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

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.

1. Introduction

According to [1, 2], it is well known that each 𝐴∈ℝ is uniquely representable as an infinite series expansion called Sylvester series expansion, which is of the form 𝐴=𝑎₀+^∞_𝑛=11𝑎_𝑛, (1) where 𝑎₀=⌊𝐴⌋𝑎if𝐴∉ℤ𝐴−1if𝐴∈ℤ,¹𝑎≥2,_𝑛+1≥𝑎_𝑛𝑎_𝑛−1+1∀𝑛≥1. (2) Moreover, 𝐴∈ℚ if and only if 𝑎_𝑛+1=𝑎_𝑛(𝑎_𝑛−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 called Engel series expansion, which is of the form 𝐴=𝑎₀+^∞_𝑛=11𝑎₁⋯𝑎_𝑛, (3) where 𝑎₀=⌊𝐴⌋𝑎if𝐴∉ℤ𝐴−1if𝐴∈ℤ,¹𝑎≥2,_𝑛+1≥𝑎_𝑛∀𝑛≥1. (4) Moreover, 𝐴∈ℚ if and only if 𝑎_𝑛+1=𝑎_𝑛 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 called Lüroth series expansion, which is of the form 𝐴=𝑎₀+1𝑎₁+^∞_𝑛=11𝑎₁𝑎₁−1⋯𝑎_𝑛𝑎_𝑛−1𝑎_𝑛+1, (5) where 𝑎₀=⌊𝐴⌋𝑎if𝐴∉ℤ𝐴−1if𝐴∈ℤ,^𝑛≥2∀𝑛≥1. (6) Moreover, 𝐴∈ℚ if and only if (𝑎_𝑛) 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 called Sylvester-Engel-Lüroth series expansion or SEL 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 number 𝐴, write it as 𝐴=𝑎₀+𝐴₁, where 𝑎₀=⌊𝐴⌋if𝐴∉ℤ𝐴−1if𝐴∈ℤ (7) and 0<𝐴₁≤1. Then recursively define 𝑎_𝑛1=1+𝐴_𝑛,𝐴_𝑛+1=𝑎_𝑛𝐴_𝑛−1𝑒_𝑛, (8)(9) where 𝑒_𝑛=𝑒_𝑛(𝑎_𝑛) is a positive rational number, which may depend on 𝑎_𝑛, for all 𝑛≥1.

Using this algorithm and the same proof as in [1, 2], we have the following.

Theorem 1. Let 𝐴∈ℝ and assume that 𝑎_𝑛−1𝑒_𝑛∈ℕ (10) for all 𝑛≥1. Then 𝐴 is uniquely representable as an infinite series expansion called SEL series expansion, which is of the form 𝐴=𝑎₀+1𝑎₁+^∞_𝑛=11𝑎₁𝑒₁⋯𝑎_𝑛𝑒_𝑛𝑎_𝑛+1, (11) where 𝑎₁≥2 and 𝑎_𝑛+1≥((𝑎_𝑛−1)/𝑒_𝑛)+1 for all 𝑛≥1.

Lemma 2. Any series 1𝑏₁+^∞_𝑛=11𝑏₁𝑓₁⋯𝑏_𝑛𝑓_𝑛𝑏_𝑛+1, (12) where 𝑓_𝑛=𝑓_𝑛𝑏_𝑛∈ℚ⁺,𝑏₁𝑏≥2,_𝑛+1≥𝑏_𝑛−1𝑓_𝑛+1≥2∀𝑛≥1, (13) converges to a real number 𝐵₁ such that 𝑏₁−1≤(1/𝐵₁)<𝑏₁. Furthermore, 𝑏₁=1+⌊1/𝐵₁⌋.

By setting 𝑒_𝑛=1/𝑎_𝑛,𝑒_𝑛=1, and 𝑒_𝑛=𝑎_𝑛−1, for all 𝑛≥1 in Theorem 1, and by setting 𝑓_𝑛=1/𝑏_𝑛,𝑓_𝑛=1, and 𝑓_𝑛=𝑏_𝑛−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. Each 𝐴∈ℝ is uniquely representable as a Sylvester series expansion; that is, 𝐴=𝑎₀+^∞_𝑛=11𝑎_𝑛, (14) where 𝑎₁≥2 and 𝑎_𝑛+1≥(𝑎_𝑛−1)𝑎_𝑛+1 for all 𝑛≥1.
Conversely, each series of the form (14) converges.

Corollary 4. Each 𝐴∈ℝ is uniquely representable as an Engel series expansion; that is, 𝐴=𝑎₀+^∞_𝑛=11𝑎₁⋯𝑎_𝑛, (15) where 𝑎₁≥2 and 𝑎_𝑛+1≥𝑎_𝑛 for all 𝑛≥1.
Conversely, each series of the form (15) converges.

Corollary 5. Each 𝐴∈ℝ is uniquely representable as a Lüroth series expansion; that is, 𝐴=𝑎₀+1𝑎₁+^∞_𝑛=11𝑎₁𝑎₁−1⋯𝑎_𝑛𝑎_𝑛−1𝑎_𝑛+1, (16) where 𝑎_𝑛≥2 for all 𝑛≥1.
Conversely, each series of the form (16) converges.

