A × B = {(a, b) : a " A, b " B}
A B
K
K × K K
(K, +, ·) K + ·
K
" x, y, z " K (x + y) + z = x + (y + z)
" x, y " K x + y = y + x
" e " K " x " K x + e = x
" x " K " y " K x + y = e
" x, y, z " K (x · y) · z = x · (y · z)
" x, y, z " K x · (y + z) = x · y + x · z
" x, y " K x · y = y · x
" f " K \ {e} " x " K f · x = x
" x " K \ {e} " y " K x · y = f
e f K
y -x e
0
y x-1 f
R
(C, +, ·) C = R × R + ·
C
" (x1, y1), (x2, y2) " C : (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)
" (x1, y1), (x2, y2) " C : (x1, y1) · (x2, y2) = (x1x2 - y1y2, x1y2 + x2y1)
 (C, +, ·) C
z w
x (x, 0)
(0, 1) i
i2 = (0, 1) · (0, 1) = -1. z = x + iy z = (x, 0) +
(0, 1) · (y, 0) z = (x, y)
z = (x, y)
z = (x, y) = x + iy.
z1 = x1 + iy1 z2 = x2 + iy2
z1 Ä… z2 = (x1 Ä… x2) + i(y1 Ä… y2),
z1 · z2 = (x1x2 - y1y2) + i(x1y2 + x2y1).
z = x + iy x z
y z
Re z = x, Im z = y.
x - iy z = x + iy
z
z = x + iy = x - iy = (x, -y).
z1 = x1 + iy1 z2 = x2 + iy2
z1
z2
z1 z1 · z2 (x1 + iy1)(x2 - iy2) x1x2 - ix1y2 + ix2y1 - i2y1y2
= = = =
2
z2 z2 · z2 (x2 + iy2)(x2 - iy2) x2 - i2y2
2
x1x2 + y1y2 + i(-x1y2 + x2y1) x1x2 + y1y2 x2y1 - x1y2
= = + i .
2 2 2
x2 + y2 x2 + y2 x2 + y2
2 2 2
z1 = 2 - 3i z2 = 3 + 4i z3 = -1 + 2i
z1 + z2 = (2 - 3i) + (3 + 4i) = 2 - 3i + 3 + 4i = 5 + i = (5, 1),
z2 - 3 · z3 = (3 + 4i) - 3 · (-1 + 2i) = (3 + 4i) - 3 · (-1 - 2i) = 3 + 4i + 3 + 6i =
= 6 + 10i = (6, 10),
z2 · z3 = (3 + 4i) · (-1 + 2i) = -3 + 6i - 4i + 8i2 = -3 + 2i + 8i2 = -3 + 2i - 8 =
= -11 + 2i = (-11, 2),
z3 -1 + 2i -1 + 2i 2 - 3i (-1 + 2i) · (2 - 3i) (-1 + 2i) · (2 + 3i)
= = · = = =
z1 2 - 3i 2 - 3i - 3i (2 - 3i) · 2 - 3i
(2 - 3i) · (2 + 3i)
2
-2 - 3i + 4i + 6i2 -2 + i - 6 -8 + i 8 1 8 1
= = = = - + i = - , .
4 - 9i2 4 + 9 13 13 13 13 13
 z = x + iy
x2 + y2 |z|
z · z = (x + iy)(x - iy) = x2 - i2y2 = x2 + y2 = |z|2
z = x + iy = 0

Õ " [0, 2Ä„)
x y
cos Õ = , sin Õ = .
|z| |z|
z z
z = { z + 2kĄ, k " N}
z
z = |z| (cos Õ + i sin Õ)
z
"
z = 2 - 2i 3 z
" 2 "
"
|z| = 22 + -2 3 = 4 + 12 = 16 = 4,
Å„Å‚ Å„Å‚ Å„Å‚
ôÅ‚cos Õ = 2 ôÅ‚cos Õ = 1 ôÅ‚cos Õ = 1
òÅ‚ òÅ‚ òÅ‚
5
4 2 2
" "
Ð!Ò! Ð!Ò! Ð!Ò! Õ = Ä„ .
