Advances in Pure Mathematics, 2013, 3, 394404 http://dx.doi.org/10.4236/apm.2013.34057 Published Online July 2013 (http://www.scirp.org/journal/apm) Copyright © 2013 SciRes. APM Representations of Each Number Type That Differ by Scale Factors Paul Benioff Physics Division, Argonne National Laboratory, Argonne, USA Email: pbenioff@anl.gov Received March 27, 2013; revised April 28, 2013; accepted May 22, 2013 Copyright © 2013 Paul Benioff. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT For each type of number, structures that differ by arbitrary scaling factors and are isomorphic to one another are de scribed. The scaling of number values in one structure, relative to the values in another structure, must be compensated for by scaling of the basic operations and relations (if any) in the structure. The scaling must be such that one structure satisfies the relevant number type axioms if and only if the other structure does. Keywords: Number Types; Structures 1. Introduction Numbers play an essential role in many areas of human endeavor. Starting with the natural numbers, N, of ari thmetic, one progresses up to integers, , rational num bers, Ra , real numbers, R, and to complex numbers, C. In mathematics and physics, each of these types of numbers is referred to as the natural numbers, the in tegers, rational numbers, real numbers, and the complex numbers. As is well known, though, “the” means “the same up to isomorphism” as there are many isomorphic representations of each type of number. In this paper, properties of different isomorphic repre sentations of each number type will be investigated. Em phasis is placed on representations of each number type that differ from one another by arbitrary scale factors. Here mathematical properties of these representations will be described. The possibility that these representations for complex numbers may be relevant to physics is de scribed elsewhere [1,2]. Here a mathematical system of each type is repre sented as a structure that satisfies a set of axioms relevant to the type of system being considered [3,4]. A structure consists of a base set, one or more basic operations, basic relations (if any), and constants. Any structure containing a base set, basic operations, relations, and constants that are relevant for the number type, and the structure satis fies the relevant axioms, is a model of the axioms. As such it is as good a representation of the number type as is any other representation. The contents of structures for the different types of numbers and the chosen axiom sets are shown below: ,,,,0,1NN Nonnegative elements of a dis crete ordered commutative ring with identity [5]. ,,,,,0,1II Ordered integral domain [6]. ,,,,,,0,1Ra Ra Smallest ordered field [7]. ,,,,,,0,1RR Complete ordered field [8]. ,,,,,,0,1CC Algebraically closed field of characteristic 0 plus axioms for complex conjugation [9,10]. Here an overline, such as in ,N denotes a structure. No overline, as for N, denotes a base set. The complex conjugation operation has been added as a basic ope ration to C as it makes the development much easier. For this work, the choice of which axioms are used for each of the number types is not important. For example, an alternate choice for N is to use the axioms of ari thmetic [11]. In this case N is changed by deleting the constant 1 and adding a successor operation. There are also other axiom choices for the real numbers [12]. The importance of the axioms is that they will be used to show that, for two structures related by a scale factor, one satisfies the axioms if and only if the other does. This is equivalent to showing that one is a structure for a given number type if and only if the other one is a struc ture for the same number type. These ideas will be expanded in the following sections. The next section gives a general treatment of fields. This applies to all the number types that satisfy the field axi
P. BENIOFF Copyright © 2013 SciRes. APM 395 oms (rational, real, complex numbers). However much of the section applies to other numbers also (natural num bers, integers). The following five sections apply the ge neral results to each of the number types. The discussions are mainly limited to properties of the number type that are not included in the description of fields. Section 8 expands the descriptions of the previous sec tions by considering ,, ,NIRaR as substructures of .C In this case the scaling factors relating two structures of the same type are complex numbers. Section 9 concludes the paper with a discussion of some aspects of number scaling, and extension to other types of mathematical systems that are based on num bers. 2. General Description of Fields It is useful to describe the results of this work for fields in general. The results can then be applied to the different number types, even those that are not fields. Let S be a field structure where ,,,,,0,1.SS (1) Here S with no overline denotes a base set, +, −, ×, ÷ denote the basic field operations, and 0,1 denote constants. Denoting S as a field structure implies that S is a structure that satisfies the axioms