A Constructive Realization of the Rational, Real, and Complex Numbers via Sequences over Finite Fields of Increasing Order

Abstract

Based on the representation of finite field elements by fractions, an algorithmic procedure is introduced to define a field ${\mathbb{Q}}_{\left(p\right)}=\left(Q_p,+,\cdot \right)$ of discrete rational numbers. The set $Q_p$ consists of fractions $m/n$ with $m$ and $n\in \mathbb{Z}$ such that $\left|m\right|<\sqrt{p}$ and $0<n<\sqrt{p}$, where $p$ is a prime number. An isomorphism with the Galois field ${\mathbb{F}}_p$ is established. Sequences $\left(q_n\right)$ such that $n\in \mathbb{N}$, $q_n\in Q_{p\left(n\right)}$, and $p\left(n\right)$ denotes the increasing sequence of primes, which are constant or convergent under suitable definitions, are used to construct fields isomorphic to $\mathbb{Q}$ and $\mathbb{R}$, respectively. Suitable addition and multiplication operations are defined on $Q_{p^{\prime }}^2$ with $p^{\prime }$, a non-Pythagorean prime, such that ${\mathbb{C}}_{p^{\prime }}=\left(Q_{p^{\prime }}^2+,\cdot \right)$ is a field. Sequences $\left(z_n\right)$ such that $n\in \mathbb{N}$ and $z_n\in Q_{p^{\prime }\left(n\right)}^2$, together with a suitably defined convergence criterion, are then used to construct a field isomorphic to $\mathbb{C}$.

1. Introduction

In computational algebra, rational reconstruction is used to recover exact rational values after performing computations in modular arithmetic. The method exploits the fact that when attention is restricted to rational numbers with bounded numerators and denominators, sufficiently, many modular images combined uniquely determine each rational number [1] (Chapter 5, Section 10) [2]. Further developments and robustness aspects of rational reconstruction have been investigated in the modern literature (see, for instance, [3,4,5]). The absolute Galois group of $\mathbb{Q}$ is a profinite group, realized as the inverse limit of the Galois groups of finite Galois extensions of $\mathbb{Q}$, and provides a natural link between arithmetic over $\mathbb{Q}$ and finite algebraic data (see [6] (Chapter I, Sections 1–3, Chapter II, Section 1.1), [7] (Chapters I, VI, and VIII), and [8] (Chapters 1–2); see also, for example, the recent literature on structural and cohomological aspects of absolute Galois groups [9,10]). This perspective extends to the adelic setting: $\mathbb{Q}$ embeds diagonally into the adele ring ${\mathbb{A}}_{\mathbb{Q}}$. For each prime $p,$the corresponding integral structure, is given by the p-adic integers ${\mathbb{Z}}_p$, which arise as the inverse limits of finite rings $\mathbb{Z}/p^n\mathbb{Z}$ [11] (Chapters II and IV) (see also, for instance, Ref. [12] for the role of adeles in the modern theory of Galois representations). In Robinson’s nonstandard analysis, every finite hyperreal has a standard part in $\mathbb{R}$, i.e., it is infinitesimally close to a unique real number [13] (Chapter I) (see also the survey on infinitesimals and modern approaches to nonstandard analysis [14]). Taken together, these constructions illustrate an interplay between rational and real numbers and finite algebraic structures. This observation motivates the constructive realization of rational, real, and complex numbers, introduced in this work, based on sequences of elements over finite fields of increasing order.

Background on finite fields can be found in [15], Chapters 1–2. Compared with previous approaches, the present framework provides an explicit link between finite-cardinality number sets and the classical fields $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$. The starting point of the constructive approach is the observation that the elements of Galois fields ${\mathbb{F}}_p$, where $p$ is a prime number, can be interpreted in terms of fractions $m/n$ with $m,n\in \mathbb{Z},$ such that $\left|m\right|<p$ and $0<n<p$, defining a set ${\stackrel{-}{Q}}_p$. The basis of such interpretation is provided by the field property stating that any linear equation with a non-zero coefficient admits a unique solution. Therefore, there exists a unique element $x$ of ${\mathbb{F}}_p$ such that

$$n·x-m=0 \left(mod p\right).$$

(1)

The following definitions are then introduced:

Definition 1.

A pair $\left(m,n\right)\in {\stackrel{-}{Q}}_p$ is called a fractional representation of an element $x\in {\mathbb{F}}_p$ if Equation (1) holds. It is also denoted as $m/n$.

Definition 2.

Two fractional representations, $\left(a,b\right) \mathit{and} \left(c,d\right),$ of the same element of ${\mathbb{F}}_p$ are said to be p-equivalent. The notation $\left(a,b\right) {~}_p \left(c,d\right)$ is used to indicate this relation.

Definition 3.

The function

$${\stackrel{-}{\mathsf{\Psi }}}_p:\left(m,n\right)\in {\stackrel{-}{Q}}_p\to x\in {\mathbb{F}}_p :n·x-m=0 \left(mod p\right)$$

 is said to be a complete representation of ${\mathbb{F}}_p$.

The function ${\stackrel{-}{\mathsf{\Psi }}}_p$ is obviously surjective, since ${\mathbb{F}}_p$ is the image of the set of pairs $\left(m,1\right)$, with $0\le m<p$, and is not injective for any $p>2$ since the finite set ${\stackrel{-}{Q}}_p$ has more elements than the finite set ${\mathbb{F}}_p$. As an illustrative example, the function ${\stackrel{-}{\mathsf{\Psi }}}_3$ is reported in Table 1.

Table 1. The values of the function ${\stackrel{-}{\mathsf{\Psi }}}_3(m,n)$ are indicated in the cells identified by the column with the label $m$ and by the row with the label $n$

	m = −2	m = −1	m = 0	m = 1	m = 2
n = 1	1	2	0	1	2
n = 2	2	1	0	2	1

The following theorem provides a criterion for determining equivalent representations of elements of ${\mathbb{F}}_p$:

Theorem 1.

$\left(a,b\right) {~}_p \left(c,d\right)$ holds if and only if

$$a·d-b·c=0 \left(mod p\right).$$

(2)

Theorem 2.

The relation ${~}_p$ is an equivalence relation in ${\stackrel{-}{Q}}_p$.

The proofs of these theorems are straightforward.

Following the study by Louboutin and Murchio [16], a minimal representation of ${\mathbb{F}}_p$ is obtained by restricting the complete representation to a suitable subset of ${\stackrel{-}{Q}}_p$. In Section 2, a specific subset $Q_p$ of this kind is constructed by means of an induction method and an algorithmic procedure.

By introducing on $Q_p$ suitable binary operations of addition and multiplication, denoted by $+$ and $\cdot$, it is shown that the restriction to $Q_p$ of ${\stackrel{-}{Q}}_p$ yields an isomorphism between ${\mathbb{Q}}_{\left(p\right)}=\left(Q_p,+, \cdot \right)$ and ${\mathbb{F}}_p$. The field ${\mathbb{Q}}_{\left(p\right)}$ is referred to as the field of discrete rational numbers of order $p$. The notation ${\mathbb{Q}}_{\left(p\right)}$ is adopted to avoid confusion with the field of p-adic numbers ${\mathbb{Q}}_p$.

Owing to the isomorphism with ${\mathbb{F}}_p$, the field ${\mathbb{Q}}_{\left(p\right)}$ does not constitute a new algebraic structure but rather a new representation of ${\mathbb{F}}_p$. The algorithmic construction ensures that the sequence $Q_{p\left(n\right)}$ is increasing in the sense that for any positive integer $n$, $Q_{p\left(n\right)}\subset Q_{p(n+1)}$, where $p\left(n\right)$ is the increasing sequence of prime numbers. In Section 3, the sequences of discrete rational numbers $\left(q_n\right)$, with $q_n\in Q_{p\left(n\right)}$, which are constant after a certain index, are introduced as the basic objects for the construction of rational numbers. An equivalence relation ${~}_K$ is defined on the set K of such sequences. Suitable addition $(+)$ and multiplication $(\cdot )$ operations are defined in $Q_G=K/{~}_K$, such that ${\mathbb{Q}}_G=(Q_G,+,·)$ is a field isomorphic to the classical field $\mathbb{Q}$ [17] (Chapter I). In Section 4, a set $H\supseteq K$ of convergent sequences of discrete rational numbers is introduced by adopting a suitable definition of convergence. These sequences are the basic objects used to construct the real field $\mathbb{R}$ [17] (Chapter V). An equivalence relation ${~}_H$ is defined in $H$ and suitable addition $(+)$ and multiplication $(\cdot )$ operations are defined in $R_G=H/{~}_H$, such that ${\mathbb{R}}_G=\left(R_G,+, \cdot \right)$ is a field isomorphic to $\mathbb{R}$. In Section 5, suitable operations of addition and multiplication are defined in$Q_{p^{\prime }}^2,$where $p^{\prime }$ is a non-Pythagorean prime, in order to obtain finite fields of discrete complex numbers ${\mathbb{C}}_{p\left(n\right)}=\left(Q_{p^{\prime }\left(n\right)}^2,+, \cdot \right)$. Sequences of elements of $Q_{p^{\prime }\left(n\right)}^2$, where $p^{\prime }\left(n\right)$ is the increasing sequence of non-Pythagorean primes, together with an appropriate convergence criterion, are used to define a field (${\mathbb{C}}_G$) isomorphic to the complex field $\mathbb{C}$ [17] (Chapter VII). Suitable distance functions are defined in ${\mathbb{Q}}_G$ and ${\mathbb{R}}_G,$such that the corresponding isomorphisms with $\mathbb{Q}$ and $\mathbb{R}$ are also isometries, with respect to the standard distance $d\left(x,y\right)=\left|x-y\right|$ [18] (Chapter 2, Section 2.1). The isomorphism between ${\mathbb{C}}_G$ and $\mathbb{C}$ preserves the Euclidean distance. A summary and discussion of the results, together with the possible future developments, are presented in Section 6.

2. The Field ${\mathbb{Q}}_{\mathit{p}}$: Algorithmic Procedure

Specific bijective functions ${\mathsf{\Psi }}_p$are introduced here, as obtained from ${\stackrel{-}{\mathsf{\Psi }}}_p$ by restriction of its domain to a suitable subset $Q_p\subseteq$ ${\stackrel{-}{Q}}_p$, which is defined by the following induction and algorithmic procedure. As a prerequisite, a bijective function ${\mathsf{\Psi }}_3$ is defined in the Table 2.

Table 2. The bijective function ${\mathsf{\Psi }}_3$ defined in $Q\left(3\right)=\left\{\left(-\mathrm{1,1}\right),\left(\mathrm{0,1}\right),\left(\mathrm{1,1}\right)\right\}\subseteq$ $\stackrel{-}{Q}\left(3\right).$

(m, n)	(−1,1)	(0,1)	(1,1)
${\mathsf{\Psi }}_3$	2	0	1

Using an inductive construction, it is assumed that a bijective function ${\mathsf{\Psi }}_p$ is defined by the restriction of the domain ${\stackrel{-}{Q}}_p$ of ${\stackrel{-}{\mathsf{\Psi }}}_p$ to a subset $Q_p\subseteq$ ${\stackrel{-}{Q}}_p$, such that if $\left(m,n\right)\in Q_p$, then also $\left(-m,n\right)\in Q_p$. This property is fulfilled by the function ${\mathsf{\Psi }}_3$ defined above. The following relation on elements of the subset $Q_p$ is defined.

Definition 4.

The relation $\left(a,b\right){\ge }_p\left(c,d\right)$ between two elements $\left(a,b\right) and (c,d)$ of the set $Q_p$ is defined by the condition that the following ordering occurs in $\mathbb{Z}$:

$$a·d\ge b·c.$$

(3)

It can be shown that ${\ge }_p$ defines a total order relation. However, since $Q_p$ is finite, the order is not compatible with the operations of addition and multiplication. Algorithm 1 is used to construct $Q_s$, where $s$ is the next prime greater than the prime $p$.

Algorithm 1

Step 1. Set $Q_c=Q_p$. Step 2. Iterate a loop in the ascending order (within the relation ${\ge }_p)$ among the couples $\left(a,b\right)\in Q_c$, with $a>0$, and remove both $\left(a,b\right)$ and $\left(-a,b\right)$ from $Q_c$ if any of them is s-equivalent to any couple $\left(c,d\right)$ or $\left(-c,d\right)$ such that $c>0$, $\left(a,b\right){\ge }_p\left(c,d\right)$, and $\left(a,b\right)\ne \left(c,d\right)$. Step 3. Find the pair $\left(n_x,1\right)\in Q_c$ with the largest value of $n_x$. Step 4. Perform a double loop in ascending order of n, such that $0<n\le n_x$, and in ascending order of m, such that $0<m<n$, and check if the couples $\left(m,n\right)$ and $\left(-m,n\right)$ are not s-equivalent to any element of $Q_c.$ In that case, add them to the set $Q_c.$ If $Q_c$ contains s elements, then set $Q_s=Q_c$ and stop. Otherwise, continue the iteration. Step 5. Perform a double loop in ascending order of m, such that $0<m\le n_x$ and in ascending order of n, such that $0<n<m$, and check if the couples $\left(m,n\right)$ and $\left(-m,n\right)$ are not s-equivalent to any element of $Q_c.$ In that case, add them to the set $Q_c.$ If $Q_c$ contains s elements, then set $Q_s=Q_c$ and stop. Otherwise, continue the iteration. Step 6. Add to $Q_c$the elements $\left(n_x^{\prime },1\right)$ and $\left(-n_x^{\prime },1\right)$, where $n_x^{\prime }=n_x+1.$ If $Q_c$ has s elements, then set $Q_s=Q_c$ and stop. Otherwise, set $n_x=$$n_x^{\prime }$ and continue from Step 4.

