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Important Information about the new real number system
Posted:
Nov 9, 2009 2:00 AM
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SOME IMPORTANT INFORMATION ABOUT THE NEW REAL NUMBER SYSTEM
We first note the sources of ambiguity in a mathematical space so that we can avoid or contain them; they are contained if their ambiguity approximated by certainty, e.g., a nonterminating decimal which is ambiguous is approximated by its initial segment at the nth decimal place at margin of error 10^-n. We consider an ambiguous concept well-defined when its ambiguity is contained. The sources of ambiguity are: infinite set, large or small number (depending on context), self-reference, e.g., the barber paradox, vacuous concept, e.g., i = the root of the equation x^2 + 1 = 0, among the real numbers, ill-defined concept and statement involving ambiguous concept.
1) The new real number system is built on the one the elements 0 and 1 defined by the addition and multiplication tables (these are the three axioms).
2) The basic digits 0, 1, ?, 9 are built first, then the integers and the terminating decimals. They are the well defined decimals.
3) Then the inverse operation to multiplication called division; the result of dividing a decimal by another if it exists is called quotient provided the divisor is not zero. Only when the integral part of the devisor is not prime other than 2 or 5 is the quotient well defined. For example, 2/7 is ill defined because the quotient is not a terminating decimal (we interpret a fraction as division).
4) Since a decimal is determined or well-defined by its digits, nonterminating decimals are ambiguous or ill-defined. Consequently, the notion irrational is ill-defined since we cannot cheeckd all its digits and verify if the digits of a nonterminaing decimal are periodic or nonperiodic.
5) Consider the sequence of decimals,
(d)^na_1a_2?a_k, n = 1, 2, ?, (1)
where d 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 d and the a_j?s, j = 1, ?, k, in (1) 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 (1) 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 < d < x.
6) As convention when d* appear in any equation or expression, it means that either is unaltered if d* is replaced by any dark number, i.e., element of d*. The dark number d* satisfies the following:
N.99? - (N ? 1)? = d*, N = 1, 2, ?; if x is any nonterminating decimal different from zero, xd* = d*x = d*; (d*)^N = d* (1)
7) Nonterminating decimals. Now we define a nonterminating decimal for the first time without contradiction and with contained ambiguity, i.e., approximable by certainty. We build them on what we know: the terminating decimals, our point of reference for all their extensions.
A sequence of terminating decimals of the form,
N.a_1, N.a_1a_2, ?, N.a_1ª_2?a_n, ? (2)
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_1a_2?a_n,?, (3)
at margin of error 10?n. The g-limit of (2) is nonterminating decimal (3) 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.
8) Decimal integers. A nonterminating decimal of the form N.99? , N = 0, 1, ?, is call decimal integer because the set of such decimals for all N is isomorphic to the integers, i.e., the integral parts of the decimals, under the mapping d* -> 0, N -> (N ? 1).99?, N ? 1, 2, ? From the kernel of this isomorphism it follows that (0.99?)^N = 0.99? and ((0.99?)10)^N = (0.99?)10^N.
9) We note these important results.
a) 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 a_ks vary along the basic digits.
b) Theorem. The g-closure (closure in the g-norm) of the decimals is a continuum R*; this the new real number system; it is a continuum, countably infinite, non-archimedian and nonhausdorff but its subspace of decimals is also countably infinite, discrete, Archimedean and hausdorff/
c) Theorem. In the lexicographic ordering R* consists of adjacent predecessor-successor pairs each joined by the continuum d*.
d) 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).
e) Theorem. The largest and smallest elements of the open interval (0,1) are 0.99? and 1 ? 0.99? = d*, respectively.
f) Theorem. An even number greater than 2 is the sum of two prime numbers (this used to be called Goldbach?s conjecture; now it has a proof in R*).
g) The counterexamples to FLT. The exact solutions of Fermat?s equation, which are the counterexamples to FLT, are given by the triples (x,y,z) = ((0.99?)10^T,d*,10^T), T = 1, 2, ?, that clearly satisfies Fermat?s equation,
x^n + y^n = z^n, (4)
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. One counterexample is, of course, sufficient to disprove a conjecture.
References
[1] Benacerraf, P. and Putnam, H. (1985) Philosophy of Mathematics, Cambridge University Press, Cambridge, 52 - 61.
[2] Brania, A., and Sambandham, M., Symbolic Dynamics of the Shift Map in R*, Proc. 5th International
Conference on Dynamic Systems and Applications, 5 (2008), 68?72.
[3] Escultura, E. E. (1997) Exact solutions of Fermat's equation (Definitive resolution of Fermat?s last theorem, Nonlinear Studies 5(2), 227 ? 2254.
[4] Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 ? 169.
[5] Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 ? 149.
[6] Escultura, E. E., The new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computations 17 (2009), 59 ? 84.
[7] Escultura, E. E., Extending the reach of computation, Applied Mathematics Letters, Applied Mathematics Letters 21(10), 2007, 1074-1081.
[8] Escultura, E. E., The mathematics of the grand unified theory, in press, Nonlinear Analysis, Series A:
Theory, Methods and Applications; online at Science Direct website
[9] Escultura, E. E., The generalized integral as dual of Schwarz distribution, in press, Nonlinear Studies.
[10] Escultura, E. E., Revisiting the hybrid real number system, Nonlinear Analysis, Series C: Hybrid Systems, 3(2) May 2009, 101-107.
[11] Escultura, E. E., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier Science, Ltd.), 2009, Paris.
[12] Counterexamples to Fermat?s last theorem, http://users.tpg.com.au/pidro/
[13] Kline, M., Mathematics: The Loss of Certainty, Cambridge University Press, 1985.
E. E. Escultura
Research Professor
V. Lakshmikantham Institute for Advanced Studies
and Departments of Mathematics and Physics
GVP College of Engineering, JNT University
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