Complexes

Table des matières
 ! ℂ " !#   $
    $
$!#   $
$%& '    $
$$(&&)   $
$*#'##') #   *
$+ ,)   *
#' "  ℂ   *
   +
$#"'   +
* -#'' '   +
'#'#.###   +
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/#'#'#) '#"  !#0"'#) '   1
""  !#   1
$(#) '"  !#   1
*/#'#'#) '#"  !#   1
0   1
+#' '"#' '#) '   2
+"   3
+$(#) '   3
+*% '#' 0   3
1(#) '' )# '   4
5/#'#  '"  !#&#"#   4
5''    4
5/#'#  '   4
5$# '   6 
5*7 #'#-&#   6 
5$8#"#   6 
5!#')'#    
5#    
5$#)##' ) )##    
5*!#''#) '#    
5+!#''# &#'    $ 
5+9# '    $ 
5+$:';'   * 
5+*''    1 
5' " "")#.&& '#"  ℂ   < 
Complexe, mais, pas compliqué...
 !#'"'=="  ""= '=
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##"#"' "'#A")#' !"5AA" '#"# 
!## ##"B
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 '#'%#'?5A@ # !#) #
';" ';!'""#'' "#-#
D20E"FGF6FE9E#E"'+D64D6

H& "5A"!'"A"'#';' I(#) "'J?
#" !#@ ' #;)'#" !#?>  !#'
## '#  '" @
 ' "## '  " ! # "#' K'×
%  !" '# '#ℕ' xB$C6 ' ' xK$C6 "
' 
 G'##ℕ#""' ? .  '# '#n  !#"n' 'Bn@
' # !ℤ" '##'&?ℕ⊂ℤ@
  "A##' $xB*C6 "' " ℤ
 G'##ℤ∗#'' ? .  '##'&n  !# -#"n'
'
n
@' # !ℚ"#' ?ℤ⊂ℚ@
' ! #'"#"#"' '
x$
C$ ## !ℝ"
# ' !##?ℚ⊂ℝ@
#' #'  
x$
 CB  '"' "  ℝ ! ℂ "
!# ' '' 
'A !# '#''0 '#-  '#''0 ';'-" '0
)A!# G)'#
 '#' '#  '
# ' '"&&# '&#0' 'L'&#)!#' 'L'&#'#) '#' 'L'
&#  '
I- L'ensemble ℂ des nombres complexes.
I-1- Le plan complexe
     " #A##'; #"#' 
OM u v
 (On ne prend pas les
vecteurs
i
et
j
, car, les lettres i et j sont réservées aux nombres complexes)
I-2- Nombre complexe
I-2-1 Définition
'
OM u v
 #A##'; #"#'
( 'A?M6@  !#
( 'B?6M@  !#
( 'M?aMb@  !#zCaKb.
I-2-2- Affixe- image
H'' !#z' '  'M"
z'&&"M. 
?  '- '
zA
&&"A@
z'&&"
OM
M' ')"z
OM
'-'#)"z
#' 'P'Q " &&"
PQ
'
';" ';!'""#'' "#-#
$D20E"FGF6FE9E#E"'+D64D6
 'M#&&zCaKb
b
a
0 1
1
x
y
M
A
B

z
PQ
C
zQ
B
zP
?
PQ
C
OQ
B
OP
@
I-2-3- Partie réelle- partie imaginaire
9' !#'##zCaKb7 a et b sont des réels.
a'la partie réelle"z 'r?z@
b'la partie imaginaire"z 'i?z@
aKb 'l'écriture algébrique" !#z
#aC6  imaginaire pur0zCB1zC$zC
– *
zC
)") ## ' '"?y'y@ ") ##
#bC6   !##0zCB6zC$zC
6
zCB*
)"# ' '"?x'x@ "#
  'ℂ !" !#
'  0ℝ⊂ℂ
I-2-4 Opposé- Conjugué.
zCaKbù a et b sont des réels.
Construction0
# '
M
"&&BaBb
M$
"&& Bb'
M*
"&&BaKb
Définitions:
'M  '"&&z
 '
M
G'#"M###'.O#&&Bz
?opposé"z@
 ' 
M$
 G'#" M ###'. (O,
u
) #
&&conjugué"z 'z.
 " 0zCaKb #zCaBb
Un complexe et son conjugué ont la même partie réelle et des parties imaginaires opposées.
II- Opérations dans ℂ
ℂ' ' '  "ℝ#A)" '"ℝ' ".ℂ #N '
$
#B
Il n'est pas nécessaire de retenir les formules,
"comprendre -' #'#sous la forme a + ib où a et b sont des réels.
';" ';!'""#'' "#-#
*D20E"FGF6FE9E#E"'+D64D6
axe des imaginaires purs
axe des réels
2 3-1-2
2
-1
-2
0 1
1
x
y
M
M
1
M
2
M
3
Propriété:
% !# ')' '  'O#'#'O#'
) #

