 b2) + 2abi a2 b2 = x . (1) 2ab = y
. (2) (a2 + b2)2 = a4 + 2a2b2 + b4 = (a4 2a2b2 + b4) + 4a2b2
= (a2 b2)2 + 4a2b2
= x2 + y2
a2 + b2 = 22 yx . (3)
(1) + (3) a2 = 2
1 ( xyx 22
a = 2
xyx 22
(3) (1) b2 = 2
1 ( xyx 22
b = 2
xyx 22
z a b (2) 2ab = y ab y a b
24
-
y 0 z
2
xyx 22 + 2
xyx 22 i 2
xyx 22 2
xyx 22 i
y < 0 z
2
xyx 22 2
xyx 22 i 2
xyx 22 +2
xyx 22 i
z = x + yi r = 22 yx z
i
2
xr
2
xr y 0
i
2
xr
2
xr y < 0
1 7 24i z = 7 24i x + yi x = 7 y = 24 r = 22 yx = 22 )24()7( =
57649 = 625 = 25
y < 0 z
i
2
xr
2
xr
7 24i
i
2
)7(25
2
)7(25
=
i
2
32
2
18
= i169 = (3 4i) 7 24i 3 4i 3 + 4i
25
-
2 3 + 5i z = 3 + 5i x + yi x = 3 y = 5 r = 22 yx = 22 59 = 259 =
34
y > 0 z
i
2
xr
2
xr
3 + 5i
i
2
334
2
334
3 + 5i i2
334
2
334
i2
334
2
334
2
i2
334
2
334
= 2
334
2
334
+ 2 i2
334
2
334
= 2
3
2
34
2
3
2
34 + 2
4
934 i
= 2
6 + 24
25 i
= 3 + 5i
2
i2
334
2
334
=
2
334
2
334
2
2
334
i
2
334
= 2
334
2
334
+ 2 i2
334
2
334
= 2
6 + 24
25 i
= 3 + 5i
z = a a z a i a i 16 16 i 16 i 4i 4i 19 19 i 19 i 50 50 i 50 i 5
2 i 5 2 i
26
-
3 ax2 + bx + c = 0 a, b c a 0 ax2 + bx + c = 0 a 0
x2 + xa
b + a
c = 0
2
22
a4
bx
a
bx
2
2
a4
b + a
c = 0 ( )
2
a2
bx
2
2
a4
ac4b = 0
b2 4ac 0 2
a2
bx
22
a2
ac4b
= 0
a2
ac4b
a2
bx
2
a2
ac4b
a2
bx
2
= 0
a2
ac4bbx
2
a2
ac4bbx
2
= 0
x = a2
ac4bb 2 x = a2
ac4bb 2
b2 4ac < 0 2
a2
bx
2
2
a4
1bac4 = 0
2
a2
bx
22
ia2
bac4
= 0
i
a2
bac4
a2
bx
2
i
a2
bac4
a2
bx
2
x = a2
ibac4b 2 x = a2
iac4bb 2
ax2 + bx + c = 0 a, b c
a 0 x = a2
ac4bb 2 b2 4ac 0
x = a2
iac4bb 2 b2 4ac < 0
27
-
4 4x2 2x + 3 = 0 4x2 2x + 3 = 0 ax2 + bx + c = 0 a = 4 , b = 2 c
= 3 b2 4ac = (2)2 4(4)(3) = 4 48 = 44 44 < 0
x = a2
iac4bb 2
x = )4(2
i44)2(
= )4(2
i1122
= 4
i111
i4
11
4
1,i
4
11
4
1
5 x2 + 9 = 0 1 x2 + 9 = 0 ax2 + bx + c = 0 a = 1 , b = 0 c = 9
b2 4ac = 02 4(1)(9) = 36 36 < 0
x = a2
iac4bb 2
x = )1(2
i36
= 2
i6
= 3i { 3i, 3i } 2 x2 + 9 = 0 x2 9i2 = 0 x2 (3i)2 = 0 (x 3i)(x +
3i) = 0 x = 3i { 3i, 3i }
28
-
1. (1) 25 25 .................................. (2) 73 73
.................................. (3) 25i z = 25i x + yi x = 0 y =
25 r = 22 yx = 22 )25(0 = 25
y < 0 z
i
2
xr
2
xr
25i
i
2
...........
2
..........
=
i
2
.....
2
.....
