On Irreversible Operations in Arithmetic: A Conjecture on Zero Multiplication

Abstract

This work proposes an intuitive conjecture about the irreversibility of transformations involving multiplication by zero. In a sequence of operations, if one involves multiplication by zero, the original input cannot be recovered from the result. This conjecture is framed as a basic arithmetic idea, with connections to broader concepts like non-invertible functions and information theory.

The Conjecture

Let $x$ be a real number transformed into $y$ by a sequence of mathematical operations. If one of the operations is a multiplication by zero, then the sequence cannot be reversed to recover $x$ from $y$.

Restated: If $f(x) = y$ involves a multiplication by $0$ at any step, then $f^{-1}(y)$ does not exist.

Examples

• Let $x = 42$. Apply the operations: $(x \times 2) \times 0 = y$.
  Result: $y = 0$. Regardless of the initial value $x$, the output is always $0$, and no amount of back-calculation can restore the original $x$.

• Let $x = -7$. Apply: $((x + 5)^2) \times 0 = y$.
  Result: $y = 0$. Again, $x$ is unrecoverable from $y$.

Discussion

The conjecture relates to the mathematical property of multiplication by zero, which annihilates all numerical information. This aligns with concepts like:

• Linear Algebra: Linear maps with non-trivial kernels are non-invertible.
• Information Theory: Irreversible transformations that erase information cannot be undone.
• Computing: Some irreversible computations involve the loss of intermediate states.

License and Intent

This paper is shared under the Creative Commons CC0 1.0 Universal (CC0 1.0) Public Domain Dedication. This means the author has waived all rights to the work, and it can be freely used, modified, and distributed for any purpose, without any restrictions.

License URL

You can read more about the CC0 1.0 Universal license here:
https://creativecommons.org/publicdomain/zero/1.0/