When Fourier Meets Binary: A Mathematical Bridge Between Numbers and Bits

When Fourier Meets Binary is a mathematical project that builds a bridge between the world of continuous mathematics and the digital world of binary code. The paper develops a complete function O(n) that can convert any integer into its binary representation using only trigonometric and logarithmic functions — no conditional or digital logic required. By combining ideas from Fourier analysis, periodic sequences, and the properties of binary bits, this work shows that even binary data can be fully expressed through pure mathematics. The final sections explore how this approach can be extended to define binary operations, bit inversion, and even the foundation for a mathematical hash function. The project aims to demonstrate that “math can express anything” — even the logic that powers computers.