for a field [13]. Let S where ,,,,,0,1. ppppppp SS (2) be another structure on the same set S that is in .S The idea is to require that p S is also a field structure on S where the field values of the elements of S in p S are scaled by ,p relative to the field values in .S Here p is a field value in .S The goal is to show that this is possible in that one can define p S so that p S satisfies the field axioms if and only if S does. To this end the notion of corres pondence is introduced as a relation between the field values of p S and .S The field value, , a in p S is said to correspond to the field value, ,pa in .S As an example, the identity value, 1, in p S corresponds to the value 1pp in .S This shows that correspondence is distinct from the concept of sameness. a is the same value in p S as a is in .S This differs from pa by the factor .p The distinction between correspondence and sameness is present only if 1.p If 1p , then the two concepts coincide, and p S and S are the same structures. So far a scaling factor has been introduced that relates field values between p S and .S This must be com pensated for by a scaling of the basic operations in p S relative to those of .S The correspondences of the basic field operations and values in p S to those in S are given by, , ,. p pp pp apa p p (3) One can use these scalings to replace the basic opera tions and constants in S and define S by, ,,, ,,0,. p SS pp p (4) Here the subscript, ,p in , S Equation (2) is re placed by p as a superscript to distinguish S from . S Both S and S can be considered as different representations or views of a structure that differs from S by a scaling factor, p. A useful expression of the relation between S and S is that S is referred to either as a representation of S on ,S or as an explicit representation of S in terms of the operations and element values of .S Besides changes in the definitions of the basic opera tions given in Equation (3) and distinguishing between correspondence and sameness, scaling introduces another change. This is that one must drop the usual assumption that the elements of the base set, ,S have fixed values, independent of structure membership. Here the field values of the elements of S, with one exception, depend on the structure containing .S In particular, to say that a in S corresponds to pa in S means that the element of S that has the value a in S has the value pa in .S This is different from the element of S that has the same value, ,a in S as a is in . S These relations are shown schematically in Figure 1. The valuations associated with elements in the base set S are shown by lines from S to the structures S and . S Figure 1. Relations between Elements in the base set and their Values in the Structures and . Here a is the same value in as a is in . The lines show that they are values for different elements of . The lines also show that the element that has the value a, as a in , has the value a in .
P. BENIOFF Copyright © 2013 SciRes. APM 396 The one exception to the structure dependence of valuations is the element of S with the value 0. This valueelement association is fixed and is independent of all values of .p In this sense it is the “number vacuum” as it is invariant under all changes pp 1. One can define isomorphic maps between the structure representations. Define the maps W and W by . pp pp pp SWSWWSFS (5) W maps S onto S and W maps S onto . S W is a map from one structure to another, and W is a map between different representations of the same structure. W and W are defined by , p pppp pppp pppp Wa pa WabWaWWbpa pb W abW aWWbpapb p WabWaWWbpap pb (6) and , . pp ppppppp pppp pp pppp ppp Wpa a Wpapb WpaWWpbab WpapbWpaWWpba b pp W pappb W paW pWpb ab (7) The following two theorems summarize the relations between ,, p SS and .S The first theorem shows the invariance of equations under the maps, W and W where ,p is a scaling factor. It also shows, that the correspondence between between element values in S and those in ,S extends to general terms. Theorem 1 Let t and u be terms in .S Let S and S be as defined in Equations (2) and (4). Then pp p tu tut u where , p tWt , pp uWu and ,. pp pp p WttWuu Proof: It follows from the properties of W and , W Equations (6) and (7), that pp tWtpt and . pp uWupu Also pp tWt and . pp uWu This gives . pp pp tutuptputu (8) From the left the first equation is in S, the second in , S the third in both S and ,S and the fourth in .S It remains to see in detail that the correspondence between element values in S and those in S extends to terms. Let ,1 . j mp pk jk pp a tb (9) The view of t in S is ,1 . pj m p k jk pa tpb (10) In the numerator, the j pa factors and 1j mul tiplications contribute factors of p and 1, j p res pectively, to give a factor p. This is canceled by a similar factor arising from the denominator. A final factor of p arises from the representation of the solidus as shown in Equation (3) for division. Using this and the fact that addition is not scaled, gives the result that ,1 ,1 . p pjj mm p kk jk jk pa a tppt pb b (11) Equation (8) and the theorem follow from the fact