Proposition 6. Let 𝑎₀+1𝑎₁+^∞_𝑛=11𝑎₁𝑒₁⋯𝑎_𝑛𝑒_𝑛𝑎_𝑛+1,𝑏₀+1𝑏₁+^∞_𝑛=11𝑏₁𝑒₁⋯𝑏_𝑛𝑒_𝑛𝑏_𝑛+1 (17) be SEL series expansions of distinct real numbers 𝐴 and 𝐵, respectively. Then the condition 𝐴<𝐵 is equivalent to the following: (i)𝑎₀<𝑏₀ if 𝑎₀≠𝑏₀;(ii)𝑎_𝑖>𝑏_𝑖 for the first 𝑖≥1 such that 𝑎_𝑖≠𝑏_𝑖 if 𝑎₀=𝑏₀.

Proof. If 𝑎₀<𝑏₀, then 𝐴=𝑎₀+𝐴₁≤𝑎₀+1≤𝑏₀<𝑏₀+𝐵₁=𝐵. (18) Hence (i) follows.
Now assume that 𝑎₀=𝑏₀ and 𝑎_𝑖>𝑏_𝑖 for the first 𝑖≥1 such that 𝑎_𝑖≠𝑏_𝑖. Applying (9) repeatedly, we obtain 𝐴_𝑛=1𝑎_𝑛+𝐴_𝑛+1𝑎_𝑛𝑒_𝑛=1𝑎_𝑛+1𝑎_𝑛𝑒_𝑛1𝑎_𝑛+1+𝐴_𝑛+2𝑎_𝑛+1𝑒_𝑛+1⋮=1𝑎_𝑛+1𝑎_𝑛𝑒_𝑛𝑎_𝑛+11+⋯+𝑎_𝑛𝑒_𝑛⋯𝑎_{𝑛+𝑘−1}𝑒_{𝑛+𝑘−1}𝑎_𝑛+𝑘+𝐴_𝑛+𝑘+1𝑎_𝑛𝑒_𝑛⋯𝑎_{𝑛+𝑘−1}𝑒_{𝑛+𝑘−1}𝑎_𝑛+𝑘𝑒_𝑛+𝑘 (19) for all 𝑛,𝑘∈ℕ. Using (8), we get 1𝑎_𝑛<𝐴_𝑛≤1𝑎_𝑛−1 (20) for all 𝑛≥1. We now prove 0<𝐴_𝑛≤1 for all 𝑛≥1 by induction on 𝑛. If 𝑛=1, then we have seen that 0<𝐴₁≤1. Assume now that 0<𝐴_𝑛≤1 for 𝑛≥1. By (8), we see that 𝑎_𝑛≥2. Since 𝐴_𝑛+1=𝑎_𝑛𝐴_𝑛−1𝑒_𝑛=𝐴_𝑛−1𝑎_𝑛𝑎_𝑛𝑒_𝑛 (21) and using (10) and (20), we have that 0<𝐴_𝑛+1=𝑎_𝑛𝐴_𝑛−1𝑒_𝑛≤𝑎_𝑛𝑎_𝑛𝑒−1−1_𝑛=𝑒_𝑛𝑎_𝑛−1≤1, (22) as desired. It follows that 𝑎_𝑛≥2 for all 𝑛≥1.
Next, we will prove that 𝐴_𝑛=1𝑎_𝑛+1𝑎_𝑛𝑒_𝑛𝑎_𝑛+1+⋯ (23) for all 𝑛≥1. For 𝑛,𝑘≥1, let 𝐵_𝑘=1𝑎_𝑛+1𝑎_𝑛𝑒_𝑛𝑎_𝑛+11+⋯+𝑎_𝑛𝑒_𝑛⋯𝑎_{𝑛+𝑘−2}𝑒_{𝑛+𝑘−2}𝑎_{𝑛+𝑘−1}. (24) Since 𝐴_𝑛,𝑒_𝑛>0 and 𝑎_𝑛∈ℕ, for all 𝑛≥1, the sequence of real numbers (𝐵_𝑘) is increasing and bounded above. Thus, lim_𝑘→∞𝐵_𝑘 exists and so lim_𝑘→∞1𝑎_𝑛𝑒_𝑛⋯𝑎_𝑛+𝑘𝑒_𝑛+𝑘𝑎_𝑛+𝑘+1=0. (25) Since (20) and 𝑎_𝑛+𝑘+1≥2, we deduce that 𝐴0<^𝑛+𝑘+1𝑎_𝑛𝑒_𝑛⋯𝑎_𝑛+𝑘𝑒_𝑛+𝑘≤1𝑎_𝑛𝑒_𝑛⋯𝑎_𝑛+𝑘𝑒_𝑛+𝑘⋅𝑎_𝑛+𝑘+1𝑎_𝑛+𝑘+1⋅1𝑎_𝑛+𝑘+1≤2−1𝑎_𝑛𝑒_𝑛⋯𝑎_𝑛+𝑘𝑒_𝑛+𝑘𝑎_𝑛+𝑘+1, (26) and so lim_𝑘→∞𝐴_𝑛+𝑘+1𝑎_𝑛𝑒_𝑛⋯𝑎_𝑛+𝑘𝑒_𝑛+𝑘=0, (27) showing that (23) holds as desired.
Using (23), (20), and 𝑎_𝑖−1≥𝑏_𝑖, we have 𝐴=𝑎₀+1𝑎₁1+⋯+𝑎₁𝑒₁⋯𝑎_𝑖−1𝑒_𝑖−1𝑎_𝑖+1𝑎₁𝑒₁⋯𝑎_𝑖𝑒_𝑖𝑎_𝑖+1+⋯=𝑎₀+1𝑎₁1+⋯+𝑎₁𝑒₁⋯𝑎_𝑖−1𝑒_𝑖−11𝑎_𝑖+1𝑎_𝑖𝑒_𝑖𝑎_𝑖+1+⋯=𝑎₀+1𝑎₁1+⋯+𝑎₁𝑒₁⋯𝑎_𝑖−1𝑒_𝑖−1𝐴_𝑖≤𝑎₀+1𝑎₁1+⋯+𝑎₁𝑒₁⋯𝑎_𝑖−1𝑒_𝑖−11𝑎_𝑖−1≤𝑎₀+1𝑎₁1+⋯+𝑎₁𝑒₁⋯𝑎_𝑖−1𝑒_𝑖−11𝑏_𝑖=𝑏₀+1𝑏₁1+⋯+𝑏₁𝑒₁⋯𝑏_𝑖−1𝑒_𝑖−1𝑏_𝑖,<𝐵, (28) and the assertion follows.

