Floating-point arithmetic

Floating-point arithmetic emerged from the practical problem of handling very large and very small numbers in scientific calculation. When Leonardo Torres y Quevedo designed his Arithmometer in 1914 Madrid, he faced a challenge that had constrained calculating machines since Babbage: fixed-point arithmetic requires deciding in advance how many digits to allocate for the integer and fractional parts of numbers. For scientific work spanning from atomic scales to astronomical distances, this was unworkable.

Torres y Quevedo's insight was to separate the significant digits (the mantissa) from the scale (the exponent), allowing numbers to 'float' to appropriate magnitudes. Just as scientific notation lets humans write 6.022 × 10²³ instead of 602,200,000,000,000,000,000,000, floating-point representation let machines handle extreme ranges without reserving billions of digit positions.

The adjacent possible for this innovation had been building since the development of logarithms in the 17th century. Scientists already thought in terms of orders of magnitude. Slide rules, which performed multiplication through addition of logarithms, embodied a similar principle. The mathematical foundation existed; the question was how to implement it mechanically.

Konrad Zuse independently developed floating-point arithmetic for his Z1 computer in 1936, demonstrating convergent evolution in computing. Working in isolation in Berlin, Zuse arrived at a binary floating-point system with a 22-bit mantissa and 7-bit exponent—a remarkably modern design. His Z3 computer of 1941 was the first machine to implement floating-point in a fully functional system.

The path dependence established by early floating-point designs persists in contemporary computing. The IEEE 754 standard, adopted in 1985 and still in use, specifies formats that descend directly from these pioneering implementations. Every scientific calculation, video game physics simulation, and machine learning model uses floating-point arithmetic—the representation that Torres y Quevedo and Zuse developed to let machines think across the scales of the universe.