The present paper aims at stating the NJIKI’s fundamental THEOREM-DEFINITION and the ONE being supported by two new operations of addition or additive operations in the mathematical set $\mathbb{Q}$, with along attached to two LEMMAS and subsidiarily one in two or two in one other nice related NJIKI’s THEOREM and some EXERCISES, and not to forget the numerous and various algebraic structures to which it leads, in the Appendix B. And the research background so far therein lacking THEM all, and not to talk about the situation of the mathematical former researches as a whole and which had straight forward jumbled into $ℝ$ and $ℂ$ forgetting THEM back. And as for the list of the problems to be solved, it can be referred to in the end of PART 4 of the paper.

There do EXIST within the mathematical set $\mathbb{Q}$ of fractions, besides the traditional addition +, for two fractions’ SUM TOTAL calculation, and mostly for COUNTING and used by Serge Lang [1] , Jean-Pierre Scoffier [2] , Nicolas Bourbaki [3] , Gilles Lachaud [4] , Michel Demazure [5] , among others, and the modular forms somewhat related to ${+}_{\alpha ,1}$ or ${+}_{1,\beta }$, and usually referred to as the 5^th operation according to Gérard Eichler [6] - [9] who might have said it on day, and whereas indeed being between $\oplus$ and ${+}_{\alpha ,\beta }$ in terms or in orders of generalisation partially and totally when the half plane of Poincarré [10] , gets involved into account, being put apart, two new other operations of addition or additive operations and namely: 1) the natural or granulometric arithmetic mean, $\oplus$, and 2) the hybrid or composite arithmetic mean, ${+}_{\alpha ,\beta }$ with $\left(\alpha ,\beta \right)\in {\mathbb{Q}}^{*2}$, and for two fractions’ SUM MEANS calculation, and mainly for MEASURING, making them be taken as NJIKI’s VECTORS or ALIKE, on one hand, and on the other hand as the 6^th and the 7^th operations due to NJIKI and as far as mathematics are concerned. They are DEFINED as follows:

1) $\oplus :\mathbb{Q}\times \mathbb{Q}\to \mathbb{Q}$

$\left(\frac{b}{a},\frac{d}{c}\right)\mapsto \oplus \left(\frac{b}{a},\frac{d}{c}\right)=\frac{b}{a}\oplus \frac{d}{c}=\frac{b+d}{a+c}$(1)

2) ${+}_{\alpha ,\beta }:\mathbb{Q}\times \mathbb{Q}\to \mathbb{Q}$

$\left(\frac{b}{a},\frac{d}{c}\right)\mapsto {+}_{\alpha ,\beta }\left(\frac{b}{a},\frac{d}{c}\right)=\frac{b}{a}{+}_{\alpha ,\beta }\frac{d}{c}=\frac{\alpha b+\beta d}{\alpha a+\beta c}$(2)

Remarks: 1) $\oplus$ and ${+}_{1,1}$ are identical and that is that ${+}_{\alpha ,\beta }$ generalizes $\oplus$ 2) $\oplus$ makes it logically clearer and more esthetic, cases generalization including $\frac{b}{a}=\frac{d}{c}=\frac{b+d}{a+c}$¹ among other rational numbers logics and not curiosities, and meaningful than the addition of DUNCE whose first user is FAREY [11] , and through Lester R. Ford Sr [12] - [14] , mathematical later works and not LOBACHEVSKI [15] - [17] , and that according to the following proof.

1) Let

$m_{a,c}\left(\frac{b}{a},\frac{d}{c}\right)=\frac{b}{a}{m}_{a,c}\frac{d}{c}$(3)

or

$m_{\frac{a}{a+c},\frac{c}{a+c}}\left(\frac{b}{a},\frac{d}{c}\right)=\frac{b}{a}{m}_{\frac{a}{a+c},\frac{c}{a+c}}\frac{d}{c}$(4)

be the natural or the granulometric arithmetic mean of $\frac{b}{a}$ and $\frac{d}{c}$ and that is to say that weighted and affected with weighing coefficients a and c, or, that other weighted and affected with weighting coefficients $\frac{a}{a+c}$ and $\frac{c}{a+c}$ respectively. It is obviously or by simple calculations SHOWN from those above that:

$\oplus =m_{a,c}\vee \oplus =m_{\frac{a}{a+c},\frac{c}{a+c}}$(5)

And so then, the EXISTENCE of $\oplus$ announced in the THEOREM and the DEFINITION following IT as well.

Now, as how to get or come about the natural or granulometric weighting coefficients a and c, or, $\frac{a}{a+c}$ and $\frac{c}{a+c}$, one and the first of the following others NJIKI’s THEOREMS enables it. And the which can be taken as a LEMMA accordingly.