The following corollaries follow immediately from Proposition 6 by setting 𝑒_𝑛=1/𝑎_𝑛,𝑒_𝑛=1, and 𝑒_𝑛=𝑎_𝑛−1, for all 𝑛≥1, respectively; the first two corollaries readily appear in [4].

Corollary 7. Let 𝑎₀+^∞_𝑛=11𝑎_𝑛,𝑏₀+^∞_𝑛=11𝑏_𝑛 (29) 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)𝑎_𝑖>𝑏_𝑖 for the first 𝑖≥1 such that 𝑎_𝑖≠𝑏_𝑖 if 𝑎₀=𝑏₀.

Corollary 8. Let 𝑎₀+^∞_𝑛=11𝑎₁⋯𝑎_𝑛,𝑏₀+^∞_𝑛=11𝑏₁⋯𝑏_𝑛 (30) 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)𝑎_𝑖>𝑏_𝑖 for the first 𝑖≥1 such that 𝑎_𝑖≠𝑏_𝑖 if 𝑎₀=𝑏₀.

Corollary 9. Let 𝑎₀+1𝑎₁+^∞_𝑛=11𝑎₁𝑎₁−1⋯𝑎_𝑛𝑎_𝑛−1𝑎_𝑛+1,𝑏₀+1𝑏₁+^∞_𝑛=11𝑏₁𝑏₁−1⋯𝑏_𝑛𝑏_𝑛−1𝑏_𝑛+1 (31) 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)𝑎_𝑖>𝑏_𝑖 for the first 𝑖≥1 such that 𝑎_𝑖≠𝑏_𝑖 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 𝒰_𝑓 be the set of all formal infinite sequences 𝒜=(𝑎₀,𝑎₁,𝑎₂,…) of integers 𝑎_𝑛 such that 𝑎₁≥2 and 𝑎_𝑛+1≥𝑓(𝑎_𝑛) for all 𝑛≥1. In other words, 𝒰_𝑓=𝑎₀,𝑎₁,𝑎₂,…∣𝑎₀,𝑎_𝑛∈ℤ,𝑎₁𝑎≥2,_𝑛+1≥𝑓𝑎_𝑛∀𝑛≥1. (32) As an analogue to Proposition 6, we define order relation on 𝒰_𝑓 as follows.

For 𝒜=(𝑎₀,𝑎₁,𝑎₂,…),ℬ=(𝑏₀,𝑏₁,𝑏₂,…)∈𝒰_𝑓, we say that𝒜<ℬif and only if(i)𝑎₀<𝑏₀ if 𝑎₀≠𝑏₀, or(ii)𝑎_𝑖>𝑏_𝑖 for the first 𝑖≥1 such that 𝑎_𝑖≠𝑏_𝑖 if 𝑎₀=𝑏₀.Note that 𝒜≤ℬ if and only if 𝒜<ℬ or 𝒜=ℬ.

Lemma 10. ≤ is a total ordering relation on 𝒰_𝑓.

Proof. It is clear that ≤ is reflexive and antisymmetric and any two elements in 𝒰_𝑓 are comparable. It remains to show that ≤ is transitive. Let 𝒜,ℬ,𝒞∈𝒰_𝑓 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 that 𝑎₀=𝑏₀=𝑐₀. Then 𝑎_𝑖>𝑏_𝑖 for the first 𝑖≥1 such that 𝑎_𝑖≠𝑏_𝑖 and 𝑏_𝑗>𝑐_𝑗 for the first 𝑗≥1 such that 𝑏_𝑗≠𝑐_𝑗. Thus, (i)if 𝑖<𝑗, then 𝑎_𝑟=𝑏_𝑟=𝑐_𝑟, for 𝑟<𝑖 and 𝑎_𝑖>𝑏_𝑖=𝑐_𝑖;(ii)if 𝑖=𝑗, then 𝑎_𝑟=𝑏_𝑟=𝑐_𝑟, for 𝑟<𝑖 and 𝑎_𝑖>𝑏_𝑖>𝑐_𝑖;(iii)if 𝑗<𝑖, then 𝑎_𝑟=𝑏_𝑟=𝑐_𝑟, for 𝑟<𝑗 and 𝑎_𝑗=𝑏_𝑗>𝑐_𝑗. Hence 𝒜<𝒞 in each case, and so 𝒜≤𝒞 as desired.

For convenience, we will denote 𝑓(𝑎̇) by the infinite sequence 𝑓(𝑎),𝑓²(𝑎),𝑓³(𝑎),…, (33) where 𝑎∈ℕ with 𝑎≥2 and 𝑓^𝑗 is the 𝑗th composite iteration of 𝑓.

