The New Real Number System
1. Introduction.
This article is excerpted from the paper, The new real number system and discrete computation and calculus, in press, Neural, Parallel and Scientific Computation, and highlights the major results of the paper. However, it took 10 years to build the new real number system in series of papers starting with the original paper, Exact solutions of Fermat’s equation (Definitive resolution of Fermat’s last theorem), Nonlinear Studies, 5(2), 1998. Simultaneous with these publications were the extensive discussion on the net in the various websites, forums, internet groups and blogs that started in 1997.
2. The terminating decimals
We denote the new real number system by R* and we build it on the additive and multiplicative identities, 0 and 1 of elementary arithmetic. That is our first axiom. Then we include as additional axioms the addition and multiplication tables that define that define the properties of 0 and 1 and the relationship betweem them. Since the elements are finite there is no ambiguity, no contradiction. Then we define the basic integers, 0, 1, … 9 using the addition table: 2 = 2 + 1, 3 = 2 + 1, …, 9 = 8 +1. Then we define the rest of the integers as base 10 place-value numerals:
anan–1…a1 = an10n + an–110n–1 + … + a1, (2)
where the ans are basic integers.
Now, we extend the integers to include the additive and multiplicative inverses −x and, if x is not 0, 1/x (reciprocal of x), respectively. Note that the reciprocal of an integer exists only if it has no prime factor other than 2 or 5. This extension requires corresponding extension of the operations + and ×, in effect, re-stating associativity, etc., as part of its axioms and something else that is new: the rules of sign that we take as part of the axioms of this extension (we need not write them as they are familiar). Then we define a new operation: division of an integer x by a nonzero integer y, or quotient, denoted by x/y and defined by:
x/y = x(1/y). (3)
This quotient exists when y has no prime factor other than 2 or 5. We similarly extend associativity and commutativity of addition and multiplication and distributivity of multiplication relative to addition and include them the axioms of the extension. We consider subtraction the inverse operation of addition and division that of multiplication, examples of duality discussed in the original paper.
We define a terminating decimal as follows:
anan–1…a1.bkbk–1…b1 = an10n + an–110n–1 +… + a1 + b1/10 + b2/102 +… + bk/10k
= an10n + an–110n–1 +… + a1 + b1(0.1) + b2(0.1)2 + … + bk(0.1)k. (4)
where anan–1…a1 is the integral part, b1b2…bk the decimal part and 0.1 = 1/10. The integral parts are defined as the integers. Note that the terminating decimals are well-defined since the reciprocal of 10 has only the factors 2 and 5. Note further that a terminating decimal is well-defined fraction. However, when the integral part has factor other than 2 or 5 its reciprocal does not exist.. Then the quotient x/y of an integer x by nonzero integer y exists only if y has no prime factor other than 2 or 5. Such quotient is called rational. We recall that in the real number system a rational is nonterminating periodic (a terminating decimal is periodic). This is ambiguous for nonterminating decimal since it is not verifiable. We call a decimal that is not rational eurrational but we do not know what that is yet because we know nothing beyond the terminating decimals.
This definition of the integers as the integral parts of the terminating decimals resolves the inadequacy of Peanos’s postulates in the development of the natural numbers for they are clearly isomorphic to them and makes them integers in the sense of [3].
3. The nonterminating decimals
Now we define the nonterminating decimals for the first time without contradiction and with contained ambiguity, i.e., approximable by certainty (a decimal segment). We build them on what we know: the terminating decimals, our point of reference for all its extensions.
A sequence of terminating decimals of the form,
N.a1, N.a1a2, …, N.a1a2…an, … (5)
where N is integer and the ans are basic integers, is called standard generating or g-sequence. Its nth g-term, N.a1a2…an, defines and approximates its g-limit, the nonterminating decimal,
N.a1a2…an…, (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.
We define the nth distance dn between two decimals a, b as the numerical value of the difference between their nth g-terms, an, bn, i.e., dn = |an − bn| and their g-distance is the g-limit of dn, Note that R* is the g-closure of the terminating decimals R in the g-norm.
A terminating decimal is a degenerate nonterminating decimal, i.e., the digits are all 0 beyond a certain value of n. The nth g-term of a nonterminating decimal repeats every preceding digit in the same order so that if finite terms are deleted or altered the nth g-term and, therefore, also the g-limit is unaltered as the remaining terms generate its g-sequence. Thus, a nonterminating decimal may have many g-sequences and we consider them equivalent for having the same g-limit.
