# SEL Series Expansion and Generalized Model Construction for the Real Number System via Series of Rationals

Authors: Patiwat Singthongla; Narakorn Rompurk Kanasri
Journal: International Journal of Mathematics and Mathematical Sciences
 (2014)

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## Abstract

We present an algorithm for constructing infinite series expansion for real numbers, which yields generalized versions of three famous series expansions, namely, Sylvester series, Engel series, and Lüroth series expansions. Using series of rationals, a generalized model for the real number system is also constructed.


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## Body

## 1. Introduction

According to [1, 2], it is well known that each A∈ℝ is uniquely representable as an infinite series expansion calledSylvester series expansion, which is of the form
(1)A=a0+∑n=1∞‍1an,
where
(2)a0={⌊A⌋ifA∉ℤA-1ifA∈ℤ,a1≥2,an+1≥an(an-1)+1∀n≥1.
Moreover, A∈ℚ if and only if an+1=an(an-1)+1 for all sufficiently large n. An analogous representation (see [1–3]) also states that every real number A has a unique representation as an infinite series expansion calledEngel series expansion, which is of the form
(3)A=a0+∑n=1∞‍1a1⋯an,
where
(4)a0={⌊A⌋ifA∉ℤA-1ifA∈ℤ,a1≥2,an+1≥an∀n≥1.
Moreover, A∈ℚ if and only if an+1=an for all sufficiently large n. For the last representation (see [1, 2]), it is also known that each A∈ℝ is uniquely representable as an infinite series expansion calledLüroth series expansion, which is of the form
(5)A=a0+1a1+∑n=1∞‍1a1(a1-1)⋯an(an-1)an+1,
where
(6)a0={⌊A⌋ifA∉ℤA-1ifA∈ℤ,an≥2∀n≥1.
Moreover, A∈ℚ if and only if (an) is periodic.In 1988, A. Knopfmacher and J. Knopfmacher [4] further derived some elementary properties of the Engel series expansion and Sylvester series expansion and then developed two new methods for constructing new models for the real number system from the ordered field of rational numbers. These methods are partly similar to the one introduced by Rieger [5] for constructing the real numbers via continued fractions.In the present work, we will first introduce an algorithm for constructing an infinite series expansion for real numbers calledSylvester-Engel-Lüroth series expansion orSEL series expansion for short which yields generalized versions of three series expansions, namely, Sylvester series expansion, Engel series expansion, and Lüroth series expansion. Then we will establish some elementary properties of the SEL series expansion and develop a method for constructing a generalized model for the real number system using series of rationals, which yields generalized versions of Knopfmachers' models.