Theorem 11. Every nonempty subset of 𝒰_𝑓 which is bounded above has a least upper bound (supremum).

Proof. Let 𝒳 be a nonempty subset of 𝒰_𝑓 which is bounded above by a sequence ℬ=(𝑏₀,𝑏₁,𝑏₂,…)∈𝒰_𝑓. Then 𝒜≤ℬ, and so 𝑎₀≤𝑏₀ for all 𝒜∈𝒳. Let 𝑚₀ be the maximum value of 𝑎₀ for all 𝒜∈𝒳. Let ℳ=𝑚₀,2,𝑓(2̇). (34) 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 𝑎_𝑛≥𝑓^𝑛−1(2) (35) 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 𝑎_𝑛≥𝑓^𝑛−1(2) for 𝑛≥1. Since 𝑓 is nondecreasing, we obtain 𝑎_𝑛+1≥𝑓𝑎_𝑛≥𝑓𝑓^𝑛−1(2)=𝑓^𝑛(2), (36) 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=sup𝒳. Then 𝒜<ℬ for all 𝒜∈𝒳, and there is the largest index 𝑘≥0 such that 𝑎₀=𝑏₀,𝑎₁=𝑏₁,…,𝑎_𝑘=𝑏_𝑘 (37) for every 𝒜∈𝒳 with 𝑎₀=𝑏₀.
Next, we define a sequence 𝒞=(𝑐₀,𝑐₁,𝑐₂,…)∈𝒰_𝑓 and then show that 𝒞=sup𝒳. Let 𝑐₀=𝑏₀,𝑐₁=𝑏₁,…,𝑐_𝑘=𝑏_𝑘, and let 𝑐_𝑘+1 be the least possible value for 𝑎_𝑘+1 of any 𝒜∈𝒳 with 𝑎₀=𝑏₀. Let 𝑐_𝑘+2 be the least possible value for 𝑎_𝑘+2 of any element of 𝒳 of the form 𝑐₀,…,𝑐_𝑘+1,𝑎_𝑘+2,𝑎_𝑘+3,…. (38) Continuing inductively to define 𝑐_𝑘+𝑖+1 as the least possible value for 𝑎_𝑘+𝑖+1 of any element of 𝒳 of the form 𝑐₀,…,𝑐_𝑘+𝑖,𝑎_𝑘+𝑖+1,𝑎_𝑘+𝑖+2,…, (39) this process eventually yields a sequence 𝒞=(𝑐₀,𝑐₁,𝑐₂,…) with 𝑐_𝑖+1≥𝑓(𝑐_𝑖) for all 𝑖=1,2,….
Finally, we prove that 𝒞=sup𝒳. 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 𝑑_𝑗>𝑐_𝑗 for the first 𝑗≥1 such that 𝑑_𝑗≠𝑐_𝑗. Hence every element of the form 𝒜=𝑐₀,𝑐₁,…,𝑐_𝑗,𝑎_𝑗+1,𝑎_𝑗+2,… (40) in 𝒳 satisfies 𝒟<𝒜≤𝒟, a contradiction.

Theorem 12. Given any element 𝒜 of 𝒰_𝑓, there exist 𝒜^(𝑛),𝒜_(𝑛)∈𝒰_𝑓 for 𝑛≥1 such that (i)𝒜_(𝑚)<𝒜_(𝑛)<𝒜≤𝒜^(𝑛)≤𝒜^(𝑚) for 𝑚<𝑛,(ii)𝒜=sup𝒜_(𝑛)=inf𝒜^(𝑛).

