A new construction of the real numbers by alternating series
Abstract.
We put forward a new method of constructing the complete ordered field of real numbers from the ordered field of rational numbers. Our method is a generalization of that of A. Knopfmacher and J. Knopfmacher. Our result implies that there exist infinitely many ways of constructing the complete ordered field of real numbers. As an application of our results, we prove the irrationality of certain numbers.
Key words and phrases:
alternating series, real number, Sylvester series, irrationality2010 Mathematics Subject Classification:
11U99, 11J721. Introduction
The purpose of this paper is to put forward a new method of constructing the complete ordered field of real numbers from the ordered field of rational numbers. Our method is similar to the method which was put forward by A. Knopfmacher and J. Knopfmacher in [5], but our method is more general. Moreover our result gives infinitely many ways of constructing the complete ordered field of real numbers. As an application of our results, we prove the irrationality of certain series.
A. Knopfmacher and J. Knopfmacher constructed the complete ordered field of real numbers by the Sylvester expansion and the Engel expansion in [4] and by the alternatingSylvester expansion and the alternatingEngel expansion in [5]. The advantages of these constructions are the fact that those are concrete and do not depend on the notion of equivalence classes. The alternatingSylvester expansion and the alternatingEngel expansion are generalizations of Oppenheim’s expansion (see [6]) and special cases of the alternating BalkemaOppenheim’s expansion (see [2]), which were introduced by A. Knopfmacher and J. Knopfmacher in [5]. The definition of the alternatingSylvester expansion and the alternatingEngel expansion are the following.
(i) AlternatingSylvester expansion
Let , and . We define, for and ,
and
Then
(1.1) 
where and for .
(ii) AlternatingEngel expansion
Let , and . We define, for and ,
and
Then
(1.2) 
where and for .
The relation
(1.3) 
is used in these expansions. We introduce a new series expansion for every real numbers by using a more general relation
Definition 1.1 (Generalized alternatingSylvester expansion).
Let , and . Let be a sequence of positive integers. We define, for ,
and
Then
(1.4) 
If we regard the alternatingSylvester series (1.1) as an analogue of the simple continued fraction
the generalized alternatingSylvester series (1.4) is an analogue of the continued fraction
Therefore we can expect that if we take some appropriate , then we can get a simple series representation for some real numbers.
The outline of this paper is the following. In Section 2 we study some fundamental properties of the generalized alternatingSylvester series. In Section 3 we take an arbitrary sequence of positive integer such that for all , and we prove that the set
(1.5) 
can be identified with the complete ordered field of real numbers by introducing the relation and the operator and . In other words we prove that becomes an ordered field which is isomorphic to . Since there exist infinitely many such that , this implies that there exist infinitely many ways of constructing the complete ordered field of real numbers. Our construction is similar to that in [5]. Therefore our construction is also concrete and does not use the notion of equivalence classes. When we prove that becomes an ordered field, we use a general lemma (see Lemma 3.4). It seems that this lemma can be used in [3], [4] and [5]. In section 4, we prove the irrationality of certain series by Proposition 2.3 and Proposition 3.1.
Remark 1.1.
It seems that we can define generalized alternatingEngel series as follows :
Let , with , . Let be a sequence of positive integers. We define, for and ,
and
Then
However, does not hold in this series. For example, if we set , and , then , and . This is a trouble. In order to simplify the argument we do not argue on this series.
2. Fundamental properties of the generalized alternating Sylvester series
In this section, we take an arbitrary sequence of positive integers and fix it.
Proposition 2.1.
The generalized alternatingSylvester series has the following properties for .

If , then we have

If , then we have

The evaluation holds. If , then we have .

The evaluation holds.

If , then we have .

If , then we have .

The evaluation holds. If , then we have .
Proof.
(1) This trivially follows from the definition of the generalized alternatingSylvester espansion.
(2) From (1) and the definition, we have
(3) In the case , we have . For , we have
(4) By (3), we have for all . Hence
holds. This implies (4).
(5) From (2), we have (5) by using (4).
(6) By (4), we have
(7) In the case , we have . For , we have
by (2) and (4). ∎
Remark 2.1.
In order to prove Proposition 2.2 we require some lemmas.
We can easily see that the following lemma holds.
Lemma 2.1.
Let and . Then

there does not exist such that

is equivalent to
Lemma 2.2.
Let , , and . If then is equivalent to .
Proof.
First, we assume . Since and hold by Lemma 2.1 (2), it is sufficient that we consider the following cases.

.

.

.
Proposition 2.2.
Let with . We define , and as , and which appear in the generalized alternating Sylvester expansion of , respectively. Let
Then is equivalent to
Proof.
First, we consider the case . If , then we have . Therefore we obtain . On the other hand, if , then we have . Therefore we obtain .
In order to consider the case we prove the next lemma.
Lemma 2.3.
Let and with . Let . Then the numerator of is less than . In other words, .
Proof.
We have
where . ∎
Proposition 2.3.
The real number is rational if and only if there exists an such that .
Proof.
If there exists an such that , then is rational. We assume , where and . Without loss of generality, we may assume that , and . By the definition of and Lemma 2.3, the numerator of is strictly monotonically decreasing. This implies the proposition. ∎
3. construction of the real numbers
In this section we take an arbitrary sequence of positive integers which satisfies the condition for all and fix it. Moreover we identify with .
Remark 3.1.
On the condition for any , the inequality in Proposition 2.1 (2) becomes
If the equality holds in the above and , then we have
This contradicts the definition of , hence or
holds.
In Section 1, we assumed the existence of the real numbers, and we defined in (1.5). In order to use for the construction of the real numbers, here we remove that assumption.
Definition 3.1.
Let be a sequence of positive integers. We define if and only if
(3.1) 
holds for all .
Let be a sequence of rational numbers. We define if and only if

,

for all ,

if , then ,

if for , then for all ,

there exists a such that for all if , and

if , then or
holds.
We can easily see that the following lemma holds.
Lemma 3.1.
Let and .

.

.

. If , then .

The series
converges.
Proposition 3.1.
.
Proof.
trivially follows by Proposition 2.1 and Remark 3.1. In order to prove , we take and assume that and or for all . Since we can set
by Lemma 3.1 (4), we have
by the generalized alternatingSylvester expansion. It is sufficient to prove that for all . Since the case is trivial, we may assume . By considering , we have and . If , then by Lemma 2.1 (2). If , then we have by (3.1) and Definition 3.1 (5). However, is impossible because of Definition 3.1 (6). Thus we obtain . This implies by Lemma 2.1 (2).
In the rest of this section, we set for simplicity, and we introduce a relation and operators , for .
First we define the binary relation on .
Definition 3.2.
Let with and
We define if and only if
Proposition 3.2.
For any , we have

does not hold (irreflexive law),

or or (trichotomy),

if and then (transitive law).
In other words, is a linear order in the strict sense on .
Proof.
We can easily see that (1) and (2) hold. In order to prove (3), we define
and . Then
and
hold. If is odd, then we have
Therefore we obtain . The other cases can be proved by the same argument. ∎
If we define
we can identify with by Proposition 2.2 and 2.3. In short, the map
is an orderisomorphism. Hence we may regard as .
Theorem 3.1.
Let be a nonempty subset of . If is bounded from above (below), then there exists a supremum (an infimum).
Proof.
Since is bounded from above, there exists a such that
If there does not exist an upper bound for such that , then is a supremum for . We assume that there exists an upper bound for such that . Since there exists a such that , we can define
from the definition of and . By the same argument, we can define
In general, if we have defined for , then we define
By the definition of and , is the supremum for . We can prove this as follows. If is not an upper bound for , then there exists a such that . By setting , we have for odd or for even . This contradicts the definition of . On the other hand, if is not minimum upper bound for , then there exists an upper bound for such that . We set . By the definition of , there exists an such that for . Then we have . This is impossible.
The case of the infimum can be proved by the same argument. ∎
In order to introduce the algebraic structure for , we require some preparations.
Definition 3.3.
Let be a sequence of rational numbers. We define if and only if, for all , there exists an such that holds for all .
Note that in the usual sense means .
The following definition and lemma are the same as in [5].
Definition 3.4.
Let with . We define
where .
We can easily see that the next lemma holds.
Lemma 3.2.
Let with . Then we have

,

,

.
In order to prove Lemma 3.4, we also require the next lemma.
Lemma 3.3.
Let be a monotonically increasing sequence of rational numbers which is bounded from above. Let . Then we have .
Proof.
By contradiction. Assume that there exists an such that
holds. Since we have by the assumption of the lemma, we have for all . On the other hand, by Lemma 3.2, there exists an such that
holds for all . Hence we have
for . This implies that is an upper bound for . Therefore we obtain
This contradicts the definition of . ∎
The following lemma is important in the proofs of algebraic properties of . It seems that this lemma can be used in the work of A. Knopfmacher and J. Knopfmacher [3], [4], [5].
Lemma 3.4.
Let be monotonically increasing sequence of rational numbers which are bounded from above. Then is equivalent to .
Proof.
Next we assume . By contradiction. Assume that . Without loss of generality, we may assume . We set . Then there exists an such that holds for all . Since for , we have
for . This contradicts . ∎
Now we define the operators on , and prove that is an ordered field. (These definitions are the same as in [5].)
Definition 3.5.
Let . We define the following symbol and operators.

.

.

.
Definition 3.6.
Let . We define the following symbol and operators.

.


Since , , and , these definitions are possible.
Now we prove that (resp. ) shares the same properties with the usual addition (resp. multiplication).
Proposition 3.3.
Let . We have

,

,

,

,

if , then .