A system based on the digital root: structure, properties and first observations

In recent months I have been working on the formalization of a mathematical idea that starts from an extremely simple concept, yet leads to structurally interesting results. I decided to publish a first version (v1) of the work on Zenodo, with the intention of developing it further in subsequent versions.

In this article I will not go into the full formal details. The goal is instead to provide a general overview of the system, highlight some of its key properties, and show why, in my view, it deserves attention.

The basic idea

The starting point is the digital root, a classical function in arithmetic.

Given a natural number, the digital root is the value obtained by iteratively summing its digits until a single digit between 1 and 9 is reached.

For example:

• 13 → 1 + 3 = 4
• 29 → 2 + 9 = 11 → 1 + 1 = 2

Starting from this function, I defined a new transformation:

take a number n and compute the remainder of n divided by its digital root.

In the full work, this transformation is formalized as an arithmetic function with precise properties

A simple function, but with non-trivial effects

At first glance, the definition may seem elementary. However, when analyzing the behavior of the function over large sets of numbers, very particular characteristics emerge.

1. A restricted set of values

The outputs of the function do not freely span all natural numbers, but are confined to a limited set.

Even more interesting is the fact that, in base 10, a specific value never appears among the results. This exclusion is not empirical, but follows directly from the structure of the system.

2. Convergence in two steps

One of the strongest properties is the following:

if the function is applied twice in succession, the result is always zero.

In more formal terms, the function is nilpotent of index 2. This means that any number, after two iterations, always collapses to the same final point.

3. Asymmetric distribution of results

When analyzing the function over very large sets of numbers, it becomes clear that the outputs are not uniformly distributed.

In particular:

• the value zero is dominant
• some values appear with noticeable frequency
• others are very rare
• one is completely absent

In the work I also show that more than half of the numbers belong to the class that directly produces zero.

Internal structure of the system

Numbers naturally organize themselves into classes determined by their digital root.

Within each class, we observe:

• regular patterns of behavior
• recurring structures
• direct connections with congruences modulo 9

This indicates that the system has a well-defined internal structure and is not random.

Generalization

The system is not limited to base 10.

It can be extended to any numerical base, redefining the digital root accordingly. In this broader setting, many of the fundamental properties still hold, including the structural exclusion of a specific value.

Motivation behind the work

This project originates from a simple question: what happens when elementary arithmetic operations are combined in a non-standard way?

The result is a system that exhibits:

• non-trivial global behavior
• extremely fast convergence
• a strongly asymmetric distribution

The first version represents a starting point for further development.

Future developments

In future versions of the work, I plan to explore:

• more precise distribution analysis
• extensions of the system
• connections with other arithmetic structures
• possible dynamic interpretations

This system shows how even very simple concepts can lead to mathematical structures with surprising properties.

It is still an evolving work, but already in its current form it highlights elements worthy of attention and further study.

To follow, read or download the complete work go to this link Zenodo .

These are the first 40 transformed numbers, just as an example, for the complete “system” follow the Zenodo link.

(and –> GitHub )

Based on this theory I created this tool for demonstration purposes only https://numblfluid.it/compressor/