that Equation (11) holds for any term, including . u ■ From this one has Theorem 2 S sa tisfies the field axioms if and only if S satisfies the field axioms if and only if S satisfies the field axioms. Proof: The theorem follows from Theorem 1 and the fact that all the field axioms are equations for terms. ■ The constructions described here can be iterated. Let p be a number value in S and qq be a number value in . S Let qp S be another field structure on the base set, ,S and let qp S be the representation of qp S using the operations and constants of S2. In more detail  ,,,,0,1 qpqp qpqpqp SS (12) and ,, ,,0,1. p qp ppp SS qq q (13) qp S is related to S by the scaling factor .q The goal is to determine the scaling factor for the representation of qp S on .S To determine this, let qp a be a value in . qp S This corresponds to a value pp qa in . S Here qp a is the same value in qp S as pp qa is in qp S as a is in . S 1Like the physical vacuum which is unchanged under all space time translations. 2An equivalent way to define qp S is as the representation of qp Son . p S
P. BENIOFF Copyright © 2013 SciRes. APM 397 The value in S that corresponds to qp a in qp S can be determined from its correspondent, , pp qa in . S The value in S that corresponds to pp qa is obtained by use of Equations (3) and (7). It is given by 1. pppp Wqa pqpapqa p (14) Here q and a are the same values in S as q and a are in . S Also 1, p W as the inverse of , W maps S onto . S This is the desired result because it shows that two steps, first with p and then with q is equivalent to one step with .qp This result shows that the represen tation of qp S on S is given by ,,,,0, 1. qp SSqp qp qp (15) Note that the steps commute in that the same result is obtained if one scales first by q and then by p as .pq qp Here qp is a value in .S Also this is equi valent to determining the scale factor for q S on S provided one accounts for the fact that q is a value in S and not in .S 3. Natural Numbers The natural numbers differ from the generic represen tation in that they are not fields [5]. This is shown by the structure representation for ,N ,,,,0,1.NN (16) The structure corresponding to n S is ,,,,0,1. nnnnnnn NN (17) Here n is any natural number 0. One can use Equation (3) to represent n N in terms of the basic operations, relations, and constants of .N It is ,, ,,0,. nn NN n n (18) Note that, as is the case for addition, the order relation is the same in n N as in n N and in .N As was the case for fields, n N and n N represent external and in ternal views of a different structure than that represented by N. For N the external and internal views coincide. The multiplication operator in , n N n has the re quisite properties. This can be seen by the equivalences between multiplication in ,, n n NN and :N . nnn n abc nanbncabc n Note that the simple verification of these equivalences takes place outside the three structures and not within any natural number structure. For this reason division by n can be used to verify the equivalences even though it is not part of any natural number structure. These equivalences show that n is the multiplicative identity in n N if and only if 1 is the multiplicative identity in .N To see this set 1.b The structure, , n N Equation (18), and that of Equa tion (17), differ from the generic description, Section 2, in that the base set n N is a subset of .N n N contains just those elements of N whose values in N are mul tiples of .n For example, the element with value n in N has value 1 in , n N and the element with value na in N has value a in . n N Elements with values na l in N where 0,ln are absent from n N3. As noted before, the choice of the representations of the basic operations and relation and constants in , n N as shown in , n N is determined by the requirement that n N satisfies the natural number axioms [5] if and only if N satisfies the axioms. For the axioms that are equa tions and do not use the ordering relation, this follows immediately from Theorem 1. For axioms that use the order relation, the requirement follows from the fact that n and for any pair of terms , nn tu, . nn n tunt nut u Here Equation (11) was used with 1.b Note that 0.n It follows from these considerations that Theorem 3 n N satisfies the axioms of arithmetic if and only if n N does if and only if N does. A simple example that illustrates the theorem is the axiom of discreteness for the ordering, 0<1>0 1aa a ∧ [5]. One has the equiva lences 0< 1>01 0<>0 1 0<1>01 . nnnnnn nn aa a nana nan aa a ∧ ∧ ∧ Subscripts are missing on 0 because the value remains the same in all structures. 4. Integers Integers generalize the natural numbers in that negative numbers are included. Axiomatically they can be characterized as an ordered integral domain [6]. As a structure, I is given by ,,,,,0,1.II (19) is a base set, ,, are the basic operations, < is an order relation, and 0,1 are the additive and 3The exclusion of elements of the base set in different representations occurs only for the natural numbers and the integers. It is a conse quence of their not being closed under division.
