Internat. J. Math. & Math. Sci. VOL. 12 NO. 3 (1989) 603-613 603 TWO CONSTRUCTIONS OF THE REAL NUMBERS VIA ALTERNATING SERIES ARNOLD KNOPFMACHER ,IOHN and KNOPFMACHER Department of Applied Mathematics Department of Mathematics University of the Witwatersrand Johannesburg. Wits 2050, S. Africa University of the Witwatersrand Johannesburg, Wits 2050,S. Africa (Received March i, 1988) ABSTRACT: Two further new methods are put forward for constructing the complete ordered field of real numbers out of the ordered field of rational numbers. The mthods are motivated by some little known results on the representation of real Amongst advantages of the numbers via alternating series of rational numbers. methods are the facts that they do not require an arbitrary choice of "base" or equivalence classes or any similar constructs. The methods bear similarities to a method of construction due to Rieger, which utilises continued fractions. Key Words and Phrases: Ordered oLLono_ numbe,. AMS (1980) Classification Number: 10A30. 1. INTRODUCTION. The series of Enge] (1913) and Sylvester (1880) (see Perron [1])for represent:g real numbers have been studied in some detail. Much less known is the fact that there are alternating series representations of real numbers in terms of rationals corresponding to the above. The only references to these alternating series that E Remez [2] and H Salzer [3]. Every real number A has a unique we are aware of in the literature are in papers of The series under discussion are as follows: representation in the form A ao + a Ii 1 a a I 2 1 + a (-1)n+1 + a a la2a3 I 2 a + (ao,a 1,a 2 n are integers such that say, where the ai+ 1 > ai + 1 >_-- 2 for il. A is rtionZ if and only if A has a finite representation (ao,a I (Compare this with the expansion of Engel (Perron [i]).) ). Furthermore. Corresponding to the series of Sylvester (Perron [1]) we have every real number A ao 1 a 1 + 1 a 3 2 a + (-1)n+1 a n + a o,a 1,a 2 )), A. KNOPFMACHER AND J. KNOPFMACHER 604 say, where the i are integers defined uniquely by >_-+ ai(ai 1) for i>--_1. Furthermore, A is ai+ has a finite representation ((ao,a an)). A such that rovt.Lonc al>_--i I In many ways, these representations may be compared with that by continued fractions. and if and only if A simple The main purpose of this note is to justify this remark by deriving some elementary properties of these alternating series representations and (with these results as an initial motivation) then developing two new methods for constructing the real number system from the ordered field of rational numbers. These methods are similar to one recently introduced by G J Rieger [4] for constructing the real numbers via continued fractions. The order relations in particular are defined in an analogous fashion. The methods share with Rieger’s method the advantage over other standard techniques that they do not require an arbitrary choice of a "base", or the use of (infinite) equivalence classes or similar such constructs. These properties are shared as well in the construction of real numbers using ordinary Sylvester and Engel series, considered in [5]. Two important differences between those and the present methods are in the definition of the order relations for the series, as well as the of terminating representations of rational numbers in place of infinite recurring representations used in [5]. use here 2. ALTERNATING SERIES REPRESENTATIONS FOR REAL NUMBERS. For the convenience of the reader, because full previous details may be inaccessible to many (including the present authors), we prove here the fundamental results concerning the representation of real numbers via infinite alternating series. It is convenient to introduce here a more general alternating series, analogous to the positive series of Oppenheim [6], out of which we can deduce the results for alternating-Engel and alternating-Sylvester series as special cases. We define the alternating-Oppenheim algorithm as follows: Given any real number A, let a Then we recursively define a A [A] A 1 o a n where An+ Herein bi b i -- [1 1 n a I a2 >_-- 1 An for n > 1, A n > 0 (Cn/bn) i) a for a c c i 0 n > i a a 2 a i) are positive numbers (usually integers). 1, The two cases of particular interest to us are those for which b 1,n >___ i (the alternatingn > 1 (alternating-Engel series) and an, Sylvester series). bn cn CONSTRUCTIONS OF REAL NUMBERS VIA ALTERNATING SERIES reaZ number A has Evy THEOREM 2.1. 605 a unique representation in the orm l=o+l-!+ aI a2 a3 where az+ > a% +i) al> i reag numbe A has a unique Fu)uthermore every 1 A=ao+l_ a ala2 I eprettion in the 6orm 1 + ala2a3 where > %+ a a o o +A 1 + bl 1 a a2 c1 1 =o + i___ aI Now c l.!_ + blb2 i 2 ClC 2 a3 a I < A n <__ 1 a n 1 An+l (_.1 a n for 0 < A 1 a+l n In particular by setting Furthermore {an} > b i. an(an + i) 1 a (a +1) is strictly increasing. if if Ar > 0 for i - 0 as 0 < A < 1 n n we obtain for all 1 cn n )(cn/bn) Cn/bn) an(an+l i Ck- 1 A n )(Cnlb n) (-n1 < An+ 1 An+ 1 < blb2..:bk-i Ak ClC2 n an+l Thus + (-i) k-i impl es an an+l i. Repeated application of the alternating-Oppenheim algorithm yields PROOF. A % +lal> n , It follows that since < n al 1 and the sequence A has an alternating-Sylvester expansion A a o + 1___ 1__ +.1__ a a a 3 2 I ((ao,al,a2 which may perhaps terminate. Secondly, by setting cn an b n 1 for all n we obtain )) A. KNOPFMACHER AND J. KNOPFMACHER 606 1 an+ a A Thus 1 a + n < An+l and so a la 2 > An+ 1 a n as 0 + 1 if =, n A.c > 0 since a n for i I n >--_ 1 has the alternating-Engel expansion i A:ao+l_ a I ala 2 1 + (a ala2a 3 o,a 1.a 2 which also may terminate. Uniqueness of the representations follows from Proposition 2.3 on order below. {Various other interesting special cases of the alternating-Oppenheim algorithm will be treated in a separate article.) We deduce now an important result on the alternating series expansions for rational numbers. PROPOSITION 2.2 t a The alternting-Sylvester and aternting-Engel sies tminate terms i an oy i A is rational. fbte numb o PROOF. Clearly any number represented by a finite expansion is rational. Conversely, since Ai,i 1, is rational let A pi/qi, (pi,qi) 1. Now i since for either algorithm ]> ai --. 1 1 it follows that qi < Pia Pi In the alternating-Sylvester case we now obtain r+ 1 eL qi/l aiqi Pi+I qi- Piai < Pi Since {pi} is a strictly decreasing sequence of non-negative integers we must eventually reach a stage at which O, whence Thus 0_<_-- Pn+1 (-1) n-1 a n + 1___ 1__ + + a I a2 The result for the alternating-Engel series follows similarly from A a o i+i qi+l qi-Piai qi We note that for rational numbers there is a possible ambiguity in the final term, analogous to that for continued fractions. We eliminate this as follows" We replace the finite sequence ((ao,a an_2,an_1+l)) in the case a n an_l(an_l+1). CONVENTION 1. ((ao,a (ao,a I ,a n we identify by A(I (ao,a I an_2,an_l+l) in the case with its finite expansion (Uo,al an)) by Similarly we replace an_l+1 Furthermore a n) or ((ao,a an)), i a n respectively. In order to be able to compare finite sequences of different lengths in size we introduce the symbol m with the following properties" For any r(I, oo+ K to r < co CONSTRUCTIONS OF REAL NUMBERS VIA ALTERNATING SERIES 607 Now we can represent finite sequences by infinite sequences as follows- CONVENTION 2 For every hence A (ao,al A (ao,a I an,m,m A ((a o,a I PROPOSITION A ((ao,a I an)) a (I let n a# Similarly represent ). (1 2.3 (On Order) )) LeA: A ((bo,b I )). B ((ao,a I A by j > n for m an,, and )). B ), or (bo,b In both o/ these case, the coditon A<B (ao,a l equivalent to" a2n < a2n+l (i i) that a. b2n or 2n+1 . > A 2n or i We shall use the notation PROOF. A where i ((ao,a 1,a 2 )), ). (ao,al,a 2 1_ An and 2n+1 1 an 1__ A’n a + arian+ 1 is the 1 fit index + for an+ an+ 2 n 1 -... anan+ l an+ 2 such iO for (Note that we do not assume at this stage that An A n as dfined by either algorithm Now suppose (i) holds. A + A < a 1 Next suppose in either case. case. a ao < b o If firstly + i a2n < b b n > O, b2n’ then + B 1 B in the alternating-Sylvester The growth condition + i), i ai( a/+ I > i, implies that 1 An a2-n since a. > 1 for a2-t __< 1 + i i a2n+l (i- a2n+2 i a 2 n+2 +i )+ i a2n+1 and by observing Convention 1. a2n+l U2n i + a2 n+i a2n 1__(i_ a2n+ l a2n i > 1, 1 A’2n a2n+2 i (i- > 1 + a2n+ 1 Furthermore, 1 a2n+2 (i i__!