Real numbers

Floating point numbers (real numbers in computer jargon) are a special form of rational numbers. The denominators of these rational numbers are powers of the base (2 or 10 for IEEE standard representations). Provided that n is relatively prime to the base, we can always compute their signatures. Notice however that there is no notion of approximation, i.e. we cannot conclude anything about |S_n(a)-S_n(b)|from $0 < \vert a-b\vert < \epsilon$.