P. BENIOFF Copyright © 2013 SciRes. APM 398 multiplicative identities. Let j be a positive integer. Let be the structure ,,,,,0,1. jjjjjjj II (20) I is the subset of containing all and only those elements of whose values in are positive or ne gative multiples of j or 0. The representation of in terms of elements, ope rations, and relations in is given by ,,, ,,0,. jj Ij j (21) This structure differs from that of the natural numbers, Equation (18), by the presence of the additive inverse, . The proof that and satisfy the integer axioms if and only if I does, is similar to that for Theorem 3. The only new operation is the additive inverse. Since the axioms for this are similar to those already present, details of the proof for axioms involving subtraction will be skipped. A new feature enters in the case that j is negative. It is sufficient to consider the case where 1j as the case for other negative integers can be described as a combination of 1j followed by scaling with a positive j. This would be done by extending the ite ration process, described for fields in Section 2, to in tegers. The integer structure representations for 1j that correspond to Equations (20) and (21) are given by 1111111 ,,,,,0,1,II (22) and 1,,,,,0,1. 1 II (23) The main thing to note here is that the order relation, 1, in 1 corresponds to in . 1 can also be described as a reflection of the whole structure, 1 I through the origin at 0. Not only are the integer values reflected but also the basic operations and order relation are reflected. Also in this case the base set, 1. I Equation (23) indicates that 1 is the identity and 1 is positive in 1. These follow from 1111 11 1 aaa a and 11 01 01. This equivalence shows that the relation, , which is interpreted as greater than in , is interpreted as less than in 1. Thus, as a relation in 1, 01 means 1 is greater than 0. As a relation in , 01 means 1 is less than 0. This is an illustration of the relation of the ordering relation 1 < to .< Integers which are positive in 1 and 1, are negative in . It follows that 11 111 01 2 is true in 1 if and only if 012 is true in 1 and in . These considerations show that 1 and 1 satisfy the integer axioms if and only if satisfies the axioms. For axioms not involving the order relation the proof is similar to that for for 0.j For axioms involving the order relation the proofs proceed by restating axioms for in terms of . An example is the axiom for transitivity .ab bcac ∧ For this axiom the validity of the equivalence 111111111 abbcac ab bcac ∧ ∧ shows that this axiom is true in 1 I if and only if it is true in 1 if and only if it is true in . These considerations are sufficient to show that a theorem equivalent to Theorem 3 holds for integers: Theorem 4 For any integer 0j, satisfies the integer axioms if and only if does if and only if I does. 5. Rational Numbers The next type of number to consider is that of the rational numbers. Let Ra denote a rational number structure ,,,,,,0,1.Ra Ra (24) For each positive rational number r let r Ra denote the structure ,,,,,,0,1. rrrrrrrr Ra Ra (25) Equations (24) and (25) show that Ra and r Ra have the same base set, .Ra This is a consequence of the fact that rational numbers are a field. As such Ra nd r Ra are special cases of the generic fields described in Section 2. Note also that Ra is the same as 1.Ra The definition of r Ra is made specific by the re presentation of its elements in terms of those of .Ra It is ,,, ,,,0,. r Ra Rarr r (26) As was seen for fields in Section 2, the number values of the elements of Ra depend on the structure con taining .Ra The element of Ra that has value ra in Ra has value a in . r Ra The element of Ra that has the value a in r Ra is different from the element that has the same value a in .Ra The only exception
P. BENIOFF Copyright © 2013 SciRes. APM 399 is the element with value 0 as this value is the same for all . r Ra Also, as noted in Section 2, r Ra and r Ra represent external and internal views of a structure that differs from that represented by .Ra As was the case for multiplication, the relation be tween rr and is fixed by the requirement that r Ra satisfy the rational number axioms [7] if and only if r Ra satisfies the axioms if and only if Ra does. this can be expressed as a theorem: Theorem 5 Let r be any nonzero rational number. Then Ra satisfies the rational number axioms if and only if r Ra satisfies the axioms if and only if r Ra does. Proof: Since the axioms for rational numbers include those of an ordered field, the proof contains a combination of that already given for fields in Section 2, Theorem 1, and for the ordering axioms for integers, as in Theorem 4. As a result it will not be repeated here. It remains to prove that r Ra is the smallest ordered field if and only if r Ra is if and only if Ra is. To show this, one uses the isomorphisms defined in Section 2. Let S be the smallest ordered field. Let r W and r W be isomorphisms whose definitions on Ra and r Ra follow that of W and W in Equations (6) and (7). That is . rr rrr Ra WRa WWRa (27) Since