## 2. SEL Series Expansion

Given any real numberA, write it as A=a0+A1, where
(7)a0={⌊A⌋ifA∉ℤA-1ifA∈ℤ
and 0<A1≤1. Then recursively define
(8)an=1+⌊1An⌋,(9)An+1=(anAn-1)en,
where en=en(an) is a positive rational number, which may depend on an, for all n≥1.Using this algorithm and the same proof as in [1, 2], we have the following.Theorem 1.
LetA∈ℝ and assume that
(10)an-1en∈ℕ
for all n≥1. Then A is uniquely representable as an infinite series expansion called SEL series expansion, which is of the form
(11)A=a0+1a1+∑n=1∞‍1a1e1⋯anenan+1,
where a1≥2 and an+1≥((an-1)/en)+1 for all n≥1.Lemma 2.
Any series(12)1b1+∑n=1∞‍1b1f1⋯bnfnbn+1,
where
(13)fn=fn(bn)∈ℚ+,b1≥2,bn+1≥bn-1fn+1≥2∀n≥1,
converges to a real number B1 such that b1-1≤(1/B1)<b1. Furthermore, b1=1+⌊1/B1⌋.By settingen=1/an,en=1, and en=an-1, for all n≥1 in Theorem 1, and by setting fn=1/bn,fn=1, and fn=bn-1, for all n≥1 in Lemma 2, we obtain the following well-known expansions for real numbers (see [1–3]), namely, Sylvester series expansion, Engel series expansion, and Lüroth series expansion, respectively, as we now record.Corollary 3.
EachA∈ℝ is uniquely representable as a Sylvester series expansion; that is,
(14)A=a0+∑n=1∞‍1an,
where a1≥2 and an+1≥(an-1)an+1 for all n≥1.
Conversely, each series of the form (14) converges.Corollary 4.
EachA∈ℝ is uniquely representable as an Engel series expansion; that is,
(15)A=a0+∑n=1∞‍1a1⋯an,
where a1≥2 and an+1≥an for all n≥1.
Conversely, each series of the form (15) converges.Corollary 5.
EachA∈ℝ is uniquely representable as a Lüroth series expansion; that is,
(16)A=a0+1a1+∑n=1∞‍1a1(a1-1)⋯an(an-1)an+1,
where an≥2 for all n≥1.
Conversely, each series of the form (16) converges.Proposition 6.
Let(17)a0+1a1+∑n=1∞‍1a1e1⋯anenan+1,b0+1b1+∑n=1∞‍1b1e1⋯bnenbn+1
be SEL series expansions of distinct real numbers A and B, respectively. Then the condition A<B is equivalent to the following: (i)
a0<b0ifa0≠b0;(ii)
ai>bifor the firsti≥1such thatai≠biifa0=b0.Proof.
Ifa0<b0, then
(18)A=a0+A1≤a0+1≤b0<b0+B1=B.
Hence (i) follows.
Now assume thata0=b0 and ai>bi for the first i≥1 such that ai≠bi. Applying (9) repeatedly, we obtain