Since addition and multiplication and their inverse operations subtraction and division are defined only on terminating decimals computing nonterminating decimals is done by approximating 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. This is standard computation, i.e., approximation by decimal segment at the nth digit. Thus, with our axioms 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 exist and 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 by its construction on the additive and multiplicative identities 0 and 1.
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, e.g., calculating f(n) = (2n4+1)/3n4, n = 1, 2, …, 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.
Consider the sequence of decimals,
(d)na1a2…ak, n = 1, 2, …, (7)
where d is any of the decimals, 0.1, 0.2, 0.3, …, 0.9, a1, …, ak, 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 d and the aj’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) where is d > 0. Moreover, while the nth term becomes smaller and smaller with indefinitely increasing n it is greater than 0 for any n is so that if x is a decimal, 0 < d < x.
Now, we allow d to vary steadily in its domain and also the ajs along the basic integers (not simultaneously 0). Then their terms trace the tail digits of all the decimals and as n increases indefinitely they become smaller and smaller and indistinguishable from each other. We call their nonstandard limits dark numbers and denote by d* which is set valued and countably infinite and includes every g-limit of the nonstandard d-sequence (7). To the extent that they are indistinguishable d* is a continuum (in the algebraic sense since no notion of open set is involved). Thus, the tail digits of the nonterminating decimals merge and form the continuum d*.
At the same time, since the tail digits of all the nonterminating decimals form a countable combination of the basic digits 0, 1, …, 9 they are countably infinite, i.e., in one-one correspondence with the integers. In fact, any set that can be labeled by integers or there is some scheme for labeling them by integers is in one-one correspondence with the integers, i.e., countably infinite. It follows that the countable union of countable set is countable. Therefore, the decimals and their tail digits are countably infinite. However, as the nth d-terms of (7) trace the tail digits of the nonterminating decimals they become unidentifiable and cannot be labeled by the integers anymore; therefore, they are no longer countable and they merge as the continuum d*.
Like a nonterminating decimal, an element of d* is unaltered if finite g-terms are altered or deleted from its g-sequence. When d = 1 and a1a2…ak = 1 (7) is called the basic or principal d-sequence of d*, its g-limit the basic element of d*; basic because all its d-sequences can be derived from it. The principal d-sequence of d* is,
(0.1)n , n = 1, 2, … (8)
obtained by the iterated difference,
N – (N – 1).99… = 1 – 0.99... = 0 with excess remainder of 0.1;
0.1 – 0.09 = 0 with excess remainder of 0.01;
0.01 – 0.009 = 0 with excess remainder of 0.001;
………………………………………………… (9)
Taking the nonstandard g-limits of the left side of (9) and recalling that the g-limit of a decimal is itself and denoting by dp the d-limit of the principal d-sequence on the right side we have,
N – (N – 1).99… = 1 – 0.99... = dp. (10)
Since all the elements of d* share its properties then whenever we have a statement “an element d of d* has property P” we may write “d* has property P”, meaning, this statement is true of every element of d*. This applies to any equation involving an element of d*. Therefore, we have,
d* = N – (N – 1).99… = 1 – 0.99... (11)
Like a decimal, we define the d-norm of d* as d* > 0.
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.