these maps, as isomorphisms, are oneone onto and are order preserving, they have inverses which are also isomorphisms. In this case one has r r rr r RaSW Ra RaS WRa RaS (28) and 1 1. r rr r r r RaSWRa RaS WRaRaS (29) This proves the theorem. ■ For rational number terms, Theorem 1 holds here. From Equation (10) one sees that for rational number structures, ,1 ,1 . r rjj mm r kk jk jk ra a trrt rb b (30) 6. Real Numbers The description for real numbers is similar to that for the rational numbers. The structures Ra and , r Ra Equa tions (24) and (25), become ,,,,,,0,1RR (31) and ,,,,,,0,1. r rrrrrrr RR (32) The external representation of the structure, whose internal representation is , r R is given in terms of the elements, operations, relations and constants of .R It is denoted by r R where ,,,,,,0,. r RRr r r (33) Here r is any positive real number value in .R. If 0r then in Equation (33) is replaced by . The axioms for the real numbers [8] are similar to those for the rational numbers in that both number types satisfy the axioms for an ordered field. For this reason the proof that r R satisfies the ordered field axioms if and only if r R satisfies the axioms if and only if R does will not be given as it is essentially the same as that for the rational numbers. Real numbers are required to satisfy an axiom of completeness. For this axiom, let :0,1,2, rr jj aaj be a sequence of real numbers in , r R Equation (32). Let r be a positive real number value in R. r a converges in r R if For all 0 rr there exists an h such that for all ,jm h . rrr rr jm r aa (34) Here rrr jm r aa denotes the absolute value, in , r R of the difference between r a and . rm a The numerical value in , r R Equation (33), that is the same as rrr jm r aa is in , r R is given by . jm ra ra This is the same value as m aa is in .R m ra ra also corresponds to a number value in R given by . mjm ra rara a (35) Theorem 6 Let 0r be a real number value in .R The sequence r a converges in r R if and only if ra converges in r R if and only if a converges in .R Proof: The proof is in two parts: first 0,r and then 0.r
P. BENIOFF Copyright © 2013 SciRes. APM 400 0:r Let r be a positive number value in r R such that . rrr rr jm r aa It follows that jm ra rar in both r R and .R Equation (35) gives the result that jm aa in .R Conversely Let jm aa be true in .R Then jm ra rar is true in both R and , r R and rrr rr jm r aa is true in . r R From this one has the equivalences, . rrr rrjmjm jm r aa raaraa It follows that r a converges in r R if and only if ra converges in r R if and only if a con verges in .R 0r: It is sufficient to set 1.r In this case 1 R and 1 R are given by Equations (32) and (33) with 1.r In this case 11111111 ,,,,,,0,1RR (36) and 1,,, ,1,,0,1. 1 RR (37) As was the case for integers, and is the case for ra tional numbers, this structure can also be considered as a reflection of the structure, ,R through the origin at 0. As before let 1 a be a convergent sequence in 1,R Equation (36). The statement of convergence is given by Equation (34) where 1.r The statement 111 11 1 jm aa says that 111 1 jm aa is a positive number in 1 R that is less than the positive number 1 and 0. The corresponding statement in 1,R Equation (37), is 1. jm aa (38) Since 1 R is a reflection of R about the origin, the absolute value, , in R becomes rx x in 1.R In 1,R the absolute value, 1, is always positive even though it is always negative in .R For example, 1 mjm aa aa where 1 and are the respective absolute values in 1 R and .R In this case Equation (38) can be recast as . jm aa (39) This can be used as the convergence condition in Equation (34). Note that 0 m aa and that de notes “less than or equal to” in 1.R It follows from this that for 1,r the sequence 1 a converges in 1,R Equation (36), if and only if the sequence a converges in 1,R Equation (37), if and only if the sequence a converges in .R Extension to arbitrary negative r can be done in two steps. One first carries out the reflection with 1r . This is followed by a scaling with a positive value of r as has already been described in Section 2. ■ Theorem 7 r R is complete if and only if r R is complete if and only if R is complete. Proof: Assume that r R is complete and that the sequence r a converges in r R. Then there is a number value r in r R such that lim . rr j a The properties of convergence, Equation (34), with rm a replaced by , r and Theorem 6, can be used to show that limlimlim . rr jj j jjj arara (40) Here r is the same number value in r R as r is in r R as is in .R Conversely assume that R is complete and that a converges in R to a number value . Repeat ing the above argument gives limlimlim . rr jj j jjj arara (41) This shows that r R is complete if and only if r R is complete if and only if R is complete. ■ Since real number structures are fields, Equation (11) holds for real number terms. That is ,1 ,1 . r rjj mm r kk jk jk ra a trrt rb b (42) These terms can be used to give relations between power series in ,, r r RR and .R Let 1 ,n rr rr jj Pnxa x be a power series in . r R Then , r Pnrx and ,Pnx are the same power series in r R and R as , rr Pnx is in . r R This means that for each real number value , r , r Pnrx and ,Pnx are the respective same number values in r R and R as , rr Pnx is in . r R Here rx and are the same number values in r R and R as r is in . r R However, the power series in R that corresponds to , rr Pnx and , r Pnrx in r R and r R is obtained from Equation (11). It is ,,,. r rr PnxP nrxrPnx (43) This shows that the element of R that has value , rr Pnx in r R has value ,rPn x in .R