The inductive construction therefore allows one to determine $Q_r$ for any prime number $r$; $Q\left(3\right)$ has been defined and, given $Q_p$ for any prime number $p$, Algorithm 1 allows for defining $Q_s$ for the next prime $s$ greater than $p$. Note that Step 4 and Step 5 may add an even number of additional elements to the initial set $Q_c$, as obtained by Step 2. If no additional elements have been provided by Step 4 and Step 5, then Step 6 will add in any case two elements to $Q_c$. Due to the iteration adopted in Step 6, $Q_c$ is thus filled by an increasing odd number of elements until it has $s$ elements. Hence, Algorithm 1 terminates after a finite number of steps. Moreover, it provides the set $Q_s$ with the required property that the function ${\mathsf{\Psi }}_s$ obtained from ${\stackrel{-}{\mathsf{\Psi }}}_s$ by restriction of its domain to $Q_s$ is bijective. The following result holds:

Theorem 3.

If $n_x<\sqrt{s}$, where $n_x$ is the maximum integer such that $\left(n_x,1\right)\in Q_s$, ${\mathsf{\Psi }}_s$ is injective.

Proof. 

The couples $\left(m,n\right)\in$ ${\stackrel{-}{Q}}_s$ and $\left(-m,n\right)\in$ ${\stackrel{-}{Q}}_s$ are not $s$-equivalent if $m\ne 0$. Their equivalence, from Theorem 1, would indeed imply that:

$$2·n·m=0 \left(mod s\right).$$

(4)

Therefore, by the field properties of ${\mathbb{F}}_s$, it follows that $n=0$ $\left(mod s\right)$ or $m=0$ $\left(mod s\right)$. This leads to a contradiction since both $m$ and $n$ are non-zero. Moreover, the two elements $\left(n_x^{\prime },1\right)$ and $\left(-n_x^{\prime },1\right)$, where $n_x^{\prime }=n_x+1$, as generated by Step 6, are not $s$-equivalent to any element of the current set $Q_c$ if $n_x^{\prime }<\sqrt{s}$. If $\left(m,n\right){~}_s\left(n_x^{\prime },1\right)$ by Theorem 1, it is indeed

$$m=n· n_x^{\prime }\left(mod s\right).$$

(5)

If $m\ge 0,$ this is impossible, since $m\le n_x$ and $n_x<n· n_x^{\prime } <s$.

If $m<0$, Equation (5) and $n ·n_x^{\prime } <s$ gives

$$m=n·n_x^{\prime }-s.$$

(6)

The largest value of the right-hand side of Equation (6) is:

$$\max \left(n· n_x^{\prime }-s\right)=\left(\sqrt{s}-\epsilon -1\right)·\left(\sqrt{s}-\epsilon \right)-s.$$

(7)

The first factor in parentheses maximizes $n$, the second factor maximizes $n_x^{\prime }$, and $\epsilon$ is defined as $\epsilon =\sqrt{s}-Int\left(\sqrt{s}\right)$, where $Int\left(x\right)$ is the largest integer such that $Int\left(x\right)\le x$ so that $0<\epsilon <1$.

Consequently,

$$\max \left(n ·n_x^{\prime }-s_g\right)=-\left(1+2\epsilon \right)·\sqrt{s}+{\epsilon }^2+\epsilon <-\sqrt{s}.$$

(8)

Therefore, Equation (6) cannot hold, since $\left|m\right|<\sqrt{s}$ while the absolute value of the r.h.s. of Equation (6) is larger than $\sqrt{s}$. If $n_x<\sqrt{s}$, where $n_x$ is the maximum integer such that $\left(n_x,1\right)\in Q_c$, then the above observations and the tests provided in both Steps 4 and 5 guarantee that ${\mathsf{\Psi }}_s$ is injective. □

In addition, the following theorem has been demonstrated in [16]:

Theorem 4.

If $p\ge 3$ is prime, $for any x\in \mathbb{N} such that x\le p-1,$ there exists a pair $\left(m, n\right)$ such that $\left|m\right|<\sqrt{p}, 0<n<\sqrt{p}$ and $m=n· x \left(mod p \right).$

Therefore, since all the couples $\left(m, n\right): \left|m\right|<\sqrt{s}, 0<n<\sqrt{s}$ are considered in Algorithm 1 to determine $Q_s$, the following theorem is derived:

Theorem 5.

If $n_x<\sqrt{s}$, where $n_x$ is the maximum integer such that $\left(n_x,1\right)\in Q_s,$ Algorithm 1 determines $Q_s\subseteq$  ${\stackrel{-}{Q}}_s$ such that the restriction ${\mathsf{\Psi }}_s$ of ${\stackrel{-}{\mathsf{\Psi }}}_s$ to the domain $Q_s$ is surjective.

Therefore, Algorithm 1 terminates with $n_x<\sqrt{s}$, providing a bijective restriction ${\mathsf{\Psi }}_s$ of ${\stackrel{-}{\mathsf{\Psi }}}_s$to the domain $Q_s$. Based on Algorithm 1, a numerical implementation of the inductive procedure to determine $Q_p$ for any prime $p$ less than an input integer value has been carried out. The values obtained for the maximum integer $n_x\left(p\right)$ such that $\left(n_x,1\right)\in Q_p$ are shown in Figure 1. For comparison, the real function $\sqrt{x}$ is also shown as a reference curve. In the left-hand side plot of Figure 1, $p$ ranges over the prime numbers in the interval [2, 9973], while $x$ varies over [2, 9973]; in the right-hand side plot, $p$ ranges over the primes in [2, 997], and $x$ varies over [2, 997]. The points ${(p,n}_x\left(p\right))$ and $(p^{\prime },n_x(p^{\prime }\left)\right)$, where $p^{\prime }$ is the next prime greater than $p,$ are joined by straight line segments, thus displaying the graph of $n_x\left(p\right)$ as a broken line for visualization purposes. The numerical results used in Figure 1 suggest the conjecture that the values $n_x\left(p\right)$, as determined by Algorithm 1, are close to the limiting value $\sqrt{p}$ indicated by Theorem 4. More precisely, they are equal to $Int\sqrt{p}$ or $Int\sqrt{p} -1$, as occurs in the interval [2, 9973] explored. It would be interesting to prove this conjecture. The graph of $n_x\left(p\right)$ exhibits an overall increasing behavior, with occasional downward jumps, as shown in Figure 1, more clearly in the right-hand plot. The downward jumps are a consequence of Step 2 of Algorithm 1, which may remove $(n_x\left(p\right),1)$ from the set ${Q_c=Q}_p$ when the set $Q_s$ is generated, where $s$ is the next prime number greater than the prime $p$.

Figure 1. In the left plot, the maximum integer $n_x\left(p\right)$ such that $\left(n_x,1\right)\in Q_p$ is plotted versus $p\le 9973$ (black, step line, joining by a straight line segment the points ${(p,n}_x\left(p\right))$ and ${(p^{\prime },n}_x\left(p^{\prime }\right))$, where $p^{\prime }$ is the next prime greater than $p$) and compared to the function $\sqrt{x}$ with $x\le 9973$ (red line). In the right plot, smaller values of $p$and $x$ are considered.

As an example, the functions ${\mathsf{\Psi }}_{47}$ and ${\mathsf{\Psi }}_{199}$are shown, respectively, in Table 3 and Table 4.

Table 3. The table shows the bijective function ${\mathsf{\Psi }}_{47}:(m,n)\in Q_{47}\to {\mathsf{\Psi }}_{47}(m,n)\in {\mathbb{F}}_{47}$. Bold elements are the additive and multiplicative identities.

$\mathit{m}/\mathit{n}$	Ψ47 (m,n)	$\mathit{m}/\mathit{n}$	Ψ47 (m,n)
−6/1	41	1/6	8
−5/1	42	1/5	19
−4/1	43	1/4	12
−3/1	44	1/3	16
−5/2	21	2/5	38
−2/1	45	1/2	24
−5/3	14	3/5	10
−3/2	22	2/3	32
−4/3	30	3/4	36
−5/4	34	4/5	29
−6/5	27	5/6	40
−1/1	46	1/1	1
−5/6	7	6/5	20
−4/5	18	5/4	13
−3/4	11	4/3	17
−2/3	15	3/2	25
−3/5	37	5/3	33
−1/2	23	2/1	2
−2/5	9	5/2	26
−1/3	31	3/1	3
−1/4	35	4/1	4
−1/5	28	5/1	5
−1/6	39	6/1	6
0/1	0

Table 4. The table shows the bijective function ${\mathsf{\Psi }}_{199}:(m,n)\in Q_{199}\to {\mathsf{\Psi }}_{199}(m,n)\in {\mathbb{F}}_{199}$. Bold elements are the additive and multiplicative identities.

$\mathit{m}/\mathit{n}$	Ψ199 (m,n)	$\mathit{m}/\mathit{n}$	Ψ199 (m,n)	$\mathit{m}/\mathit{n}$	Ψ199 (m,n)	$\mathit{m}/\mathit{n}$	Ψ199 (m,n)	$\mathit{m}/\mathit{n}$	Ψ199 (m,n)
−14/1	185	−7/6	32	−4/13	15	5/14	43	6/5	41
−13/1	186	−8/7	141	−3/10	139	4/11	127	5/4	51
−12/1	187	−9/8	173	−2/7	85	3/8	75	9/7	115
−11/1	188	−11/10	178	−3/11	54	5/13	31	4/3	134
−10/1	189	−1/1	198	−1/4	149	2/5	80	7/5	81
−9/1	190	−11/12	82	−3/13	61	5/12	17	10/7	172
−8/1	191	−9/10	19	−2/9	44	3/7	171	3/2	101
−7/1	192	−8/9	176	−1/5	159	4/9	111	8/5	121
−13/2	93	−7/8	24	−2/11	36	5/11	109	5/3	68
−6/1	193	−6/7	56	−1/6	33	6/13	77	12/7	87
−11/2	94	−5/6	165	−2/13	107	1/2	100	7/4	151
−5/1	194	−9/11	162	−1/7	142	7/13	123	9/5	161
−9/2	95	−4/5	39	−1/8	174	6/11	91	2/1	2
−13/3	62	−7/9	154	−1/9	22	5/9	89	11/5	42
−4/1	195	−3/4	49	−1/10	179	4/7	29	9/4	52
−11/3	129	−8/11	144	−1/11	18	7/12	183	7/3	135
−7/2	96	−5/7	113	−1/12	116	3/5	120	5/2	102
−10/3	63	−7/10	59	−1/13	153	8/13	169	8/3	69
−13/4	146	−2/3	132	−1/14	71	5/8	125	11/4	152
−3/1	196	−7/11	126	0/1	0	7/11	73	3/1	3
−11/4	47	−5/8	74	1/14	128	2/3	67	13/4	53
−8/3	130	−8/13	30	1/13	46	7/10	140	10/3	136
−5/2	97	−3/5	79	1/12	83	5/7	86	7/2	103
−7/3	64	−7/12	16	1/11	181	8/11	55	11/3	70
−9/4	147	−4/7	170	1/10	20	3/4	150	4/1	4
−11/5	157	−5/9	110	1/9	177	7/9	45	13/3	137
−2/1	197	−6/11	108	1/8	25	4/5	160	9/2	104
−9/5	38	−7/13	76	1/7	57	9/11	37	5/1	5
−7/4	48	−1/2	99	2/13	92	5/6	34	11/2	105
−12/7	112	−6/13	122	1/6	166	6/7	143	6/1	6
−5/3	131	−5/11	90	2/11	163	7/8	175	13/2	106
−8/5	78	−4/9	88	1/5	40	8/9	23	7/1	7
−3/2	98	−3/7	28	2/9	155	9/10	180	8/1	8
−10/7	27	−5/12	182	3/13	138	11/12	117	9/1	9
−7/5	118	−2/5	119	1/4	50	1/1	1	10/1	10
−4/3	65	−5/13	168	3/11	145	11/10	21	11/1	11
−9/7	84	−3/8	124	2/7	114	9/8	26	12/1	12
−5/4	148	−4/11	72	3/10	60	8/7	58	13/1	13
−6/5	158	−5/14	156	4/13	184	7/6	167	14/1	14
−13/11	35	−1/3	66	1/3	133	13/11	164

In Figure 2, the inverse functions ${\mathsf{\Psi }}_{997}^{-1}\left(x\right)$ and ${\mathsf{\Psi }}_{9973}^{-1}\left(x\right)$ are plotted using the representation of the discrete rational numbers $(m,n)$ in terms of the fractions $m/n$. Figure 2 illustrates the progressive filling of the set of the fractions. This is due to the Aufbau principle embedded in Algorithm 1, producing an increasing sequence of sets $Q_p$. The functions ${\mathsf{\Psi }}_{997}^{-1}\left(x\right)$ and ${\mathsf{\Psi }}_{9973}^{-1}\left(x\right)$ exhibit a similar characteristic pattern. The points of the graphs $\left(x,{\mathsf{\Psi }}_{997}^{-1}\left(x\right)\right)$ and $\left(x,{\mathsf{\Psi }}_{9973}^{-1}\left(x\right)\right)$ are denser near the axis of the abscissae. The numerical results shown in Figure 2 suggest the conjecture that the density of the points is larger near the axis of the abscissae and increases with $p$. This appears to be a direct consequence of the ordering of Step 4 and Step 5 in Algorithm 1, so that the pairs $(m,n)$ with $\left|m\right|<n$ are selected for the set of discrete rational numbers before selecting the pairs $(m,n)$ with $\left|m\right|>n$. It would be interesting to confirm this conjecture. Even more interesting would be to calculate in closed form the set $Q_p$ instead of deriving it via an algorithmic procedure, providing the bijective mapping ${\mathsf{\Psi }}_p$. A closed-form expression for ${\mathsf{\Psi }}_p$ might also characterize quantitatively the pattern shown in Figure 2.