___) a2n+l +I i i (i a2n+3 +I U2n+3 1 < a2n Thus An > a2n+i It now follows from A that A < B b2 n ao + 2n a-l a- l + A’2n, In the alternating-Engel case, from B ao+ ai+ I a- l l_ _a2 + >_-- a. + i, i > i, B’2n A. KNOPFMACHER AND J. KNOPFMACHER 608 A’2n >__ 1 1 2n a2na2n+l 1___ (1- a2na2n+la2n+2 1 a2n a2n+l > 1__ 1 1 a2n 1 + Thus again . < 1 1 a2n a2na2n+l 1 1 a2n a2 ha2 n+ 1 a2n+2 +1 i a2n+l as in the alternating-SyIvester case. A’2n (i- a2na2n+la2n+ 2 )_ a2n+l 1 + Also 1 + a2na2n+la2n+2 (1- 1 a2n+ 1 + 1 1 a2n > > a2n+ l 1__>= B’ 2n b2n A < B now follows from and the result ao + a 1I A 1 ala2 1 ao + 1_ a ala2 B 1 +..._ ala2 a2n_l ..._ + An 1 ala2..a2n_1 BI,’n Note that if 0 and the result remains valid in this case m then result is proved in a similar fashion if (ii) holds. B’2n b2n 3. The CONSTRUCTIONS AND ORDER PROPERTIES In the constructions below, standard facts about the ordered field Q of all rational numbers are taken as understood. With the results of Section 1 as initial and order relations on them as follows" motivation, we now define two sets E* and * Let E* be the set of all formal infinite sequences A >_--a.+r (ao,al,a2 for i>_--1,a11. integers Also, let i such that ai+ 1 all formal infinite sequences A of integers )) ((ao,al,a 2 al>_-- 1 and ai+l>--_ai(ai+l) for i 1. of the set of * suchbe that x Finite sequences (rational numbers) are included in our sets E* and use of the property all sequences in E* We will frequently mae Convention 2. using and satisfy a. implies a. j > i we shall use corresponding lower-case letters * "digits" of the elements of the respective sets, and we define In both the sets E* and to denote the for all CONSTRUCTIONS OF REAL NUMBERS VIA ALTERNATING SERIES A < B 609 if and only if a2n < b2n or a2n+l > b2n+l (i) (ii) where 2n or i i 2n+1 >_-- 0 is the first index such that a. b. In both cases, < is a "total ord%ing" 6ation, i.e., it is traitive and satisfies the trichotomy law. LEMMA 3.1. We use the same argument in both cases. PROOF. Next let A < B and B < C. Suppose ar for r < b r or and for r < c If i < j then a r r a. > b. c. (i odd). Firstly, trichotomy is b obvious. ,b and (i) If i (ii) a then j > bz > ci c a r r ( odd). c , r < j for a and for r < b c (i even) or ai < bi < ci (i even) or a b < c ( even) or for r < j, and If Z > j then a r cr b. > c. ( odd). Thus A < C in each case. (iii) a. We may now introduce symbols <--_, > and and (9reotest) lome bound/, in the usual way. LEMMA 3.2. Evg non-emptw ube;t has a ast uppe boun (supremum). o * B by a sequence which is bounded above a isb.an Now A <B b AX with a since otherwise there is nothing to prove. such that every AEX, and there is a largest index We may assume a bI I for elements of X of E*, (bo,b I Be. Assume and define (east) upp bound (respeetive!W, *) mich is First consider a non-empty subset PROOF. (o,l,m,m , b a upper bound for X, AX for some where o for every has since otherwise is the maximum value of X be the least If + 1 is odd let We now define c o bo 1 + 1 is even b o. If of any AX with a possible value for the digit 1 of any AX with let be the greatest possible value for the digit +1 1 In either case has no largest value. a m if b o, where we take 1 +1 ). Otherwise we continue if m we are done, and put C (Co,C 1 1 to define as the least possible value or greatest possible value depending on 2 of all elements of + 2 is odd or even, respectively, for the digit whether 2 m we are done and in X. Again if the form (Co,C C+l,a+2,a+ 3 2 I as the define to inductively, Continue put C ). (Co,C 1 1 C+l,m,m for the value (+i+l even) least possible value (++i odd) or.greatest possible ). of an element of X of the form (Co,C digit c+Z,a++l,+Z+ 2 a+ c+ c+ c b c+ c+ c,m,m c+ a+ c+ a++ c++ 1 1 The m. If, when ++1 is even, a+i+ 1 has no largest value, we take c+Z+ m. We then take process terminates if at any stage 1 process constructs a non-terminating this Otherwise C ). (Co,C 1 c+,m,m c++ A. KNOPFMACHER AND J. KNOPFMACHER 6 i0 In either case we have CE* since Cr+l>--_c +i for C A then C >A since either i I c > a. (i even) or e. < a. (xL odd) for the first index a. >k such that c. C sequence I (c o Cl, i. Also if c (by the definition of the sequence C Lastly, c-,O d r. < i < m,dm A (eo,C of the form since otherwise sup)< em If is odd then m D< C. Then every element Hence > Cm satisfies D < A --<_ D has an upper bound )< in Cm,am+l,am+ 2 dm X In the cases m 0 or contradiction. If m is even we have d < c (e e < m (m > O) every element of the form A 1,....cm, m satisfies D < A <_-- D In the case em m am m for every A we can choose )X with arbitrarily large am. For am > dm we A (c e I, 1 choose have m for some A m and a < A <_-- D. Finally, if c o,c am_ am,am+ A (Co,C Cm_l, )) ((Co,Cl,C 2 saisfy 4. and m, The argument for C , S* D < A I) once more. is almost identical to the above, except that the sequence defined inductively via suitable elements of ci+ >_--ci(i+l),Cl S* will now >_-- 1. EMBEDDING AND DENSITY OF RATIONALS The following proposition justifies our use of Convention 1. PROPOSITION 4.1 The tntZng-EngZ and ntng-Sylvt ogoCtm dne 1 1 ord-prving maps PE* (I whose images oe dense in E* E* and and * PS* (I * respectively. - It is an immediate consequence of the results quoted earlier that the By 1 maps two algorithms define 1 pE (1-- E* and PS* are then maps these Proposition 2.3 and the definition of order in E* and PROOF. , order-preserving. Now let A < B in E Let k * be the least index for which a k bk satisfying We show now in every possible case that we can find a rational number D Now let A ( we take D- A+B B A < D < B. For A if we choose D If k is even then k < (Oo,Ol,...Ok,Ok+l+l,m,m if m’ i.e., m’ or D ak,ak+l,k+2+l,m, m (ao,a I choose we then > odd is If instead k Bm{,A ak Ok; if ak+l m m’ or D D,6k+1,O+2 +1, (o,1 D (=o,al ak,ak+l+l,m, in the case of works A similar argument m and A{,B if m,m , Ok+l . Ok+l . ai+l have the intuitively desirable property m’s) that rational numbers can be represented only by finite sequences (excluding for (see series Sylvester and Engel ordinary This des not hold in the case of We note that sequences in E* example [1] ). and x CONSTRUCTIONS OF REAL NUMBERS VIA ALTERNATING SERIES APPROXIMATION LEMMA A n there exist ationols or n >--_ 0 A (iii) PROOF. A (n) A(2n) A(2n+l) Given (rpecLve.y, *), E* such A (2m+1) A (2n+1) A A (2n) (2n+1) (2n) inf A sup A A (2m) (i) (ii) o eenent A Given anq 4.2 611 or m < n i (2n+1)! (ao,a I A an,,m define the rational sequences Next suppose that part(i) follows. Then EE*, A (n) by ). (2n+1) if m is odd, for all n. In that case, we must have a > b A A< B Then m b or a < b if m is even, for the first index m such that am m m m (m) (m) contradiction while m even gives the A odd gives the contradiction A < B (2n) =<C<A (2n+l) Similrly, suppose that A Thus A infA A (m+l) A (m+l) < B Consider the first index m for which om# cm. If m is even we must for all n (m) If m is odd we which yields the contradiction A(m)<_-- C < A have a > c m m (m+l) The same argument <C < A have a < Cm, which gives the contradiction A m for and parts leads to (ii) (i) (ao,a I m+li * For part (iii) in Ex, the formula for alternating-Engel series for rationals leads to A(2n) A(2n+l) 1 <__ alaz...a2n+l 1 (2n+l)! For *, the corresponding formula for alternatinga 1,ai+ 1 >_-I Sylvester series of rationals gives instead a/+l since A(2n+l)_ A(2n) 1 a2n+l < 1 (2n+l)! ->_- >_-- 1 and ai+ 1 ai(ai+l) I ALGEBRAIC OPERATIONS IN E* AND since 5. a * * by Convention and as an actual subset of 1, it will simplify the discussion on algebraic operations below if we now