 Advances in Pure Mathematics, 2013, 3, 394-404 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, I, 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 .  ,,,,,0,1II Ordered integral domain .  ,,,,,,0,1Ra Ra Smallest ordered field .  ,,,,,,0,1RR Complete ordered field .  ,,,,,,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 . 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 . 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 395oms (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 . Let pS where ,,,,,0,1.pppppppSS (2) be another structure on the same set S that is in .S The idea is to require that pS is also a field structure on S where the field values of the elements of S in pS 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 pS so that pS 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 pS and .S The field value, ,pa in pS is said to correspond to the field value, ,pa in .S As an example, the identity value, 1,p in pS corresponds to the value 1pp in .S This shows that correspondence is distinct from the concept of sameness. pa is the same value in pS 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 pS and S are the same structures. So far a scaling factor has been introduced that relates field values between pS and .S This must be com- pensated for by a scaling of the basic operations in pS relative to those of .S The correspondences of the basic field operations and values in pS to those in S are given by, ,,.pppppapapp  (3) One can use these scalings to replace the basic opera- tions and constants in pS and define pS by, ,,, ,,0,.pSS ppp  (4) Here the subscript, ,p in ,pS Equation (2) is re- placed by p as a superscript to distinguish pS from .pS Both pS and pS 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 pS and pS is that pS is referred to either as a representation of pS on ,S or as an explicit representation of pS 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 pa in pS corresponds to pa in S means that the element of S that has the value pa in pS has the value pa in .S This is different from the element of S that has the same value, ,a in S as pa is in .pS 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 .pS Figure 1. Relations between Elements in the base set S and their Values in the Structures S and pS. Here pa is the same value in pS as a is in S. The lines show that they are values for different elements of S. The lines also show that the S element that has the value a, as pa in pS, has the value pa in S. 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 value-element 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 pW and pW by .pppp ppSWSWWSFS  (5) pW maps S onto pS and pW maps pS onto .pS pW is a map from one structure to another, and pW is a map between different representations of the same structure. pW and pW are defined by    ,pppppppppppppWa paWabWaWWbpa pbW abW aWWbpapbpWabWaWWbpap pb    (6) and    ,.pppppppppppppppppppppppWpa aWpapb WpaWWpbabWpapbWpaWWpba bppW pappb W paW pWpbab      (7) The following two theorems summarize the relations between ,,ppSS and .S The first theorem shows the invariance of equations under the maps, pW and pW where ,p is a scaling factor. It also shows, that the correspondence between between element values in pS and those in ,S extends to general terms. Theorem 1 Let t and u be terms in .S Let pS and pS be as defined in Equations (2) and (4). Then pppptu tut u  where ,pptWt ,ppuWu and ,.ppppp pWttWuu Proof: It follows from the properties of pW and ,pW Equations (6) and (7), that pptWtpt and .ppuWupu Also ppptWt and .pppuWu This gives .pppptutuptputu  (8) From the left the first equation is in pS, the second in ,pS the third in both pS and ,S and the fourth in .S It remains to see in detail that the correspondence between element values in pS and those in S extends to terms. Let ,1.jmppkjk pppatb (9) The view of pt in pS is ,1.ppjmpkjkpatpb (10) In the numerator, the j pa factors and 1j mul- tiplications contribute factors of jp and 1,jp 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.ppjjmmpkkjk jkpa atpptpb b    (11) Equation (8) and the theorem follow from the fact that Equation (11) holds for any term, including .pu ■ From this one has Theorem 2 pS sa tisfies the field axioms if and only if pS 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 pqq be a number value in .pS Let |qpS be another field structure on the base set, ,S and let |qpS be the representation of |qpS using the operations and constants of pS2. In more detail |||||,,,,0,1qpqp qpqpqpSS (12) and |,, ,,0,1.pqp pppSS qqq  (13) |qpS is related to pS by the scaling factor .q The goal is to determine the scaling factor for the representation of |qpS on .S To determine this, let |qpa be a value in |.qpS This corresponds to a value pppqa in .pS Here |qpa is the same value in |qpS as pppqa is in |qpS as pa is in .pS 1Like the physical vacuum which is unchanged under all space time translations. 