Figure 2. Plot of the functions ${\mathsf{\Psi }}_{997}^{-1}\left(x\right)$ (left) and (right) ${\mathsf{\Psi }}_{9973}^{-1}\left(x\right)$ vs. the integer $x$. The image pairs $(m,n)$ are shown as rational numbers $m/n$. Most points are located near the horizontal axis $m/n=0$ owing to the structure of Algorithm 1.

To define the algebra of the sets $Q_p$, the following binary operations of addition and multiplication, denoted by $+$ and $\cdot$, respectively, are defined:

Definition 5.

For any $\left(a,b\right)\in Q_p$ and $\left(c,d\right)\in Q_p$,

$$\left(a,b\right)+\left(c,d\right)={\mathsf{\Psi }}_p^{-1}\left\{\left[{\mathsf{\Psi }}_p\left(a,b\right)+{\mathsf{\Psi }}_p\left(c,d\right)\right] (mod p)\right\};$$

(9)

$$\left(a,b\right)\cdot \left(c,d\right)={\mathsf{\Psi }}_p^{-1}\left\{\left[{\mathsf{\Psi }}_p\left(a,b\right)\cdot {\mathsf{\Psi }}_p\left(c,d\right)\right] \left(mod p\right)\right\}.$$

(10)

This definition ensures that ${\mathsf{\Psi }}_p$ is an isomorphism, and ${\mathbb{Q}}_{\left(p\right)}=\left(Q_p,+, \cdot \right)$ is a field isomorphic to the Galois field ${\mathbb{F}}_p$. The following definitions are introduced:

Definition 6.

Any element of $Q_p$ is defined as a discrete rational number.

Definition 7.

${\mathbb{Q}}_{\left(p\right)}$ is defined as the field of discrete rational numbers of order $p$.

The term discrete rational number is not standard in the mathematical literature; it is used here to convey the idea of the fractional representation of the field ${\mathbb{F}}_p$ provided by ${\mathbb{Q}}_{\left(p\right)}$.

As observed in the introduction, ${\mathbb{Q}}_{\left(p\right)}$ is not a new field, being merely a different representation of its isomorphic field ${\mathbb{F}}_p$.

A link with the ordinary operations of addition and multiplication in $\mathbb{Q}$ is provided, respectively, by the following Theorems 7 and 8, which will be used in the next section devoted to the construction of $\mathbb{Q}$ based on the fields ${\mathbb{Q}}_{\left(p\right)}$. To this end, the equivalence relation, denoted as $~$, is defined between pairs of integer numbers, with $Z_o=Z-\left\{0\right\}$:

Definition 8.

$\left(a,b\right)\in Z\times Z_o~ \left(c,d\right)\in Z\times Z_o \iff a·d=b·c$.

Theorem 6.

$\mathrm{F}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{n}\mathrm{y} \left(a,b\right)\in Q_p$ and $\left(c,d\right)\in Q_p, such that \left(e,f\right)\in Q_p, with$

$$\left(e,f\right) ~ \left(a·d+b·c,b·d\right),$$

(11)

it follows that:

$$\left(a,b\right)+\left(c,d\right)=\left(e,f\right).$$

(12)

Proof. 

$\left(a,b\right)\in Q_p$ and $\left(c,d\right)\in Q_p\Rightarrow$ $a=m·b+\alpha ·p,c=n·d+\beta ·p,$ so that it is also ${\mathsf{\Psi }}_p\left(a,b\right)=m, {\mathsf{\Psi }}_p\left(c,d\right)=n$. Therefore,

$$\begin{gathered} \left(a,b\right)+\left(c,d\right)={\mathsf{\Psi }}_p^{-1}\left\{\left(m+n\right) (mod p)\right\}={\mathsf{\Psi }}_p^{-1}\left\{\left({\displaystyle \frac{a-\alpha ·p}{b}}+{\displaystyle \frac{c-\beta ·p}{d}}\right)(mod p)\right\} \\ ={\mathsf{\Psi }}_p^{-1}\left\{\left({\displaystyle \frac{a-\alpha ·p}{b}}+{\displaystyle \frac{c-\beta ·p}{d}}\right)·{\displaystyle \frac{b·d}{b·d}} (mod p)\right\} \\ ={\mathsf{\Psi }}_p^{-1}\left\{{\displaystyle \frac{\left(a-\alpha ·p\right)·d}{b·d}}+{\displaystyle \frac{\left(c-\beta ·p\right)·b}{b·d}} (mod p)\right\} \\ ={\mathsf{\Psi }}_p^{-1}\left\{{\displaystyle \frac{\left(a-\alpha ·p\right)·d+\left(c-\beta ·p\right)·b}{b·d}} (mod p)\right\}. \end{gathered}$$

(13)

The above equalities hold in $Z$, since $\left(a-\alpha ·p\right)·d/\left(b·d\right)$ and $\left(c-\beta ·p\right)·b/\left(b·d\right)$ are divisions with zero remainder and $b·d\ne 0$. It is then obtained from Equation (14), being $f\ne 0$:

$$\begin{array}{cc}\hfill \left(a,b\right)+ \left(c,d\right)& ={\mathsf{\Psi }}_p^{-1}\left\{{\displaystyle \frac{a·d+b·c-\alpha ·p-\beta ·p}{b·d}} (mod p)\right\}={\mathsf{\Psi }}_p^{-1}\left\{{\displaystyle \frac{a·d+b·c-\alpha ·p-\beta ·p}{b·d}}·{\displaystyle \frac{f}{f}} (mod p)\right\}\hfill \\ & ={\mathsf{\Psi }}_p^{-1}\left\{{\displaystyle \frac{\left(a·d+b·c\right)·f-\alpha ·p·f-\beta ·p·f}{b·d·f}} (mod p)\right\}\hfill \\ & ={\mathsf{\Psi }}_p^{-1}\left\{{\displaystyle \frac{e·b·d-\alpha ·p·f-\beta ·p·f}{b·d·f}} (mod p)\right\}. \hfill \end{array}$$

(14)

The last equality holds due to the equivalence $\left(12\right)$. If the quantity in the curly brackets is indicated as $q$, one obtains:

$${\displaystyle \frac{e·b·d-\alpha ·p·f-\beta ·p·f}{b·d·f}}=q+\gamma ·p.$$

(15)

Hence, since $b·d\ne 0$ has an inverse in ${\mathbb{F}}_p$, one obtains:

$$e=q·f \left(mod p\right).$$

(16)

In conclusion, since $\left(e,f\right)\in Q_p$

$$\left(a,b\right)+\left(c,d\right)={\mathsf{\Psi }}_p^{-1}\left(q\right)={\mathsf{\Psi }}_p^{-1}{\mathsf{\Psi }}_p\left(e,f\right)=\left(e,f\right).$$

(17)

□

Theorem 7.

$\mathrm{F}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{n}\mathrm{y} \left(a,b\right)\in Q_p$ and$\left(c,d\right)\in Q_p, \mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h} \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t} \left(g,h\right)\in Q_p \mathrm{a}\mathrm{n}\mathrm{d}$

$$\left(g,h\right)~\left(a·c,b·d\right),$$

(18)

 it is also:

$$\left(a,b\right)· \left(c,d\right)=\left(g,h\right).$$

(19)

Proof. 

$\left(a,b\right)\in Q_p$ and$\left(c,d\right)\in Q_p\Rightarrow$ $a=m·b+\alpha ·p,c=n·d+\beta ·p,$ so that it is also ${\mathsf{\Psi }}_p\left(a,b\right)=m, {\mathsf{\Psi }}_p\left(c,d\right)=n$. Therefore,

$$\left(a,b\right)· \left(c,d\right)={\mathsf{\Psi }}_p^{-1}\left\{\left(m·n\right) mod p\right\}={\mathsf{\Psi }}_p^{-1}\left\{\left({\displaystyle \frac{a-\alpha ·p}{b}}·{\displaystyle \frac{c-\beta ·p}{d}}\right)\left(mod p\right)\right\}={\mathsf{\Psi }}_p^{-1}\left\{\left({\displaystyle \frac{a·c-\alpha ·p-\beta ·p}{b·d}}\right) \left(mod p\right)\right\}.$$

(20)

The final equality holds in $Z$ only since $\left(a-\alpha ·p\right)/b$ and $\left(c-\beta ·p\right)/d$ are divisions with zero remainder. It is then obtained from Equation (14) since $h\ne 0$:

$$\begin{array}{cc}\hfill \left(a,b\right)+ \left(c,d\right)& ={\mathsf{\Psi }}_p^{-1}\left\{\left({\displaystyle \frac{a·c-\alpha ·p-\beta ·p}{b·d}}\right)·{\displaystyle \frac{h}{h}} (mod p)\right\}={\mathsf{\Psi }}_p^{-1}\left\{{\displaystyle \frac{a·c·h-\alpha ·p·h-\beta ·p·h}{b·d·h}} (mod p)\right\}\hfill \\ & ={\mathsf{\Psi }}_p^{-1}\left\{{\displaystyle \frac{b·d·g-\alpha ·p·h-\beta ·p·h}{b·d·h}} (mod p)\right\}.\hfill \end{array}$$

(21)

The last equality holds due to the equivalence $\left(18\right)$. If the quantity in the curly brackets is indicated as $r$, one obtains:

$${\displaystyle \frac{b·d·g-\alpha ·p·h-\beta ·p·h}{b·d·h}}=r+\gamma ·p.$$

(22)

Hence, since $b·d\ne 0$, it has an inverse in ${\mathbb{F}}_p$, so that one obtains:

$$g=r·h \left(mod p\right).$$

(23)

In conclusion, since $\left(g,h\right)\in Q_p$, it is:

$$\left(a,b\right)· \left(c,d\right)={\mathsf{\Psi }}_p^{-1}\left(r\right)={\mathsf{\Psi }}_p^{-1}{\mathsf{\Psi }}_p\left(g,h\right)=\left(g,h\right).$$

(24)

□

Owing to the properties of Algorithm 1, the following theorem follows (the proof is straightforward):

Theorem 8.

If $(a, b)\in Q_p$, then $\left|a\right|$ and $b$ are coprime.

A sufficient condition for $(a, b)\in Q_p$ is provided by the following theorem:

Theorem 9.

If $\left|a\right|,b are coprime and less than \sqrt{p/2}$, then $(a, b)\in Q_p$.

Proof. 

If $\left|a\right|,b<\sqrt{p/2}$ and $(a, b)\notin Q_p$ by Theorem 1, there exists $(c, d)\in Q_p$ such that Equation (2) is fulfilled and $\left(a, b\right)\ge (c, d)$. This condition follows from the fact that the set of couples with elements less than $\sqrt{p/2}$ in absolute value cannot provide a surjective mapping in ${\mathbb{F}}_p$, as its cardinality is less than $p$. Therefore, Algorithm 1 cannot stop before checking if $(a, b)\in Q_p$ so that the above condition must be verified to have that $(a, b)\notin Q_p$. Moreover, since all the elements of the couple of integer numbers belonging to $Q_p$ have absolute values less than $\sqrt{p}$, Equation (2) can be fulfilled only if the products $ab$ and $cd$ have opposite signs and

$$a·d=b·c\pm p.$$

(25)

If $\left|a\right|,b<\sqrt{p/2}$, the last equation is impossible since the products $a·d$ and $b·c$ both have absolute values less than $p/2$. □

The field ${\mathbb{Q}}_{\left(p\right)}$ can be endowed with the following definition of local distance function $d_p$:

Definition 9.

The local distance function $d_p$ is given by

$${d_p:(x,y)\in Q_p^2 \to d}_p\left(x,y\right)=\left|x-y\right|\in Q_p.$$

(26)

The local distance function $d_p$ does not satisfy the triangle inequality so that ${\mathbb{Q}}_{\left(p\right)}$ is not a metric space with respect to it. However, the triangle inequality for $x,y,z\in Q_p$ holds in subsets of $Q_p^3.$In this sense, ${\mathbb{Q}}_{\left(p\right)}$ is a locally metric space with respect to $d_p$.

