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Universit´e Blaise Pascal
U.F.R. Sciences et Technologies
D´epartement de Math´ematiques et Informatique
2011-2012

Chapitre 1: les nombres entiers naturels
1 Propri´et´es de base de l’ensemble Ndes entiers naturels
1.1 La d´efinition des entiers naturels
N
∈N
n∈Ns(n)
s(n) = s(m)n=m
n∈N=s(n)
P⊆N∈P[n∈P⇒s(n)∈P]P=N
n n =
N
1.2 Les op´erations sur les entiers naturels
+×N
[∀n∈Nn+ = n] [∀n m ∈Nn+s(m) = s(n+m)]
[∀n∈Nn×= ] [∀n m ∈Nn×s(m) = n×m+n]
a b c ∈N
a+ (b+c)=(a+b) + c a +b=b+a
a×(b×c)=(a×b)×c a ×b=b×a
a×(b+c) = a×b+a×c
1.3 La relation d’ordre sur les entiers naturels
6
a b a b
a6b n ∈Na+n=b
a∈Na6a
a b ∈Na6b b 6a a =b
a b c ∈Na6b b 6c a 6c
+×
a b ∈Na6b c ∈Na+c6b+c a ×c6b×c

•total a b a 6b b 6a
•bon ordre N
•propri´et´e d’Archim`ede a b ∈Nb6=n∈Nnb >a
•toute partie non-vide et major´ee de Nadmet un plus grand ´el´ement.
1.4 La division euclidenne des entiers naturels
a6b a 6=b a < b a b
a6b b −a c b =a+c
Proposition. a b Nb6=q r N
a=bq +r r < b
(q r) (q0r0)
a=bq +r r < b a =bq0+r0r0< b
q q0q6q0
q0=q+ (q0−q)bq0+r0=bq +r b(q0−q) + r0=r
r06r r −r0=b(q0−q)r−r0< b r < b r0< b
b(q0−q)< b Nq0−q=q0=q
r=r0
B={k∈Nkb 6a}N
∈B a q
q qb 6a < (q+ )b r =a−qb r < b ut
2 Bases de num´eration
2.1 Syst`emes d’´ecriture des entiers naturels
s( ) s(s( )) s(s(s( )))
s( ) = s(s( )) = s( ) = s(s(s( ))) = s( ) =
s( ) = s( ) = s( ) =

a) Le syst`eme d´ecimal
s( ) = s( ) = s( ) = s( ) s( ) = s( ) =
×+×+×+×=
b) Le syst`eme binaire
s( ) = s( ) = s( ) = s( ) = s( ) =
s( ) = s( ) = s( ) = s( ) =
= + + + = = + ×
c)
♦ ♥
= = = = ♦=♥===
=×+×+ = = =
♦=×+ = ♥=×+×+ =
2.2 Existence et unicit´e de la repr´esentation en une base donn´ee
a) Point de d´epart.
s( ) =
s(n) = n+n∈N>
=s( ) + = s(s( )) + + = s(s(s( ))) + + + = s(s(s(s( ))))
IN∗N
I[[n m]] n6mN
a n 6a6m
b) Lemme. b∈Nb >
m∈N∗n∈Nbn6m < bn+
bn6m<bn+bn06m<bn0+
bn6m < bn0+n<n0+n6n0n06n n =n0
f:N→N∗x7→ bx
E=f(N)N∗
bxx∈Nb6=b6=x6=yN

bx6=byfNE
E m ∈N∗
Am={x∈Nbx> m}z
bz> m >z>z=n+n∈Nbn+> m
z Amn=z−/∈Ambn6m
bn6m < bn+ut
c) Th´eor`eme et d´efinitions. b∈Nb >
m∈N∗n∈N(a a an)
N
6ai< b 6i6n an6=m=
n
P
i=
aibi
Im=anabm b
Ib >
In+m b
b=b
(a a an)
(a a a +a+ + an)
m=akak−ab=chch−cb
h k ai6b−
m=
k
P
i=
aibi6
k
P
i=
(b− )bi= (b− )
k
P
i=
bi=bk+−
m < bk+ak6=ak>
m=
k
P
i=
aibi>akbk>bk
bk6m<bk+bh6m<bh+k=h
m=akak−ab=chch−cb
m=
k
P
i=
aibi=
k
P
i=
cibi(?)
p∈[[ k]] ap6=cpE={p∈[[ k]] ap6=cp}
q
< q < k (?)
aqbq+
k
P
i=q+
aibi=cqbq+
k
P
i=q+
cibi
bqA=Pk
i=q+aibi−q−C=Pk i=q+cibi−q−
aq+bA =cq+bC aq< b cq< b N
aq=cqq∈E
q=k(?)aqbq=cqbqaq=cq
q= (?)
a+
k
P
i=
aibi=c+
k
P
i=
cibi
a=c k =k > a +bA =c+bC
E6=∅
ap=cpp∈[[ k]]