(19)An=1an+An+1anen=1an+1anen(1an+1+An+2an+1en+1)⋮=1an+1anenan+1+⋯+1anen⋯an+k-1en+k-1an+k+An+k+1anen⋯an+k-1en+k-1an+ken+k
for all n,k∈ℕ. Using (8), we get
(20)1an<An≤1an-1
for all n≥1. We now prove 0<An≤1 for all n≥1 by induction on n. If n=1, then we have seen that 0<A1≤1. Assume now that 0<An≤1 for n≥1. By (8), we see that an≥2. Since
(21)An+1=(anAn-1)en=(An-1an)anen
and using (10) and (20), we have that
(22)0<An+1=(anAn-1)en≤(anan-1-1)en=enan-1≤1,
as desired. It follows that an≥2 for all n≥1.
Next, we will prove that(23)An=1an+1anenan+1+⋯
for all n≥1. For n,k≥1, let
(24)Bk=1an+1anenan+1+⋯+1anen⋯an+k-2en+k-2an+k-1.
Since An,en>0 and an∈ℕ, for all n≥1, the sequence of real numbers (Bk) is increasing and bounded above. Thus, lim⁡k→∞Bk exists and so
(25)lim⁡k→∞1anen⋯an+ken+kan+k+1=0.
Since (20) and an+k+1≥2, we deduce that
(26)0<An+k+1anen⋯an+ken+k≤1anen⋯an+ken+k·an+k+1an+k+1·1an+k+1-1≤2anen⋯an+ken+kan+k+1,
and so
(27)lim⁡k→∞An+k+1anen⋯an+ken+k=0,
showing that (23) holds as desired.
Using (23), (20), and ai-1≥bi, we have
(28)A=a0+1a1+⋯+1a1e1⋯ai-1ei-1ai+1a1e1⋯aieiai+1+⋯=a0+1a1+⋯+1a1e1⋯ai-1ei-1(1ai+1aieiai+1+⋯)=a0+1a1+⋯+1a1e1⋯ai-1ei-1Ai≤a0+1a1+⋯+1a1e1⋯ai-1ei-1(1ai-1)≤a0+1a1+⋯+1a1e1⋯ai-1ei-1(1bi)=b0+1b1+⋯+1b1e1⋯bi-1ei-1bi,<B,
and the assertion follows.The following corollaries follow immediately from Proposition6 by setting en=1/an,en=1, and en=an-1, for all n≥1, respectively; the first two corollaries readily appear in [4].Corollary 7.
Let(29)a0+∑n=1∞‍1an,b0+∑n=1∞‍1bn
be the Sylvester series expansions of real numbers A and B, respectively. Then the condition A<B is equivalent to the following: (i)
a0<b0 if a0≠b0;(ii)
ai>bi for the first i≥1 such that ai≠bi if a0=b0.Corollary 8.
Let(30)a0+∑n=1∞‍1a1⋯an,b0+∑n=1∞‍1b1⋯bn
be the Engel series expansions of real numbers A and B, respectively. Then the condition A<B is equivalent to the following: (i)
a0<b0 if a0≠b0;(ii)
ai>bi for the first i≥1 such that ai≠bi if a0=b0.Corollary 9.
Let(31)a0+1a1+∑n=1∞‍1a1(a1-1)⋯an(an-1)an+1,b0+1b1+∑n=1∞‍1b1(b1-1)⋯bn(bn-1)bn+1
be the Lüroth series expansions of real numbers A and B, respectively. Then the condition A<B is equivalent to the following: (i)
a0<b0 if a0≠b0;(ii)
ai>bi for the first i≥1 such that ai≠bi if a0=b0.


