Nombres complexes

x3= 15x+ 4
√−1R
ix2= 1 √
√−1
C
Ca+ ib a b
x2=−1
a b z a + ib
b= 0 z=a a = 0 z= ib
R⊂C
N⊂Z⊂Q⊂R⊂C
R C i2=−1
z+z0=... +
C
C
C
z a + ib=<(z)+i=(z)
z
•a+ ib= 0 ⇐⇒ (a= 0 b= 0) i ∈R
z=z0⇐⇒ z−z0= 0 ⇐⇒ ...
z a + ib a
z<(z)b=(z)
<(1 + 2i) = 1 =(1 + 2i) = 2 2i
z=z0⇐⇒ (<(z) = <(z0)=(z) = =(z0))

(O;~u, ~v)M(x;y)
(O;~u, ~v)z=x+iy
M(x;y)
z=x+ iy
C
−→
wa
b−→
w
z−→
w=a+ib
M−−→ OM
M−−→ OM
A B zAzBz−−→
AB =zB−zA
A B C
~u ~v
zA=−3+i zB= 2i zC= 2 z~u = 1 + 2i z~v =−2i
z¯z
A z A0
z0A A0(Ox)z0= ¯z
z+z0zz0znz
z0
z∈R⇐⇒ ¯z=z z ∈iR⇐⇒ ¯z=−z
•z= ¯z⇐⇒ ··· ⇐⇒ 2ib= 0 ⇐⇒ b= 0 ⇐⇒ z∈R¯ λz =λ¯z
λ∈R
<(z) = z+¯z
2=(z) = z−¯z 2i

z¯z∈R+z¯z=a2+b2
|z|√z¯z
R
|z|=pa2+b2
z z0
zz0= 1 z0=1
|z|¯z1/z
z
1/z ¯z/¯z z
A zAB
zB
AB =k−−→ ABk=|z−−→
AB |=|zB−zA|
(O;~u, ~v)
OAB A B
zA= 1 + 2i zB=−2 + i
O
zz0z/z0
C
C
az2+bz +c= 0 a∈R∗, b ∈R, c ∈R
z2+ 2z+ 5 = 0
z2+ 2z+ 5 = (z+ 1)2−1 + 5 = (z+ 1)2+ 4 = (z+ 1)2−(2i)2
z2+ 2z+ 5 = 0 ⇐⇒ (z+ 1 −2i) ·(z+ 1 + 2i) = 0 ⇐⇒ z∈ {−1 + 2i ; −1−2i}
C
z2−4z+ 5 = 0 z2+ 6z+ 11 = 0 2z2−4z+ 3 = 0

az2+bz +c∆ = b2−4ac
∆<0∆
4a2
az2+bz +c
az2+bz +c=a(z−z1)(z−z1)z1∈C
az2+bz +c
Rb z
z2+ 3 = 0 ⇐⇒ z2−(i√3)2= 0 ⇐⇒ z∈ {i√3 ; −i√3}
C
2z2+ 4z+ 5 = 0 −2z2+ 6z−5 = 0 −5z2+ 2z+ 2 = 0
z2+z+ 1 = 0 (z2+2)(z2−4z+4) = 0 (z+ 1)2=−(2z+ 1)2
2z4−9z2+ 4 = 0 z2+ 12= 1 z−1
z+ 1 =z+ 2
(E)z4−5z3+ 6z2−5z+ 1 = 0
(E)(u2−5u+ 4 = 0
u=z+1
z
u2−5u+ 4 = 0 (E)C
C
z0(z−z0)
z P (z) = z3−12z2+ 48z−128 P(8)
a b c z
P(z) = (z−8)(az2+bz +c)
CP(z) = 0