ôÅ‚sin Õ = -2 3 ôÅ‚sin Õ = - 3 ôÅ‚Õ " 3Ä„, 2Ä„ 3
ół ół ół
2
4 2
5 5
z = 4 cos Ä„ + i sin Ä„, .
3 3
z = |z| (cos Õ + i sin Õ) n " N
zn = |z|n (cos nÕ + i sin nÕ) .
"
5 5
z = 2 - 2i 3 z = 4 cos Ä„ + i sin Ä„
3 3
2011 · 5 2011 · 5 10055Ä„ 10055Ä„
z2011 = 42011 cos Ä„ + i sin Ä„ = 24022 cos + i sin =
3 3 3 3
1675 · 3 · 2Ä„ + 5Ä„ 1675 · 3 · 2Ä„ + 5Ä„
= 24022 cos + i sin =
3 3
5Ä„ 5Ä„
= 24022 cos 1675 · 2Ä„ + + i sin 1675 · 2Ä„ + =
3 3
"
"
5Ä„ 5Ä„ 1 3
= 24022 cos + i sin = 24022 - i = 24021 - 24021 3i.
3 3 2 2
n z" n " N
n
w wn = z z w wn = z
n z
 z = |z| (cos Õ + i sin Õ) n " N
"
n
z = {w0, w1, . . . , wn-1},
Õ + 2kÄ„ Õ + 2kÄ„
n
wk = |z| cos + i sin , k = 0, 1, . . . , n - 1.
n n
{w0, w1, . . . , wn-1} n
n
|z|
"
5 5
z = 2 - 2i 3 z = 4 cos Ä„ + i sin Ä„
3 3
"
5
z = {w0, w1, w2, w3, w4},
5Ä„ 5Ä„
"
+ 2kĄ + 2kĄ
5
3 3
wk = 4 cos + i sin , k = 0, 1, 2, 3, 4,
5 5
5Ä„ 5Ä„ 5Ä„ 5Ä„
" "
+ 2 · 0 · Ä„ + 2 · 0 · Ä„
5 5
3 3 3 3
w0 = 4 cos + i sin = 4 cos + i sin =
5 5 5 5
" " " "
5 5
" "
Ä„ Ä„ 1 3 4 4 · 3
5 5
= 4 cos + i sin = 4 + i = + i ,
3 3 2 2 2 2
5Ä„ 5Ä„ 5Ä„ 6Ä„ 5Ä„ 6Ä„
" "
+ 2 · 1 · Ä„ + 2 · 1 · Ä„ + +
5 5
3 3 3 3 3 3
w1 = 4 cos + i sin = 4 cos + i sin =
5 5 5 5
"
11Ä„ 11Ä„
5
= 4 cos + i sin ,
15 15
5Ä„ 5Ä„ 5Ä„ 12Ä„ 5Ä„ 12Ä„
" "
+ 2 · 2 · Ä„ + 2 · 2 · Ä„ + +
5 5
3 3 3 3 3 3
w2 = 4 cos + i sin = 4 cos + i sin =
5 5 5 5
"
17Ä„ 17Ä„
5
= 4 cos + i sin ,
15 15
5Ä„ 5Ä„ 5Ä„ 18Ä„ 5Ä„ 18Ä„
" "
+ 2 · 3 · Ä„ + 2 · 3 · Ä„ + +
5 5
3 3 3 3 3 3
w3 = 4 cos + i sin = 4 cos + i sin =
5 5 5 5
"
23Ä„ 23Ä„
5
= 4 cos + i sin ,
15 15
5Ä„ 5Ä„ 5Ä„ 24Ä„ 5Ä„ 24Ä„
" "
+ 2 · 4 · Ä„ + 2 · 4 · Ä„ + +
5 5
3 3 3 3 3 3
w4 = 4 cos + i sin = 4 cos + i sin =
5 5 5 5
"
29Ä„ 29Ä„
5
= 4 cos + i sin .