2An equivalent way to define |qpS is as the representation of |qpSon .pS P. BENIOFF Copyright © 2013 SciRes. APM 397The value in S that corresponds to |qpa in |qpS can be determined from its correspondent, ,pppqa in .pS The value in S that corresponds to pppqa is obtained by use of Equations (3) and (7). It is given by  1.ppppWqa pqpapqap  (14) Here q and a are the same values in S as pq and pa are in .pS Also 1,pW as the inverse of ,pW maps pS onto .pS 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 |qpS on S is given by |,,,,0, 1.qpSSqp qpqp  (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 qS on S provided one accounts for the fact that q is a value in pS and not in .S 3. Natural Numbers The natural numbers differ from the generic represen- tation in that they are not fields . This is shown by the structure representation for ,N ,,,,0,1.NN  (16) The structure corresponding to nS is ,,,,0,1.nnnnnnnNN  (17) Here n is any natural number 0. One can use Equation (3) to represent nN in terms of the basic operations, relations, and constants of .N It is ,, ,,0,.nnNN nn (18) Note that, as is the case for addition, the order relation is the same in nN as in nN and in .N As was the case for fields, nN and nN 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 ,nN n has the re- quisite properties. This can be seen by the equivalences between multiplication in ,,nnNN and :N .nnn nabc nanbncabcn  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 nN if and only if 1 is the multiplicative identity in .N To see this set 1.b The structure, ,nN Equation (18), and that of Equa- tion (17), differ from the generic description, Section 2, in that the base set nN is a subset of .N nN 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 ,nN and the element with value na in N has value a in .nN Elements with values na l in N where 0,ln are absent from nN3. As noted before, the choice of the representations of the basic operations and relation and constants in ,nN as shown in ,nN is determined by the requirement that nN satisfies the natural number axioms  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 ,nntu, .nnntunt nut u  Here Equation (11) was used with 1.b Note that 0.n It follows from these considerations that Theorem 3 nN satisfies the axioms of arithmetic if and only if nN 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∧ . One has the equiva- lences 0< 1>010<>0 10<1>01 .nnnnnn nnaa anana nanaa 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 . As a structure, I is given by ,,,,,0,1.II (19) I 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 jI be the structure ,,,,,0,1.jjjjjjjjII (20) jI is the subset of I containing all and only those elements of I whose values in I are positive or ne- gative multiples of j or 0. The representation of jI in terms of elements, ope- rations, and relations in I is given by ,,, ,,0,.jjIIjj (21) This structure differs from that of the natural numbers, Equation (18), by the presence of the additive inverse, . The proof that jI and jI 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,,,,,0,1.1II  (23) The main thing to note here is that the order relation, 1, in 1I corresponds to  in .I 1I can also be described as a reflection of the whole structure, 1II 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.II Equation (23) indicates that 1 is the identity and 1 is positive in 1.I These follow from 1111111aaa a   and 1101 01. This equivalence shows that the relation, , which is interpreted as greater than in ,I is interpreted as less than in 1.I Thus, as a relation in 1,I 01 means 1 is greater than 0. As a relation in I, 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 1I and 1,I are negative in .I It follows that 11 11101 2  is true in 1I if and only if 012  is true in 1I and in .I These considerations show that 1I and 1I satisfy the integer axioms if and only if I satisfies the axioms. For axioms not involving the order relation the proof is similar to that for jI for 0.j For axioms involving the order relation the proofs proceed by restating axioms for I in terms of . An example is the axiom for transitivity .ab bcac∧ For this axiom the validity