使用四則運算中的除法，就可以把分數化成小數。在下面，我們會學習把循還小數化成最簡的分數的方法。
1. |
設 |
\(x \) |
\( = 0. \dot{5}\) |
|
即 |
\(x \) |
\( = 0.555 \; 555 \dotso\) |
(1) |
|
\(10\)\(x \) |
\( = 5.555 \; 555 \dotso\) |
(2) |
||
(2) \(-\) (1) |
\(9x \) |
\( = 5\) |
||
\(x \) |
\( = \displaystyle{\frac{5}{9}}\) |
|||
\(\therefore\) |
\(0. \dot{5} \) |
\( = \displaystyle{\frac{5}{9}}\) |
2. |
設 |
\(x \) |
\( = 0. \dot{3}1\dot{5}\) |
|
即 |
\(x \) |
\( = 0.315 \; 315 \dotso\) |
(3) |
|
\(1000\)\(x \) |
\( = 315.315 \; 315 \dotso\) |
(4) |
||
(4) \(-\) (3) |
\(999x \) |
\( = 315\) |
||
\(x \) |
\( = \displaystyle{\frac{315}{999}}\) |
|||
|
\( = \displaystyle{\frac{35}{111}}\) |
|||
\(\therefore\) |
\(0. \dot{3}1\dot{5} \) |
\( = \displaystyle{\frac{35}{111}}\) |
3. |
設 |
\(x \) |
\( = 0. 4\dot{2}\) |
|
即 |
\(x \) |
\( = \; 0.422 \; 222 \dotso\) |
(5) |
|
\(10x \) |
\( = 4.222 \; 222 \dotso\) |
(6) |
||
(6) \(-\) (5) |
\(9x \) |
\( = 3.8\) |
||
\(x \) |
\( = \displaystyle{\frac{3.8}{9}}\) |
|||
|
\( = \displaystyle{\frac{19}{45}}\) |
|||
\(\therefore\) |
\(0. 4\dot{2} \) |
\( = \displaystyle{\frac{19}{45}}\) |
4. |
設 |
\(x \) |
\( = \;\; 0. \dot{4}\dot{2}\) |
|
即 |
\(x \) |
\( = 0.42 \; 42 \; 42 \dotso\) |
(7) |
|
\(100\)\(x \) |
\( = 42.42 \; 42 \; 42 \dotso\) |
(8) |
||
(8) \(-\) (7) |
\(99x \) |
\( = 42\) |
||
\(x \) |
\( = \displaystyle{\frac{42}{99}}\) |
|||
|
\( = \displaystyle{\frac{14}{33}}\) |
|||
\(\therefore\) |
\(0. \dot{4}\dot{2} \) |
\( = \displaystyle{\frac{14}{33}}\) |
從例題三和例題四，我們得到以下的結果：
1.\(0. 4\dot{2} = \displaystyle{\frac{19}{45}}\)
2.\(0. \dot{4}\dot{2} = \displaystyle{\frac{14}{33}}\)
因此，我們可以看到
\(0. 4\dot{2} \ne 0. \dot{4}\dot{2}\)。