25i
..............................................................................................................
(4) 49i z = 49i x + yi x = 0 y = 49 r = 22 yx = 22 490 = 49
y > 0 z
i
2
xr
2
xr
49i
i
2
...........
2
..........
=
i
2
.....
2
.....
49i
...............................................................................................................
11..22
29
-
(5) 1 + 4i z = 1 + 4i x + yi x = . y = r = 22 yx = = ..
y > 0 z
i
2
xr
2
xr
1 + 4i
i
2
.............
2
............
=
i
2
.............
2
............
1 + 4i
...........................................................................................................
(6) 3 + 2i z = 3 + 2i x + yi x = .. y = r = 22 yx = = ..
y > 0 z
i
2
xr
2
xr
3 + 2i
i
2
.............
2
............
=
i
2
.............
2
............
3 + 2i
............................................................................................................
(7) 1 7i z = 1 7i x + yi x = . y = r = 22 yx = = ..
y < 0 z
i
2
xr
2
xr
1 7i
i
2
.............
2
............
1 7i
...........................................................................................................
30
-
2. (1) x2 + 7 = 0 1 x2 + 7 = 0 ax2 + bx + c = 0 a = 1 , b = 0 c
= 7 b2 4ac = 02 4(1)(7) = 28 28 < 0
x = a2
iac4bb 2
x = (....)2
i........
= ..
= ..
........................................................................................
2 x2 + 7 = 0 x2 () 2 = 0 (x .)(x + ..) = 0 x =
.........................................................................................
(2) 2x2 + 3x + 5 = 0 2x2 + 3x + 2 = 0 ax2 + bx + c = 0 a = .. , b =
.. c = b2 4ac =
..................................................................................
x = a2
iac4bb 2
x =
=...
=...
........................................................................................
31
-
1. (1) 8 6i
..........................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
(2) 5 + 12i
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
(3) 1 2 2 i
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
11.2.2
32
-
2. (1) x2 = 72
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
(2) 5x2 + 2 = 0
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
(3) x2 2x + 40 = 0
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
(4) x2 + 2x + 1 = 0
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
33
-
(5) x2 + 7x + 2 = 0
........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
(6) x2 6x + 10 = 0
...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................
(7) x2 x + 5 = 0
.............................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................................
34
-
(8) (x + 1)2 + 49 = 0
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
(9) 2x2 + 5x + 25 = 0
.............................................................................................................................................................................
.............................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................................
(10) 3x2 + 5x 16 = 0
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
35
-
(11) 4x2 x + 1 = 0
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
(12) 3x2 + 5x + 3 = 0
.............................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................................
(13) x2 + ix + 2 = 0
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................
(14) x2 + 2ix 1 = 0
.............................................................................................................................................................................
.............................................................................................................................................................................
.............................................................................................................................................................................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
36
-
11 .3 .3 (a, b) a + bi a b (a, b) (real axis) (imaginary axis)
(complex plane) X Y z = (a, b) = a + bi z 2 1. (a, b)
2. (0, 0) (a, b)
O
Y ()
X ()
a O
Y
X
b (a, b)
a O
Y
X b
(a, b)
37
-
1 z1 = (4, 3), z2 = (2, 4), z3 = 3 2i, z4 = 3i, z5 = 4 4i z6 =
2
2 z1 = (4, 3), z2 = (2, 4), z3 = 3 2i, z4 = 3i, z5 = 4 4i z6 =
2
z6 = (2, 0)
z5 = (4, 4) z4 = (0, 3)
z3 = (3, 2)
z2= (2, 4) z1 = (4, 3)
5 O 5 X
Y
5
5
z6 = (2, 0)
z5 = (4, 4) z4 = (0, 3)
z3 = (3, 2)
z2= (2, 4) z1 = (4, 3)
5 O 5 X
Y
5
5
38
-
(absolute value modulus) a + bi | a + bi | | a + bi | = 22
ba
a + bi (0, 0) (a, b)
z = (a, b) = a + bi | z | = 22 ba
3 (1) z = (4, 3) | (4, 3) | = 22 34 = 25 = 5
(2) z = 2 5i |2 5i | = 22 )5(2 = 29
(3) z = 7i | 7i | = 22 )7(0 = 49 = 7
(4) z = 1 2 i | 1 2 i | = 22 )2()1( = 3
(5) z = 8 | 8| = 22 0)8( = 64 = 8
(6) z = 21 +
2
5 i
i2
5
2
1 =
22
2
5
2
1
= 4
5
4
1
= 2
6
39
-
2 z, z1, z2 1. | z |2 = z z 2. | z | = | z | 3. | z | = | z | 4.