Proof. Let 𝒜=(𝑎₀,𝑎₁,…) be any sequence in 𝒰_𝑓. We define 𝒜^(𝑛) and 𝒜_(𝑛) for 𝑛≥1 as follows: 𝒜^(𝑛)=𝑎₀,𝑎₁,…,𝑎_𝑛−1,𝑎_𝑛,𝑓𝑎̇_𝑛,𝒜_(𝑛)=𝑎₀,𝑎₁,…,𝑎_𝑛−1,𝑎^′_𝑛,̇𝑓𝑎^′_𝑛, (41) where 𝑎^′_𝑛=𝑎_𝑛+1. It is clear that 𝒜^(𝑛),𝒜_(𝑛)∈𝒰_𝑓 for all 𝑛≥1. (i)For positive integers 𝑚,𝑛, with 𝑚<𝑛, we have 𝒜^(𝑚)=𝑎₀,𝑎₁,…,𝑎_𝑚−1,𝑎_𝑚,𝑓𝑎_𝑚,𝑓²𝑎_𝑚,𝒜,…_(𝑛)=𝑎₀,𝑎₁,…,𝑎_𝑚−1,𝑎_𝑚,𝑎_𝑚+1𝑎,…,_𝑛−1,𝑎_𝑛,𝑓𝑎_𝑛,𝑓²𝑎_𝑛,𝑎,…𝒜=₀,𝑎₁,…,𝑎_𝑚−1,𝑎_𝑚,𝑎_𝑚+1𝑎,…,_𝑛−1,𝑎_𝑛,𝑎_𝑛+1,𝑎_𝑛+2,𝒜,…_(𝑛)=𝑎₀,𝑎₁,…,𝑎_𝑚−1,𝑎_𝑚,𝑎_𝑚+1𝑎,…,_𝑛−1,𝑎^′_𝑛,𝑓𝑎^′_𝑛,𝑓²𝑎^′_𝑛,𝒜,…_(𝑚)=𝑎₀,𝑎₁,…,𝑎_𝑚−1,𝑎^′_𝑚,𝑓𝑎^′_𝑚,𝑓²𝑎^′_𝑚,…. (42) It is clear that 𝒜_(𝑚)<𝒜_(𝑛)<𝒜. Since 𝑎_𝑖+1≥𝑓(𝑎_𝑖) for all 𝑖≥1 and the property of 𝑓, we have 𝒜≤𝒜^(𝑛)≤𝒜^(𝑚). Thus part (i) follows.(ii)Suppose that there exists ℬ∈𝒰_𝑓 such that 𝒜_(𝑛)≤ℬ<𝒜 for all 𝑛≥1. Then we must have 𝑎₀=𝑏₀ and 𝑏_𝑘>𝑎_𝑘 for the first 𝑘 such that 𝑎_𝑘≠𝑏_𝑘. Thus ℬ<𝒜_(𝑘+1)=𝑎₀,𝑎₁,…,𝑎_𝑘,𝑎^′_𝑘+1,𝑓𝑎̇_′𝑘+1, (43) a contradiction. Hence 𝒜=sup𝒜_(𝑛). Similarly, suppose that there is 𝒞∈𝒰_𝑓 such that 𝒜<𝒞≤𝒜^(𝑛) for all 𝑛≥1. Then we must have 𝑎₀=𝑐₀ and 𝑎_𝑙>𝑐_𝑙 for the first 𝑙 such that 𝑎_𝑙≠𝑐_𝑙. This gives the contradiction 𝒜^(𝑙)=𝑎₀,𝑎₁,…,𝑎_𝑙−1,𝑎_𝑙,𝑓𝑎̇_𝑙<𝒞. (44) Thus 𝒜=inf𝒜^(𝑛) and part (ii) follows.

4. Algebraic Operations in 𝒰_𝑓

For a nondecreasing function 𝑓∶[2,∞)→ℝ such that 𝑓(𝑥)≥2 for all 𝑥≥2, let 𝜌_𝑓 be any 1-1 order-preserving map from ℝ onto 𝒰_𝑓. To show that 𝜌_𝑓⁻¹ is order-preserving, let 𝒜,ℬ∈𝒰_𝑓 be such that 𝒜≤ℬ. Then 𝒜=𝜌_𝑓(𝐴) and ℬ=𝜌_𝑓(𝐵) for some 𝐴,𝐵∈ℝ. Since 𝜌_𝑓 is order-preserving, we obtain 𝜌_𝑓⁻¹(𝒜)=𝐴≤𝐵=𝜌_𝑓⁻¹(ℬ), (45) as desired.

For 𝒜∈𝒰_𝑓, let 𝐴_(𝑛) and 𝐴^(𝑛) be real numbers such that 𝜌_𝑓𝐴_(𝑛)=𝒜_(𝑛),𝜌_𝑓𝐴^(𝑛)=𝒜^(𝑛), (46) for 𝑛≥1, where 𝒜_(𝑛) and 𝒜^(𝑛) are defined as in Theorem 12. Since 𝜌_𝑓 is order-preserving, we deduce that 𝐴_(𝑚)<𝐴_(𝑛)<𝐴≤𝐴^(𝑛)≤𝐴_(𝑚) (47) for all positive numbers 𝑚,𝑛 with 𝑚<𝑛, where 𝐴=𝜌_𝑓⁻¹(𝒜).

Now, for any 𝒜,ℬ∈𝒰_𝑓, define 𝒜⊕ℬ=𝜌_𝑓sup𝐴_(𝑛)+𝐵_(𝑛)−𝒜=𝜌_𝑓sup−𝐴^(𝑛). (48) Note that both operators are well defined in 𝒰_𝑓 because 𝐴_(𝑛)+𝐵_(𝑛)≤𝐴+𝐵,−𝐴^(𝑛)≤−𝐴(𝑛≥1) (49) and (𝐴_(𝑛)+𝐵_(𝑛)),(−𝐴^(𝑛)) are increasing sequences of real numbers. It follows that 𝐴+𝐵=sup𝐴_(𝑛)+𝐵_(𝑛),−𝐴=sup−𝐴^(𝑛). (50) Using (48), we deduce that 𝜌_𝑓(𝐴+𝐵)=𝜌_𝑓(𝐴)⊕𝜌_𝑓(𝐵),𝜌_𝑓(−𝐴)=−𝜌_𝑓(𝐴) (51) for all 𝐴,𝐵∈ℝ.

Theorem 13. (𝒰_𝑓,⊕) is an abelian group. Furthermore, for 𝒜,ℬ,𝒞∈𝒰_𝑓, one has the following: (i)if 𝒜<ℬ, then 𝒜⊕𝒞<ℬ⊕𝒞;(ii)𝒜<ℬ if and only if −𝒜>−ℬ.

