[iovoid]

Wouldn't there be exactly twice (or twice + 1, if you allow negative fractions) as much fractions, since fractions are represented as two positive numbers (plus a bit if you consider the sign).

(The encoding could be "represent both numbers in binary, put the denominator in the odd bits (LSB = first bit), and the numerator in the even bits" so 2/3 => 10/11 => 1110 => 14)