## 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 $\pi$: