Chapitre 7 – Les nombres complexes – Algèbre et géométrie

C
i i2=−1
Cz z =a+ ib a b
zRe(z)a
zIm(z)b
z z = Re(z) + i Im(z)
zIm(z)=0
z= Re(z)
R⊂C
zRe(z)=0
z= i Im(z)
•2 + i 0.5−2
3i−π+√2i
•3i −871i 21i
4

i j
z=a+bj
C
z1= 4 −iz2=−3−5i
•z1+z2= (4 −i) + (−3−5i) = 4 −3+(−1−5)i = 1 −6i
•z1−z2= (4 −i) −(−3−5i) = 4 + 3 + (−1 + 5)i = 7 + 4i
i2= 1
z1= 4 −iz2=−3−5i
z1×z2= (4 −i) ×(−3−5i)
= 4 ×(−3) + 4 ×(−5i) −i×(−3) −i×(−5i)
=−12 −20i + 3i + 5(i)2
=−12 −20i + 3i −5
=−17 −17i

z=a+ ib
z=a−ib
z= 14 −78i z= 14 + 78i
z=a+ ib z1=a1+ ib1z2=a2+ ib2
z=z
z1+z2=z1+z2
z1×z2=z1×z2
z1
z2=z1 z2
z×z=a2+b2
z=a+ ib=a−ib=a+ ib
z1+z2=a1+ ib1+a2+ ib2=a1+a2+ i(b1+b2) = a1+a2−i(b1+b2) = a1−ib1+a2−ib2
z1×z2= (a1+ ib1)×(a2+ ib2) = a1a2−b1b2+ i(a1b2+a2b1)
=a1a2−b1b2−i(a1b2+a2b1)
z1×z2=a1+ ib1×a2+ ib2= (a1−ib1)×(a2−ib2) = a1a2−b1b2−i(a1b2+a2b1)
z1×z2=z1×z2
z1
z2=a1+ib1 a2+ib2=(a1+ib1)(a2−ib2)
a2
2+b2 2=a1a2+b1b2
a2
2+b2 2
+a2b1−a1b2
a2
2+b2 2
i=a1a2+b1b2
a2
2+b2 2−a2b1−a1b2
a2
2+b2 2
i
z1
z2=a1+ib1
a2+ib2=a1−ib1
a2−ib2=(a1−ib1)(a2+ib2) (a2−ib2)(a2+ib2)=a1a2+b1b2
a2
2+b2 2
+a1b2−a2b1
a2
2+b2 2
i
z1
z2=z1 z2
z×z= (a−ib)(a+ ib) = a2+abi−abi−b(i)2=a2+b2
z= 3+2i z×z= (3+2i)×(3−2i) = 32+22= 9+4 = 13
z z ×z

z1=a1+ ib1z2=a2+ ib2
z1= 5 −4i z2=−2−3i
z1
z2
=5−4i
−2−3i
=(5 −4i)(−2 + 3i)
(−2−3i)(−2 + 3i)
=−10 + 15i + 8i −12i2
(−2)2+ 32
=−10 + 23i + 12
4+9
=2 + 23i
13
=2
13 +23 13i

(O;~u;~v)
z=a+ ib M (a;b)
a z b
M(x;y)z=x+ iy
M z z M
−−→
OM
OM M z =a+ ib OM =√a2+b2=√z×z
z|z|
−−→
OM
[OM]a
b OM2=a2+b2
OM OM =√a2+b2
a2+b2=z×z OM =√z×z
z=a+ ib a ∈Rb∈R
|z|=|z|=| − z|=| − z|