This means that the decimals are joined together at their tails digits by the continuum d*. If x is nonzero decimal, terminating or nonterminating, there is no difference between (0.1)n and x(0.1)n as they become indistinguishably small as n increases indefinitely. Similarly, we may write, xdp = dp (where dp is the principal element of d*), and since the elements of d* share this property we may write xd* = d*, meaning, that xd = d for every element d of d*. We consider d* the equivalence class of its elements. In the case of x + (0.1)n and x, we look at the nth g-terms of each and, as n increases indefinitely, x + (0.1)n and x become indistinguishable. Now, since (0,1)n > ((0.1)m)n > 0 and the extreme terms both tend to 0 as n increases indefinitely, so must the middle term tend to 0 so that they become indistinguishably small (the reason d* is called dark for being indistinguishable form 0 yet greater than 0): We summarize our discussion as follows:
If x is not a new integer, x + d* = x; otherwise, if x = N.99… x + d* = N+1, x – d* = x; if x ¹ 0,
xd* = d*; (d*)n = d*, n = 1, 2, …, N = 0, 1, … (12)
1 – d* = 0.99…, N – (N – 1).99… 1 – 0.99… = d*, N = 1, 2, … (13)
It follows that the g-closure of R, i.e., its closure in the g-norm, is R* which includes the additive and multiplicative inverses and d*. We also include in R* the upper bounds of the divergent sequences of terminating decimals and integers (a sequence is divergent if the nth terms are unbounded as n increases indefinitely, e.g., the sequence 9, 99, …) called unbounded number u* which is countably infinite since the set of sequences is. We follow the same convention for u*: whenever we have a statement “u has property P for every element u of u*” we can simply say “u* has property P). Then u* satisfies these dual properties:
for all x, x + u* = u*; for x ¹ 0, xu* = u*. (14)
Neither d* nor u* is a decimal and their properties are solely determined by their sequences. Then d* and u* have the following dual or reciprocal properties and relationship:
0d* = 0, 0/d* = 0, 0u* = 0, 0/u* = 0, 1/d* = u*, 1/u* = d*. (15)
Numbers like u* - u*, d*/d* and u*/u* are still indeterminate but indeterminacy is avoided by computation with the g- or d-terms.
The decimals are linearly ordered by the lexicographic ordering “<” defined as follows: two elements of R are equal if corresponding digits are equal. Let
N.a1a2..., M.b1b2... Î R. (16)
Then,
N.a1a2. . . < M.b1b2 if N < M or if N = M, a1 < b1; if a1 = b1, a< b2; …, (17)
and, if x is any decimal we have,
0 < d* < x < u* (18)
The trichotomy axiom follows from lexicographic ordering. This is the natural ordering mathematicians sought among the real numbers but it does not exist there because it contradicts the trichotomy axiom.
4. Duals and their reciprocals
Mathematical systems are better understood by bringing in the notion of dual systems because it introduces some symmetry that may be useful. We can look at divergent sequences, i.e., sequences whose terms become bigger and bigger that we can no longer comprehend them and become indistinguishable from each other, as dual of convergent sequences. In this sense the divergent sequences also form a continuum. We denote their upper bounds by u* which satisfies (12) and (15). Then we look at d* as the dual of u* and R* that of the system of additive and multiplicative inverses (which has holes, namely, the nonexistent multiplicative inverses of integers). Thus R* is a semi-field, the nonzero integers forming a semi-ring since some of them have no multiplicative inverses. Like d*, u* cannot be separated from the decimals, i.e., there is no boundary between either of them and the decimals and between finite and infinite, i.e., we cannot separate d* from a decimals and there is no boundary to cross between finite and infinite so that beyond a certain finite decimal everything else is infinite. The latter is what is meant by the expression u* + x = u* for any decimal x. Duality is also seen in this case: Let h > 1 be terminating decimal then the sequence hn, n = 1, 2, …, diverges to u* but (1/h)n, n = 1, 2, … converges, d-lim (1/h)n = d*.
5. Isomorphuism between the integers and the decimal integers
To find out more about the structure of R* we show the isomorphism between the integers and the decimal integers, i.e., integers of the form,
N.99…, N = 0, 1, … (19)
but before doing so we first note that 1 + 0.99… is not defined in R since 0.99… is nonterminating but we can write 0.99. . . = 1 – d* so that 1 + 0.99… = 1 + 1 – d* = 2 – d* =1.99… and we now define 1 + 0.99… = 1.99… or, in general, N – d* = (N-1).99… The pairs (N,(N–1).99…), N = 1, 2,..., are called twin integers because they are isomorphic:
Let f be the mapping N → (N – 1).99… then we show that f is an isomorphism between the integers and decimal integers.
= N – 1 + M – 1 + 1.99… = N – 1 + 0.99… + M – 1 + 0.99…
= (N–1).99… + (M–1).99… = f(N) + f(M). (20)
Equation (20) means that addition of integers is the same as addition of decimal integers.
Next, we show that multiplication is also an isomorphism.
