A potential relation between the algebraic approach to calculus and rational functions

This paper considers: (1) Colignatus (2011), "Conquest of the Plane" (COTP), with its new algebraic approach to calculus, and (2) the theory of rational functions (RF).

This paper assumes that most readers will come from a background with RF. They might be interested whether there might be some (re-) design by using some notions from COTP.

The derivative df / dx = f ' [x] concerns the slope of the function. If df is polynomial then we have a rational function r = df / dx. The theory of rational functions at the fundamental level (RF-FL) recognises domains and singularities. Conventionally singularities must be resolved by using limits. For polynomials there is the possibility of factoring, df = f ' [x] dx. Multiplicative factoring can be proven by use of coefficients only, which leads to Ruffini's Rule.

The major conceptual issue w.r.t. factoring is whether the multiplicative form df = f ' [x] dx can still be recognised as the slope df / dx (since a slope is given by the tangent in trigonometry). Ruffini's Rule factors and solves df / dx by "synthetic division", but to what extent is "synthetic" also proper division, so that "eliminating" the factor dx generates a result that can be understood as the slope of the function at that point ?

This conceptual problem is resolved as follows. We better state explicitly that the domain must be manipulated. Let y // x be the following process or program, called dynamic division or dynamic quotient, with numerator y and denominator x:

y // x ≡ { y / x, unless x is a variable and then: assume x ≠ 0, simplify the expression y / x, declare the result valid also for the domain extension x = 0 }

The algebraic definition of the derivative then follows directly:

f ’[x] = {Δf // Δx, then set Δx = 0}

This implies that the expression "df / dx" only has proper meaning as an operator "d / dx" applied to f, without proper division. This also means that we finally have a sound interpretation for differentials. These would not be infinitesimals. The differentials df and dx are better seen as variables, so that, when f ' [x] has been found by other methods of algebraic manipulation, we can define df = f ' [x] dx for the incline (tangent) to f.

This gives:    df // dx = f ' [x] dx // dx = f ' [x]

The group theory approach to rational functions (RF-GT) (the version that we looked at) appears to have limited value, because of the assumption that these "functions" don't have domains. If its results are to be useful, they must be translated, and domains and singularities come into consideration anyhow. The notion of an equivalence class relies on limits and continuity, and the manipulation of the domain is not explicit enough.

An algebraic approach to calculus is possible that relies on algebra and expressions only, and that manipulates the domain to find the slope of the function. The formal continuity given by the expression is sufficient, and there is no need for numerical continuity and limits.