## 3. Constructions and Ordered Properties

In this section, we construct a generalized model for the real number system using series of rationals, which yields generalized versions of Knopmachers' models in [4] as follows: let f:[2,∞)→ℝ be a nondecreasing function such that f(x)≥2 for all x≥2 and let 𝒰f be the set of all formal infinite sequences 𝒜=(a0,a1,a2,…) of integers an such that a1≥2 and an+1≥f(an) for all n≥1. In other words,
(32)𝒰f={(a0,a1,a2,…)∣a0,an∈ℤ,a1≥2,an+1≥f(an)∀n≥1}.
As an analogue to Proposition 6, we define order relation on 𝒰f as follows.For𝒜=(a0,a1,a2,…),ℬ=(b0,b1,b2,…)∈𝒰f, we say that𝒜<ℬ
if and only if
(i)
a0<b0 if a0≠b0, or(ii)
ai>bi for the first i≥1 such that ai≠bi if a0=b0.Note that 𝒜≤ℬ if and only if 𝒜<ℬ or 𝒜=ℬ.Lemma 10.
≤ is a total ordering relation on 𝒰f.Proof.
It is clear that≤ is reflexive and antisymmetric and any two elements in 𝒰f are comparable. It remains to show that ≤ is transitive. Let 𝒜,ℬ,𝒞∈𝒰f be such that 𝒜≤ℬ and ℬ≤𝒞. If 𝒜=ℬ=𝒞,𝒜=ℬ<𝒞, or 𝒜<ℬ=𝒞, then 𝒜≤𝒞. Assume that 𝒜<ℬ and ℬ<𝒞, so a0≤b0≤c0. If a0<b0<c0,a0=b0<c0, or a0<b0=c0, then it is clear that 𝒜<𝒞.
Now assume thata0=b0=c0. Then ai>bi for the first i≥1 such that ai≠bi and bj>cj for the first j≥1 such that bj≠cj. Thus, (i)
ifi<j, then ar=br=cr, for r<i and ai>bi=ci;(ii)
ifi=j, then ar=br=cr, for r<i and ai>bi>ci;(iii)
ifj<i, then ar=br=cr, for r<j and aj=bj>cj. Hence 𝒜<𝒞 in each case, and so 𝒜≤𝒞 as desired.For convenience, we will denotef(a)˙ by the infinite sequence
(33)f(a),f2(a),f3(a),…,
where a∈ℕ with a≥2 and fj is the jth composite iteration of f.Theorem 11.
Every nonempty subset of𝒰f which is bounded above has a least upper bound (supremum).Proof.
Let𝒳 be a nonempty subset of 𝒰f which is bounded above by a sequence ℬ=(b0,b1,b2,…)∈𝒰f. Then 𝒜≤ℬ, and so a0≤b0 for all 𝒜∈𝒳. Let m0 be the maximum value of a0 for all 𝒜∈𝒳. Let
(34)ℳ=(m0,2,f(2)˙).
We will first show that ℳ is an upper bound of 𝒳. For 𝒜=(a0,a1,a2,…)∈𝒳, if a0<m0, then it is clear that 𝒜<ℳ. If a0=m0, then we prove that
(35)an≥fn-1(2)
for all n≥1 by induction on n, where f0 is the identity map. If n=1, we have a1≥2=f0(2). Assume now that an≥fn-1(2) for n≥1. Since f is nondecreasing, we obtain
(36)an+1≥f(an)≥f(fn-1(2))=fn(2),
as desired. It then follows that 𝒜≤ℳ; that is, ℳ is an upper bound of 𝒳. Hence we may assume that b0=m0, and so b0=a0 for some 𝒜=(a0,a1,a2,…)∈𝒳. Moreover, we may assume that ℬ∉𝒳, since otherwise ℬ=sup⁡𝒳. Then 𝒜<ℬ for all 𝒜∈𝒳, and there is the largest index k≥0 such that