2) Let

$m_{\alpha a,\beta c}\left(\frac{b}{a},\frac{d}{c}\right)=\frac{b}{a}{m}_{\alpha a,\beta c}\frac{d}{c}$(6)

or

$m_{\frac{\alpha a}{a+c},\frac{\beta c}{a+c}}\left(\frac{b}{a},\frac{d}{c}\right)=\frac{b}{a}{m}_{\frac{\alpha a}{a+c},\frac{\beta c}{a+c}}\frac{d}{c}$(7)

be the hybrid or the composite arithmetic mean of $\frac{b}{a}$ and $\frac{d}{c}$ and that is to say that weighted and affected with weighing coefficients $\alpha a$ and $\beta c$, or, that other weighted and affected with weighting coefficients $\frac{\alpha a}{a+c}$ and $\frac{\beta c}{a+c}$ respectively. It is obviously or by simple calculations SHOWN from those above that:

${+}_{\alpha ,\beta }=m_{\alpha a,\beta c}\vee {+}_{\alpha ,\beta }=m_{\frac{\alpha a}{a+c},\frac{\beta c}{a+c}}$(8)

And so then, the EXISTENCE of ${+}_{\alpha ,\beta }$ announced in the THEOREM and the DEFINITION THAT follows IT as well.

Now, as how to get or come about the hybrid or composite weighting coefficients $\alpha a$ and $\beta c$, or, $\frac{\alpha a}{a+c}$ and $\frac{\beta c}{a+c}$, one and the first of the following others NJIKI’s THEOREMS enables it. And the which can be taken as a LEMMA accordingly and along with the observation preceding IT.

That above can so be taken as a very nice welcome back to the Euclidian geometry after almost 3 centuries from now back to N. I. LOBACHEVSKI, to be considered as the father of the non-Euclidian or hyperbolic geometries, and 20 centuries from LOBACHEVSKI to Euclid [18] [19] . And all things making NJIKI to become the leader or the pioneer of the neo Euclidian school in mathematics.

Show the following:

(A) Given $\left(a,b,c,d\right)\in {\mathbb{Z}}^4\wedge \left(a\ne 0\wedge c\ne 0\right)\wedge \left(\alpha ,\beta \right)\in {\mathbb{Z}}^{*2}$,

$\left(\frac{b}{a}\oplus \frac{d}{c}\right)+\frac{c}{a+c}\frac{b}{a}+\frac{a}{a+c}\frac{d}{c}=\frac{bc+ad}{ac}=\frac{b}{a}+\frac{d}{c}$

$\left(\frac{b}{a}{+}_{\alpha ,\beta }\frac{d}{c}\right)+\frac{\beta c}{\alpha a+\beta c}\frac{b}{a}+\frac{\alpha a}{\alpha a+\beta c}\frac{d}{c}=\frac{b}{a}+\frac{d}{c}$

(B) $\forall \left(\frac{b}{a},\frac{d}{c}\right)\in \left(\mathbb{Q},+,\times \right)$

(1)	$\frac{\alpha b+\beta d}{\alpha a+\beta c}=\frac{b}{c}+\frac{d}{c}$		$\alpha a^{2}d+\beta b{c}^2=0$
(2)	$\frac{\alpha b+\beta d}{\alpha a+\beta c}=\frac{1}{2}\left(\frac{b}{a}+\frac{d}{c}\right)$		$\left(bc-ad\right)\left(\alpha a-\beta c\right)=0$
(2)'	$\frac{b+d}{a+c}=\frac{1}{2}\left(\frac{b}{a}+\frac{d}{c}\right)$		$\left(bc-ad\right)\left(a-c\right)=0$
(3)	$\frac{b+d}{a+c}=\frac{1}{\alpha +\beta }\left(\alpha \frac{b}{a}+\beta \frac{d}{c}\right)$		$\left(bc-ad\right)\left(\beta a-\alpha c\right)=0$
(3)'	$\frac{\alpha b+\beta d}{\alpha a+\beta c}=\frac{1}{\alpha +\beta }\left(\alpha \frac{b}{a}+\beta \frac{d}{c}\right)$		$\left(bc-ad\right)\left(\alpha \beta a-\alpha \beta c\right)=0$
			$\alpha \beta \left(bc-ad\right)\left(a-c\right)=0$
(4)	$\frac{1}{2}\left(\frac{b}{a}+\frac{d}{c}\right)=\frac{1}{\alpha +\beta }\left(\alpha \frac{b}{a}+\beta \frac{d}{c}\right)$		$\left(bc-ad\right)\left(\alpha ac-\beta ac\right)$
			$ac\left(bc-ad\right)\left(\alpha -\beta \right)=0$
(4)'	$\frac{\alpha b+\beta d}{\alpha a+\beta c}=\frac{b+d}{a+c}$		$\begin{array}{l}\left(\alpha -\beta \right)\left(bc-ad\right)\\ =0\vee \left(\beta -\alpha \right)\left(bc-ad\right)=0\end{array}$
(0)	$\frac{\alpha b+\beta d}{\alpha a+\beta c}=\frac{b}{a}+\frac{d}{c}$		$bc-ad\ne 0$
(3)(bis)	$\frac{b+d}{a+c}=\frac{1}{\alpha +\beta }\left(\alpha \frac{b}{a}+\beta \frac{d}{c}\right)$		$\alpha =\frac{a}{a+c}\wedge \beta =\frac{c}{a+c}$
(4)'(bis)	$\frac{\alpha b+\beta d}{\alpha a+\beta c}=\frac{b+d}{a+c}$		$\alpha =\beta =1$

