Ecriture décimale illimitée d'un rationnel

2
3= 0,66666...
2
3= 0,[6]
4
5= 0,8 0,80 0,800 0,8000 0,8[0]
a
bb6= 0 a b
b
7
4= 1,75 = 1,75[0]
b

155
198 = 0,7[82] [82]
0,7
b b b −1
b−1
b−1
15
7= 2,[142857]
x a
b k 10−n
b
x= 12,23[47] a= 12,23 b= 47 k= 2 n= 4
x=a+b.10−n+b.10−n−k+b.10−n−2k+b.10−n−3k+...
x=a+b.10−n.[1 + (10−k) + (10−k)2+ (10−k)3+...]
10−k
1
1−10−k=10k 10k−1
x=a+b.10k−n
10k−1x
x= 12,23[47] = 12,23 + 47 ×102−4
102−1=1223 100 +47 ×10−2 99 =1223 100 +47 9900
x=30281
2475
x= 4,029029029... = 4,[029]
a= 4 b= 29 k= 3 n= 3
x= 4 + 29 ×103−3
103−1= 4 + 29 999 =4025 999

x= 33,33333...
n
a= 0 b= 3 k= 1 n=−1
x= 0 + 3×102
10 −1=300 9=100 3x= 33,[3]
a= 33 b= 3 k= 1 n= 1
x= 33 + 3×100
10 −1= 33 + 3 9= 33 + 1 3=100 3a
b= 9 k= 1 x=a+9×101−n
10 −1=a+ 101−n
2,3[9] = 2,3 + 10−1= 2,4
2,4[0] 2,4 2,3[9]
0,99999... = 1