(37)a0=b0,a1=b1,…,ak=bk
for every 𝒜∈𝒳 with a0=b0.
Next, we define a sequence𝒞=(c0,c1,c2,…)∈𝒰f and then show that 𝒞=sup⁡𝒳. Let c0=b0,c1=b1,…,ck=bk, and let ck+1 be the least possible value for ak+1 of any 𝒜∈𝒳 with a0=b0. Let ck+2 be the least possible value for ak+2 of any element of 𝒳 of the form
(38)(c0,…,ck+1,ak+2,ak+3,…).
Continuing inductively to define ck+i+1 as the least possible value for ak+i+1 of any element of 𝒳 of the form
(39)(c0,…,ck+i,ak+i+1,ak+i+2,…),
this process eventually yields a sequence 𝒞=(c0,c1,c2,…) with ci+1≥f(ci) for all i=1,2,….
Finally, we prove that𝒞=sup⁡𝒳. It is clear that 𝒜<𝒞 for every 𝒜∈𝒳 with a0<b0=c0. By the construction of 𝒞, we see that 𝒜≤𝒞 for every 𝒜∈𝒳 with a0=b0, so 𝒞 is an upper bound of 𝒳. If 𝒳 has an upper bound 𝒟<𝒞, then d0≤c0. If d0<c0=a0, then 𝒟<𝒜 for all 𝒜∈𝒳 with a0=b0, which is impossible. Thus d0=c0 and dj>cj for the first j≥1 such that dj≠cj. Hence every element of the form
(40)𝒜=(c0,c1,…,cj,aj+1,aj+2,…)
in 𝒳 satisfies 𝒟<𝒜≤𝒟, a contradiction.Theorem 12.
Given any element𝒜 of 𝒰f, there exist 𝒜(n),𝒜(n)∈𝒰f for n≥1 such that (i)
𝒜(m)<𝒜(n)<𝒜≤𝒜(n)≤𝒜(m) for m<n,(ii)
𝒜=sup⁡𝒜(n)=inf⁡𝒜(n).Proof.
Let𝒜=(a0,a1,…) be any sequence in 𝒰f. We define 𝒜(n) and 𝒜(n) for n≥1 as follows:
(41)𝒜(n)=(a0,a1,…,an-1,an,f(an)˙),𝒜(n)=(a0,a1,…,an-1,an′,f(an′)˙),
where an′=an+1. It is clear that 𝒜(n),𝒜(n)∈𝒰f for all n≥1. (i)
For positive integersm,n, with m<n, we have
(42)𝒜(m)=(a0,a1,…,am-1,am,f(am),f2(am),…),𝒜(n)=(f2(an)a0,a1,…,am-1,am,am+1,…,an-1,an,f(an),f2(an),…),𝒜=(a0,a1,…,am-1,am,am+1,…,an-1,an,an+1,an+2,…),𝒜(n)=(f2(an′)a0,a1,…,am-1,am,am+1,…,an-1,an′,f(an′),f2(an′),…),𝒜(m)=(a0,a1,…,am-1,am′,f(am′),f2(am′),…).
It is clear that 𝒜(m)<𝒜(n)<𝒜. Since ai+1≥f(ai) for all i≥1 and the property of f, we have 𝒜≤𝒜(n)≤𝒜(m). Thus part (i) follows.(ii)
Suppose that there existsℬ∈𝒰f such that 𝒜(n)≤ℬ<𝒜 for all n≥1. Then we must have a0=b0 and bk>ak for the first k such that ak≠bk. Thus
(43)ℬ<𝒜(k+1)=(a0,a1,…,ak,ak+1′,f(ak+1′)˙),
a contradiction. Hence 𝒜=sup⁡𝒜(n). Similarly, suppose that there is 𝒞∈𝒰f such that 𝒜<𝒞≤𝒜(n) for all n≥1. Then we must have a0=c0 and al>cl for the first l such that al≠cl. This gives the contradiction
(44)𝒜(l)=(a0,a1,…,al-1,al,f(al)˙)<𝒞.
Thus 𝒜=inf⁡𝒜(n) and part (ii) follows.


