What CFs are particularly good at

– representing irrational numbers, in a way independent of a particular base.
– approximating irrational numbers, finding the best approximating fraction.

What they are not good at

– being added and multiplied. That is very complicated. Although possible, and maybe fine for computers.

Gosper’s batting average problem

Q. If a baseball player’s batting average is 0.334, what’s the smallest number of at-bats they could have? (batting average = number of hits/at-bats)

A. 287.
0.334 corresponds to an actual average in the range 0.3335-0.3345. The CFs for these values are
0.3335=667/2000=[0;2,1,666] and
0.3345=669/2000=[0;2,1,94,1,1,3].
This implies that the “simplest” number within the range is
[0,2,1,95]=69/287=0.334495.

references

The Elementary Arithmetic Operators of Continued Fraction.pdf