Fractional representations in RNS

Combinatorial Precision in Modular Fractions

One of the most fascinating and underexplored properties of modular computation is the ability to construct fractional representations with exact rational denominators—something that fixed-radix systems like binary or decimal cannot do beyond simple powers of their radix. In modular arithmetic using the Residue Number System (RNS), it is possible to insert a “fraction point” to divide the moduli set into integer and fractional domains. By carefully selecting the moduli in the fractional range as powers of distinct small primes, RNS enables exact representation of a rich set of rational numbers like 1/2, 1/3, 1/5, 1/7, and even more complex combinations like 1/30 or 1/45.

Unlike fixed-radix systems, where only one base (e.g., 2 in binary) contributes to the denominator space, RNS fractions compose multiple distinct base primes, enabling combinatorial richness in the exact fractional space.

Let:

$\mathcal{M} = { p_1^{n_1}, p_2^{n_2}, \dots, p_P^{n_P} }$

be the set of moduli in the fractional range, where

$p_i$

is a unique base prime, and

$n_i$

is the number of power levels of that base prime.

Then the total number of unique exact fractional denominators is given by:

$D_{\text{total}} = \sum_{\emptyset \neq S \subseteq {1, 2, \dots, P}} \prod_{i \in S} n_i$

This expression accounts for all non-empty combinations of base primes, multiplied by the number of available powers in each group. The combinatorial structure enables a wide coverage of rational space—far richer than is possible in binary systems with the same number of fractional digits.

A New Frontier in Arithmetic Precision

To our knowledge, this property of modular fractions—namely the ability to form combinatorially rich and exact rational denominators—has not been previously analyzed or documented in numerical mathematics literature. Yet it opens a potentially transformative avenue for high-precision arithmetic, especially in numerically sensitive domains.

For example, in chaotic systems or orbital simulations, the difference between a rounded and an exact constant can mean divergence versus stability. One known case involves Newton’s formula for selecting the best initial estimate in an iterative method, which includes a constant with a denominator of 17. If the prime 17 is included in the fractional moduli set, that constant becomes exact in RNS representation—offering potentially improved accuracy in convergence or root-finding.

The foundational structure of mathematics is built not just on numbers, but on relations — and the most fundamental of those are low-order ratios composed of small primes. RNS, when constructed from those very primes, doesn’t just represent numbers efficiently — it mirrors the structure of rationality itself. Meanwhile, binary fractions — powerful as they are — only approximate that deep lattice by repeated halving, never quite reaching the rich internal geometry of the rational plane that small primes define.

This suggests that modular computation may offer a form of arithmetic selectivity, where critical constants can be deliberately represented with zero error. Over time, this may provide a new approach to controlling numerical drift, rounding artifacts, or error propagation—especially in simulation, scientific computing, and cryptographic applications.