15 15
{w0, w1, w2, w3, w4}
"
5
(0, 0) 4
z1 = |z1| (cos Õ1 + i sin Õ1) z1 = |z1| (cos Õ1 + i sin Õ1)
z1 · z2 = |z1||z2| [cos(Õ1 + Õ2) + i sin(Õ1 + Õ2)]
z2 = 0

z1 |z1|
= [cos(Õ1 - Õ2) + i sin(Õ1 - Õ2)] .
z2 |z2|
"
z1 = 2 - 2i 3 z2 = 1 + i
5 5
z1 = 4 cos Ä„ + i sin Ä„ z2
3 3
"
(1, 1) 2
Ä„
y = x
4
"
1 1
0X z2 = 2 cos Ä„ + i sin Ä„
4 4
z1 z2
5 5
4 cos Ä„ + i sin Ä„
z1 3 3 4 5 1 5 1
"
= = cos Ä„ - Ä„ + i sin Ä„ - Ä„ =
z2 " 1 1 3 4 3 4
2
2 cos Ä„ + i sin Ä„
4 4
"
17 17
= 2 2 cos Ä„ + i sin Ä„ .
12 12
z1
z2
z1 z2
" " " "
z1 2 - 2i 3 (2 - 2i 3)(1 - i) 2 - 2i - 2i 3 + 2i2 3)
= = = =
z2 1 + i (1 + i)(1 - i) 1 - i2
" "
" "
2 - 2 3 - i(2 + 2 3)
= 1 - 3 - i(1 + 3)
2
" " 2 " " " "
2
1 - 3 + 1 + 3 = 1 - 2 3 + 3 + 1 + 2 3 + 3 = 8 = 2 2
Å„Å‚ " " "
ôÅ‚cos Õ = 1 - 3 2 - 6
ôÅ‚
"
òÅ‚ =
4
2 2" " " .
ôÅ‚
ôÅ‚sin Õ = -1 + 3 = - 2 + 6
ół
"
4
2 2
ez = ex+iy = ex · eiy = ex(cos y + i sin y).
"
z5 - 2 + 2i 3 = 0
"
z5 = 2 - 2i 3,
"
5
z = 2 - 2i 3.
"
2 - 2i 3
" " "
"
5 5
5
4 4· 3 11Ä„ 11Ä„
4 cos + i sin
15 15
" 17Ä„ 17Ä„ " z = 2 + i 2 z = "
5 5 5
23Ä„ 23Ä„ 29Ä„ 29Ä„
z = 4 cos + i sin z = 4 cos + i sin z = 4 cos + i sin
15 15 15 15 15 15
w3 - 1 = 0
w3 - 1 = (w - 1)(w2 + w + 1) = 0 Ð!Ò! w = 1 w2 + w + 1 = 0 .
 w2
"
" " -1+"3i+ w + 1 = 0 3 -1-"3i 1 "3 " = -3 = 3i2
1
" = Ä… 3i w = = -"+ w = = -
2 2 2 2
" "-2 2
"
" = 3i " = - 3i
"
3
z = 1 z = -1 +
2 2
"
3
z = -1 -
2 2
"
3
w = 1.
z = 1 = 1 + 0i
OX
1 = cos 0 + i sin 0,
"
0 + 2kĄ 0 + 2kĄ 2kĄ 2kĄ
3
1 = 1 · cos + i sin = cos + i sin , k " {0, 1, 2}.
