Andrzej Odrzywołek, a postdoctoral researcher at the Institute of Theoretical Physics, Jagiellonian University, has proposed a mathematical framework that reduces all elementary functions to a two-button system.
In a recently updated, yet-to-be-peer-reviewed paper, Odrzywołek suggests that a single binary operator can perform the same tasks as a standard scientific calculator. The proposed system relies on just two inputs: the operator itself and the constant 1.
The core of the proposal is the 'exponential-minus-log' function, or eml(x, y), defined as exp(x) - ln(y). Odrzywołek claims this single operator can generate everything from basic arithmetic to complex trigonometry.
A single operator for complex functions
According to the paper, the eml operator can produce sine, cosine, and tangent functions, as well as algebraic operations like addition, subtraction, multiplication, and division. It also allows for the derivation of fundamental constants including π, e, and the imaginary unit i.
Odrzywołek argues that this simplicity could transform scientific computing. "A calculator with just two buttons, EML and the digit 1, can compute everything a full scientific calculator does," the paper states.
He suggests that because one repeatable element suffices, mathematical expressions can be treated as uniform circuits, similar to how electronics are built from identical transistors. This could lead to new methods for encoding and discovering formulas.
To address the lack of a universal definition for elementary functions, the researcher focused on a list of the 36 most commonly used functions. The goal was to prove that every function on this list can be expressed through a finite composition of the eml operator and the number 1.
While the concept has gained traction in discussions on platforms like Hacker News, the paper acknowledges certain limitations. Odrzywołek noted that direct symbolic verification via formal proof remains "intractable."
