# Continued Fraction

- math. continued fraction.
- Mathematician Bill Gosper made significant contribution.

- Continued fractions.
- For rational numbers, it's finite.
- For irrationals, it's infinite.
- If it repeats, it's a quadratic irrational. That is, number of the form
`(a + b * Sqrt[c])/d`

- interesting property of continued fraction for rationals.

now, the incomprehensible part of continued fractions.

- Stil, interesting is quadratic rationals, dyadic rationals (binary exact numbers), Minkowski question mark function.
- And that something with Möbius transformations is super interesting.

- the cumulation of my math study today
- https://twitter.com/xah_lee/status/1717784384273785250

ContinuedFraction[ 6961/9976 ] === {0, 1, 2, 3, 4, 5, 6, 7} FromContinuedFraction[ {0, 1, 2, 3, 4, 5, 6, 7} ] === 6961/9976 Fold[ Function[{a,b}, HoldForm[ b + 1/a ] ], Reverse[ ContinuedFraction[ 6961/9976 ] ] ] (* typeset Continued Fraction *) Fold[ Function[{a,b}, b + 1/a ], Reverse[ {a,b,c,d} ] ] (* a + (b + (c + d^(-1))^(-1))^(-1) *) ContinuedFraction[ Pi//N ] (* {3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14} *)

continued fractions