(b) f(NM) = (NM–1).999... = NM – 1 + 0.99…
(c) = NM – N M + 1 + N + –1 + M + –1 + 0.99…
= NM – N – M + 1 + (N–1).99… + (M–1).99… + (–1)(0.99…
= NM – N – M + 1 + N(0.99… + (–1)(0.99… + M(0.99…
+ (–1)(0.99…) + 0.99… = (N – 1)(M – 1) + (N–1)(0.99…)
+ (M – 1)(0.99… + (0.99…)2 = ((N – 1) + 0.99… M – 1) + 0.99…
= (((N – 1).99…)((M – 1).99…) = (f(N))(f(M)). (21)
We have now established the isomorphism between the integers and the decimal integers with respect to both operations. We include in this isomorphism the map d* ® 0, so that its kernel is the set {d*,0.99…} from which follows equations (22):
(d*)n = d* and (0.99…)n = 0.99…, n = 1, 2, …. (22)
(The second equation can be proved also by mathematical induction)
For the curious reader we exhibit other properties of 0.99… Let K be an integer, M.99… and N.99… decimal integers. Then
(a) K + M.99… = (K+M).99…
(b) K(M.99…) = K(M + 0.99…) = KM + K(0.99…) = KM + (K–1).99…
(c) M.99… + N.99… = M + N + 0.99… + 0.99…). (23)
To verify that 2(0.999...) = 1.99…, we note that (1.99…)/2 = 0.99…
(d) (M.99…)(N.99…) = (M + 0.99…)(N + 0.99…)
= MN + M(0.99… + N(0.99…) + (0.99…)2
= MN + (M–1).999… + (N–1).99… + 0.99…
= MN + (M + N–2).99… + 0.99…
= MN + (M + N–.1).99… = (MN+M+N–1).99…
(e) 0.99… + 0.99… = 2(0.99…) = 1.99… (24)
We extended the isomorphism to include d* by the mapping f(0) = d*, even if d* is neither a decimal nor an integer, because d* behaves like 0 and 0.99… like 1. The isomorphism makes the decimal integers also integers (i.e., equivalent and behave alike) in the sense of [3].
6. Adjacent decimals and recurring 9s
Two decimals are adjacent if they differ by d*. Predecessor-successor pairs and twin integers are adjacent. In particular, 74.5700… and 74.5699… are adjacent.
Since the decimals have the form N.a1a2…an,…, N = 0, 1, 2, …, the digits are identifiable and, in fact, countably infinite and they are linearly ordered by lexicographic ordering. Therefore, they are discrete or digital and the adjacent pairs are also countably infinite. However, since their tail digits form a continuum, R* is a continuum with the decimals its countably infinite discrete subsystem.
A decimal is called recurring 9 if its tail decimal digits are all equal to 9. For example, 4.3299… and 299.99… are recurring 9s; so are the decimal integers. (In an isomorphism between two algebraic systems, their operations are interchangeable, i.e., they have the same algebraic structure but differ only in notation).
The recurring 9s have interesting properties. For instance, the difference between the integer N and the recurring 9, (N – 1).99…, is d*; such pair of decimals are called adjacent because there is no decimal between them and they differ by d*. In the lexicographic ordering the smaller of the pair of adjacent decimals is the predecessor and the larger the successor. The average between them is the predecessor. Thus, the average between 1 and 0.99… is 0.99… since (1.99…)/2 = 0.99…; this is true of any recurring 9, say, 34.5799… whose successor is 34.5800… Conversely, the g-limit of the iterated or successive averages between a fixed decimal and another decimal of the same integral part is the predecessor of the former.
Since adjacent decimals differ by d* and there is no decimal between them, i.e., we cannot split d* into nonempty disjoint sets, we have another proof that d* is a continuum (in the algebraic sense). Then we have another proof that R* is a continuum (also in the algebraic sense).
It follows from the counterexample to the trichotomy axiom that an irrational number cannot be expressed as limit of sequence of rationals since the closest it can get to it is some rational interval which still contains some rational whose relationship to it is unknown.
Now we know what the eurrationals are; they are the nonterminating decimals, periodic and nonperiodic. The g-sequence of an eurrational, which is a sequence of rationals, gets directly to its g-limit, digit by digit. We note further that an eurrational is an infinite series in terms of its digits as follows:
N.a1a2…an…, = N + .a1 + .0a2 + … + .00…0an +…; 0.99… = 0.9 + 0.09 + … (25)
7. The structure of R* and its subspacess
We add the following results to the information we now have about the various subspaces of R* to provide a full picture of the structure of the new real number system. The next theorem is a definitive result about the continuum R*.
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].
However, the decimals form countably infinite discrete subspace of R* since there is a scheme for labeling them by the integers.
