From Complex Number to Hypercomplex Algebra: An Evolution from Non-Relativistic Quantum Theory to Particle Physics in 16D Spacetime

The evolution of the number system development intricately encodes the development of mathematics and physis— from the philosophical unity of Tai-Chi to the 16D sedenion algebra, via the construction of 1 to natural numbers, fractional numbers, irrational number, 1D real numbers, 2D complex numbers, 4D quaternions, 8D octonions, and 1D sedenions through the Cayley–Dickson construction. Each step introduces deeper layers of symmetry, duality, and structure, mirroring the emergence of physical phenomena such as space-time, spin, charge, and gauge interactions. Notably, the progression from complex to quaternion and octonion algebras corresponds precisely to the internal gauge symmetries of the Standard Model: ℂ ↔ U(1), ℍ ↔ SU(2), ↔ SU(3), providing an elegant algebraic origin for electromagnetism, weak, and strong interactions. We argue that this layered algebraic development not only parallels the hierarchy of physical laws but may also provide the foundation for a unified framework of matter and force — one that transcends traditional field theory and bridges the mathematical and physical worlds. This hypercomplex framework provides a path from classical dynamics, through layer upon layer of non-relativistic quantum mechanics, special relativity, quantum electrodynamics, unified electroweak theory, quantum chromodynamics, and quantum gravity, to grand unification theory and quantum cosmology.