THE FINAL STRETCH IN THE CONSTRUCTION OF THE NEW REAL NUMBER SYSTEM R*: WELL DEFINING THE NONTERMINATING DECIMALS (for the first time)

First we note that since a decimal is defined by its digits the only well defined decimals are the terminating ones. Nonterminating decimals are ill-defined or ambiguous because not all their digits are known. Therefore, the concept rational (and also irrational) is ambiguous because it is impossible to verify if its decimal representation is periodic since we cannot check all its digits being infinite. However, ambiguity can be contained by approximating it with certainty, e.g., by a terminating decimal (which has no ambiguity); such approximation is valid if the margin of error is known and can be made small as desired. Thus, while nonterminating decimals cannot be well defined we can contain its ambiguity to the point where we do algebraic operations with them and approximate the result with desired margin of error. Now we introduce the generating or g-sequence and its g-limit, a nonterminating decimal which has contained ambiguity (approximable by certainty).

A sequence of terminating decimals of the form,

N.a_1, N.a_1a_2, …, N.a_1a_2…a_n, … (5)

where N is integer and the a_ns are basic integers, is called standard generating or g-sequence. Its nth g-term, N.a_1a_2…a_n, defines and approximates its g-limit, the nonterminating decimal,

N.a_a_2…a_n,…, (6)

at margin of error 10^-n. The g-limit of (5) is nonterminating decimal (6) provided the nth digits are not all 0 beyond a certain value of n; otherwise, it is terminating. As in standard analysis where a sequence converges, i.e., tends to a specific number, in the standard norm, a standard g-sequence, converges to its g-limit in the g-norm where the g-norm of a decimal is itself. Note that a nonterminating decimal is well defined by its g-sequence although it is ambiguous.

Since addition and multiplication and their inverse operations subtraction and division are defined only on terminating decimals computing nonterminating decimals is done by approximation each by its nth g-terms (called n-truncation) and using their approximation to find the nth g-term of the result as its approximation at the same margin of error. (Note that the g-nth term is a terminating decimal whose last digit is the nth digit) This is standard computation, i.e., approximation by decimal segment at the nth digit. Thus, we have retained standard computation but avoided the contradictions and paradoxes of the real numbers. We have also avoided vacuous statement, e.g., vacuous approximation, because nonterminating decimals are g-limits of g-sequences which belong to R*. Moreover, we have contained the inherent ambiguity of nonterminating decimals by approximating them by their nth g-terms which are not ambiguous being terminating decimals. In fact, the ambiguity of R* has been contained altogether.

As we raise n, the tail digits of the nth g-term of any decimal recedes to the right indefinitely, i.e., it becomes steadily smaller until it is unidentifiable. While it tends to 0 in the standard norm it never reaches 0 and is not a decimal since its digits are not fixed; ultimately, they are indistinguishable from the similarly receding tail digits of the other nonterminating decimals. In iterated computation when we are trying to get closer and closer approximation of a decimal the tail digits may vary but recede to the right indefinitely and become steadily smaller leaving fixed digits behind that define a decimal. We approximate the result by taking its initial segment, the nth g-term, to desired margin of error.
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Consider the sequence of decimals,

(delta^n(a_1a_2…a_k), n = 1, 2, …, (7)

where delta is any of the decimals, 0.1, 0.2, 0.3, …, 0.9, a_1, …, a_k, basic integers (not all 0 simultaneously). We call the nonstandard sequence (7) d-sequence and its nth term nth d-term. For fixed combination of delta and the a_j’s, j = 1, …, k, in (7) the nth term is a terminating decimal and as n increases indefinitely it traces the tail digits of some nonterminating decimal and becomes smaller and smaller until we cannot see it anymore and indistinguishable from the tail digits of the other decimals (note that the nth d-term recedes to the right with increasing n by one decimal digit at a time). The sequence (7) is called nonstandard d-sequence since the nth term is not standard g-term; while it has standard limit (in the standard norm) which is 0 it is not a g-limit since it is not a decimal but it exists because it is well-defined by its nonstandard d-sequence. We call its nonstandard g-limit dark number and denote by d. Then we call its norm d-norm (standard distance from 0) which is d > 0. Moreover, while the nth term becomes smaller and smaller with indefinitely increasing n it is greater than 0 no matter how large n is so that if x is a decimal, 0 < delta =" 1" a_k =" 1" n =" 1," 09 =" 0" 009 =" 0"> 0.

We state some theorems about R*.
Theorem. The d-limits of the indefinitely receding (to the right) nth d-terms of d* is a continuum that coincides with the g-limits of the tail digits of the nonterminating decimals traced by those nth d-terms as the aks vary along the basic digits.
Theorem. In the lexicographic ordering R* consists of adjacent predecessor-successor pairs (each joined by d*); therefore, the g-closure R* of R is a continuum [9].
Corollary. R* is non-Archimedean and non-Hausdorff in both the standard and the g-norm and the subspace of decimals are countably infinite, hence, discrete but Archimedean and Hausdorff.
Theorem. The rationals and irrationals are separated, i.e., they are not dense in their union (this is the first indication of discreteness of the decimals) [7].
Theorem. The largest and smallest elements of the open interval (0,1) are 0.99… and 1 – 0.99…, respectively [6].
Theorem. An even number greater than 2 is the sum of two prime numbers.

Remark. Gauss’ diagonal method proves neither the existence of nondenumerable set nor a continuum; it proves only the existence of countably infinite set, i.e., the off-diagonal elements consisting of countable union of countably infinite sets. The off-diagonal elements are not even well-defined because we know nothing about their digits (a decimal is determined by its digits). We state the following corollaries from our discussion: (1) Nondenumerable set does not exist; (2) Only discrete set has cardinality; a continuum has none.

(This article is excerpted from Escultura, E. E., The new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computations 17 (2009), 59 – 84)

E. E. Escultura
Research Professor
GVP - V. Lakshmikantham Institute for Advanced Studies
and Departments of Mathematics and Physics
GVP College of Engineering, JNT University
Madurawada, Visakhapatnam. AP, India