We can imagine the terminating decimals as forming a right triangle with one edge horizontal and the vertical one extending without bounds. The integral parts are lined up on the vertical edge and they are joined together by their branching digits between the hypotenuse and the horizontal and extend to d* which is adjacent to 0 (i.e., differs from 0 by a dark number) at the vertex of the horizontal edge.
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.
The following theorem is standard in the real in the real number system with the standard norm. Therefore, we do not bring in d* in the proof so that this is really a theorem about the decimals with the standard norm which is not true in the g-norm because the decimals merge into a continuum at their tail digits and cannot be separated.
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].
This means that every decimal is separated from the rest, the terminating
decimals from the eurrationals and from each other.
The next theorem has standard proof (in R); it raised eyebrows in internet forums.
Theorem. The largest and smallest elements of the open interval (0,1) are 0.99… and 1 – 0.99…, respectively [6].
This theorem is true in the real number system and follows from the properties of the terminating decimals but it was not known because neither 0.99… nor 1 – 0.99… was well-defined; it was assumed all along that 1 = 0.99…the right side being ill-defined.
The next theorem used to be called Goldbach’s conjecture [4,7] but now has a proof in R*.
Theorem. An even number greater than 2 is the sum of two prime numbers.
This is unsolved because, like Fermat’s equation (FLT) [5], it is indeterminate. Before proving the theorem, we first note that an integer is a prime if it leaves a positive remainder when divided by another integer other than 1. We retain this definition in R*.
We now have a sense of how the decimals are arranged by the lexicographic ordering. Consider the decimals with integral part N:
N.99…………….
………………….
N.4800………….
N.4799………….
………………….
N.10……………..
…………………..
N.00…0100…
…………………..
N.00… (26)
The largest decimal in the set is the decimal integer N.99… and the smallest is the terminating decimal N.00… From the bottom up the decimals with integral part N are arranged as predecessor-successor pairs each joined by d*.Each gap indicated by the ellipses is filled by countably infinite adjacent predecessor-successor pairs each also joined by d* so that their union is a continuum. We now have a clear picture of how R* is arranged on the new real line linearly ordered by <, the lexicographic ordering.
9. Important results; resolution of a paradox
(1) Every convergent sequence has a g-subsequence that defines a decimal adjacent to its limit. If the decimal is terminating it is the limit itself.
(2) It follows from (1) that the limit of a sequence of terminating decimals can be found by evaluating the g-limit of its g-subsequence which is adjacent to it. We can use this as alternative way of computing the limit of ordinary sequence.
(3) In [10] several counterexamples to the generalized Jourdan curve theorem for n-sphere are shown where a continuous curve has points in both the interior and exterior of the n-sphere, n = 2, 3,. . . , without crossing the n-sphere. The explanation is: the functions cross the n-sphere through dark numbers.
(5) Given two decimals and their g-sequences and respective nth g-terms An, Bn we define the nth g-distance as the g-normôAn- Bnôof the difference between their nth g-terms. Then their g-distance is the g-limôAn- Bnô, as n ® ¥, which is adjacent to the standard norm of the difference [3]. The advantage here is that the g-distance is the g-norm of their decimal difference and the difference between nonterminating decimals cannot be evaluated otherwise. Moreover, this notion of distance can be extended to n-space, n – 2, 3, .., and the distance between two points can be evaluated digit by digit in terms of their components without the need for evaluating roots. In fact, any computation in the g-norm yields the results directly, digit by digit, without the need for intermediate computation such as evaluation of roots in standard computation.
10. THE COUNTEREXAMPLES TO FLT
Given the contradiction in negative statement, we use Fermat’s equation in place of the statement of Fermat’s last theorem (FLT) so that its solutions are counterexamples to FLT. We first summarize the properties of the basic digit 9.
(1) String of 9s differs from nearest power of 10 by 1, e.g., 10100 – 99…9 = 1.
(2) If N is an integer, then (0.99…)N = 0.99… and, naturally, both sides of the equation have the same g-sequence. Therefore, for any integer N, ((0.99,..)10)N = (9.99…)10N.
(3) (d*)N = d*; ((0.99,..)10)N + d* = 10N, N = 1, 2, …
Then the exact solutions of Fermat’s equation are given by the triple (x,y,z) = ((0.99…)10T,d*,10T), T = 1, 2, …, that clearly satisfies Fermat’s equation,
xn + yn = zn, (26)
for n = NT > 2. Moreover, for k = 1, 2, …, the triple (kx,ky,kz) also satisfies Fermat’s equation. They are the countably infinite counterexamples to FLT that prove the conjecture false [5]. One counterexample is, of course, sufficient to disprove a conjecture.
