循環小數的條件

其實ａ並不會影響ｘ／ｙ是否循環小數，只有ｂ會影響，所以我們只會考慮ｂ

假設　１／ｂ不是一個循環小數
，這樣１／ｂ可以寫成

1/b = D1D2D3......Dk．N1N2.......Nj where D,N is digits　”．”是分數點

我們每邊乘　10^j
因此
10^j/b = D1D2.....DkN1N2........Nj

b=10^j/Z where Z = D1D2.......Nj

因為b是整數，所以
10^j必須可以寫成　Zx10^i
所以
b=10^i=2^mx5^n

2006-11-17 14:09:26 補充：
對唔住，因剛才趕住出去，故沒有清楚說明，現在補充其實，因為不考慮ａ故1/b=0.N1N2N3......Nj我們每邊乘　10^j因此10^j/b = N1N2........Nj=Zbxz=10^j=2^j x 5^j所以設z=2^k．5^ithen b =2^(j-k)．5^(j-i) 是　２＾ｍ．５＾ｎ的形式

The decimal expansion of non-negative real number x will end in zeros (or in nines) if, and only if, x is a rational number whose denominator is of the form 2n5m, where m and n are non-negative integers.
Proof:
If the decimal expansion of x will end in zeros, or
圖片參考：http://upload.wikimedia.org/math/5/5/a/55a5eb3ea295d08e430df806323381c3.png
for some n, then the denominator of x is of the form 10n = 2n5n.
Conversely, if the denominator of x is of the form 2n5m,
圖片參考：http://upload.wikimedia.org/math/4/6/2/462e64d31f6706d8de9167bfb22377e4.png
, x will end in zeros.