The Role of Decimals

Cauchy in the 1800s developed his ideas of convergence in terms of decimal numbers and error control. The epsilon in his computations stood for an error E > 0 that could be made in principle as small as possible.

The decimal-free set theoretic view of mathematics reached it almost final form in the 1920s. It took another 30 years, that is, until the 1950s, for the set theoretic view of mathematics to be adopted in mathematics departments. The modern mathematics movement in the 1960s was intended to spread or provide a setting for the teaching of the set theoretic perspective.

The set theoretic perspective began about the mid 1800s, and it was used in the period 1900 -1930 to provide a strict thought-based foundation for computations --- the arithmetic based part of mathematics --- a foundation (hopefully) free of contradictions and inconsistencies. This set theoretic perspective was not developed for ease of exposition. The initial aim in studying sets was not to provide a foundation for arithmetic based mathematics. In the set theoretic approach to mathematics after arithmetic (counting included), the decimal perspective of real numbers was not necessary. So it was put aside.

In contrast, the common knowledge of mathematics is based on counting, a decimal knowledge of arithmetic and real numbers, and the use of simple formulas. This common knowledge is introduced and hopefully explained in elementary school in a thought-based manner. The common knowledge presently encompasses counting, arithmetic and the use of simple formulas.

The decimal expansion of real numbers provides a concrete sense of convergence. Unfortunately, in the zeal to derive the set theoretic perspective from first set-theoretic principles or assumptions about real numbers in our high schools and colleges, the decimal perspective was put aside at least partially. That is, while the decimal representation of whole numbers and real numbers was employed in computational examples in algebra, trig, chemistry, physic, business and calculus, the chains of reasoning emphasized in algebra and calculus typically made no mention of decimals (nor units). Decimals (and sometimes units) were used in many computational subjects yet not recognized nor sanctioned in math courses axioms.

Courses on analysis (advanced calculus) could be made more accessible to students by detailing in them a set-theoretic justification of decimal expansions and their convergence of the latter. Before and after this, courses that discuss the decimal and decimal-free perspective would be agreeable both to students of analysis and students who just assume the convergence of decimal expansion. Ease of exposition is the motivation for this suggestion.

Remark: A mathematics or science student could follow the more accessible decimal perspective in a calculus and then in a later analysis (or advance calculus) course, meet the set theoretic perspective justification and/or reformulation of the decimal arguments. Does rigor in haste lead to rigor mortis?

Remark (for advance students): Appendices in Volume 3, Why Slopes and More Math, provide the decimal and decimal free perspective of the basic theorems in calculus. For instance, the Bolzano Weierstrass Theorem that every infinite set in a closed interval has a limit point can be viewed as consequence of the Pigeon hole principle. The leftmost limit point has a decimal expansion computed to k decimals by covering the interval with nicely aligned subintervals of length 10**(-k) and locating the leftmost one with infinitely points (or concluding that such an interval most exist. The latter must contain the leftmost interval of length 10^-k-1. Thus a sequence of nested intervals with left end points converging to the "lim inf" (technical expression) of the set is obtained.

Remark (also for advanced students): Lexicographic ordering of points and intervals with sides of length 10^-k should extend this argument to bounded infinite sets or sequences in Rⁿ

If a bounded region B is covered and partitioned by intervals whose sides have length 10^-k, and an infinite S in B is given, then there must be a lexicographically least interval with infinitely many points of S inside it. As k increases, these intervals will be nested, and the lexicographical least corners of the k-th interval will yield the decimal expansion of a limit point -- approach the limit in a lexicographically increasing fashion. Exercise: Verify the details and show that writer has not made any mistakes. (In this process we could define the lim inf of a set in R^{^}n with respect to the lexicographic ordering of points.

Here (x₁, x₂, ..., x_n) is lexicographically > (y₁, y₂, ..., y_n) iff there is a whole number k (1 < k < n) such that x_m = y_m if m < k and x_k <  y_k.