## 4. Algebraic Operations in𝒰f

For a nondecreasing functionf:[2,∞)→ℝ such that f(x)≥2 for all x≥2, let ρf be any 1-1 order-preserving map from ℝ onto 𝒰f. To show that ρf-1 is order-preserving, let 𝒜,ℬ∈𝒰f be such that 𝒜≤ℬ. Then 𝒜=ρf(A) and ℬ=ρf(B) for some A,B∈ℝ. Since ρf is order-preserving, we obtain
(45)ρf-1(𝒜)=A≤B=ρf-1(ℬ),
as desired.For𝒜∈𝒰f, let A(n) and A(n) be real numbers such that
(46)ρf(A(n))=𝒜(n),ρf(A(n))=𝒜(n),
for n≥1, where 𝒜(n) and 𝒜(n) are defined as in Theorem 12. Since ρf is order-preserving, we deduce that
(47)A(m)<A(n)<A≤A(n)≤A(m)
for all positive numbers m,n with m<n, where A=ρf-1(𝒜).Now, for any𝒜,ℬ∈𝒰f, define
(48)𝒜⊕ℬ=ρf(sup⁡(A(n)+B(n)))-𝒜=ρf(sup⁡(-A(n))).
Note that both operators are well defined in 𝒰f because
(49)A(n)+B(n)≤A+B,-A(n)≤-A(n≥1)
and (A(n)+B(n)),(-A(n)) are increasing sequences of real numbers. It follows that
(50)A+B=sup⁡(A(n)+B(n)),-A=sup⁡(-A(n)).
Using (48), we deduce that
(51)ρf(A+B)=ρf(A)⊕ρf(B),ρf(-A)=-ρf(A)
for all A,B∈ℝ.Theorem 13.
(
𝒰
f
,
⊕
) is an abelian group. Furthermore, for 𝒜,ℬ,𝒞∈𝒰f, one has the following: (i)
if𝒜<ℬ, then 𝒜⊕𝒞<ℬ⊕𝒞;(ii)
𝒜<ℬ if and only if -𝒜>-ℬ.Proof.
First, we will show that (𝒰f,⊕) is an abelian group. Let 𝒜=ρf(A),ℬ=ρf(B),𝒞=ρf(C)∈𝒰f for some A,B,C∈ℝ. Using (51), we obtain the following: (1)
𝒜⊕ℬ=ρf(A)⊕ρf(B)=ρf(A+B)∈𝒰f;(2)
𝒜⊕(ℬ⊕𝒞)=ρf(A+(B+C))=ρf((A+B)+C)=(𝒜⊕ℬ)⊕𝒞;(3)
let𝒪f:=ρf(0)∈𝒰f; then 𝒜⊕𝒪f=ρf(A)⊕ρf(0)=ρf(A+0)=𝒜 for all 𝒜∈𝒰f;(4)
𝒜⊕(-𝒜)=ρf(A)⊕ρf(-A)=ρf(0)=𝒪f for all 𝒜∈𝒰f; and(5)
𝒜⊕ℬ=ρf(A)⊕ρf(B)=ρf(A+B)=ρf(B+A)=ρf(B)⊕ρf(A)=ℬ⊕𝒜.From (1)–(5), we conclude that (𝒰f,⊕) is an abelian group.
Lastly, assume that𝒜<ℬ. Since ρf-1 is order-preserving, we have A<B. Using (51), we have
(52)𝒜⊕𝒞=ρf(A+C)<ρf(B+C)=ℬ⊕𝒞,-𝒜=-ρf(A)=ρf(-A)>ρf(-B)=-ρf(B)=-ℬ.
Hence (ii) follows since -(-𝒜)=-(-ρf(A))=-(ρf(-A))=ρf(A)=𝒜.Next, we define the binary operation⊙ on 𝒰f as follows: for any 𝒜,ℬ∈𝒰f, let