3 3 3 3
k = 0 w = cos 0 + i sin 0 = 1
"
2Ä„ 2Ä„ 3
k = 1 w = cos + i sin = -1 +
3 3 2 2
"
4Ä„ 4Ä„ 3
k = 2 w = cos + i sin = -1 -
3 3 2 2
"
Å„Å‚ Å„Å‚
1
2
ôÅ‚ ôÅ‚
òÅ‚ òÅ‚
sin x =
sin x = -
sin x = 0
2
"
2
"
3
cos x = -1
ôÅ‚ ôÅ‚ 2
ół ół
cos x =
cos x =
2"
2
"
Å„Å‚ Å„Å‚ Å„Å‚
1
2
ôÅ‚ 3 ôÅ‚ ôÅ‚
òÅ‚ òÅ‚ òÅ‚
sin x = -
sin x = -
sin x = -
2
"
2
"
2
1 3
ôÅ‚ ôÅ‚ 2 ôÅ‚
ół ół ół
cos x = - cos x =
cos x = -
2 2
2
z1 = 2 + 3i z2 = 1 - 2i z3 = 5 + 12i z4 = (-2, 1)
z1 + 3z2 2z4 - z1 z3 · z1
z1 "
z12 + z23 z2
z2
x y
(2 + 3i)x + (5 - 2i)y = -8 + 7i 2x2 + y2 - 2yi = 12 - 4i
x y 1 + yi
+ = 1 = 3i - 1
2 - 3i 3 + 2i x - 2i
2
3 + i 4 - i
(2 + yi) · (x - 3i) = 7 - i x + y = 1 + i
3 - i 1 - 3i
x = 1 y = -2 x = 2 y = 2 x = -2 y = 2 x = 2 y = 3 x = 5 y = 17
x = 2 y = 3
" " "
1 3 2 2
z = 1 + i z = i - z = - - i
2 2 2 2
z = 2010 - 2010 i z = -16 z = -2i
"
Ä„ Ä„
z = -1 - i 3 z = 1 + itgÄ…, Ä… " [0, ] z = tgÄ… + i, Ä… " [0, ]
2 2
" Ä„ Ä„ "
5Ä„ 5Ä„ 5Ä„ 5Ä„ 7Ä„ 7Ä„
2(cos + i sin ) cos + i sin cos + i sin 2010 2(cos + i sin )
4 4 6 6 4 4 4 4
3Ä„ 3Ä„ 4Ä„ 4Ä„ 1
16(cos Ä„ + i sin Ä„) 2(cos + i sin ) 2(cos + i sin ) (cos Ä… + i sin Ä…)
2 2 3 3 cos Ä…
1
(cos(Ä„ - Ä…) + i sin(Ä„ - Ä…))
sin Ä… 2 2
" 11 "
1 3 (1 + 3 i)14
(1 + i)2010 - i
2 2 (-1 - i)20
"
10
"
3 + 3i 3 3 - i
" -2 + 2 3 i
-2 + 2i
- 3 + i
" "
"
3 4
2i -4 1
" "
z3 + i 6 + 2 = 0 É3 + 1 = 0
x2 + 2x + 5 = 0 z4 + z2 + 1 = 0
"
(i - 1)3z3 = (1 + 3 i)6 zz + (z - z) = 3 + 2i
2 + i 1 - i
z2 + (1 + 4i)z - (5 + i) = 0 =
z - 1 + 4i 2z + i
"
z4 - 4i 3z2 - 16 = 0 (z2 - 6z + 11)(z3 + 1) = 0
" " " "
7 1
z = 1 - i z = -2 - 3i - i 1 + i 3 -1 - i 3 3 + i - 3 - i
6 6
" "
" "
1 3 1 3
3 + 2i 3 - 2i -1 + i - i
2 2 2 2
A = {z " C : zz + (Im z)2 e" 1
Ä„
A = z " C : |z - i| > 1 '" d" z < Ä„
4
4 + 3i
A = z " C : |z - 2| d" Re
2 + i
A = {z " C : z2 - 2i Re (z - i) Im (z + 4) e" 1}
sin 3x = 3 sin x - 4 sin3 x
cos 4x = cos4 x - 6 sin2 cos2 x + sin4 x
sin 6x = 6 cos5 x sin x - 20 cos3 x sin3 x + 6 cos x sin5 x
ctg4x - 6ctg2x + 1
ctg4x =
4ctg3x - 4ctgx