% '"#)#;zCaKb'z'Ca'Kb'7aa'bb' '"#
II- 1 Somme
zz '
C
aa ' 
bb '
M"&&z'M'"&&z'# 'M""&&
zz'
'
 ''
OM ' '
C
OM
K
OM '
OMM ' ' M '
' #)#?'# 'OM'
M' ') @
?5# '##'' " )#@
II-2 Produit
zz '
C
aa ' – bb ' ab 'a ' b
Ne pas apprendre par coeur
Exemple: ?$B1@?*K$@C<K+B1B6PC<B
   '##''   )'#  "  #"'  # '  '"  " 
#'#"  #; #)#;?5#5+*@
II-3 Inverse et quotient
z≠6'."#0?aMb@≠?6M6@
z
C
aib
C
a – ib
a$b$
C
a
a$b$
– b
a$b$
Ne pas apprendre par
coeur, mais, comprendre que
aiba – ib
=
a2b2
est un réel non nul
z '
z
=
z '×
z
=#'"" #' '&#"#'#)!#" 
Exemples et cas particuliers à repérer.
*–$
+–
C
*–$+
+–+
C
$* –3$
<
C
+
2
B
1
2
*
CB
+
C
1
C
) # '0
+
C
+n
C
n
C
+ 
C
+ ×
C
+ $
C
+ ×$
CB
+ *
C
+ $×
CB
CB
z
C$B
z$
CB*K+
zz$
CBK*
z×z$
CB$K
iz
CK$?###'"'#"#'@
z× z$
C
$––*–+ =–6
B1
z
z$
C
$–
–*+
C
$––*–+
$1
C
–6 –1
$1
C
–$
1
–
1
II-4 Opérations et conjugués.
" '#'  '"'  ';!#
zz '
C
z
K
z '
z×z '
C
z
×
z '
';" ';!'""#'' "#-#
+D20E"FGF6FE9E#E"'+D64D6
2-1-2
2
3
4
5
6
-1
0 1
1
x
y
M
M'
O
M
"

z
z '
C
z
z'
z× z
C
a$b$
QK
z
C$aC$r?z@
QB
z
C$bC$i?z@
III- Écriture trigonométrique d'un nombre complexe: Module et argument
 '##'#A##'; #"#'
OM u v
;'#.&#";
  #  '#) '#  # "  !  "#'   '     )  "  #&#   "- '  O'#  ##
#" ' #) '#) '#
 -#" #-"- #-" '
III-1- Module d'un nombre complexe
'z  !#&&"  'M
""z '
∣ z ∣
' )#OM?
∣ ∣
OM
∣ ∣
(- '' "#)#;
∣ z ∣
C
a$b$
C
z z
"#A';#A"G';)#
Conséquence0'r #'#' ''& !" 'M"&&z'
∣ z ∣
Cr'
#" '#O'"#G r
III-2- Argument d'un nombre complexe
'z  !#non nul &&"  'M?6'# )# '"-'#  '
"& # -'#' @
> #) '"z '
#)z
' #" )# '?
u
OM
@
Rappel0# '"& "$
Conséquences0
@'  # !" 'M"&&z'#)?z@CR$S'""#'"#) O
-O ' ' -'#"#'#" 0?
u
@CR$S
&&'""'# #  'A'?
u
OA
@CR$SCSOA@
!@'  # !" 'M"&&z'#)?z@CRS'"#' '#O-
O ' ' -'#"#'#" 0?
u
@CRS
&&'""'# #  'A'?
u
OA
@CRSC?OA@ETOU
III-3- Écriture trigonométrique d'un nombre complexe
'  'M"&&   zCaKb
#" #"M '
[ ∣ z ∣ ,]
' 0zC
∣ z ∣  i 
Résumé:
z≠6zCaKbC
ri 
- r > 0 M)"z
#" #' 
MaMb
#" #"M
[r ,  ]
∣ z ∣
CrC
a$b$
#)z
CKk$7k∈ℤ
C
a
r
' C
b
r
';" ';!'""#'' "#-#
1D20E"FGF6FE9E#E"'+D64D6