When a problem of physical science appears it's either surprisingly observed data or a textbook (scholastic) problem, be it of the natural type (solar, geological, astronomical), the solution lies in identifying the system in subject as one which has an orthodoxy, we need the motivations of the empirical, along with the mathematical rigor giving us logical structure, which is physics + math.

• An introduction to these articles, with some set notation and the field of real numbers.

• An end to the mathematical frustration of man comes with analysis of the binomial (1620), and the generalization of the formula to a series solution of the binomials with non-natural number exponent (1665).

• The oldest series, being described in Euclid's Elements, has a wide range of applicability.

• Exponents are a simple algebraic construction and notation; they are used in scientific notation and are implicit in the decimal notation. Roots are an intrinsic part of exponents (the fractional exponents). So, in this section the "original" exponent value (zero) is explored.

• The logarithm is presented as an 18th century revelation. The natural exponential is compared to other exponential bases, graphically.

• If the summation of a series grows faster than some rate, then it diverges. Up until the 17th century, no one knew whether the Harmonic series diverged, or converged, as it grows very slowly, but faster than the Geometric series.

• Tangents in general are considered, particularly those of curves which come to the minds of early modern natural philosophers.

• The inverse operation of the tangent of a curve, is the summed area under such function.

• In the modern frameworks, a ubiquitously appearing tool, is the representation of a any curve in terms of a polynomial with differential coefficients.

• Trigonometry is rounded out with the Sine and Cosine functions, and the Taylor Series of the cofunctions are considered.

• Here, the Trigonometric Taylor Series' are combined with the Exponential Series to derive,

• A theorem of Mertzbacher is presented, using Fourier analysis to outline a direct route from wave-particle duality to their complex existence.