the.com/continued fraction
A number confessing itself one reciprocal at a time, forever if it has to.
means A way of writing a number as a whole part plus 1 divided by (another whole part plus 1 divided by (another whole part plus...)), nesting fractions inside fractions instead of using decimals.
from The structure was worked with by mathematicians in India and the Islamic world centuries before Europe caught on, but the term and its systematic theory took shape in 17th-century Europe, with figures like Cataldi and later Euler formalizing it — 'continued' simply marking that the fraction-within-a-fraction process keeps going, continuing rather than terminating in a single division.
- pi's version
- [3;7,15,1,292,...] — no pattern, ever found
- golden ratio
- all 1s forever — the slowest, purest fraction
- best approximations
- truncating it beats any same-size decimal guess
- rational giveaway
- the expansion simply stops, unlike irrationals