Proof. First, we will show that (𝒰_𝑓,⊕) is an abelian group. Let 𝒜=𝜌_𝑓(𝐴),ℬ=𝜌_𝑓(𝐵),𝒞=𝜌_𝑓(𝐶)∈𝒰_𝑓 for some 𝐴,𝐵,𝐶∈ℝ. Using (51), we obtain the following: (1)𝒜⊕ℬ=𝜌_𝑓(𝐴)⊕𝜌_𝑓(𝐵)=𝜌_𝑓(𝐴+𝐵)∈𝒰_𝑓; (2)𝒜⊕(ℬ⊕𝒞)=𝜌_𝑓(𝐴+(𝐵+𝐶))=𝜌_𝑓((𝐴+𝐵)+𝐶)=(𝒜⊕ℬ)⊕𝒞;(3)let 𝒪_𝑓∶=𝜌_𝑓(0)∈𝒰_𝑓; then 𝒜⊕𝒪_𝑓=𝜌_𝑓(𝐴)⊕𝜌_𝑓(0)=𝜌_𝑓(𝐴+0)=𝒜 for all 𝒜∈𝒰_𝑓;(4)𝒜⊕(−𝒜)=𝜌_𝑓(𝐴)⊕𝜌_𝑓(−𝐴)=𝜌_𝑓(0)=𝒪_𝑓 for all 𝒜∈𝒰_𝑓; and(5)𝒜⊕ℬ=𝜌_𝑓(𝐴)⊕𝜌_𝑓(𝐵)=𝜌_𝑓(𝐴+𝐵)=𝜌_𝑓(𝐵+𝐴)=𝜌_𝑓(𝐵)⊕𝜌_𝑓(𝐴)=ℬ⊕𝒜.From (1)–(5), we conclude that (𝒰_𝑓,⊕) is an abelian group.
Lastly, assume that 𝒜<ℬ. Since 𝜌_𝑓⁻¹ is order-preserving, we have 𝐴<𝐵. Using (51), we have 𝒜⊕𝒞=𝜌_𝑓(𝐴+𝐶)<𝜌_𝑓(𝐵+𝐶)=ℬ⊕𝒞,−𝒜=−𝜌_𝑓(𝐴)=𝜌_𝑓(−𝐴)>𝜌_𝑓(−𝐵)=−𝜌_𝑓(𝐵)=−ℬ. (52) Hence (ii) follows since −(−𝒜)=−(−𝜌_𝑓(𝐴))=−(𝜌_𝑓(−𝐴))=𝜌_𝑓(𝐴)=𝒜.

Next, we define the binary operation ⊙ on 𝒰_𝑓 as follows: for any 𝒜,ℬ∈𝒰_𝑓, let 𝜌𝒜⊙ℬ=^𝑓sup𝐴_(𝑛)⋅𝐵_(𝑛)if𝒜>𝒪_𝑓,ℬ>𝒪_𝑓,−(𝒜⊙(−ℬ))if𝒜>𝒪_𝑓,ℬ<𝒪_𝑓,−((−𝒜)⊙ℬ)if𝒜<𝒪_𝑓,ℬ>𝒪_𝑓,(−𝒜)⊙(−ℬ)if𝒜<𝒪_𝑓,ℬ<𝒪_𝑓,𝒪_𝑓if𝒜=𝒪_𝑓orℬ=𝒪_𝑓. (53) Also define 𝒜⁻¹=𝜌_𝑓sup𝐴^(𝑛)−1if𝒜>𝒪_𝑓−(−𝒜)⁻¹if𝒜<𝒪_𝑓. (54) To see that these definitions are unambiguous, first we note that 𝐴_(𝑛)⋅𝐵_(𝑛)≤𝐴⋅𝐵,𝐴^(𝑛)−1≤𝐴⁻¹ (55) for all sufficiently large 𝑛, where 𝒜,ℬ>𝒪_𝑓, 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<𝒪_𝑓 if and only if −𝒜>𝒪_𝑓. Using (53), we deduce that 𝜌_𝑓(𝐴⋅𝐵)=𝜌_𝑓(𝐴)⊙𝜌_𝑓(𝐵) (56) for all real numbers 𝐴 and 𝐵. In addition, using (54), we have 𝜌_𝑓(𝐴)⁻¹=𝜌_𝑓𝐴⁻¹ (57) for all nonzero real number 𝐴.

Theorem 14. (𝒰_𝑓,⊕,⊙) is an ordered field. Furthermore, if 𝒜<ℬ and 𝒞>𝒪_𝑓, then 𝒜⊙𝒞<ℬ⊙𝒞.

