[Doctor Stochastic]

Note that all rational numbers have a finite continued-fraction (qv) expansion. Quadratic irrationalities have periodic continued fractions. Other algebraic irrationalities and most transcendentals have non-repeating continued fraction expansions.

The decimal (or binary, etc.) expansions are not all there is.

[betty boop]

The decimal (or binary, etc.) expansions are not all there is.

Understood, Doc. But my own experience does not extend much beyond working in terms of the decimal and binary systems (and I'm still "binary light"). And so I am thankful for your discussion of finite-continued, periodic-continued, and non-repeating-continued fraction expansions. There is "no decimal in the binary system," and so I wondered what binary expansions describing, say, a transcendental would look like. I think you may have given me a clue.

[RightWhale]

It is beneath the dignity and interest-level of philosophers to discuss math. Or science. They will construct arguments for a fee.

[The_Reader_David]

"Most transcendentals?" How about all? There is a construction for a continued fraction expansion from a decimal or binary expansion (for positive reals: take integer part, reciprocate the fractional part, iterate). In any event the problems persist: there are uncountably many continued fractions expansions. Most of them are not only transcendental but can't be generated algorithmically and thus require an infinite amount of data to specify, etc. I'm thus not sure what the point of confusing the rhetorical and philosophical issue by moving from the familiar from school mathematics decimal and binary expansions to continued fractions was.

Sure you get rid of base-dependence, but if one treats 'terminating' decimal, binary, ternary, . . . expansions as ending in a repeated string of zeros, the set of numbers with repeating place-value expansions is base independent--the rationals--and the really intersting break, between numbers which admit place-value expansions generated by a Turing machine and those which don't is the same as the break between numbers which admit continued fractions expansions generated by a Turing machine and those which don't.