In vector analysis the concepts of addition and multiplication were developed and then applied successfully among the physical sciences, engineering, and technology. It was first applied to the theory of electromagnetism. However, where and when Maxwell started his profound idea he actually used the algebra of quaternions to describe both the electric field and the magnetic field. On the other hand, years earlier, where and when Hamilton developed the mathematics of a quaternion he intended it to be the sum of a scalar quantity and a directed (vector) quantity. Later, Gibbs and Heaviside both contributed to the separate theories of scalars and vectors. The former remains Abelian while the latter becomes non-Abelian.

In the beginning, Hamilton also intended to develop quaternions as extension of the 2-dimensional complex domain of one real component and one imaginary component into the 4-dimensional quaternion domain of 1-component scalars and 3-component vectors. Although a 2D complex number can be divided simply by multiplying both the dividend and the divisor by the complex conjugate of the divisor which really tantamount to changing the directional divisor into a real scalar divisor, 4D analogue of this transformation does not exist for vectors until the new definitions of both inner and outer or scalar and vector or dot and cross products. Advanced resolutions laid the foundation of an idea of gauge invariance and its theory was responsible for the success of both quantum electrodynamics and quantum chromodynamics. Coincidentally, a gauge theory has to inevitably imply the absolute existence of a scalar gauge boson called the spin-zero scalar Higgs boson for the origin of physical mass. On the other hand, the idea of relative gauges using Hadamard matrices together with a principle of directional invariance at the infinitesimal domain of space-time allow scalars and rational solutions of physical mass simply using matrix multiplications while matrix additions provide calculated values of quantized space-time charges and their transformations into the usual visualizable electric charges, weak charges, and color charges. This realizes a topological transformation between directional divisor and scalar divisor without destroying their intrinsic or extrinsic properties.