of the equivalence 111111111abbcacab bcac  ∧∧ shows that this axiom is true in 1I if and only if it is true in 1I if and only if it is true in .I These considerations are sufficient to show that a theorem equivalent to Theorem 3 holds for integers: Theorem 4 For any integer 0j, jI satisfies the integer axioms if and only if jI 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 rRa denote the structure ,,,,,,0,1.rrrrrrrrRa Ra (25) Equations (24) and (25) show that Ra and rRa have the same base set, .Ra This is a consequence of the fact that rational numbers are a field. As such Ra nd rRa are special cases of the generic fields described in Section 2. Note also that Ra is the same as 1.Ra The definition of rRa is made specific by the re- presentation of its elements in terms of those of .Ra It is ,,, ,,,0,.rRa Rarrr (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 .rRa The element of Ra that has the value a in rRa is different from the element that has the same value a in .Ra The only exception P. BENIOFF Copyright © 2013 SciRes. APM 399is the element with value 0 as this value is the same for all .rRa Also, as noted in Section 2, rRa and rRa 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 rRa satisfy the rational number axioms  if and only if rRa 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 rRa satisfies the axioms if and only if rRa 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 rRa is the smallest ordered field if and only if rRa 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 rW and rW be isomorphisms whose definitions on Ra and rRa follow that of pW and pW in Equations (6) and (7). That is .rrrrrRa WRa WWRa (27) Since these maps, as isomorphisms, are one-one onto and are order preserving, they have inverses which are also isomorphisms. In this case one has rrrrrRaSW Ra RaSWRa RaS  (28) and 11.rrrrrrRaSWRa RaSWRaRaS  (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.rrjjmmrkkjk jkra atrrtrb b    (30) 6. Real Numbers The description for real numbers is similar to that for the rational numbers. The structures Ra and ,rRa Equa- tions (24) and (25), become ,,,,,,0,1RR (31) and ,,,,,,0,1.r rrrrrrrRR (32) The external representation of the structure, whose internal representation is ,rR is given in terms of the elements, operations, relations and constants of .R It is denoted by rR where ,,,,,,0,.rRRr rr (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  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 rR satisfies the ordered field axioms if and only if rR 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,rrjjaaj be a sequence of real numbers in ,rR Equation (32). Let r be a positive real number value in R. rja converges in rR if For all 0rr there exists an h such that for all ,jm h  .rrr rrjmraa (34) Here rrrjmraa denotes the absolute value, in ,rR of the difference between rja and .rma The numerical value in ,rR Equation (33), that is the same as  rrrjmraa is in ,rR is given by .jmra ra This is the same value as jmaa is in .R jmra ra also corresponds to a number value in R given by .jmjmra rara a (35) Theorem 6 Let 0r be a real number value in .R The sequence rja converges in rR if and only if jra converges in rR if and only if ja 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 rR such that  .rrr rrjmraa It follows that jmra rar in both rR and .R Equation (35) gives the result that jmaa in .R Conversely Let jmaa be true in .R Then jmra rar is true in both R and ,rR and  rrr rrjmraa is true in .rR From this one has the equivalences,  .rrr rrjmjmjmraa raaraa It follows that rja converges in rR if and only if jra converges in rR if and only if ja con- verges in .R 0r: It is sufficient to set 1.r In this case 1R and 1R are given by Equations (32) and (33) with 1.r In this case 11111111,,,,,,0,1RR  (36) and 1,,, ,1,,0,1.1RR  (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 1ja be a convergent sequence in 1,R Equation (36). The statement of convergence is given by Equation (34) where 1.r The statement  111 111jmaa  says that  1111jmaa is a positive number in 1R that is less than the positive number 1 and 0. The corresponding statement in 1,R Equation (37), is 1.jmaa  (38) Since 1R 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, 1jmjmaa aa  where 1 and  are the respective absolute values in 1R and .R In this case Equation (38) can be recast as .jmaa  (39) This can be used as the convergence condition in Equation (34). Note that 0jmaa  and that  de- notes “less than or equal to” in 1.R It follows from this that for 1,r the