P. BENIOFF Copyright © 2013 SciRes. APM 401 These relations extend to convergent power series. Theorem 6 gives the result that , rr Pnx is convergent in r R if and only if , r Pnrx is convergent in r R if and only if ,rPn x is convergent in .R It follows from Theorem 6 and Equations (40) and (41) that, lim , lim , lim ,. rrrr n rr n n Pnxfx Pnrxf rx rPn xrfx (44) Here rr x is the same analytic function [14] in r R as r rx is in r R as x is in .R Equations (43) and (44) give the result that, for any analytic function , . r rr xfrxrf x (45) Here r and are functions in r R and R and rr x is the same number value in r R as r rxrf x is in r R as x is in .R Simple examples are ee e rrx xrx rr for the exponential and sin sinsin r rr rx rx for the sine function. Caution: 22 sinsin , rr rx not 22 sin .rx 7. Complex Numbers The descriptions of structures for complex numbers is si milar to that for the real numbers. Let C denote the complex number structure ,,,,,,0,1.CC (46) For each complex number c let c C be the internal representation of another structure where ,,,,,,0,1. c ccccccc CC (47) The external representation of the structure, in terms of operations and constants in ,C is given by ,,, ,,,0,1. c CCcc c c (48) The relations for the field operations are the same as those for the real numbers except that c replaces .r It follows that Equations (42)(45) hold with c replacing .r These equations show that analytic functions, cc x in , c C have corresponding functions in C given by . cc xcfx (49) Here c denotes the same number in c C as is in .C The relation for complex conjugation is given by c c aca It is not . c c aca One way to show this is through the requirement that the relation for complex conjugation must be such that 1c is a real number value in c C if and only if 1 is a real number value in .C This requires that the equivalences 11111 111 c c cc cccc be satisfied. These equivalences show that 11 c cc or more generally . cc c acaca (50) Note that any value for a is possible including c or powers of .c For example, . c c nnn c ccccc As values of elements of the base set, C, Equation (50) shows that the element of C that has value c c a in c C has value ca in .C This is different from the ele ment of C that has the same value, ,a in C as c c a is in . c C Another representation of the relation of complex conjugation in c C to that in C is obtained by writing i e.cc Here c is the absolute value of .c This can be used to write 2i e. cc c aca ca (51) That is 2i e. c cc For most of the axioms, proofs that c C satisfies an axiom if and only if C does are similar to those for the number types already treated. However, it is worth dis cussing some of the new axioms. For example the complex conjugation axiom [10] x has an easy proof. It is based on Equation (50), which gives . cc cc c c acacaca From this one has the equivalences, . c c cc aacacaaa This shows that x is valid for c C if and only if it is valid for c C and .C The other axiom to consider is that for algebraic clo sure. Theorem 8 c C is algebraically closed if and only if c C is algebraically closed if and only if C is alge braically closed. Proof: The proof consists in showing that any polynomial equation has a solution in c C if and only if the same po lynomial equations in c C and C have the same solu tions. Let 00 nj j jbx denote a polynomial equation that has a solution c a in . c C Then 00 j n cc jj ba in . c C Carrying out the replacements , cj j bcb , c aca , cc and c gives the implications
P. BENIOFF Copyright © 2013 SciRes. APM 402 00 000 00 000. c nn c jj cc j j jj c nnn jjj jjj jjj bacb ca cbacbaba From the left, the first equation is in , c C the second is in c C, and the other three are in .C This shows that if c a is the solution of a polynomial in c C then ca is the solution of the same polynomial equation in , c C and a is the solution of the same polynomial equation in .C The proof in the other direction consists in assuming that 00 nj j jba is valid in C for some number ,a and reversing the implication directions to obtain 00 j n cc jj cba in . c C ■ 8. Number Types as Substructures of c C, c C, and C As is well known each complex number structure con tains substructures for the real numbers, the rational num bers, the integers, and the natural numbers. The struc tures are nested in the sense that each real number struc ture contains substructures for the rational numbers, the integers, and the natural numbers, etc. Here it is suffi cient to limit consideration to the real number substruc tures of , c C , c C