11. ADVANTAGES OF THE G-NORM
Here are the advantages of the g-norm over other norms.
(a) It avoids indeterminate forms.
(b) Since the g-norm of a decimal is itself, computation yields the answer directly as decimal, digit by digit, and avoids the intermediate approximations of standard computation. This means significant savings in computer time for large computations.
(c) Since the standard limit is adjacent to the g-limit of some g-sequence, evaluating it reduces to finding some nonterminating decimal adjacent to it; the decimal is computed using the g. Both the computation and approximation are precise. In fact, the exact margin of error is d*. This applies to the result of any computation: it is adjacent to some nonterminating decimal and the latter is found using the g-norm.
(d) In iterated computation along successive refinements of sequence xj that tends to a as j ® ¥. The iteration is simplified by taking midpoints or averages between the sequence of points xj and the g-limit s.
(e) Approximation by nth g-term or n-truncation contains the ambiguity of nonterminating decimals.
(f) Calculation of distance between two decimals is direct, digit by digit, and requires no square root. In fact, calculation by the g-norm involves no root or radical at all.
(i) In general radicals in computation, e.g., taking root of a prime, is avoided by nth g-term approximation or n-truncation to any desired margin of error where accuracy is measured by number of digits of the result obtained.
The g-norm is the natural norm for purposes of computation for it does three things: (a) it puts rigor in computation since every step is in accordance with the new definition of the previously ill-defined nonterminating decimals (as meaningless infinite arrays of digits most of which unknown) in terms of the well-defined terminating decimals, (b) the margin of error is precisely determined and (c) the result of the computation is obtained digit by digit and avoids intermediate unnecessary approximations of standard computation. For large computation (c) provides significant saving in computing time as it avoids these intermediate approximations and proceeds directly to the calculation of digits of the resulting decimal.
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). Therefore, we raise these conjectures:
Conjectures. (1) Nondenumerable set does not exist; (2) Only discrete set has cardinality; a continuum has none.
References
[1] Benacerraf, P. and Putnam, H., Philosophy of Mathematics,
Cambridge University Press, Cambridge, 1985.
[2] Bhaskar, T. G., Kovak, D., Lakshmikantham V. (2006) The Hybrid
Set Theory,
[3] Corporate Mathematical Society of Japan, Kiyosi Ito, ed.,
Encyclopedic Dictionary of Mathematics (2nd ed.), MIT Press,
Cambridge, MA, 1993.
[4] Davies, P. J. and Hersch, R., The Mathematical Experience,
Birkhäuser, Boston, 1981.
[5] Escultura, E. E., Exact solutions of Fermat’s equation (A definitive
resolution of Fermat’s last theorem, Nonlinear Studies, 5(2), 1998.
[6] Escultura, E. E., The mathematics of the new physics, J.
Applied Mathematics and Computations, 130(1), 2002.
[7] Escultura, E. E., The new mathematics and physics, J. Applied
Mathematics and Computation, 138(1), 2003.
[8] Escultura, E. E., From macro to quantum gravity, Problems of
Nonlinear Analysis in Engineering Systems, 7(1), 2001.
[9] Escultura, E. E., The mathematics of the grand unified theory,
in press, Nonlinear Analysis.
[10] Kline, M. Mathematics: The Loss of Certainty, Oxford University
Press, New York, 1980.
[11] Royden, H. L., Real Analysis, MacMillan, 3rd ed., New York, 1983.
[12] Young, L. C., Lectures on the Calculus of Variations and Optimal
Control Theory, W. B. Saunders, Philadelphia, 1969.
[13] Zeigler, B. P., “An Introduction to Calculus” Course Based on DEVS:
Implications of a Discrete Reformulation of Mathematical Continuity”.
Presented at International Conference on Simulation in Educaton
ICSiE’05, January 23 – 25, New Orleans, USA.
[14] Zeigler, B. P., Continuity and Change (Activity) are Fundamentally
Related in DEVS Simulation Of Continuous Systems”, Keynote Talk
at AI, Simulation and Planning AIS’04, October 4 – 6, Kor.
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