3. Rational Numbers

The basic objects for defining the set $\mathbb{Q}$ are the constant sequences of discrete rational numbers. These are defined by:

Definition 10.

A sequence of discrete rational numbers $\left(q_n\right)$ such that $q_n\in Q_{p\left(n\right)},$ where $p\left(n\right)$ is the increasing sequence of primes, is called a constant sequence if there exists $m\in \mathbb{N}$ such that $q_k= q_m$ for$k\ge m$. The constant value $q_m$ is said to be the limit of the sequence $\left(q_n\right)$, indicated by the symbol $\mathrm{l}\mathrm{i}\mathrm{m}\left(q_n\right).$

An equivalence relation ${~}_K$ is introduced on the set $K$ of the constant sequences of discrete rational numbers by the following:

Definition 11.

Two sequences $\left(q_n\right)\in K$ and $\left(r_n\right)\in K$ are said to be $K\text{-}equivalent,$ and the relation $\left(q_n\right) {~}_K\left(r_n\right)$ holds if $there exists m\in \mathbb{N}$ such that $q_s=r_s$  $for s\ge m$.

It is straightforward to verify that two sequences of discrete rational numbers are $K\text{-}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{t}$ if and only if they have the same limit value.

A set of Galois rational numbers is now introduced:

Definition 12.

The quotient set $Q_G=K/{~}_K$ is called the set of Galois rational numbers.

Addition and multiplication, denoted, respectively, by $+$ and $\cdot$, are defined on $Q_G$:

Definition 13.

Given$x,y\in Q_G$, let $\left(x_n\right)\in x$ and $\left(y_n\right)\in y$ and define the constant sequences

$$\left(a_n\right)=\left(x_n+y_n\right);$$

(27)

$$\left(b_n\right)=\left(x_n\cdot y_n\right),$$

(28)

 where $+$ and $\cdot$ denote the addition and multiplication in $Q_{p\left(n\right)}$ as provided by Definition 5.

The addition and multiplication on $Q_G$ are defined, respectively, as:

$$x+y=a;$$

(29)

$$x\cdot y=b,$$

(30)

where $a\in Q_G$, $\left(a_n\right)\in a$, $b\in Q_G$, and $\left(b_n\right)\in b$.

To show that the operations defined above descend to the quotient, let us define $x_m=\left(\alpha ,\beta \right)$ and $y_m=\left(\gamma ,\delta \right)$. Here, $m$ is such that $x_n=x_m$ and $y_n=y_m$ $\forall n\ge m.$ The pairs $\left(\alpha ,\beta \right)$ and $\left(\gamma ,\delta \right)$ do not depend on the specific elements $\left(x_n\right)$ and $\left(y_n\right)$ of the respective classes $x$ and $y.$ Concerning the addition, let $e\in \mathbb{Z}$ and $f\in \mathbb{Z}$ such that $\left|e\right|$ and $f>0$ are coprime and

$${\displaystyle \frac{e}{f}}~{\displaystyle \frac{\alpha ·\delta +\gamma ·\beta }{\beta ·\delta }}.$$

(31)

By Theorem 9, $\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{n}\mathrm{y} n\ge s$ such that $\left|e\right|<\sqrt{p\left(s\right)/2}$ and $f<\sqrt{p\left(s\right)/2}$, $\left(e,f\right)\in Q_{p\left(n\right)}$. Due to Theorem 6, it follows that $x_n+y_n=\left(e,f\right)$ $\forall n\ge s$. Therefore, the result of the addition $x+y$ does not depend on the choice of representatives of the classes $x$ and $y$ used in Definition 13.

Concerning multiplication, define $g\in \mathbb{Z}$ and $h\in \mathbb{Z}$ such that $\left|g\right|$ and $h>0$ are coprime and

$${\displaystyle \frac{g}{h}}~{\displaystyle \frac{\alpha ·\gamma }{\beta ·\delta }}.$$

(32)

Due to Theorem 9, $\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{n}\mathrm{y} n\ge s$ such that $\left|g\right|<\sqrt{p\left(s\right)/2}$ and $h<\sqrt{p\left(s\right)/2}$, $\left(g,h\right)\in Q_{p\left(n\right)}$. Due to Theorem 7, it is then $x_n\cdot y_n=\left(g,h\right)$ $\forall n\ge s$. Therefore, the result of the multiplication $x\cdot y$ does not depend on the choice of representatives of the classes $x$ and $y$ used in Definition 13. The above-defined addition and multiplication inherit the properties of the addition and multiplication of the fields ${\mathbb{Q}}_{\left(p\right)}$. The proof that ${\mathbb{Q}}_G=\left(Q_G,+, \cdot \right)$ is a field is straightforward. The additive identity is the class denoted as $\mathbf{0}$, which contains the sequence $\left(o_n\right)$, $o_n=\left(\mathrm{0,1}\right)$ for any $n$. The multiplicative identity is the class denoted as $\mathbf{1}$, which contains the sequence $\left(u_n\right)$, where $u_n=\left(\mathrm{1,1}\right)$ for any $n$. As noted above, the elements of any equivalence class $x$ have the same limit, which is denoted by $\lim \left(x\right)$. The following order is defined on the field ${\mathbb{Q}}_G$:

Definition 14.

For $x,y\in Q_G$, the order $x{\ge }_{G}y$ holds if said $\left(\alpha ,\beta \right)=\mathrm{l}\mathrm{i}\mathrm{m}\left(x\right)$ and $\left(\gamma ,\delta \right)=\mathrm{l}\mathrm{i}\mathrm{m}\left(y\right)$

$$\alpha ·\delta \ge \beta ·\gamma .$$

It can be shown that the order is a total order and compatible with the addition and multiplication. Moreover, it is straightforward to prove that the field ${\mathbb{Q}}_G$ is a metric space, endowed with the distance $d_G$:

Definition 15.

For $x,y\in Q_G$, the distance $d_G$ is defined as $d_G\left(x,y\right)=\mathrm{l}\mathrm{i}\mathrm{m}\left|x_n-y_n\right|$, where $\left(x_n\right)\in x$ and $\left(y_n\right)\in x$.

One readily verifies that the definition of the distance $d_G$ is meaningful, i.e., it descends to the quotient. An ordering between the distances is established by the following:

Definition 16.

If $d_G\left(x,y\right)=\left(\alpha ,\beta \right)$ and $d_G\left(x^{\prime },y^{\prime }\right)=\left({\alpha }^{\prime },{\beta }^{\prime }\right)$, the order $d_G\left(x,y\right)\ge d_G\left(x^{\prime },y^{\prime }\right)$ holds if $\alpha ·{\beta }^{\prime }\ge \beta · {\alpha }^{\prime }$.

To compare the field ${\mathbb{Q}}_G$ with the field of rational numbers $\mathbb{Q}$, the latter can be defined as $\left(Q,+,\cdot \right)$, where $Q=(Z\times Z_o)/~$. The equivalence relation $~$ is defined by:

$$\left(a,b\right)\in Z\times Z_o~ \left(c,d\right)\in Z\times Z_o \iff ad=bc.$$

Given $x,y\in Q$ and any pairs $\left(x_1,x_2\right)\in x$and $\left(y_1,y_2\right)\in y$, the binary operations of addition $\left(+\right)$and multiplication $\left(\cdot \right)$ are defined as:

$$x+y=z \ni \left(x_1·y_2+x_2·y_1,x_2·y_2\right),$$

$$x·y=w \ni \left(x_1·y_1,x_2·y_2\right)$$

A detailed definition and discussion of the properties of the rational numbers can be found in classical textbooks, e.g., [17] (Chapter I). The following theorem is stated:

Theorem 10.

The function $\mathsf{\Psi }:Q_G\to Q$ defined by:

$$\mathsf{\Psi }:x\in Q_G:\left(x_n\right)\in x, \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} m: x_n=\left(a,b\right) \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{n}\mathrm{y} n\ge m \to \mathsf{\Psi }\left(x\right)\in Q: \left(a,b\right)\in \mathsf{\Psi }\left(x\right),$$

is an isomorphism, and ${\mathbb{Q}}_G=\left(Q_G,+, \cdot \right)$ is a field isomorphic to the field $\mathbb{Q}$.

Proof. 

First, observe that the function $\mathsf{\Psi }$ is well defined, i.e., it descends to the quotient since all representatives of the class $x$ have an identical constant value above a certain index. The function $\mathsf{\Psi }$ is injective. If $x,y \in Q_G$ and $\mathsf{\Psi }\left(x\right)=\mathsf{\Psi }\left(y\right)$, then there exist two sequences ($\left(x_n\right)\in x$ and $\left(y_n\right)\in y$) such that there exists an index $m \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} x_n=\left(a,b\right) \mathrm{a}\mathrm{n}\mathrm{d} y_n=(c,d) \forall n\ge m$ and $\left(a,b\right)\in \mathsf{\Psi }\left(x\right), \left(c,d\right)\in \mathsf{\Psi }\left(y\right)= \mathsf{\Psi }\left(x\right)$. The two pairs $\left(a,b\right)$ and $\left(c,d\right)$ are thus equivalent with respect to the equivalence relation $~$. Moreover, by Theorem 9, $\left|a\right|$ and $b$ are coprime, and $\left|c\right|$ and $d$ are coprime. Therefore $\left(a,b\right)=\left(c,d\right)$, so that $x=y$. The function $\mathsf{\Psi }$ is surjective. Let $q\in Q$ be arbitrary and a representative $\left(a,b\right)\in q$ such that $\left|a\right|$ and $b$ are coprime. Consider a prime $p_q$ such that $\left|a\right|,b<\sqrt{p_q/2}$; hence, by Theorem 9, $\left(a,b\right)\in Q_{p\left(n\right)}$ for any $n$ larger or equal to $n_q$, where $p_q=p\left(n_q\right)$. The class $x\in Q_G$ that has the element $\left(x_n\right)$ defined by

$$x_n=\left(\mathrm{0,1}\right) for n<n_q, x_n=\left(a,b\right) for n\ge n_q,$$

is such that $\mathsf{\Psi }\left(x\right)=q$.

The function $\mathsf{\Psi }$ preserves addition. Let $x,y \in Q_G$ and two sequences $\left(x_n\right)\in x$ and $\left(y_n\right)\in y$. There exists an index $m \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} x_n=\left(a,b\right) \mathrm{a}\mathrm{n}\mathrm{d} y_n=(c,d) \forall n\ge m$. Consider the integer numbers $e,f>0$ such that $\left|e\right|$ and $f$ are coprime and

$$\left(e,f\right)~\left(a·d+b·c,b·d\right).$$

Consider the prime number $p_{+}$ such that $\left|e\right|,f<\sqrt{p_{+}/2}$, so that $(e, f)\in Q_{p(n)}$ for $n\ge n_{+}$, where $p_{+}=p\left(n_{+}\right)$. Owing to Theorem 6, $x_n+y_n=(e, f)$ for $n\ge n_{+}$. Hence, $(e, f)\in \mathsf{\Psi }(\mathrm{x}+\mathrm{y})$. Moreover, $\left(a,b\right)\in \mathsf{\Psi }\left(\mathrm{x}\right)$ and $(c,d)\in \mathsf{\Psi }\left(\mathrm{y}\right)$. Consequently, it is also $\left(a·d+b·c,b·d\right)\in \mathsf{\Psi }\left(\mathrm{x}\right)+\mathsf{\Psi }\left(\mathrm{y}\right)$. The classes $\mathsf{\Psi }(\mathrm{x}+\mathrm{y})$ and $\mathsf{\Psi }\left(\mathrm{x}\right)+\mathsf{\Psi }\left(\mathrm{y}\right)$ are thus identical since they contain, respectively, the elements $(e, f)$ and $\left(a·d+b·c,b·d\right)$ which are equivalent with respect to $~$. □

The function $\mathsf{\Psi }$ preserves multiplication. Let $x,y \in Q_G$ and two sequences $\left(x_n\right)\in x$ and $\left(y_n\right)\in y$. There exists an index $m \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} x_n=\left(a,b\right) \mathrm{a}\mathrm{n}\mathrm{d} y_n=(c,d) \forall n\ge m$. Let us consider the integer numbers $g,h>0$ such that $\left|g\right|$ and $h$ are coprime and

$$\left(g,h\right)~\left(a·c,b·d\right).$$

Consider the prime number $p_{\times }$ such that $\left|g\right|,h<\sqrt{p_{\times }/2}$, so that $(g, h)\in Q_{p\left(n\right)}$ for $n\ge n_{\times }$, where $p_{\times }=p\left(n_{\times }\right)$. Owing to Theorem 7, $x_n·y_n=(g, h)$ for $n\ge n_{\times }$.

Therefore, it follows that $(g, h)\in \mathsf{\Psi }(\mathrm{x}·\mathrm{y})$. Moreover, $\left(a,b\right)\in \mathsf{\Psi }\left(\mathrm{x}\right)$, $(c,d)\in \mathsf{\Psi }\left(\mathrm{y}\right)$, and $\left(a·c,b·d\right)\in \mathsf{\Psi }\left(\mathrm{x}\right)+\mathsf{\Psi }\left(\mathrm{y}\right)$. The classes $\mathsf{\Psi }(\mathrm{x}·\mathrm{y})$ and $\mathsf{\Psi }\left(\mathrm{x}\right)·\mathsf{\Psi }\left(\mathrm{y}\right)$ are thus identical since they contain, respectively, the elements $(g, h)$ and $\left(a·c,b·d\right)$, which are equivalent with respect to the relation $~$.

It is easy to show that $\mathsf{\Psi }$ preserves the order and provides an isometry between $d_G$ and the distance $d_Q$ defined in $Q$ as $d_Q\left(x,y\right)=\left|x-y\right|$. By abuse of notation, it might be indicated as ${\mathsf{\Psi }}^{-1}\left(x\right)$ the pair $\left(a,b\right)\in Q_{p\left(m\right)}$ if the sequence $\left(x_n\right)\in {\mathsf{\Psi }}^{-1}\left(x\right)$ is such that $x_n=\left(a,b\right) \forall n\ge m$, and the element $\mathsf{\Psi }\left(x\right)\in Q$ such that $\left(x_n\right)\in x$ might be indicated as $\mathsf{\Psi }\left(\left(x_n\right)\right)$.

4. Real Numbers

The basic objects for defining the set $\mathbb{R}$ are the convergent sequences of discrete rational numbers. To define them, a preliminary definition is introduced:

Definition 17.

Given a sequence of discrete rational numbers $\left(q_n\right)$ such that $n\in \mathbb{N}$ and $q_n\in Q_{p\left(n\right)}$, any sequence $\left(q_n^m\right)\in K$ such that $q_s^m=q_m$ for $s\ge m$ is said to be $m\text{-}associated$ to $\left(q_n\right)$.

Definition 18.

A sequence of discrete rational numbers $\left(q_n\right)$ such that $n\in \mathbb{N}$, $q_n\in Q_{p\left(n\right)}$, and $p\left(n\right)$ is the increasing sequence of primes is said to be a convergent sequence if for any positive discrete rational number$\epsilon$, there exists an index $m$ such that $\epsilon \in Q_{p\left(m\right)}$ and

$$\epsilon \ge d_G(\left(q_n^s\right),\left(q_n^t\right))\mathrm{for} \mathrm{all} s,t\ge m.$$

The sequences $\left(q_n^s\right)$ and $\left(q_n^t\right)$ are, respectively, s-associated and t-associated with $\left(q_n\right)$. The set $H$ of the convergent sequences includes the set $K$. An equivalence relation ${~}_H$ is introduced on the set $H$ of the convergent sequences of discrete rational numbers:

Definition 19.

Two sequences $\left(q_n\right),\left(r_n\right)\in H$ are said to be $H\text{-}equivalent,$ and the relation $\left(q_n\right) {~}_H \left(r_n\right)$ holds if for any discrete rational number $\epsilon$ there exists a $m_{\epsilon }\in \mathbb{N}$ such that

$$\epsilon >d_G(\left(q_n^s\right),\left(\left(r_n^t\right)\right))\mathrm{for} \mathrm{any} s,t\ge m_{\epsilon }.$$

Note that the sequences $\left(q_n^s\right)$ and $\left(r_n^t\right)$ are, respectively, s-associated and t-associated with $\left(q_n\right)$ and $\left(r_n\right)$. The following Lemma 1 and Lemma 2 allow one to prove that, introducing the binary operations of addition $+$and multiplication $·$ of convergent sequences, as defined in the following, one obtains a field $R_G=\left(H,+,·\right)$, which is isomorphic to the field $\mathbb{R}$ of the real numbers.

Lemma 1.

For any convergent sequence $\left(q_n\right)$, there exists an $H\text{-}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{t}$ convergent sequence $\left({\dot{q}}_n\right)$ and $m_k\in \mathbb{N}$ such that if $\left(a_n,b_n\right)={\dot{q}}_n$, both $\left|a_n\right|$ and $b_n$ are bounded by $\sqrt[k]{p\left(n\right)/2}$ $\mathrm{f}\mathrm{o}\mathrm{r} n\ge m_k.$

Proof. 

For any convergent sequence $\left(q_n\right)$, let $q_m=\left({\alpha }_m,{\beta }_m\right)\in Q_{p\left(m\right)}$ and denote as $S_m^k$ the set of the discrete rational numbers $r=\left(\alpha , \beta \right)\in Q_{p\left(m\right)},$with $\left|\alpha \right|,\beta$less than $\sqrt[k]{p\left(m\right)/2}$. Choose in $S_m^k$ the element ${\dot{q}}_m$ such that:

$$d_G(\left({\dot{q}}_n^m\right),\left(q_n^m\right))=\underset{r\in S_m^k}{\min }{d}_G\left(\left(r_n^m\right),\left(q_n^m\right)\right).$$

(33)

Here, $\left({\dot{q}}_n^m\right)$ is a constant sequence of discrete rational numbers such that ${\dot{q}}_n^m={\dot{q}}_m$ for $n\ge m$; $\left(q_n^m\right)$ is a constant sequence $m\text{-}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{c}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}$ with $\left(q_n\right)$; $\left(r_n^m\right)$ is a constant sequence such that $r_n^m=r$ for $n\ge m$. For any discrete rational number $\epsilon =\left(a_{\epsilon },b_{\epsilon }\right)>0$, let $m^{\prime }$ such that ${\epsilon }^{\prime }=\left(a_{\epsilon },{4·b}_{\epsilon }\right)\in Q_{p\left(m^{\prime }\right)}$ and ${\epsilon }^{\prime }\ge d_G\left(\left(q_n^m\right),\left(q_n^t\right)\right) \mathrm{i}\mathrm{f} m,t\ge m^{\prime }$. Since the set $K$ is dense, there exists $\left(x_n\right)\in K$ such that ${\epsilon }^{\prime }\ge d_G\left(\left(q_n^m\right),\left(x_n\right)\right)$. Let $\mathrm{l}\mathrm{i}\mathrm{m}\left(x\right)=a/b$ with $\left|a\right|$ and $b>0$ coprime integer numbers and $\sigma$ such that they are both less than $\sqrt[k]{p\left(\sigma \right)/2}$. Therefore, $\left(a,b\right)\in S_j^k \mathrm{i}\mathrm{f} j\ge \sigma$ and $d_G(\left(x_n\right),\left(q_n^t\right))\le \epsilon \mathrm{i}\mathrm{f}t\ge m^{\prime }$. Indeed,

$$d_G\left(\left(x_n\right),\left(q_n^t\right)\right)\le d_G\left(\left(x_n\right),\left(q_n^m\right)\right)+d_G\left(\left(q_n^m\right),\left(q_n^t\right)\right)\le {2·\epsilon }^{\prime }=\epsilon /2.$$

Therefore, $d_G\left(\left({\dot{q}}_n^s\right),\left(q_n^t\right)\right)\le d_G\left(\left({\dot{q}}_n^s\right),\left(q_n^s\right)\right)+d_G\left(\left(q_n^s\right),\left(q_n^t\right)\right)\le \epsilon$if $s,t\ge \max (\sigma ,m^{\prime })$, owing to Equation (33). This shows that $\left({\dot{q}}_n\right)$is a convergent sequence and $\left({\dot{q}}_n\right) {~}_H \left(q_n\right)$. □

Lemma 1 leads to the following definition:

Definition 20.

For any convergent sequence $\left(q_n\right)$, any convergent sequence $\left({\dot{q}}_n\right){ ~}_H \left(q_n\right)$ such that ${\dot{q}}_n=\left(a_n,b_n\right)$, where $a_n$ and $b_n$ are less than $\sqrt[k]{p\left(n\right)/2}$ for $n\ge m$, is said to be a $k-normal sequence$ associated with $\left(q_n\right)$.

Lemma 2.

For any convergent sequence $\left(q_n\right)$, there exists a constant sequence $\left({\stackrel{-}{q}}_n\right)$ and $m\in \mathbb{N}$ such that $\left|q_n\right|\le {\stackrel{-}{q}}_n$ for $n\ge m$.

The proof is straightforward. Lemma 2 leads to the following definition:

Definition 21.

For any convergent sequence $\left(q_n\right)$, the limit of a constant sequence $({\stackrel{-}{q}}_n)$ such that $\left|q_n\right|\le {\stackrel{-}{q}}_n$ for$n\ge m$ is said to be an absolute bound of $\left(q_n\right)$.

The following definitions allow for the constructive realization of the field of real numbers.

Definition 22.

The quotient $R_G=H/{~}_H$ is said to be the set of Galois real numbers.

Definition 23.

For any $x\in R_G$ and $y\in R_G$, with $\left(x_n\right)\in x$ and $\left(y_n\right)\in y$, the operations of addition and multiplication are defined, respectively, as

$$x+y=a;$$

(34)

$$x\cdot y=b,$$

(35)

with $\left(x_n+y_n\right)\in a$ and $\left(x_n·y_n\right)\in b$.

Definition 23 descends to the quotient. Concerning the addition, note that $\left(x_n+y_n\right)$ is a convergent sequence since

$$d_G\left(\left(x_n^s+y_n^s\right),\left(x_n^t+y_n^t\right)\right)\le d_G\left(\left(x_n^s\right),\left(x_n^t\right)\right)+d_G\left(\left(y_n^s\right),\left(y_n^t\right)\right).$$

(36)

Moreover, if $\left({\dot{x}}_n\right){~}_H\left(x_n\right)$ and $\left({\dot{y}}_n\right){~}_H\left(y_n\right)$, it is true that $\left({\dot{x}}_n+{\dot{y}}_n\right){~}_H\left(x_n+y_n\right)$ since

$$d_Q\left(\left({\dot{x}}_n^s+{\dot{y}}_n^s\right),\left(x_n^t+y_n^t\right)\right)\le d_G\left(\left({\dot{x}}_n^s\right),\left(x_n^t\right)\right)+d_Q\left(\left({\dot{y}}_n^s\right),\left(y_n^t\right)\right).$$

(37)

Concerning the multiplication, $\left(x_n·y_n\right)$ is a convergent sequence since

$$\begin{array}{cc}\hfill d_G\left(\left(x_n^s·y_n^s\right),\left(x_n^t·y_n^t\right)\right)& =\lim \left|x_n^s·y_n^s-x_n^t·y_n^t\right|=\lim \left|\left(x_n^s-x_n^t\right)·y_n^s+x_n^t·\left(y_n^s-y_n^t\right)\right|\hfill \\ & \le \lim \left|\left(x_n^s-x_n^t\right)·y_n^s\right|+\lim \left|x_n^t·\left(y_n^s-y_n^t\right)\right|\le \lim \left|\left(x_n^s-x_n^t\right)\right|·\lim \left|y_n^s\right|+h_x·\lim \left|\left(y_n^s-y_n^t\right)\right|\hfill \\ & =d_G\left(x_n^s,x_n^t\right)·h_y+h_x·d_G\left(y_n^s,y_n^t\right).\hfill \end{array}$$

(38)

Here, $h_x$ and $h_y$ are, respectively, an absolute bound of $\left(x_n\right)$ and $\left(y_n\right)$. Moreover, if $\left({\dot{x}}_n\right){~}_H\left(x_n\right)$ and $\left({\dot{y}}_n\right){~}_H\left(y_n\right)$, then $\left({\dot{x}}_n·{\dot{y}}_n\right){~}_H\left(x_n·y_n\right)$ since

$$\begin{array}{cc}\hfill d_G\left(\left({\dot{x}}_n^s·{\dot{y}}_n^s\right),\left(x_n^t·y_n^t\right)\right)& =\lim \left|{\dot{x}}_n^s·{\dot{y}}_n^s-x_n^t·y_n^t\right|=\lim \left|\left({\dot{x}}_n^s-x_n^t\right)·{\dot{y}}_n^s+x_n^t·\left({\dot{y}}_n^s-y_n^t\right)\right|\hfill \\ & \le \lim \left|\left({\dot{x}}_n^s-x_n^t\right)·{\dot{y}}_n^s\right|+\lim \left|x_n^t·\left({\dot{y}}_n^s-y_n^t\right)\right|\le \lim \left|\left({\dot{x}}_n^s-x_n^t\right)\right|·\lim \left|{\dot{y}}_n^s\right|+h_x·\lim \left|\left({\dot{y}}_n^s-y_n^t\right)\right|\hfill \\ & =d_G\left({\dot{x}}_n^s,x_n^t\right)·h_{\dot{y}}+h_x·d_G\left({\dot{y}}_n^s,y_n^t\right). \hfill \end{array}$$

(39)

Here, $h_x$ and $h_{\dot{y}}$ denote, respectively, the absolute bound of $\left(x_n\right)$ and $\left({\dot{y}}_n\right)$. It is easy to show that both the equivalences $\left({\dot{x}}_n+{\dot{y}}_n\right){~}_H\left(x_n+y_n\right)$ and $\left({\dot{x}}_n·{\dot{y}}_n\right){~}_H\left(x_n·y_n\right)$ hold, based on the inequalities provided, respectively, by Equations (38) and (39).

The addition and multiplication defined above inherit the properties of the addition and multiplication of the fields ${\mathbb{Q}}_{\left(p\right)}$ and the proof that ${\mathbb{R}}_G=\left(R_G,+, \cdot \right)$ is a field is straightforward. The additive identity is the class denoted as $\mathbf{0}$, which contains the sequence $\left(o_n\right)$, where $o_n=\left(\mathrm{0,1}\right)$ for any $n$. The multiplicative identity is the class denoted as $\mathbf{1}$, which contains the sequence $\left(u_n\right)$, where $u_n=\left(\mathrm{1,1}\right)$ for any $n$. A comprehensive definition and discussion of the properties of the field $\mathbb{R}$ of the real numbers is provided in classical textbooks, e.g., [17] (Chapter V). A brief summary is now given for a comparison with the field ${\mathbb{R}}_G$ defined above. First, the following definitions are introduced:

Definition 24.

A sequence $\left(q_n\right)$ of rational numbers is said to be a Cauchy sequence if, for any positive $\epsilon \in Q$, there exists $m\in \mathbb{N}$ such that $\left|q_s-q_t\right|\le \epsilon \mathrm{f}\mathrm{o}\mathrm{r} s,t\ge m.$

Definition 25.

Two Cauchy sequences$\left(a_n\right),\left(b_n\right)$ are said to be equivalent, with the notation $\left(a_n\right){~}_c\left(b_n\right)$, if for any $\epsilon >0\in Q$, there exists $m\in \mathbb{N}$ such that $\left|a_n-b_n\right|\le \epsilon \mathrm{f}\mathrm{o}\mathrm{r} n\ge m.$

The field of the real numbers $\mathbb{R}$ can be defined as $\left(R,+,\cdot \right)$, where $R=S_c/{~}_c$, $S_c$ is the set of the Cauchy sequences, and the binary operations of addition and multiplication of two elements$a,b\in R$are defined, respectively, as:

$$a+b=c;$$

(40)

$$a\cdot b=d,$$

(41)

where $\left(a_n\right)\in a,\left(b_n\right)\in b$, $\left(a_n\right)+\left(b_n\right)\in c$, and $\left(a_n\right)\cdot \left(b_n\right)\in d$.

The following theorem is stated:

Theorem 11.

The function $\tilde{\mathsf{\Psi }}:R_G\to R$ defined by

$$\tilde{\mathsf{\Psi }}:x\in R_G:\left(x_m\right)\in x\to \tilde{\mathsf{\Psi }}\left(x\right)\in R: \left(\mathsf{\Psi }\left(x_m^n\right)\right)\in \tilde{\mathsf{\Psi }}\left(x\right),$$

 is an isomorphism, and ${\mathbb{R}}_G=\left(R_G,+, \cdot \right)$ is a field isomorphic to the field $\mathbb{R}$.

For clarity, it should be noted that the function $\tilde{\mathsf{\Psi }}$ is defined by the following steps:

(1)

For $x\in R_G$, a representative of $x$ is chosen as the convergent sequence $\left(x_m\right)$

(2)

For each integer $n\ge 1,$ consider the sequence $\left(x_m^n\right)\in K$, such that $x_m^n=x_n$ for $m\ge n$

(3)

For each integer $n\ge 1$, $q_n\in Q$ is obtained given $q_n=\mathsf{\Psi }\left(x_m^n\right)$

(4)

The sequence $\left(q_n\right)$ is a Cauchy sequence, which is an element of the equivalence class, with respect to the relation ${~}_c$, denoted as $\tilde{\mathsf{\Psi }}\left(x\right)\in R$

(5)

The function $\tilde{\mathsf{\Psi }}$ maps $x\in R_G$ to $\tilde{\mathsf{\Psi }}\left(x\right)\in R.$

Proof. 

It is first proved that $\left(q_n\right)$ is a Cauchy sequence. For any positive $\epsilon \in Q$, there are two positive coprime integer numbers $a_{\epsilon },b_{\epsilon }$ and a positive integer $l$ such that $a_{\epsilon },b_{\epsilon }$are less than $\sqrt{p\left(l\right)/2}$ and $\epsilon =(a_{\epsilon },b_{\epsilon })$. Therefore, $\epsilon \in Q_{p\left(l\right)}$ is also a discrete rational number. Since $\left(x_m\right)$ is a convergent sequence, there exists $l^{\prime }\ge l$ such that $d\left(\left(x_m^s\right),\left(x_m^t\right)\right)\le \epsilon$ for $s,t\ge l^{\prime }$. The definition of the distance implies that $\mathrm{l}\mathrm{i}\mathrm{m}\left|x_m^s-x_m^t\right|\le \epsilon$ for $s,t\ge l^{\prime }$. The definition of the isomorphism $\mathsf{\Psi }$ thus implies $\left|q_s-q_t\right|\le \epsilon$ for $s,t\ge l^{\prime }$. It is then proved that $\tilde{\mathsf{\Psi }}$ descends to the quotient of its domain. Let $\left({\dot{x}}_m\right)\in x$ and $\left({\dot{q}}_n\right)=\left(\mathsf{\Psi }\left({\dot{x}}_m^n\right)\right)$. Since $\left({\dot{x}}_m\right){~}_H\left(x_m\right)$, there exists $l^{″}\ge l$ such that $d\left(\left({\dot{x}}_m^s\right),\left(x_m^t\right)\right)\le \epsilon$ for $s,t\ge l^{″}$. The definition of the distance implies that $\mathrm{l}\mathrm{i}\mathrm{m}\left|{\dot{x}}_m^s-x_m^t\right|\le \epsilon$ for $s,t\ge l^{″}$. The definition of the isomorphism $\mathsf{\Psi }$ thus implies $\left|{\dot{q}}_s-q_t\right|\le \epsilon$ for $s,t\ge l^{″}$. Hence, $\left(q_n\right)$ and $\left({\dot{q}}_n\right)$ are elements of the same equivalence class $\tilde{\mathsf{\Psi }}\left(x\right)$. The function $\tilde{\mathsf{\Psi }}$ is surjective. Let us consider a Cauchy sequence $\left(q_n\right)$ and a positive $\epsilon \in Q$ such that $\epsilon =(1,m)$ with positive integer $m$. There exists a positive integer number $\mu \left(m\right)$ such that both ${(a_m,b_m)\in q}_{\mu \left(m\right)} \mathrm{a}\mathrm{n}\mathrm{d} \epsilon \in Q_{p\left(\mu \left(m\right)\right)}$ and $\left|q_s-q_t\right|\le \epsilon \mathrm{f}\mathrm{o}\mathrm{r} s,t\ge \mu \left(m\right)$. Consider the sequence of rational numbers $\left(x_n\right)$ such that it is $x_n=(a_n,b_n)$ if $\mu \left(n\right)\le n\le \mu (n+1)$. The sequence $\left(x_n\right)$ is convergent. For any positive discrete rational number $\delta$, there exists a positive integer number $m$ such that $\delta \ge \epsilon$ and $\epsilon =(1,m)\in Q_{p\left(\mu \left(m\right)\right)}$. Therefore, it is $\delta \ge \epsilon \ge d(\left(x_n^s\right),\left(x_n^t\right))$ $\mathrm{f}\mathrm{o}\mathrm{r} s,t\ge \mu \left(m\right)$. The function $\tilde{\mathsf{\Psi }}$ maps $x\ni \left(x_n\right)$ to $\left(\mathsf{\Psi }\left(x_m^n\right)\right)\ni \left({\dot{q}}_n\right)=(a_n,b_n)$ if $\mu \left(n\right)\le n\le \mu (n+1)$. It can then be verified that $\left({\dot{q}}_n\right)$ ${~}_c$ $\left(q_n\right)$. The function $\tilde{\mathsf{\Psi }}$ preserves the addition and multiplication. Let us consider $x,y$ $\in R_G$ and, based on Lemma 1, two representative elements $\left({\dot{x}}_l\right)\in x$ and $\left({\dot{y}}_l\right)\in x$ as $8-\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}$ sequences. This implies that there exists $m\in \mathbb{N}$ such that if $\left(a_n,b_n\right)={\dot{x}}_n$ and $\left(c_n,d_n\right)={\dot{y}}_n$, being ${\dot{x}}_n,{\dot{y}}_n\in Q_{p\left(n\right)}$, the integer numbers$\left|a_n\right|,b_n,\left|c_n\right|,d_n$ are less than $\sqrt[8]{p\left(n\right)/2}$ for $n\ge m$. Hence, within $Q_{p\left(n\right)}$, by Theorems 6 and 7, for $n\ge m\ge 12,$

$${\dot{x}}_n+{\dot{y}}_n=\left(a_n,b_n\right)+\left(c_n,d_n\right)=\left(a_n·d_n+b_n·c_n,b_n·d_n\right);$$

(42)

$${\dot{x}}_n·{\dot{y}}_n=\left(a_n,b_n\right)·\left(c_n,d_n\right)=\left(a_n·c_n,b_n·d_n\right).$$

(43)

Indeed, $\left|a_n·d_n\right|,\left|a_n·c_n\right|,\left|b_n·c_n\right|,b_n·d_n\le \sqrt[4]{p\left(n\right)/2}<\sqrt{p\left(n\right)/2}$ for $n\ge m$. Therefore, by Theorem 9, they are elements of $Q_{p\left(n\right)}$. In addition, for $n\ge m$ and if $m\ge 12,$

$$\left|a_n·d_n+b_n·c_n\right|\le \left|a_n·d_n\right|+\left|b_n·c_n\right|\le 2·\sqrt[4]{p\left(n\right)/2}<\sqrt{p\left(n\right)/2}.$$

The last inequality follows from the condition $m\ge 12$ so that $p\left(n\right)\ge 37>2^5$. It ensures, by Theorem 9, that $\left(a_n·d_n+b_n·c_n\right)\in$ $Q_{p\left(n\right)}$. In conclusion, for $n\ge m\ge 12$, it is:

$$\mathsf{\Psi }\left(x_l^n+y_l^n\right)=\mathsf{\Psi }\left(x_l^n\right)+\mathsf{\Psi }\left(y_l^n\right);$$

(44)

$$\mathsf{\Psi }\left(x_l^n·y_l^n\right)=\mathsf{\Psi }\left(x_l^n\right)·\mathsf{\Psi }\left(y_l^n\right).$$

(45)

Hence, $\tilde{\mathsf{\Psi }}\left(x+y\right)=\tilde{\mathsf{\Psi }}\left(x\right)+\tilde{\mathsf{\Psi }}\left(y\right)$ and $\tilde{\mathsf{\Psi }}\left(x·y\right)=\tilde{\mathsf{\Psi }}\left(x\right)·\tilde{\mathsf{\Psi }}\left(y\right)$. □

The order and the distance function $d_{\tilde{G}}$ on $R_G$ are defined, respectively, as:

Definition 26.

For $x,y\in R_G$ the order $x{\ge }_{\tilde{G}}y$ holds if, for $\left(x_n\right)\in x$ and $\left(y_n\right)\in x$, there exists $m\in \mathbb{N}$ such that $\left(x_n^s\right){\ge }_G\left(y_n^s\right)$ for $s\ge m.$

Definition 27.

The distance function $d_{\tilde{G}}$ is defined on $R_G$ by:

$$(x,y)\in R_G^2\to d_{\tilde{G}}\left(x,y\right)=\left|x-y\right|\in R_G.$$

It can be verified that $\tilde{\mathsf{\Psi }}$ preserves the order and is an isometry between $d_{\tilde{G}}$ and the distance function $d_R$ defined in $R$ by $d_R\left(x,y\right)=\left|x-y\right|$. With respect to these metrics, the fields ${\mathbb{R}}_G$ and $\mathbb{R}$ are complete.

5. Complex Numbers

In this section, non-Pythagorean primes $p$ are used to define the field of discrete complex numbers ${\mathbb{C}}_p=\left(Q_p^2, +, \cdot \right)$. The addition and multiplication are defined by algebraic rules analogous to those of the complex numbers, as follows:

Definition 28.

For any $x=(a,b)\in Q_p^2$ and $y=(c,d)\in Q_p^2$, the operations of addition and multiplication are defined, respectively, as:

$$\left(a,b\right)+\left(c,d\right)=\left(a+c,b+d\right);$$

(46)

$$\left(a,b\right)\cdot \left(c,d\right)=\left(a\cdot c-b\cdot d, a\cdot d+b\cdot c\right).$$

(47)

The operations on the right-hand side in Equations (46) and (47) are performed in $Q_p$. It can be shown that ${\mathbb{C}}_p$ is a field. The additive and multiplicative identities are $\left(\mathrm{0,0}\right)=(\left(\mathrm{0,1}\right),\left(\mathrm{0,1}\right))$and $\left(\mathrm{1,0}\right)=(\left(\mathrm{1,1}\right),\left(\mathrm{0,1}\right))$, respectively. It is worth noting that the need of non-Pythagorean primes $p$ emerges when considering the solutions of the linear system in the field $Q_p$, which arises in the computation of the inverse of $\left(a,b\right)\ne \left(0, 0\right)$ with respect to the multiplication defined in $Q_p^2$ by Equation (47), namely:

$$\left(a,b\right)\cdot \left(x,y\right)=\left(\mathrm{1,0}\right).$$

(48)

This equation can also be written in matrix form:

$$\left( \begin{array}{cc}a& -b\\ b& a\end{array} \right)\left( \begin{array}{c}x\\ y\end{array} \right)=\left( \begin{array}{c}1\\ 0\end{array} \right).$$

(49)

It has a unique solution provided that $a^2+b^2\ne 0 mod\left(p\right)$, which implies that $p$ must be a non-Pythagorean prime to ensure that ${\mathbb{C}}_p$ is a field, since the inverse with respect to the multiplication for any element $\left(a,b\right)\ne \left(0, 0\right)$ must exist and be unique. If $z=\left(a,b\right)\in Q_p^2$, the elements $a,b$ are said, respectively, to be the real part and the imaginary part of $z$, indicated as $Re\left(z\right)$ and $Im\left(z\right)$. It should be emphasized that this is an abuse of notation, as it concerns elements of $Q_p^2$ and not the real and imaginary parts of complex numbers. The discrete complex number $i=\left(\mathrm{0,1}\right)\in Q_p^2$ is called the imaginary unit of ${\mathbb{C}}_p$, such that $i·i=\left(-\mathrm{1,0}\right)$.

Owing to Dirichlet’s theorem [19] (Chapter 4), the set of non-Pythagorean prime numbers is infinite, so that sequences of discrete complex numbers can be defined. In particular, constant sequences are introduced by the following:

Definition 29.

A sequence of discrete complex numbers $\left(z_n\right)$, such that $z_n\in Q_{p\left(n\right)}^2$ and $p\left(n\right)$ is the increasing sequence of non-Pythagorean prime numbers, is said to be a constant sequence if there exists a positive integer number $m$ such that $z_k= z_m$ for$k\ge m$. The constant value $z_m$ is said to be the limit value of the sequence $\left(z_n\right)$, indicated by the symbol $\mathrm{l}\mathrm{i}\mathrm{m}\left(z_n\right).$

Moreover, an equivalence relation ${~}_{K2}$ is introduced in the set $K2$ of the constant sequences of discrete complex numbers by the following:

Definition 30.

Two sequences $\left(z_n\right)\in K2$ and $\left(w_n\right)\in K2$ are said to be $K2\text{-}equivalent,$ and the relation $\left(z_n\right) {~}_{K2}\left(w_n\right)$ holds if $\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} \mathrm{e}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{s} m\in \mathbb{N}$ such that $z_s=w_s$$\mathrm{f}\mathrm{o}\mathrm{r} s\ge m$.

One readily verifies that two sequences of discrete complex numbers are $K2\text{-}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{t}$ if and only if they have the same limit value.

This allows one to introduce a set of Galois r-complex numbers:

Definition 31.

The quotient set $C_{rG}=K2/{~}_{K2}$ is said to be the set of Galois r-complex numbers.

Binary operations of addition and multiplication are defined on $C_{rG}$, denoted, respectively, as $+$ and $\cdot$, namely:

Definition 32.

Given$x,y\in C_{rG}$, let $\left(x_n\right)\in x$ and $\left(y_n\right)\in y$ and define the constant sequences:

$$\left(a_n\right)=\left(x_n+y_n\right);$$

(50)

$$\left(b_n\right)=\left(x_n\cdot y_n\right),$$

(51)

 where $+$ and $\cdot$ denote, respectively, the addition and multiplication in ${\mathbb{C}}_p$. The addition and multiplication in $C_{rG}$ are thus defined, respectively, as:

$$x+y=a;$$

(52)

$$x\cdot y=b,$$

(53)

 where $a,b\in C_{rG}$, $\left(a_n\right)\in a$, and $\left(b_n\right)\in b$.

The operations defined above descend to the quotient. The proof is analogous to that provided concerning the addition and multiplication in $Q_G$. The addition and multiplication defined above inherit the properties of the addition and multiplication of the fields ${\mathbb{C}}_p$, and the proof that ${\mathbb{C}}_{rG}=\left(C_{rG},+, \cdot \right)$ is a field is straightforward.

The additive identity is the class denoted as $\stackrel{-}{\mathbf{0}}$, which contains the sequence $\left({\stackrel{-}{o}}_n\right)$, and ${\stackrel{-}{o}}_n=\left(\left(\mathrm{0,1}\right),\left(\mathrm{0,1}\right)\right)$ for any $n$. The multiplicative identity is the class denoted as $\stackrel{-}{\mathbf{1}}$, which contains the sequence $\left({\stackrel{-}{u}}_n\right)$, with ${\stackrel{-}{u}}_n=\left(\left(\mathrm{1,1}\right),\left(\mathrm{0,1}\right)\right)$ for any $n$. All the elements of an equivalence class $x$ have the same limit value, which is denoted as $\lim \left(x\right)$.

It is easy to show that the field ${\mathbb{C}}_{rG}$ is a metric space, endowed with the distance function $d_{rG}$:

Definition 33.

For $x,y\in C_{rG}$, $\left(a_n,b_n\right)\in x$, and $\left(c_n,d_n\right)\in y$, the distance function $d_{rG}$ is defined as

$$d_{rG}\left(x,y\right)=\lim \left|a_n-c_n\right|+\mathrm{l}\mathrm{i}\mathrm{m}\left|b_n-d_n\right|.$$

(54)

One readily verifies that the definition of the distance $d_{rG}$ is meaningful, i.e., it descends to the quotient. From a formal point of view, to avoid any reference to the field ${\mathbb{Q}}_G$ and the relevant notion of limit, one might define identically:

$$d_{rG}\left(x,y\right)=\left|Re\left(\mathrm{l}\mathrm{i}\mathrm{m}\left(x-y\right)\right)\right|+\left|Im\left(\mathrm{l}\mathrm{i}\mathrm{m}\left(x-y\right)\right)\right|.$$

(55)

The adoption of this distance [20] (Chapter 1) is useful in the present context. The Euclidean distance is not suitable since the square root cannot be performed as an internal operation. An ordering between the distances is established by the following:

Definition 34.

If $d_{rG}\left(x,y\right)=\left(\alpha ,\beta \right)$ and $d_{rG}\left(x^{\prime },y^{\prime }\right)=\left({\alpha }^{\prime },{\beta }^{\prime }\right)$, the order $d_{rG}\left(x,y\right)\ge d_{rG}\left(x^{\prime },y^{\prime }\right)$ holds if $\alpha ·{\beta }^{\prime }\ge \beta · {\alpha }^{\prime }$.

The field ${\mathbb{C}}_{rG}$ is not a new field; rather, it is a different representation of the complex quadratic field $\mathbb{Q}\left(i\right)$ [21] (Chapter 1). The latter is defined as $\left(Q^2,+,·\right),$ with addition and multiplication given by:

Definition 35.

For any $x=(a,b)\in Q^2$ and $y=(c,d)\in Q^2$, the operations of addition and multiplication are defined, respectively, as:

$$\left(a,b\right)+\left(c,d\right)=\left(a+c, b+d\right),$$

(56)

$$\left(a,b\right)\cdot \left(c,d\right)=\left(a\cdot c-b\cdot d, a\cdot d+b\cdot c\right).$$

(57)

The operations on the right-hand side in Equations (56) and (57) are performed in $Q$.

Consider the following function:

$${\mathsf{\Psi }}_2:x\in C_{rG},(\left(a_n\right),\left(b_n\right))\in x\to {\mathsf{\Psi }}_2\left(x\right)=\left(\mathsf{\Psi }\left(\left(a_n\right)\right),\mathsf{\Psi }\left(\left(b_n\right)\right)\right)\in Q^2.$$

It is well defined in the sense that it descends to the quotient. Based on analog proofs performed in previous sections, it is easy to prove that ${\mathsf{\Psi }}_2$ is an isomorphism, and ${\mathbb{C}}_{rG}$ is a field isomorphic to $\mathbb{Q}\left(i\right)$. Moreover, one can define in $\mathbb{Q}\left(i\right)$ the distance function:

$${d_1:\left(x,y\right)\in Q^2, \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} x=\left(a,b\right),y=\left(c,d\right) \to d}_1\left(x,y\right)=\left|a-c\right|+\left|b-d\right|.$$

It can be shown that ${\mathsf{\Psi }}_2$ is also an isometry, with respect to $d_{rG}$ and $d_1$.

The basic objects used to obtain a field that is isomorphic to the field $\mathbb{C}$ of the complex numbers are the convergent sequences of discrete complex numbers. These are introduced by the following definitions:

Definition 36.

Given a sequence of discrete complex numbers $\left(z_n\right)$ such that $n\in \mathbb{N}$ and $z_n\in Q_{p\left(n\right)}^2$, any sequence $\left(z_n^m\right)\in K2$ such that $z_s^m=q_m$ for $s\ge m$ is said to be $m-associated$ with $\left(z_n\right)$.

Definition 37.

A sequence of discrete complex numbers $\left(z_n\right)$ such that $n\in \mathbb{N}$ and $z_n\in Q_{p\left(n\right)}^2$ is said to be a convergent sequence if for any positive discrete rational number$\epsilon$ there exists $m\in \mathbb{N}$ such that $\epsilon \in Q_{p\left(m\right)}$ and

$$\epsilon \ge d_{rG}(\left(z_n^s\right),\left(z_n^t\right))\mathrm{if} s,t\ge m.$$

The sequences $\left(z_n^s\right)$ and $\left(z_n^t\right)$ are, respectively, s-associated and t-associated with $\left(z_n\right)$. The set $H2$ of the convergent sequences includes the set $K2$. An equivalence relation ${~}_{H2}$ is introduced in the set $H2$ of the convergent sequences of discrete complex numbers:

Definition 38.

Two sequences $\left(z_n\right),\left(w_n\right)\in H2$ are said to be $H2\text{-}equivalent,$ and the relation $\left(z_n\right) {~}_{H2} \left(w_n\right)$ holds if for any discrete rational number $\epsilon$ there exists $m_{\epsilon }\in \mathbb{N}$ such that

$$\epsilon \ge d_{rG}(\left(z_n^s\right),\left(\left(w_n^t\right)\right))\mathrm{for} \mathrm{any} s,t\ge m_{\epsilon }.$$

Note that the sequences $\left(z_n^s\right)$ and $\left(w_n^t\right)$ are, respectively, s-associated and t-associated with $\left(z_n\right)$ and $\left(w_n\right)$. The following Lemma 3 and Lemma 4 allow one to prove that, introducing the binary operations of addition $+$and multiplication $·$ of convergent sequences, as defined in the following, one obtains a field $C_G=\left(H2,+,·\right),$ which is isomorphic to the field $\mathbb{C}.$

Lemma 3.

For any convergent sequence $\left(z_n\right)$, there exists an $H2\text{-}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{t}$ convergent sequence $\left({\dot{z}}_n\right)$ and $m_k\in \mathbb{N}$ such that letting $\left(\left(a_n,b_n\right),(c_n,d_n)\right)={\dot{z}}_n$, where $a_n,b_n,c_n,d_n$ are less than $\sqrt[k]{p\left(n\right)/2}$  $\mathrm{f}\mathrm{o}\mathrm{r} n\ge m_k.$

The proof can be easily obtained by adopting the method used for the proof of Lemma 1. Lemma 3 leads to the following:

Definition 39.

For any convergent sequence of discrete complex numbers $\left(z_n\right)$, any convergent sequence $\left({\dot{z}}_n\right){ ~}_{H2} \left(z_n\right)$ such that ${\dot{z}}_n=\left(\left(a_n,b_n\right),(c_n,d_n)\right)$, with $a_n,b_n,c_n,d_n$ less than $\sqrt[k]{p\left(n\right)/2}$ for $n\ge m$, is said to be a $k-normal sequence$ associated with $\left(z_n\right)$.

Lemma 4.

For any convergent sequence $\left(z_n\right)$, there exists a constant sequence $\left({\stackrel{-}{z}}_n\right)$ and $m\in \mathbb{N}$ such that

$$\left|Re\left(z_n\right)\right|+|Im\left(z_n\right)|\le |Re\left({\stackrel{-}{z}}_n\right)|+|Im\left({\stackrel{-}{z}}_n\right)|\mathrm{for} n\ge m.$$

The proof is straightforward. Lemma 4 leads to the following:

Definition 40.

For any convergent sequence $\left(z_n\right)$, a constant sequence $\left({\stackrel{-}{z}}_n\right)$, such that

$$\left|Re\left(z_n\right)\right|+\left|Im\left(z_n\right)\right|\le \left|Re\left({\stackrel{-}{z}}_n\right)\right|+\left|Im\left({\stackrel{-}{z}}_n\right)\right|\mathrm{for} n\ge m,$$

is said to be a bounding sequence of $\left(z_n\right)$, and $lim\left(\left|Re\left({\stackrel{-}{z}}_n\right)\right|+\left|Im\left({\stackrel{-}{z}}_n\right)\right|\right)$ is called an absolute bound of $\left(z_n\right)$.

The following definitions allow for the constructive realization of a field isomorphic to $\mathbb{C}$:

Definition 41.

The quotient $C_G=H2/{~}_{H2}$ is said to be the set of Galois complex numbers.

Definition 42.

For any $x\in C_G$ and $y\in C_G$, letting $\left(x_n\right)\in x$ and $\left(y_n\right)\in y$, the operations of addition and multiplication are defined, respectively, as:

$$x+y=a;$$

(58)

$$x\cdot y=b,$$

(59)

with $\left(x_n+y_n\right)\in a$ and $\left(x_n·y_n\right)\in b$.

Definition 42 is meaningful, i.e., it descends to the quotient. The relevant proof can be easily obtained, based on Lemma 4 and Definition 40, by the same method used above to prove that Definition 23 descends to the quotient. The addition and multiplication defined above inherit the properties of the addition and multiplication of the fields ${\mathbb{C}}_p$, and the proof that ${\mathbb{C}}_G=\left(C_G,+, \cdot \right)$ is a field is straightforward. The additive identity is the class denoted as $\stackrel{-}{\mathbf{0}}$, which contains the sequence $({\stackrel{-}{o}}_n)$. The multiplicative identity is the class denoted as $\stackrel{-}{\mathbf{1}}$, which contains the sequence $({\stackrel{-}{u}}_n)$. The field ${\mathbb{C}}_G$ is now compared with the field of complex numbers $\mathbb{C}$. A comprehensive definition and discussion of the properties of the field $\mathbb{C}$ of the complex numbers is provided in classical textbooks, e.g., [17] (Chapter VII). Here, a less common definition is adopted, which is based on Cauchy sequences of pairs of rational numbers within the metric $d_1$ defined above. However, this is equivalent to the standard definition based on pairs of Cauchy sequences of rational numbers, i.e., on pairs of real numbers. As a preliminary step, the following definitions are introduced:

Definition 43.

A sequence $\left(z_n\right)$ of elements of $Q^2$ is said to be a Cauchy sequence in $Q^2$ if, given any $\epsilon >0\in Q$, there exists $m\in \mathbb{N}$ such that $d_1\left(z_s,z_t\right)\le \epsilon \mathrm{f}\mathrm{o}\mathrm{r} s,t\ge m.$

Definition 44.

Two Cauchy sequences $\left(z_n\right),\left(w_n\right)$ are said to be equivalent, with the notation $\left(z_n\right){~}_{c1}\left(w_n\right)$, if for any positive $\epsilon \in Q$ there exists $m\in \mathbb{N}$ such that $d_1\left(z_n,w_n\right)\le \epsilon \mathrm{f}\mathrm{o}\mathrm{r} n\ge m.$

The field of the complex numbers $\mathbb{C}$ can be defined as $\left(C,+,\cdot \right)$, where $C=S_{c1}/{~}_{c1}$, with $S_{c1}$ being the set of the Cauchy sequences in $Q^2$, and the binary operation of addition and multiplication of two elements$a,b\in C$are defined, respectively, as:

$$a+b=c;$$

(60)

$$a\cdot b=d.$$

(61)

Here, $\left(a_n\right)\in a,\left(b_n\right)\in b$, $\left(a_n+b_n\right)\in c$, and $\left(a_n\cdot b_n\right)\in d$, and the relevant operations are defined as in $\mathbb{Q}\left(i\right)$.

The following theorem is now stated:

Theorem 12.

The function $\hat{\mathsf{\Psi }}:C_G\to C$ defined by:

$$\hat{\mathsf{\Psi }}:z\in C_G:\left(z_m\right)\in z\to \hat{\mathsf{\Psi }}\left(z\right)\in C: \left({\mathsf{\Psi }}_2\left(z_m^n\right)\right)\in \hat{\mathsf{\Psi }}\left(x\right),$$

is an isomorphism, and${\mathbb{C}}_G$ is a field isomorphic to the field $\mathbb{C}$.

For clarity, it should be noted that the function $\hat{\mathsf{\Psi }}$ is defined by means of the following steps:

(1)

For $z\in C_G$, a representative of $z$ is chosen as the convergent sequence $\left(z_m\right)$

(2)

For each integer $n\ge 1,$ consider the sequence $\left(z_m^n\right)\in K2$, such that $z_m^n=z_n$ for $m\ge n$

(3)

For each integer $n\ge 1$, $z_n\in Q^2$ is obtained given $z_n={\mathsf{\Psi }}_2\left(z_m^n\right)$

(4)

The sequence $\left(z_n\right)$ is a Cauchy sequence in $Q^2$, which is an element of the equivalence class $\hat{\mathsf{\Psi }}\left(z\right)$$\in C$, with respect to the relation ${~}_{c2}$

(5)

The function $\hat{\mathsf{\Psi }}$ maps $z\in C_G$ to $\hat{\mathsf{\Psi }}\left(z\right)\in C.$

The proof of Theorem 12 can be obtained adopting the same method used to prove Theorem 11. After having established that ${\mathbb{C}}_G$ and $\mathbb{C}$ are isomorphic via $\hat{\mathsf{\Psi }}$, the Euclidean distance is now introduced:

Definition 45.

For $x,y\in C_G$, their Euclidean distance is defined as:

$$d_{G2}\left(x,y\right)=\sqrt{{\left[Re\left(x-y\right)\right]}^2+{\left[Im\left(x-y\right)\right]}^2}.$$

One readily verifies that $\hat{\mathsf{\Psi }}$ is an isometry between $d_{G2}$ and the Euclidean distance defined in $C$:

$$d_2\left(x,y\right)=\sqrt{{\left[Re\left(x-y\right)\right]}^2+{\left[Im\left(x-y\right)\right]}^2},$$

for $x,y\in C$. With respect to these metrics, the fields ${\mathbb{C}}_G$ and $\mathbb{C}$ are complete.

6. Summary and Discussion

An algorithmic procedure (Algorithm 1) has been introduced as the basis for an explicit realization of a finite field ${\mathbb{Q}}_{\left(p\right)}=(Q_p,+,·)$, which is isomorphic to the Galois field ${\mathbb{F}}_p=\left(F_p,+,·\right)$, for any prime number $p$. Its elements, called discrete rational numbers, are given as a set $Q_p$ of fractions $a/b$, such that $a,b>0\in \mathbb{Z}$, and $\left|a\right|,b$ are coprime. ${\mathbb{Q}}_{\left(p\right)}$ is not a new field, being merely a different representation in terms of fractions of the isomorphic field ${\mathbb{F}}_p$. This representation was made possible thanks to the theorem due to Laboutin and Murchio [1], here reported as Theorem 4, which allows for a minimal representation of the elements of ${\mathbb{F}}_p$ in terms of fractions of integer numbers with absolute values less than $\sqrt{p}$. Actually, numerical results, as shown in Figure 1, suggest that the specific representation obtained by Algorithm 1 is achieved with integer numbers with maximum absolute values equal to $Int\sqrt{p}$ or $Int\sqrt{p} -1$. An interesting direction for future work is to confirm this conjecture. Even more interesting would be to calculate the set $Q_p$ and the relevant bijective mapping into $F_p$ in closed form instead of deriving it via an algorithmic procedure. Discrete rational numbers are used as basic objects to construct fields isomorphic to the classical fields of the rational, real, and complex numbers. Sequences $\left(q_n\right)$ of discrete rational numbers such that $q_n\in Q_{p\left(n\right)}$, where $q_n=q_m$ for $n\ge m\in \mathbb{N}$, are introduced, considering the increasing sequence $p\left(n\right)$ of the prime numbers. Such constant sequences, provided with an equivalence relation, naturally based on the equality of the limit values, and with addition and multiplication term by term, define a field ${\mathbb{Q}}_G$ isomorphic to $\mathbb{Q}$. The isomorphism established between ${\mathbb{Q}}_G$ and $\mathbb{Q}$ is also an isometry between the distance functions, defined on ${\mathbb{Q}}_G$ as the limit value of the absolute value of the differences term by term and on $\mathbb{Q}$ as the absolute value of the difference of rational numbers. The definition of distance on ${\mathbb{Q}}_G$ also allows one to introduce a suitable definition of Cauchy sequences $\left(q_n\right)$ of discrete rational numbers and the relevant equivalence relation. To this end, the constant sequences $\left(q_n^m\right)$, derived from $\left(q_n\right)$ by the condition $q_n^m=q_n$ for $n\ge m\in \mathbb{N}$, have been used.

A field ${\mathbb{R}}_G$ is obtained by defining the addition and multiplication term by term within the equivalence class $R_G$ of the Cauchy sequences of discrete rational numbers. It is proven that ${\mathbb{R}}_G$ is isomorphic to $\mathbb{R}$. The isomorphism established in Theorem 11 is also an isometry with respect to the function distance defined in ${\left(R_G\right)}^2$ and in ${\left(R\right)}^2$ by the absolute value of the difference $\left|x-y\right|$ for any given pair $\left(x,y\right)$. The fields ${\mathbb{R}}_G$ and $\mathbb{R}$ are complete with respect to such a metric. An analogous method is used to define a field ${\mathbb{C}}_p$ of pairs of discrete rational numbers, provided with the definitions of addition and multiplication algebraically equivalent to those adopted in $\mathbb{C}$, and then to define a field ${\mathbb{C}}_{rG}$, based on constant sequences, which is isomorphic to $\mathbb{Q}\left(i\right)$. An important point is that the prime numbers involved must be non-Pythagorean. This is required to ensure that any non-zero pair in the set of pairs of discrete rational numbers has a unique inverse with respect to the multiplication. The isomorphism between ${\mathbb{C}}_{rG}$ and $\mathbb{Q}\left(i\right)$ is also an isometry with respect to distance functions defined on ${\mathbb{C}}_{rG}$ and $\mathbb{Q}\left(i\right)$ as $d_{rG}\left(x,y\right)=\left|Re\left(\mathrm{l}\mathrm{i}\mathrm{m}\left(x-y\right)\right)\right|+\left|Im\left(\mathrm{l}\mathrm{i}\mathrm{m}\left(x-y\right)\right)\right|$ and $d_1\left(x,y\right)=\left|Re\left(x-y\right)\right|+\left|Im\left(x-y\right)\right|$, respectively. Here, $Re\left(z\right)$ and $Im\left(z\right)$ give, respectively, the first and second elements of the pair $z=(a,b)$. The distance $d_{rG}$ is then used to define Cauchy sequences of pairs of discrete rational numbers, the relevant equivalence relation, and, adopting addition and multiplication term by term, a field ${\mathbb{C}}_G$isomorphic to $\mathbb{C}$. The isomorphism, as defined in Theorem 12, is also an isometry with respect to the Euclidean metric in ${\mathbb{C}}_G$ and $\mathbb{C}$, which are complete metric spaces with respect to it. In summary, a constructive realization of rational, real, and complex numbers based solely on finite fields has been obtained. The construction provides an explicit link between finite cardinality number sets and the classical fields $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$. The applications of the field ${\mathbb{Q}}_{\left(p\right)}$, briefly discussed in the following, represent potential avenues of research rather than a concrete plan of work. ${\mathbb{Q}}_{\left(p\right)}$ might have applications in the framework of computational physics. Modeling of physical systems can be performed by means of approximations of differential and integral operators in terms of finite differences and sums. By appropriate normalization, the spectrum of values of the physical quantities involved might be limited to a suitably small range, such that the discrete rational numbers are sufficiently dense to capture the essential physical values. In this vein, it should be noted that the mappings ${\mathsf{\Psi }}_p^{-1}$ are concentrated around small fractions, as shown in Figure 2. Therefore, a computation based on ${\mathbb{Q}}_{\left(p\right)}$might benefit from modular arithmetic combined with a rich spectral coverage, possibly leading to efficient calculations. However, this requires controlling the effects of aliasing and discretization arising from the adoption of finite fields, in analogy with signal processing based on the discrete Fourier transform (see [22] (Chapters 2, 4, and 8)). It should be noted that operational definitions of the measure of a physical quantity do not in practice require the use of infinitesimal or infinite numbers. Measurement outcomes are conventionally described by a pair of quantities expressed in a given unit of measure: an expected value and an experimental uncertainty. Both quantities can be represented as elements of discrete subsets of rational numbers. This observation suggests that physical measurements may be consistently modeled within discrete rational numbers, opening the possibility for applications of the fields ${\mathbb{Q}}_{\left(p\right)}$ in physical theories. In this vein, a finite structure of space–time with coordinates in ${\mathbb{Q}}_{\left(p\right)}^4$ might be considered, following the framework of general or special relativity [23] (Chapters 2 and 8–11). Within this approach, owing to the adoption of finite fields, no global causal structure is allowed. Consequently, the assumptions required to formulate Bell inequalities [24] cannot be applied globally. This allows for a classical-like formulation of quantized fields that would otherwise be forbidden by Bell’s theorem, based on finite space–time. This possibility has been explored in previous approaches such as causal set theory [25] and causal dynamical triangulation [26], which formulate a consistent quantum field theory on discrete or finite space–times.

A further possible application of the fields ${\mathbb{Q}}_{\left(p\right)}$ and ${\mathbb{C}}_p$ concerns the approximation of real or complex functions by sequences of functions (see [18] (Chapter 7)). It is known that every mapping in a finite field can be expressed by a polynomial, as reminded by Paul Moritz Cohn in his preface to [15]. However, rational functions might be used to extend the spectrum of functions defined in ${\mathbb{C}}_p$ to approximate analytic functions. This appears to be a promising direction, since Runge’s theorem ensures that any function analytic on a domain can be uniformly approximated on compact subsets by rational functions with poles outside the domain [27] (Chapter VIII).