and C. To this end let c be a complex number and let c R and R be real number substructures in c C and .C In this case ,,,,,,0,1 c cccccccc RR (52) and ,,,,,,0,1.RR (53) The field operations in c R and R are the same as those in c C and C respectively, restricted to the num ber values of the base sets, c R and .R Expression of the field operations and constants of c R in terms of those of C gives a representation of c R that corresponds to Equation (48). It is ,,, ,,,0,1. cc c RRcc c (54) Recall that the number values associated with the ele ments of a base set are not fixed but depend on the struc ture containing them. R contains just those elements of C that have real values in .C c R contains just those values of C that that have real values in c C and . c C Let be an element of R that has a real value, ,a in .R This is different from another element, ,y of c R that has the same real value, , c a in c R as a is in .R The element, ,y also has the value, ,ca in , c R which is the same real value in c R as a is in .R This shows that y cannot be an element of R if c is complex. The reason is that ca is a complex number value in ,C and R cannot contain any elements of C that have complex values in .C It follows that, if c is complex, then c R and R have no elements in common, except for the element with value 0, which is the same in all structures. The order relations , c , c and are defined re lations as ordering is not a basic relation for complex numbers. A simple definition of in R is provided by defining ab in C by if and are real number values in and there exists a positive real number value, , in such that . ab C ab dC adb (55) Here d is a positive real number in C if there exists a number in C such that .dgg The definition for c in c R is obtained by putting c sub scripts everywhere in Equation (55). The order relation c in c R is if and are real number sin, and there exists a positive real number such that . c c ca cbC ca cbcd ca cdcb (56) Here cd is a positive real number in c C if there exists a number cg in c C such that . c cdcg cg One still has to prove that these definitions of ordering are equivalent: Theorem 9 Let ,a b be real number values in .R Then . cccc abcacbab Proof: Replace the three order statements by their definitions, Eqs. 55 and 56, to get . ccc c ad bcacd cbadb Here dgg , c cd cgcg , and . c ccc dgg Let ,,abg be real numbers values in .R Then ,,ca cb cg and ,, cc c abg are the same real number values in c R and c R as ,,ab g are in .R Define the positive number value d by .dgg Then by Equation (48), g is the same number value in R as c cgcgcg gcd c is in c R as c ccc gd is in . c R Thus cd and c d are the same positive number values in c R and c R as d is in .R From this one has , cccc ad bcacdcbadb
P. BENIOFF Copyright © 2013 SciRes. APM 403 which gives . cccc ab cacbab The reverse implications are proved by starting with , cc ab, and c ccc dgg and using an argument similar to that given above. ■ 9. Discussion So far the existence of many different isomorphic struc ture representations of each number type has been shown. Some properties of the different representations, such as the fact that number values, operations, and relations, in one representation are related to values in another repre sentation by scale factors have been described. Here some additional aspects of these structure repre sentations and their effect on other areas of mathematics will be briefly summarized. One aspect worth noting is the fact that the rational, real, and complex numbers are each both additive and multiplicative groups. These group properties of the fields for rational, real, and complex numbers, induce corresponding properties in the collection of structures for each of these number types. The discussion will be limited to complex numbers since extension to other number types is similar. The map for complex numbers is a complex number. c cCc For the additive group, the map needs to be extended to include 0r. This is done by including the empty structure, 0.C Let C G denote the collection of complex number structures that differ by arbitrary nonzero complex scaling factors. Define the operation, , by . cd cd CC C (57) Justification for this definition is provided by the description of iteration that includes Equations (12)(15). GC is defined relative to the identity group structure, ,C as ,cd and cd are values in .C Every structure c C has an inverse, 1c C as 1. cc CC C (58) This shows that GC is a multiplicative group of complex number structures induced by the multiplicative group of complex number values in .C Its properties mirror the properties of the group of number values. The additive group is defined similarly by an operation where . cd cd CCC The empty structure, 0 C, is the additive identity and c C is the additive inverse of . c C Another consequence of the existence of representa tions of number types that differ by scale factors is that they affect other mathematical systems that are based on different number types as scalars. Examples include any system type that is closed