re-deJrine Since we alreaady regard A (n) for any routional For any A a (n>_-0) A. sup(A (2n) + A,B( EX (or A,B*) we now define A + B (respectively, I*) because which exist in sup(-A(2n+l)), A (2n) + B (2n) * A (1) + B (1) A(2n+1) < B(2n)), A (0) x are very At this stage we note that the formal structures of the sets E* and similar to the set K, (based on continued fractions) used by Rieger [4] to construct the real numbers. Thus to avoid repetition, we will refer the reader to the corresponding result of Rieger whenever the proof of the algebraic property there is the same. The above operations make E (respectively, *) into an aelian group cow.raining ((I,+) as a dense subgroup. Fuuther LEMMA 5.1 A. KNOPFMACHER AND J. KNOPFMACHER 612 A < B (i) () => => < - A+ C < B + C A > PROOF. Obviously A + B B + A and A + 0 A. The proof of the associative law of addition can be found in Theorem 3.1 of [4]. Also the proof that A + (-A) 0 is found in Theorem 3.2 of [4]. The only changes that must be made are that we use A(2n+l) A(2n) in place of inequality (1.4) of [4]. (2n+l) It now follows that E (respectively, *) forms an abelian group with (Q,+) A B and hence the strict monotone law A + C B + C (i) follows from the weak one proved below" as a dense subgroup. Then By the definitions of order and rational approximations we have for n sufficiently large A < B =) A (2n) < B (2n) A (2n+l) < B (2n+l) + Hence A < B= A + C C B Finally, let A B. Then A (2n+I) < B (2n+I) or -A (2n+I) >-B (2n+I) < for n -A sup(-A sufficiently large. (2n+I}) Thus sup(-B HenCe (i i) follows, (2n+I)) -(-X) since -B -A > -B giving since A B. X Next, for any A,B E* (respectively, *), define sup(A (2n) .B (2n)) A.B (-A). (-B) -((-A).B) -(A.(-B)) A >_-- O,B >_-- 0 if A O,B =< 0 0 if A O,B 0 if A >_-- O, B A -1 sup((A(2n+l)) -1) if if Also define -((-A) A > 0 if A < 0 -i The definitions are unambiguous, since we have A (2n) for B (2n) A > O,B > 0 < A (I) .B (1) (A(2n+l)) -1 :< (A(O)) -1 and the theorem of the supremum is applicable. cases, we use the fact that A < 0 if and only if -A > O, To cover all the by Lemma 5.1(ii). LEMMA 5.2 The above denons t.ogethe)t wfth the eov.ier operoons make o a dense subfield, into a feld containing E* (Respectively, * A < B,C> 0 PROOF. Clearly C (n) > 0 for all n, 0 A.B B.A =, and A.C < B.C A.1 A. Also, since C > 0 implies we obtain easily < A < B, C > 0:=> A.C <--_ B.C (with strict inequality to be shown later). is In order to verify that E*(X) is a field it remains only to verify that -i The O. .A 1, A associative, and distributive relative to +, and that A CONSTRUCTIONS OF REAL NUMBERS VIA ALTERNATING SERIES 613 associative law appears in Theorem 4.1 of [4], the distributive law in Theorem 4.2 of -1 0 is shown in Theorem 5.1 of [4]. In each case we must 1, A [4], and A .A replace Rieger’s inequality (1.4} by A(Zn+1) A(2n) 1 (2n+l) The strict monotone law for multiplication of positive elements now follows B.C= A B (for positive elements). By from the weak one, and the law A.C simple manipulation of the "sign" cases the result follows for all elements in E* (or *). * form ordered fields E* and By standard theorems, treated for example in form concrete and Chapter 5 of Cohen and Ehrlich [7], it then follows that The models are in many ways equivalent new models for the real number system to that of Rieger [4], except that they arise from the (simpler) representation of real numbers as infinite series rather than as infinite continued fractions. The above discussion has shown that both with the ]east upper bound property. * * REFERENCES i. 2. 3. 4. 5. 6. 7. Irratonzahlen Chelsea Publ. Co., 1951). (New York On series with alternating sign which may be connected with two algorithms of M V Ostragradskii for the approximation of irrational numbers. Uspe Mtem. Nauk (N S) 6,No. 5(45), 33-42(1951); Mth. Rev. IB, 444(1952). The approximation of numbers as sums of reciprocals. SALZER, H.E. Ame. Mouth. 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