Designing a Minimalist Calculator with Only Two Buttons
Physicist Andrzej Odrzywolek demonstrates that scientific calculators can function using only one operator and number.
1Theoretical Minimalist Computing
Physicist Andrzej Odrzywolek proposed a radical simplification for scientific calculators in a March 2026 preprint. By utilizing only the number 1 and a specific operator, eml(x, y) = exp(x) ln(y), he demonstrated that all 36 standard scientific functions can be reconstructed. This theoretical model challenges the necessity of complex, multi-button hardware interfaces.
2Mathematical Reduction Logic
Odrzywolek analyzed 36 common operations found on school calculators, including trigonometric functions and exponentiation. He discovered that these functions are often redundant, as many can be reduced to complex exponentials. By systematically searching for a universal operation, he identified the eml function as a sufficient basis for generating all required mathematical constants.
3Historical Technical Precedents
The concept of reducing complex systems to a single operator mirrors Henry Sheffer’s 1913 proof regarding NAND gates. Just as digital logic can be performed using only one type of transistor, Odrzywolek’s work suggests that arithmetic operations can be derived from a single binary operator. This approach simplifies the underlying instruction set architecture significantly.
4Implementation and Verification
To validate his hypothesis, Odrzywolek used a computer program to explore all possible combinations for deriving elementary functions. The results were verified by a proof checker, ensuring logical correctness. Generating simple values like Euler’s number requires minimal steps, while more complex functions like sine necessitate nesting the eml operator more than ten times.
5Practicality and User Experience
While theoretically sound, a two-button calculator is highly impractical for human users. Generating the number zero requires a lengthy, nested calculation sequence involving multiple eml operations. The design prioritizes mathematical elegance over usability, making it an unlikely candidate for consumer hardware where speed and efficiency are essential for daily arithmetic tasks.
6Neural Network Applications
Beyond handheld devices, Odrzywolek suggested applying this structure to neural networks. By replacing various functional units like ReLU or tanh with identical eml cells, researchers could standardize the architecture of AI algorithms. This mirrors the efficiency of using uniform logic gates on computer chips, potentially streamlining the design of complex machine learning systems.
7Computational Efficiency Challenges
Researcher Eymen Ipek tested this eml structure in a neural network designed for electric vehicle powertrains in April 2026. Findings indicated that while the architecture appears simpler, it requires significantly more computational operations. Specifically, executing an eml function demands 111 operations, compared to a single operation for standard functions like ReLU or tanh.
8Future of Minimalist Systems
The research highlights a significant trade-off between architectural simplicity and computational performance. Although Odrzywolek’s model offers a fascinating theoretical framework for reducing mathematical complexity, the high operational cost makes it impractical for current technology. Future developments will determine if such unified structures can ever be optimized for real-world, high-speed computing applications.