James Propp <jamespropp@gmail.com>
There's a unique way to 3-color the rationals in [0,1] using the colors red, blue, and green so that 0 is red, 1 is blue, and the fractions a/b, c/d, and (a+c)/(b+d) all have distinct colors whenever ad-bc=1.
Puzzle (no spoilers till Sunday please!): What color is 355/113?
Argh: 355/113 isn't in [0,1]; replace it by 113/355.
Either way, it's blue. Odd/odd is blue. Even/odd is red. Odd/even is green. Even/even can be reduced to one of the above. I have no idea how to prove that this coloring is unique, however. Do you have a proof? I noticed, years ago, that the rationals have these three parities, so that was the first thing I tried. I agree with RCS that this is unambiguous over the whole of the rationals. I don't understand why you think there's any ambiguity with negatives. Just as negative integers have the same parity as if they were positive, so do negative rationals. Extra credit: Is there any color or pairs of colors that form a group under multiplication? Is there any color or any pair of colors that forms a group under addition? Can this coloring be extended in any natural way to any irrationals? You mention 355/113, which is best known as one of the convergents to pi. Are there any irrational numbers whose convergents are all the same color? (Pi isn't one.) If so, it might make sense to regard the number as having that color. Especially if it wouldn't break any group of rationals with that color.