sequence 1ja converges in 1,R Equation (36), if and only if the sequence ja converges in 1,R Equation (37), if and only if the sequence ja 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 rR is complete if and only if rR is complete if and only if R is complete. Proof: Assume that rR is complete and that the sequence rja converges in rR. Then there is a number value r in rR such that lim .jrrja  The properties of convergence, Equation (34), with rma replaced by ,r and Theorem 6, can be used to show that limlimlim .rr jjjjjjarara  (40) Here r is the same number value in rR as r is in rR as  is in .R Conversely assume that R is complete and that ja converges in R to a number value . Repeat- ing the above argument gives limlimlim .rr jjjjjjarara  (41) This shows that rR is complete if and only if rR 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.rrjjmmrkkjk jkra atrrtrb b    (42) These terms can be used to give relations between power series in ,,rrRR and .R Let  1,njrr rrjjPnxa x be a power series in .rR Then ,rPnrx and ,Pnx are the same power series in rR and R as ,rrPnx is in .rR This means that for each real number value ,rx ,rPnrx and ,Pnx are the respective same number values in rR and R as ,rrPnx is in .rR Here rx and x are the same number values in rR and R as rx is in .rR However, the power series in R that corresponds to ,rrPnx and ,rPnrx in rR and rR is obtained from Equation (11). It is ,,,.rrrPnxP nrxrPnx (43) This shows that the element of R that has value ,rrPnx in rR has value ,rPn x in .R P. BENIOFF Copyright © 2013 SciRes. APM 401These relations extend to convergent power series. Theorem 6 gives the result that ,rrPnx is convergent in rR if and only if ,rPnrx is convergent in rR if and only if ,rPn x is convergent in .R It follows from Theorem 6 and Equations (40) and (41) that,    lim ,lim ,lim ,.rrrrnrrnnPnxfxPnrxf rxrPn xrfx (44) Here rrfx is the same analytic function  in rR as rfrx is in rR as fx is in .R Equations (43) and (44) give the result that, for any analytic function f,  .rrrfxfrxrf x (45) Here rf and f are functions in rR and R and rrfx is the same number value in rR as  rfrxrf x is in rR as fx is in .R Simple examples are ee errxxrxrr for the exponential and  sin sinsinrrrxrx rx for the sine function. Caution:  22sinsin ,rrxrx not 22sin .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 cC be the internal representation of another structure where ,,,,,,0,1.ccccccccCC  (47) The external representation of the structure, in terms of operations and constants in ,C is given by ,,, ,,,0,1.cCCcc cc (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, ccfx in ,cC have corresponding functions in C given by .ccfxcfx (49) Here cx denotes the same number in cC as x is in .C The relation for complex conjugation is given by ccaca It is not .ccaca 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 cC if and only if 1 is a real number value in .C This requires that the equivalences 11111 111cccc cccc be satisfied. These equivalences show that 11ccc or more generally .cccacaca (50) Note that any value for a is possible including c or powers of .c For example, .ccnnncccccc As values of elements of the base set, C, Equation (50) shows that the element of C that has value cca in cC has value ca in .C This is different from the ele- ment of C that has the same value, ,a in C as cca is in .cC Another representation of the relation of complex conjugation in cC to that in C is obtained by writing ie.cc Here c is the absolute value of .c This can be used to write 2ie.cccaca ca (51) That is  2ie.ccc For most of the axioms, proofs that cC 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  xx has an easy proof. It is based on Equation (50), which gives .ccccccacacaca From this one has the equivalences, .ccccaacacaaa This shows that xx is valid for cC if and only if it is valid for cC and .C The other axiom to consider is that for algebraic clo- sure. Theorem 8 cC is algebraically closed if and only if cC 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 cC if and only if the same po- lynomial equations in cC and C have the same solu- tions. Let 00njjjbx denote a polynomial equation that has a solution ca in .cC Then 00jnccjjba in .cC Carrying out the replacements ,cjjbcb ,caca ,cc and c gives the implications P. BENIOFF Copyright © 2013 SciRes. APM 402 0000000000.cnncjjcc jjjjcnnnjjjjjjjjjbacb cacbacbaba     From the left, the first equation is in ,cC the second is in cC, and the other three are in .C This shows that if ca is the solution of a polynomial in cC then