# 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 $$\pi$$:

\begin{aligned} \pi &=& 3 + \frac{1}{0 + \frac{10}{1 + \frac{1}{0 + \frac{10}{4 + ...}}}} \\ &=& 3 + \frac{1}{0 + \frac{1000}{141 + \frac{1}{0 + \frac{1000}{592 + ...}}}} \end{aligned}

Ben Lynn blynn@cs.stanford.edu 💡