Converting to Decimal
Suppose we want to output a decimal representation of \(\alpha = [a_1;a_2, ...]\). Then we start computing convergents via a table with one change: when the last two convergents have the same integer part \(n\), we output and subtract \(n\), then multiply the numerator by 10. We demonstrate this on \(\pi\), using the space-saving "/"-notation.
3 |
7 |
15 |
1 |
292 |
… |
|||
0/1 |
1/0 |
3/1 |
22/7 |
The last two convergents both floor to 3, so we output 3, subtract it from both convergents to get \(0/1, 1/7\), and multiply by 10:
Output: \(3. ...\)
3 |
7 |
15 |
1 |
292 |
… |
|||
0/1 |
1/0 |
0/1 |
10/7 |
We’ve deleted the old values for clarity, but in future we shall preserve them. Continuing for a few more steps:
Output: \( 3.141...\)
3 |
7 |
15 |
1 |
292 |
… |
|||
0/1 |
1/0 |
0/1 |
10/7 |
150/106 |
180/113 |
|||
30/7 |
440/106 |
|||||||
20/7 |
160/106 |
Conversion From Decimal
Converting to a nonsimple continued fraction is immediate, as seen by the example of \(\pi\):