ca is the solution of the same polynomial equation in ,cC and a is the solution of the same polynomial equation in .C The proof in the other direction consists in assuming that 00njjjba is valid in C for some number ,a and reversing the implication directions to obtain 00jnccjjcba in .cC ■ 8. Number Types as Substructures of cC, cC, 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 ,cC ,cC and C. To this end let c be a complex number and let cR and R be real number substructures in cC and .C In this case ,,,,,,0,1c ccccccccRR (52) and ,,,,,,0,1.RR (53) The field operations in cR and R are the same as those in cC and C respectively, restricted to the num- ber values of the base sets, cR and .R Expression of the field operations and constants of cR in terms of those of C gives a representation of cR that corresponds to Equation (48). It is ,,, ,,,0,1.cccRRccc (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 cR contains just those values of C that that have real values in cC and .cC Let x be an element of R that has a real value, ,a in .R This is different from another element, ,y of cR that has the same real value, ,ca in cR as a is in .R The element, ,y also has the value, ,ca in ,cR which is the same real value in cR 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 cR 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 CabdC adb (55) Here d is a positive real number in C if there exists a number g in C such that .dgg The definition for c in cR is obtained by putting c sub- scripts everywhere in Equation (55). The order relation c in cR is if and are real number sin, andthere exists a positive real number such that .ccca cbCca cbcdca cdcb (56) Here cd is a positive real number in cC if there exists a number cg in cC such that .ccdcg 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 .ccccabcacbab  Proof: Replace the three order statements by their definitions, Eqs. 55 and 56, to get .ccc cad bcacd cbadb  Here dgg, ccd cgcg, and .ccccdgg Let ,,abg be real numbers values in .R Then ,,ca cb cg and ,,cc cabg are the same real number values in cR and cR as ,,ab g are in .R Define the positive number value d by .dgg Then by Equation (48), gg is the same number value in R as  ccgcgcg gcdc is in cR as ccccggd is in .cR Thus cd and cd are the same positive number values in cR and cR as d is in .R From this one has ,ccccad bcacdcbadb P. BENIOFF Copyright © 2013 SciRes. APM 403which gives .ccccab cacbab  The reverse implications are proved by starting with ,ccab, and ccccdgg 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.ccCc For the additive group, the map needs to be extended to include 0r. This is done by including the empty structure, 0.C Let CG denote the collection of complex number structures that differ by arbitrary nonzero complex scaling factors. Define the operation, , by .cd cdCC 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 cC has an inverse, 1cC as 1.ccCC 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 cdCCC The empty structure, 0C, is the additive identity and cC is the additive inverse of .cC 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, ,.ccVC The scalars for cV are number values in .cC The representation of cV as a structure with the basic operations expressed in terms of those in ,V with C as scalars, is similar to that for ,cC Equation (48). For Hilbert spaces, the structure ,H based on ,C is given by ,,,, ,,.HH (59) Here  and , denote scalar vector multiplica- tion and scalar product.  denotes a general state in .H The representation of another Hilbert space structure that is based on ,cC is given by ,,,,, ,.cccc ccHH (60) The representation that expresses the basic operations of cH in terms of those for H is given by ,,,, ,,.cHH ccc  (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 xV is associated with each space time point .x So far just one complex num- ber field, C, serves as the scalars for all the .xV Expansion of this approach by replacing C with dif- ferent complex number fields, xC at each point x has been explored in [1,2]. If the freedom of basis choice in the vector spaces xV, introduced by Yang , is ex- tended to include freedom of scaling of the different xC, then yC and xC are related by a scaling factor ,e.xyxyxr This factor scales the numbers of yC relative to those in xC. A number value ya in yC corresponds to the number value ,yxxra in xC. xa is the same number value in xC as ya is in .yC The scalar field x 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. 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