Proof. By Theorem 13, we know that (𝒰_𝑓,⊕) is an abelian group. To prove that (𝒰_𝑓,⊕,⊙) is a field, it remains to show that (𝒰_𝑓⧵{𝒪_𝑓},⊙) is an abelian group. Let 𝒜=𝜌_𝑓(𝐴),ℬ=𝜌_𝑓(𝐵),𝒞=𝜌_𝑓(𝐶)∈𝒰_𝑓 for some 𝐴,𝐵,𝐶∈ℝ. Using (56) and (57), we obtain the following: (1)𝒜⊙ℬ=𝜌_𝑓(𝐴)⊙𝜌_𝑓(𝐵)=𝜌_𝑓(𝐴⋅𝐵)∈𝒰_𝑓; (2)𝒜⊙(ℬ⊙𝒞)=𝜌_𝑓(𝐴⋅(𝐵⋅𝐶))=𝜌_𝑓((𝐴⋅𝐵)⋅𝐶)=(𝒜⊙ℬ)⊙𝒞;(3)let ℐ∶=𝜌_𝑓(1)∈𝒰_𝑓; then 𝒜⊙ℐ=𝜌_𝑓(𝐴⋅1)=𝒜 for all 𝒜=𝜌_𝑓(𝐴)∈𝒰_𝑓;(4)𝒜⊙𝒜⁻¹=𝜌_𝑓(𝐴)⊙𝜌_𝑓(𝐴⁻¹)=𝜌_𝑓(1)=ℐ for all 𝒜=𝜌_𝑓(𝐴)∈𝒰_𝑓; and(5)𝒜⊙ℬ=𝜌_𝑓(𝐴⋅𝐵)=𝜌_𝑓(𝐵⋅𝐴)=ℬ⊙𝒜.From (1)–(5), we conclude that (𝒰_𝑓⧵{𝒪_𝑓},⊙) is an abelian group; thus (𝒰_𝑓,⊕,⊙) is a field.
Next, we will show that (𝒰_𝑓,⊕,⊙) is an ordered field (see [6]). Let 𝒫_𝑓=𝒜∈𝒰_𝑓∣𝒜>𝒪_𝑓 (58) and 𝒜,ℬ∈𝒫_𝑓. Then 𝒜=𝜌_𝑓(𝐴) and ℬ=𝜌_𝑓(𝐵) for some positive real numbers 𝐴 and 𝐵. (1)Since 𝐴+𝐵>0, we have 𝒜⊕ℬ=𝜌_𝑓(𝐴+𝐵)>𝜌_𝑓(0)=𝒪_𝑓, and so 𝒜⊕ℬ∈𝒫_𝑓.(2)Since 𝐴⋅𝐵>0, we have 𝒜⊙ℬ=𝜌_𝑓(𝐴⋅𝐵)>𝜌_𝑓(0)=𝒪_𝑓, and so 𝒜⊙ℬ∈𝒫_𝑓.(3)Since 𝐴>0, we have −𝐴<0 and −𝒜=−𝜌_𝑓(𝐴)=𝜌_𝑓(−𝐴)<𝜌_𝑓(0)=𝒪_𝑓. Hence −𝒜∉𝒫_𝑓.(4)For 𝒜=𝜌_𝑓(𝐴)∈𝒰_𝑓, we have 𝐴=0 or 𝐴>0 or −𝐴>0. It follows that 𝒜=𝒪_𝑓, 𝒜>𝒪_𝑓, or −𝒜>𝒪_𝑓. Hence 𝒜=𝒪_𝑓, 𝒜∈𝒫_𝑓, or −𝒜∈𝒫_𝑓.
Finally, assume that 𝒜<ℬ and 𝒞>𝒪_𝑓. Since 𝜌_𝑓⁻¹ is order-preserving, then 𝐴<𝐵 and 𝐶>0, and so 𝐴⋅𝐶<𝐵⋅𝐶. Thus 𝒜⊙𝒞=𝜌_𝑓(𝐴⋅𝐶)<𝜌_𝑓(𝐵⋅𝐶)=ℬ⊙𝒞 (59) since 𝜌_𝑓 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 (𝒰_𝑓,⊕,⊙), 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. Let 𝑓_𝑆∶[2,∞)→ℝ be defined by 𝑓_𝑆(𝑥)=𝑥(𝑥−1)+1, (60) for all 𝑥≥2, 𝒰_𝑓𝑆=𝑎₀,𝑎₁,𝑎₂,…∣𝑎_𝑛∈ℤ,𝑎₁𝑎≥2,_𝑛+1≥𝑎_𝑛𝑎_𝑛−1+1∀𝑛≥1 (61) and 𝜌_𝑓𝑆∶ℝ→𝒰_𝑓𝑆 defined by 𝜌_𝑓𝑆(𝐴)=𝑎₀,𝑎₁,𝑎₂,…, (62) for all 𝐴∈ℝ, where 𝐴=𝑎₀+^∞_𝑛=11𝑎_𝑛 (63) is the Sylvester series expansion of 𝐴. Then (𝒰_𝑓𝑆,⊕,⊙) is an ordered field containing (𝜌_𝑓𝑆(ℚ),⊕,⊙) as a dense subfield. Furthermore, for 𝒜,ℬ,𝒞∈𝒰_𝑓𝑆, one has the following: (i)if 𝒜<ℬ, then 𝒜⊕𝒞<ℬ⊕𝒞;(ii)𝒜<ℬ if and only if −𝒜>−ℬ; (iii)if 𝒜<ℬ and 𝒞>𝒪_𝑓𝑆, then 𝒜⊙𝒞<ℬ⊙𝒞, where 𝒪_𝑓𝑆 is the zero element in 𝒰_𝑓𝑆.

Proof. It is clear that 𝑓_𝑆 is a nondecreasing function such that 𝑓(𝑥)≥2 for all 𝑥≥2. It is an immediate consequence of Corollary 3 that 𝜌_𝑓𝑆 is 1-1 map from ℝ onto 𝒰_𝑓𝑆. By Corollary 7 and the definition of order in 𝒰_𝑓𝑆, this map is order-preserving. Using Theorems 13 and 14, we can conclude that (𝒰_𝑓𝑆,⊕,⊙) is an ordered field containing 𝜌_𝑓𝑆(ℚ) and satisfies properties (i), (ii), and (iii). To complete the proof of this proposition, it remains to show that 𝜌_𝑓𝑆(ℚ) is a dense subfield of 𝒰_𝑓𝑆.
It is clear that 𝜌_𝑓𝑆(ℚ) is a subfield of 𝒰_𝑓𝑆. Now, let 𝒜,ℬ∈𝒰_𝑓𝑆 be such that 𝒜<ℬ. If 𝑎₀<𝑏₀, let 𝒞=𝑏₀,𝑏^′₁,̇𝑓_𝑆𝑏^′₁; (64) then it is easy to see that 𝒜<𝒞<ℬ and 𝒞∈𝜌_𝑓𝑆(ℚ). On the other hand, if 𝑎₀=𝑏₀, then 𝑎_𝑚>𝑏_𝑚 for the first 𝑚 such that 𝑎_𝑚≠𝑏_𝑚. In that case, we have 𝒜<𝒟<ℬ, where 𝒟=𝑏₀,…,𝑏_𝑚,𝑏^′_𝑚+1,̇𝑓_𝑆𝑏^′_𝑚+1. (65) Then 𝒟∈𝜌_𝑓𝑆(ℚ). This shows that 𝜌_𝑓𝑆(ℚ) is dense in 𝒰_𝑓𝑆 as desired.