(53)𝒜⊙ℬ={ρf(sup⁡(A(n)·B(n)))if𝒜>𝒪f,ℬ>𝒪f,-(𝒜⊙(-ℬ))if𝒜>𝒪f,ℬ<𝒪f,-((-𝒜)⊙ℬ)if𝒜<𝒪f,ℬ>𝒪f,(-𝒜)⊙(-ℬ)if𝒜<𝒪f,ℬ<𝒪f,𝒪fif𝒜=𝒪forℬ=𝒪f.
Also define
(54)𝒜-1={ρf(sup⁡((A(n))-1))if𝒜>𝒪f-(-𝒜)-1if𝒜<𝒪f.
To see that these definitions are unambiguous, first we note that
(55)A(n)·B(n)≤A·B,(A(n))-1≤A-1
for all sufficiently large n, where 𝒜,ℬ>𝒪f, since A(n) and B(n) are positive for all sufficiently large n. Secondly, in order to cover all cases, we use the fact from Theorem 13 (ii) that 𝒜<𝒪f if and only if -𝒜>𝒪f. Using (53), we deduce that
(56)ρf(A·B)=ρf(A)⊙ρf(B)
for all real numbers A and B. In addition, using (54), we have
(57)ρf(A)-1=ρf(A-1)
for all nonzero real number A.Theorem 14.
(
𝒰
f
,
⊕
,
⊙
) is an ordered field. Furthermore, if 𝒜<ℬ and 𝒞>𝒪f, then 𝒜⊙𝒞<ℬ⊙𝒞.Proof.
By Theorem13, we know that (𝒰f,⊕) is an abelian group. To prove that (𝒰f,⊕,⊙) is a field, it remains to show that (𝒰f∖{𝒪f},⊙) is an abelian group. Let 𝒜=ρf(A),ℬ=ρf(B),𝒞=ρf(C)∈𝒰f for some A,B,C∈ℝ. Using (56) and (57), we obtain the following: (1)
𝒜⊙ℬ=ρf(A)⊙ρf(B)=ρf(A·B)∈𝒰f;(2)
𝒜⊙(ℬ⊙𝒞)=ρf(A·(B·C))=ρf((A·B)·C)=(𝒜⊙ℬ)⊙𝒞;(3)
letℐ:=ρf(1)∈𝒰f; then 𝒜⊙ℐ=ρf(A·1)=𝒜 for all 𝒜=ρf(A)∈𝒰f;(4)
𝒜⊙𝒜-1=ρf(A)⊙ρf(A-1)=ρf(1)=ℐ for all 𝒜=ρf(A)∈𝒰f; and(5)
𝒜⊙ℬ=ρf(A·B)=ρf(B·A)=ℬ⊙𝒜.From (1)–(5), we conclude that (𝒰f∖{𝒪f},⊙) is an abelian group; thus (𝒰f,⊕,⊙) is a field.
Next, we will show that(𝒰f,⊕,⊙) is an ordered field (see [6]). Let
(58)𝒫f={𝒜∈𝒰f∣𝒜>𝒪f}
and 𝒜,ℬ∈𝒫f. Then 𝒜=ρf(A) and ℬ=ρf(B) for some positive real numbers A and B. (1)
SinceA+B>0, we have 𝒜⊕ℬ=ρf(A+B)>ρf(0)=𝒪f, and so 𝒜⊕ℬ∈𝒫f.(2)
SinceA·B>0, we have 𝒜⊙ℬ=ρf(A·B)>ρf(0)=𝒪f, and so 𝒜⊙ℬ∈𝒫f.(3)
SinceA>0, we have -A<0 and -𝒜=-ρf(A)=ρf(-A)<ρf(0)=𝒪f. Hence -𝒜∉𝒫f.(4)
For𝒜=ρf(A)∈𝒰f, we have A=0 or A>0 or -A>0. It follows that 𝒜=𝒪f, 𝒜>𝒪f, or -𝒜>𝒪f. Hence 𝒜=𝒪f, 𝒜∈𝒫f, or -𝒜∈𝒫f.
Finally, assume that𝒜<ℬ and 𝒞>𝒪f. Since ρf-1 is order-preserving, then A<B and C>0, and so A·C<B·C. Thus
(59)𝒜⊙𝒞=ρf(A·C)<ρf(B·C)=ℬ⊙𝒞
since ρf is order-preserving. This completes the proof of the theorem.


