Le corps des nombres complexes

C
R2
Ri i2=−1
C
R
i2=−1iR⊂C
{1, i}C R
E=R2
z= (a, b)E×E→E z = (a, b)z0= (a0, b0)
zz0= (aa0−bb0, ab0+a0b).
E E
z= (a, 0) R
Ra(a, 0) i:= (0,1)
z= (a, b)=(a, 0) + (0, b) = a+bi .
i2=−1E=R2
C
z=a+bi z=a−bi
z∈Rz=z z1+z2=z1+z2z1z2=z1z2.
|z|=√zz z =a+bi |z|=√a2+b2
|z| ≥ 0|z|= 0 z= 0 |·|:C→R+
z|z| 6= 0 z(|z|−2z)=(|z|−2z)z= 1 z
|z|−2zC
C1C2
R2R
i1i2(i1)2=−1 (i2)2=−1
Rf:C1→C2f(1) = 1 f(i1) = i2

z=a+bi
ez=ea(cos(b) + isin(b))
zexp(z)
eib = cos(b) + isin(b)
z1, z2ez1+z2=ez1ez2
ez1ez2=ea1(cos(b1) + isin(b1))ea2(cos(b2) + isin(b2))
=ea1+a2cos(b1) cos(b2)−sin(b1) sin(b2)+icos(b1) sin(b2) + cos(b2) sin(b1)
=ea1+a2(cos(b1+b2) + isin(b1+b2)) = ez1+z2
z ρ
α∈R/2πZθ∈Rα z =ρeiθ
α θ θ 2πZ
α
θ∈R
θ θ0a k ∈Z
θ0=θ+ 2kπ θ0≡θ(2π)
y=ρ−1z1y=a+bi
a2+b2= 1 θ a = cos(θ)b= sin(θ)
θ, θ02π
α=θ∈R/2πZ
z=ρy =ρ(a+bi) = ρ(cos(θ) + isin(θ)) = ρeiθ 
α z Arg(z)
z=ρeiθ ρ=|z|θ= Arg(z)z θ
2π
n≥1C C
n