Proposition 16. Let 𝑓_𝐸∶[2,∞)→ℝ be defined by 𝑓_𝐸(𝑥)=𝑥, (66) for all 𝑥≥2, 𝒰_𝑓𝐸=𝑎₀,𝑎₁,𝑎₂,…∣𝑎_𝑛∈ℤ,𝑎₁𝑎≥2,_𝑛+1≥𝑎_𝑛∀𝑛≥1 (67) and 𝜌_𝑓𝐸∶ℝ→𝒰_𝑓𝐸 defined by 𝜌_𝑓𝐸(𝐴)=𝑎₀,𝑎₁,𝑎₂,…, (68) for all 𝐴∈ℝ, where 𝐴=𝑎₀+^∞_𝑛=11𝑎₁𝑎₂⋯𝑎_𝑛 (69) is the Engel series expansion of 𝐴. Then (𝒰_𝑓𝐸,⊕,⊙) is an ordered field containing (𝜌_𝑓𝐸(ℚ),⊕,⊙) as a dense subfield. Furthermore, for 𝒜,ℬ,𝒞∈𝒰_𝑓𝐸, one has the following: (i)if 𝒜<ℬ, then 𝒜⊕𝒞<ℬ⊕𝒞;(ii)𝒜<ℬ if and only if −𝒜>−ℬ; (iii)if 𝒜<ℬ and 𝒞>𝒪_𝑓𝐸, then 𝒜⊙𝒞<ℬ⊙𝒞, where 𝒪_𝑓𝐸 is the zero element in 𝒰_𝑓𝐸.

Proof. It is clear that 𝑓_𝐸 is a nondecreasing function such that 𝑓(𝑥)≥2 for all 𝑥≥2. It is an immediate consequence of Corollary 4 that 𝜌_𝑓𝐸 is 1-1 map from ℝ onto 𝒰_𝑓𝐸. By Corollary 8 and the definition of order in 𝒰_𝑓𝐸, this map is order-preserving. Using Theorems 13 and 14, we can conclude that (𝒰_𝑓𝐸,⊕,⊙) is an ordered field containing 𝜌_𝑓𝐸(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 𝜌_𝑓𝐸(ℚ) is a dense subfield of 𝒰_𝑓𝐸.

Proposition 17. Let 𝑓_𝐿∶[2,∞)→ℝ be defined by 𝑓_𝐿(𝑥)=2, (70) for all 𝑥≥2, 𝒰_𝑓𝐿=𝑎₀,𝑎₁,𝑎₂,…∣𝑎_𝑛∈ℤ,𝑎_𝑛≥2∀𝑛≥1 (71) and 𝜌_𝑓𝐿∶ℝ→𝒰_𝑓𝐿 defined by 𝜌_𝑓𝐿(𝐴)=𝑎₀,𝑎₁,𝑎₂,…, (72)
for all 𝐴∈ℝ, where 𝐴=𝑎₀+1𝑎₁+^∞_𝑛=11𝑎₁𝑎₁−1⋯𝑎_𝑛𝑎_𝑛−1𝑎_𝑛+1 (73) is the Lüroth series expansion of 𝐴. Then (𝒰_𝑓𝐿,⊕,⊙) is an ordered field containing (𝜌_𝑓𝐿(ℚ),⊕,⊙) as a dense subfield. Furthermore, for 𝒜,ℬ,𝒞∈𝒰_𝑓𝐿, one has the following: (i)if 𝒜<ℬ, then 𝒜⊕𝒞<ℬ⊕𝒞;(ii)𝒜<ℬ if and only if −𝒜>−ℬ; (iii)if 𝒜<ℬ and 𝒞>𝒪_𝑓𝐿, then 𝒜⊙𝒞<ℬ⊙𝒞, where 𝒪_𝑓𝐿 is the zero element in 𝒰_𝑓𝐿.

Proof. It is clear that 𝑓_𝐿 is a nondecreasing function such that 𝑓(𝑥)≥2 for all 𝑥≥2. It is an immediate consequence of Corollary 5 that 𝜌_𝑓𝐿 is 1-1 map from ℝ onto 𝒰_𝑓𝐿. By Corollary 9 and the definition of order in 𝒰_𝑓𝐿, this map is order-preserving. Using Theorems 13 and 14, we conclude that (𝒰_𝑓𝐿,⊕,⊙) is an ordered field containing 𝜌_𝑓𝐿(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 𝜌_𝑓𝐿(ℚ) is a dense subfield of 𝒰_𝑓𝐿.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, and the Research and Academic Affairs Promotion Fund, Faculty of Science, Khon Kaen University, Fiscal year 2014 (RAAPF), Thailand.