无限循环小数如何化分数?

如：
0.1111……=1/9
0.2222……=2/9
0.232323……=23/99
0.234234234……=234/999
0.02222……=2/90
0.00222……=2/900
自己整理吧！

如：
0.xxxxxxxxxx.....=x/9
0.xyxyxyxyxy.....=xy/99
0.xyzxyzxyz......=xyz/999
.......
依次类推

证明：0.xxxxxxxxxx.....=x/9
设: y=0.xxxxxxxxxx.....
10Y=x.xxxxxxxxxx.....
=x+0.xxxxxxxxx.....
即10y=x+y
9y=x
y=x/9 即0.xxxxxxxxxx.....=x/9

其它依次类推

由于水平有限,只能举例说明了,有不详之处,还请见谅!

纯循环:
0.1111111... == 1/9
0.5555555... == 5/9

混循环:
1.010101010... == 1+1/100+1/10000+1/1000000+...
== 1/( 1 -1/100 )
== 100/99
1.001001001... == ... == 1000/999
1.0001000100...== ... == 10000/9999
... ... ... ...
0.511511511... == 0.511 * 1.001001001...
== 0.511 * 1.001001001...
== 0.511 * 1000/999
== 511/999
2.7182818281828... == 2