| z1 z2 | = | z1 | | z2 | 5. | z-1 | = | z |-1
6. 21
zz
= 2
1
zz
, z2 0
7. | z1 + z2 | | z1 | + | z2 | 8. | z1 z2 | | z1 | | z2 | 2 1, 2
4 1. | z |2 = z z z = a + bi z = a bi z z = (a + bi)(a bi) = a2 +
b2 = ( 22 ba )2
= | z |2 z z = | z |2
2. | z | = | z | z = a + bi z = a bi | z | = 22 ba | z | = 22
)b()a( = 22 ba | z | = | z |
40
-
4. | z1 z2 | = | z1 | | z2 | z1 = a + bi z2 = c + di z1 z2 = (a
+ bi)(c + di) = (ac bd) + (ad + bc)i | z1 | = 22 ba | z2 | = 22 dc
| z1 z2 | = 22 )bcad()bdac( = 22222222 cbabcd2dadbacbd2ca =
22222222 cbdadbca = )dc)(ba( 2222 = 22 ba 22 dc = | z1 | | z2 | |
z1 z2 | = | z1 | | z2 |
4 z (1) | z | = 3 z = x + yi | z | = 3 | x + yi | = 3 22 yx = 3
x2 + y2 = 32 | z | = 3 (0, 0) 3
(2) | z + 2 | = 3 z = x + yi | z + 2 | = 3 | x + yi + 2 | = 3
|(x + 2) + yi | = 3 22 y)2x( = 3 (x + 2)2 + y2 = 32 | z + 2 | = 3
(2, 0) 3
X
YX
(0, 3)
(0, 3)
(3, 0) O (3, 0)
X
Y Y
(2, 3)
(2, 3)
(5, 0) (2,0) O (1, 0)
41
-
(3) | z 5i | = 4 z = x + yi | z 5i | = 4 | x + yi 5i | = 4 | (x
+ (y 5)i | = 4 22 )5y(x = 4 x2 + (y 5)2 = 42 | z 5i | = 4 (0, 5) 4
(4) | z + 3 2i | = 2 z = x + yi | z + 3 2i | = 2 | x + yi + 3 2i |
= 2 | (x + 3) + (y 2)i | = 2 22 )2y()3x( = 2 (x + 3)2 + (y 2)2 =
22
|z + 3 2i | = 2 (3, 2) 2 (5) z z = 2i z = x + yi z = x yi z z =
2i (x yi) (x + yi) = 2i x yi x yi = 2i 2yi = 2i y = 1 z z = 2i X (
X )
(0, 1) X
X
YX (0, 9)
(4, 5) (0, 5) (4, 5)
O
(3, 4)
(3, 0) X
X (5, 2) (3, 2) (1, 2)
O
Y Y
y = 1 (0, 1) O X
Y
42
-
5 z z | z | 4 | z | 4 z = x + yi | x + yi | 4 22 yx 4 x2 + y2 42
| z | 4 (0, 0) 4
6 z z | z | 4 | z + 3 | < 2 | z | 4 z = x + yi | x + yi | 4
22 yx 4 x2 + y 2 42 | z | 4 (0, 0) 4 | z + 3 | < 2 z = x + yi |
z + 3 | < 2 | x + yi + 3 | < 2 |(x + 3) + yi | < 2 22
y)3x( < 2 (x + 3)2 + y2 < 22 | z + 3 | < 2 () (3, 0) 2 )
z
(0, 4)
(0, 1)
X
YX
(4, 0) (0, 5) (4, 0) O
X
YX
4 O 3
4
5
4
43
-
1. z1 = (5, 0), z2 = (3, 6), z3 = 4 + 2i, z4 = 6, z5 = 4 3i, z6
= 5i z7 = 5 i 2. z1 = (5, 0), z2 = (3, 6), z3 = 4 + 2i, z4 = 6, z5
= 4 3i, z6 = 5i z7 = 5 i
11..33
X
Y
O
X
Y
O
44
-
3. (1) z = (2, 3) | (2, 3) | = .. .
(2) z = 15 + 12i | 15 + 12i | = .. .
