Richard Guy wrote: << Let me put in my 9.9999999... cents worth, though I'm not sure it's suitable for the classroom. Define the reals as (infinite) negative continued fractions. Each real has a unique expansion. The rationals are those with only finitely many nontwos. There's no trouble of the 1.000000... =? 0.9999999... kind. Proofs of Cantor's theorems are very simple. Incidentally, if you want to pursue the theory, you don't have to worry about whether it's (-1)^n or (-1)^{n+1} -- the signs are always the same. R.
Found on the Web: << According to the mathematician Richard Guy, the theory of continued fractions can be developed equally well or better by using bi = −1 or equivalently, reversing the + signs to −. This modification is not completely trivial and approximates a given number with different convergents than the regular version. However, references on using continued fractions this way are scarce, and the first part of the project will be to review what is known about this kind of continued fraction and to decipher Richard Guy’s conversational remarks.
--Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele