Continued Fractions - Converting to Decimal

Converting to Decimal

Suppose we want to output a decimal representation of $α = [a_{1}; a_{2}, . . .]$ . 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 $π$ , using the space-saving "/"-notation.

\begin{aligned} 3 & 7 & 15 & 1 & 292 & . . . \\ 0 / 1 & 1 / 0 & 3 / 1 & 22 / 7 \end{aligned}

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: $3 . . . .$

\begin{aligned} 3 & 7 & 15 & 1 & 292 & . . . \\ 0 / 1 & 1 / 0 & 0 / 1 & 10 / 7 \end{aligned}

We've deleted the old values for clarity, but in future we shall preserve them. Continuing for a few more steps:

Output: $3.141 . . .$

\begin{aligned} 3 & 7 & 15 & 1 & 292 & . . . \\ 0 / 1 & 1 / 0 & 0 / 1 & 10 / 7,30 / 7,20 / 7 & 150 / 106,440 / 106,160 / 106 & 180 / 113 \end{aligned}

Conversion From Decimal

Converting to a nonsimple continued fraction is immediate, as seen by the example of $π$ :

\begin{aligned} π & = & 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}