under multiplication by nu merical scalars. Specific examples are vector spaces based on either real or complex scalars, and operator al gebras. For vector spaces based on complex scalars, one has for each complex number c a corresponding pair of structures, ,. cc VC The scalars for c V are number values in . c C The representation of c V as a structure with the basic operations expressed in terms of those in ,V with C as scalars, is similar to that for , c C Equation (48). For Hilbert spaces, the structure , based on ,C is given by ,,,, ,,.HH (59) Here and , denote scalar vector multiplica tion and scalar product. denotes a general state in . The representation of another Hilbert space structure that is based on , c C is given by ,,,,, ,. cccc c c HH (60) The representation that expresses the basic operations of c in terms of those for is given by , ,,, ,,. c HH c cc (61) Additional details with c restricted to be a real num ber are given in [1,2]. A possible use of these structures in physics is based on an approach to gauge theories [15,16] in which a fi nite dimensional vector space V is associated with each space time point . So far just one complex num ber field, C, serves as the scalars for all the . V Expansion of this approach by replacing C with dif ferent complex number fields, C at each point has been explored in [1,2]. If the freedom of basis choice in the vector spaces V, introduced by Yang [17], is ex tended to include freedom of scaling of the different C, then C and C are related by a scaling factor ,e. x x yx r This factor scales the numbers of C relative to those in C. A number value a in C corresponds to the number value , xx ra in C. a is the same number value in C as a is in . C The scalar field appears in gauge theories as a scalar boson that may or may not have mass. At present it is an open question if is a candidate for any of the proposed scalar fields in physics. In any case, the scaling of number structures, and other types of mathematical structures based on numbers,
P. BENIOFF Copyright © 2013 SciRes. APM 404 might be useful for development of a coherent theory of physics and mathematics together [18,19]. More work needs to be done to see if these ideas have any merit. 10. Acknowledgements This work was supported by the US Department of Energy, Office of Nuclear Physics, under Contract No. DEAC0206CH11357. REFERENCE [1] P. Benioff, “New Gauge Field from Extension of Space Time Parallel Transport of Vector Spaces to the Underly ing Number Systems,” International Journal of Theore tical Physics, Vol. 50, No. 6, 2011, pp. 18871907. doi:10.1007/s1077301107043 [2] P. Benioff, “Effects on Quantum Physics of the Local Availability of Mathematics and Space Time Dependent Scaling Factors for Number Systems,” In: I. Ion Cotaescu, Ed., Advances in Quantum Theory, InTech, 2012. http://www.intechopen.com/ doi:10.5772/36485 [3] J. Barwise, “An Introduction to First Order Logic,” In: J. Barwise, Ed., Handbook of Mathematical Logic, North Holland Publishing Co., New York, 1977, pp. 546. doi:10.1016/S0049237X(08)710978 [4] H. J. Keisler, “Fundamentals of Model Theory,” In: J. Barwise, Ed., Handbook of Mathematical Logic, North Holland Publishing Co., New York, 1977, pp. 47104. doi:10.1016/S0049237X(08)71098X [5] R. Kaye, “Models of Peano Arithmetic,” Clarendon Press, Oxford, 1991, pp. 1621. [6] Wikipedia: Integral Domain. [7] A. J. Weir, “Lebesgue Integration and Measure,” Cambri dge University Press, New York, 1973, p. 12. [8] J. Randolph, “Basic Real and Abstract Analysis,” Acade mic Press, Inc., New York, 1968, p. 26. [9] J. Shoenfield, “Mathematical Logic,” Addison Weseley Publishing Co. Inc., Reading, 1967, p. 86. [10] Wikipedia: Complex Conjugate. [11] R. Smullyan, “Gödel’s Incompleteness Theorems,” Ox ford University Press,, New York, 1992, p. 29. [12] S. Lang, “Algebra,” 3rd Edition, Addison Weseley Publi shing Co., Reading, 1993, p. 272. [13] I. Adamson, “Introduction to Field Theory,” 2nd Edition, Cambridge University Press, New York, 1982, Chapter 1. [14] W. Rudin, “Principles of Mathematical Analysis,” 3rd Edition, International Series in Pure and Applied Mathe matics, McGraw Hill Book Co., New York, 1976, p. 172. [15] G. Mack, “Physical Principles, Geometric Aspects, and Lo cality Properties of Gauge Field Theories,” Fortshritte der Physik, Vol. 29, No. 4, 1981, pp. 135185. doi:10.1002/prop.19810290402 [16] I. Montvay and G. Münster, “Quantum Fields on a Lat tice,” Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1994. doi:10.1017/CBO9780511470783 [17] C. N. Yang and R. L. Mills, “Conservation of Isotopic Spin and Isotopic Gauge Invariance,” Physic al Revi ew , Vol. 96, No. 1, 1954, pp. 191195. doi:10.1103/PhysRev.96.191 [18] P. Benioff, “Towards a Coherent Theory of Physics and Mathematics,” Foundations of Physics, Vol. 32, No. 7, 2002, pp. 9891029. doi:10.1023/A:1016561108807 [19] P. Benioff, “Towards a Coherent Cheory of Physics and Mathematics: The TheoryExperiment Connection,” Foun dations of Physics, Vol. 35, No. 11, 2005, pp. 18251856. doi:10.1007/s1070100573516
