Digital System Research

Fractional Representations

Researchers at DSR develop new methods for representing and processing fractional quantities.

We promise that everything is different with residues! For example, when performing general purpose calculations, we usually turn to fractional formats such as floating point numbers. If we have N binary bits dedicated to the fractional portion of a binary value, then we have N number of distinct denominators for our fractions. On the other hand, if we have P number of digits dedicated to the fractional portion of a residue value, we get $latex 2^P-1&bg=f1f1f1&fg=000000&s=-2$ number of distinct denominators for our fractions!

For fixed radix systems, the number of denominators is the number of distinct powers of the fractional range. For residue fractions, the number of denominators is the total number of combinations of digit modulus, which yields a much larger number of fractional denominators. This means that residue fractions exactly represent ratios according to their modulus, such as $latex \frac{1}{2}&bg=f1f1f1&fg=000000&s=1$, $latex \frac{1}{3}&bg=f1f1f1&fg=000000&s=1$, $latex \frac{1}{5}&bg=f1f1f1&fg=000000&s=1$, $latex \frac{1}{7}&bg=f1f1f1&fg=000000&s=1$, $latex \frac{1}{P}&bg=f1f1f1&fg=000000&s=1$, etc., and also combinations of its modulus, like $latex \frac{1}{6}&bg=f1f1f1&fg=000000&s=1$, $latex \frac{1}{10}&bg=f1f1f1&fg=000000&s=1$, $latex \frac{1}{14}&bg=f1f1f1&fg=000000&s=1$, $latex \frac{1}{30}&bg=f1f1f1&fg=000000&s=1$, etc…

The bottom line is that residue fractions can exactly represent many more common ratios of integers than an equivalently ranged binary fraction.

Sorry, the comment form is closed at this time.

Delivering the next leap in computation™