## 5. Examples

In this section, we will give three models for the real number system which are special examples of the ordered field(𝒰f,⊕,⊙), where f is chosen from three different nondecreasing functions from [2,∞) into ℝ such that f(x)≥2 for all x≥2. The first two propositions were readily mentioned in Knopmachers’ results [4].Proposition 15.
LetfS:[2,∞)→ℝ be defined by
(60)fS(x)=x(x-1)+1,
for all x≥2,
(61)𝒰fS={(a0,a1,a2,…)∣an∈ℤ,a1≥2,an+1≥an(an-1)+1∀n≥1}
and ρfS:ℝ→𝒰fS defined by
(62)ρfS(A)=(a0,a1,a2,…),
for all A∈ℝ, where
(63)A=a0+∑n=1∞‍1an
is the Sylvester series expansion of A. Then (𝒰fS,⊕,⊙) is an ordered field containing (ρfS(ℚ),⊕,⊙) as a dense subfield. Furthermore, for 𝒜,ℬ,𝒞∈𝒰fS, one has the following: (i)
if𝒜<ℬ,then𝒜⊕𝒞<ℬ⊕𝒞;(ii)
𝒜<ℬif and only if-𝒜>-ℬ;(iii)
if𝒜<ℬand𝒞>𝒪fS,then𝒜⊙𝒞<ℬ⊙𝒞, where 𝒪fS is the zero element in 𝒰fS.Proof.
It is clear thatfS is a nondecreasing function such that f(x)≥2 for all x≥2. It is an immediate consequence of Corollary 3 that ρfS is 1-1 map from ℝ onto 𝒰fS. By Corollary 7 and the definition of order in 𝒰fS, this map is order-preserving. Using Theorems 13 and 14, we can conclude that (𝒰fS,⊕,⊙) is an ordered field containing ρfS(ℚ) and satisfies properties (i), (ii), and (iii). To complete the proof of this proposition, it remains to show that ρfS(ℚ) is a dense subfield of 𝒰fS.
It is clear thatρfS(ℚ) is a subfield of 𝒰fS. Now, let 𝒜,ℬ∈𝒰fS be such that 𝒜<ℬ. If a0<b0, let
(64)𝒞=(b0,b1′,fS(b1′)˙);
then it is easy to see that 𝒜<𝒞<ℬ and 𝒞∈ρfS(ℚ). On the other hand, if a0=b0, then am>bm for the first m such that am≠bm. In that case, we have 𝒜<𝒟<ℬ, where
(65)𝒟=(b0,…,bm,bm+1′,fS(bm+1′)˙).
Then 𝒟∈ρfS(ℚ). This shows that ρfS(ℚ) is dense in 𝒰fS as desired.Proposition 16.
LetfE:[2,∞)→ℝ be defined by
(66)fE(x)=x,
for all x≥2,
(67)𝒰fE={(a0,a1,a2,…)∣an∈ℤ,a1≥2,an+1≥an∀n≥1}
and ρfE:ℝ→𝒰fE defined by
(68)ρfE(A)=(a0,a1,a2,…),
for all A∈ℝ, where
(69)A=a0+∑n=1∞‍1a1a2⋯an
is the Engel series expansion of A. Then (𝒰fE,⊕,⊙) is an ordered field containing (ρfE(ℚ),⊕,⊙) as a dense subfield. Furthermore, for 𝒜,ℬ,𝒞∈𝒰fE, one has the following: (i)
if𝒜<ℬ,then𝒜⊕𝒞<ℬ⊕𝒞;(ii)
𝒜<ℬif and only if-𝒜>-ℬ;(iii)
if𝒜<ℬand𝒞>𝒪fE,then𝒜⊙𝒞<ℬ⊙𝒞, where 𝒪fE is the zero element in 𝒰fE.Proof.
It is clear thatfE is a nondecreasing function such that f(x)≥2 for all x≥2. It is an immediate consequence of Corollary 4 that ρfE is 1-1 map from ℝ onto 𝒰fE. By Corollary 8 and the definition of order in 𝒰fE, this map is order-preserving. Using Theorems 13 and 14, we can conclude that (𝒰fE,⊕,⊙) is an ordered field containing ρfE(ℚ) and satisfies properties (i), (ii), and (iii). For density of rationals, we can prove in a similar way to the proof of Proposition 15 that ρfE(ℚ) is a dense subfield of 𝒰fE.Proposition 17.
LetfL:[2,∞)→ℝ be defined by
(70)fL(x)=2,
for all x≥2,
(71)𝒰fL={(a0,a1,a2,…)∣an∈ℤ,an≥2∀n≥1}
and ρfL:ℝ→𝒰fL defined by
(72)ρfL(A)=(a0,a1,a2,…),
for allA∈ℝ, where
(73)A=a0+1a1+∑n=1∞‍1a1(a1-1)⋯an(an-1)an+1
is the Lüroth series expansion of A. Then (𝒰fL,⊕,⊙) is an ordered field containing (ρfL(ℚ),⊕,⊙) as a dense subfield. Furthermore, for 𝒜,ℬ,𝒞∈𝒰fL, one has the following: (i)
if𝒜<ℬ,then𝒜⊕𝒞<ℬ⊕𝒞;(ii)
𝒜<ℬif and only if-𝒜>-ℬ;(iii)
if𝒜<ℬand𝒞>𝒪fL,then𝒜⊙𝒞<ℬ⊙𝒞, where 𝒪fL is the zero element in 𝒰fL.Proof.
It is clear thatfL is a nondecreasing function such that f(x)≥2 for all x≥2. It is an immediate consequence of Corollary 5 that ρfL is 1-1 map from ℝ onto 𝒰fL. By Corollary 9 and the definition of order in 𝒰fL, this map is order-preserving. Using Theorems 13 and 14, we conclude that (𝒰fL,⊕,⊙) is an ordered field containing ρfL(ℚ) and satisfies properties (i), (ii), and (iii). For density of rationals, we can prove in a similar way to the proof of Proposition 15 that ρfL(ℚ) is a dense subfield of 𝒰fL.



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