(3) z = 2 2 i | 2 2 i | = .. .
4. z (1) | z | = 5 z = x + yi | z | = 5 | x + yi | = 5 22 yx = 5
..=..
......................................................................................................................................
(2) | z 5| < 3 z = x + yi | z 5| < 3 | x + yi 5 | < 3 .
< . < . < ...
......................................................................................................................................
..
45
-
5. z (1) (0, 0) 2 x2 + y2 = r2 x2 + y2 = 22 22 yx = 2 22 yx = |
x + yi | | x + yi | = 2 | z | = 2
(2) (0, 0) 4 x2 + y2 = r2 x2 + y2 = . 22 yx = 22 yx = | x + yi |
| x + yi | = .. (3) (3, 0) 5 (x h) 2 + (y k) 2 = r2 (x 3) 2 + (y 0)
2 = 52 22 y)3x( = 5 22 y)3x( = | (x 3) + yi | | (x 3) + yi | = | (x
+ yi) 3 | =. ..
X
YX
(0, 2)
(0, 2)
(2, 0) O (2, 0)
X
YX
(0, 4)
(0, 4)
(4, 0) O (4, 0)
X
Y Y
(3, 5)
(3, 5)
(2, 0) O (3,0) (8, 0)
46
-
1. (1) z = (4 5i) (1 + 7i) . . . . . (2) z = (4 + 3i)(4 3i) . .
. . .
(3) z = i1
1
. . . . . .
(4) z = i3
i52
. . . . . . .
11.3.3
47
-
2. z (1) | z 3 | = 2
............................................ (2) | z 4 + 3i | = 3
................................... (3) | z 1| = | z + 2 |
.................................................................
48
-
(4) | z | = z + z
.................................................... (5) Im(i + z )
= 4
....................................................
3. ),(),(
yx514 = (4, 1) | (x, y) |
.....................................................................
.....
49
-
4. z = x + yi 3z2z
= 1 x
.................................................... 5. z | z 4
| = 2| z 1| | z |
....................................................
6. z = a + bi a 0 b 0 z-1 = 2zz
................................................................
.. ...... ..
50
-
7. z | z | = | z |
.................................................... 8. z | z-1 | =
| z |-1 ....................................................
9. z 21
zz
= 2
1
zz
, z2 0
.............................................................
51
-
10. z (1) | z 2 | 3 . . . . . (2) | z + i | 2 . . . . . (3) | z
+ 2 3i | < 4 . . . ........... (4) | z 2 + 3i | 4 . . . . .
........... .....
52
-
(5) Im (z) > 5 . . . . . (6) Re(z) < 3 . . . . . . (7)
Re(z i) 5 . . . . . . . (8) | z 3 | | z | . . . . . . . .
53
-
11. z z | z | > 4 | z + 3i | 2 . . . . . ...... . . . . . 12.
z z | z 2 2i | 3 | z 3 2i | < 2 . . . ...... . . . . . . . .
54
-
11 .4 .4 (( PP oo ll aa rr ff oo rr mm )) z = (a, b) = a + bi
z
z 0 X oz r = | oz | r = | oz | = 22 ba
tan = a
b
sin = r
b b = r sin
cos = r
a a = r cos
z = a + bi z = r cos + ri sin
z = a + bi r (cos + i sin ) tan = a
b
r (cos + i sin ) (polar form) a + bi z (arqument of z) cos =
cos( + 2k ) sin = sin( + 2k ) k r[cos( + 2k ) + i sin( + 2k )]
(polar form) a + bi
1 (1) z = 1 i3 (2) z = 3 3i (3) z = 4i (4) z = 3 (1) z = 1 i3 a
= 1 b = 3 r = | z | = 22 ba = 22 )3(1 = 31 = 4 = 2
tan = a
b = 1
3 = 3
b z = (a, b)
a O
Y
X
55
-
(1 , 3 ) 4 = 300 = 3
5
z = 1 i3 2(cos 300 + i sin 300)
= 2(cos3
5 + i sin 3
5 )
1 i3 2
k2
3
5sinik2
3
5cos k I
(2) z = 3 3i a = 3 b = 3 r = | z | = 22 ba = 22 )3()3( = 99 = 18
= 3 2
tan = a
b = 3
3
= 1
(3, 3) 3 = 225 = 4
5
z = 3 3i 3 2 (cos 225 + i sin 225)