(C) ${+}_{\alpha ,\beta }\left(\frac{b}{a},\frac{d}{c}\right)=m_{\frac{\alpha a}{\alpha a+\beta c},\frac{\beta c}{\alpha a+\beta c}}\left(\frac{b}{a},\frac{d}{c}\right)$

And A, B and C being referred to as the NJIKI’s mathematical IDENTITIES, the NJIKI’s logical EQUIVALENCES and the NJIKI’s 3^rd value of ${+}_{\alpha ,\beta }$ named the “weighted-granulometric” one, ${+}_{\alpha ,\beta }=m_{\frac{\alpha a}{\alpha a+\beta c},\frac{\beta c}{\alpha a+\beta c}}$, and besides: 1) ${+}_{\alpha ,\beta }=m_{\alpha a,\beta c}$ and, 2) ${+}_{\alpha ,\beta }=m_{\frac{\alpha a}{a+c},\frac{\beta c}{a+c}}$, respectively. And definitely: ${+}_{\alpha ,\beta }\in \left\{m_{\alpha a,\beta c},m_{\frac{\alpha a}{a+c},\frac{\beta c}{a+c}},m_{\frac{\alpha a}{\alpha a+\beta c},\frac{\beta c}{\alpha a+\beta c},\cdots }\right\}$.

It can be done in two parts and as follows:

1) A mathematical study of the relation of equality, marked ${=}_{1,1}$, or, ${=}_{\alpha ,\beta }$, and knowing that ${=}_{1,1}={=}_{\alpha ,\beta }$, within the $\mathbb{Q}$ equivalence classes or classes of equivalence $\frac{\stackrel{\dot{}}{b}}{a}$ or $\frac{\stackrel{\dot{}}{d}}{c}$.

The relation of equality, marked ${=}_{1,1}$, or, ${=}_{\alpha ,\beta }$, within the $\mathbb{Q}$ equivalence classes or classes of equivalence $\frac{\stackrel{\dot{}}{b}}{a}$ or $\frac{\stackrel{\dot{}}{d}}{c}$, is an ADDITIVE internal composition law and as an application defined as follows:

${=}_{1,1}$, or, ${=}_{\alpha ,\beta }$: $\frac{\stackrel{\dot{}}{b}}{a}\times \frac{\stackrel{\dot{}}{b}}{a}\to \frac{\stackrel{\dot{}}{b}}{a}$

$\left(\frac{b}{a},\frac{d}{c}\right)\mapsto {=}_{\alpha ,\beta }\left(\frac{b}{a},\frac{d}{c}\right)=\frac{b}{a}{=}_{\alpha ,\beta }\frac{d}{c}=\frac{b+d}{a+c}=\frac{\alpha a+\beta d}{\alpha a+\beta c}$

or

${=}_{1,1}$, or, ${=}_{\alpha ,\beta }$: $\frac{\stackrel{\dot{}}{d}}{c}\times \frac{\stackrel{\dot{}}{d}}{c}\to \frac{\stackrel{\dot{}}{d}}{c}$

$\left(\frac{b}{a},\frac{d}{c}\right)\mapsto {=}_{1,1}\left(\frac{b}{a},\frac{d}{c}\right)=\frac{b}{a}{=}_{1,1}\frac{d}{c}=\frac{b+d}{a+c}=\frac{\alpha a+\beta d}{\alpha a+\beta c}$

From the geometry view point, the relation of equality $\frac{b}{a}=\frac{d}{c}$, algebraically defining an ADDITIVE operation or an ADDITIVE internal composition law and as an application $\left(\frac{b}{a},\frac{d}{c}\right)\mapsto \frac{b}{a}=\frac{d}{c}=\frac{b+d}{a+c}$, of $\frac{\stackrel{\dot{}}{b}}{a}\times \frac{\stackrel{\dot{}}{b}}{a}\to \frac{\stackrel{\dot{}}{b}}{a}$ or $\frac{\stackrel{\dot{}}{d}}{c}\times \frac{\stackrel{\dot{}}{d}}{c}\to \frac{\stackrel{\dot{}}{d}}{c}$ in $\mathbb{Q}$, does reflect the theorem of THALES studied by Rudolf Bkouche [20] , and as it can be found in the “Algèbre et geometrie” of Michel Demazure [5] , and as well as the NJIKI’s suites $s_k=\frac{kb}{ka}$ or $s_h=\frac{hd}{hc}$, in $\mathbb{Q}$, do reflect the extension of the very theorem, in the real affine plane ${ℝ}^2$.

In analysis or calculus, the equality or proportion $\frac{b}{a}=\frac{d}{c}$, algebraically also meaning that $\frac{b}{a}=\frac{d}{c}=\frac{b-d}{a-c}=\frac{d-b}{c-a}$, can be linked to the theorem of ROLLS [21] - [25] , found into Hervé Queffélec [26] , and Jacques Baranger’s [27] , books.

And that so from the NJIKI’s values a and c of a certain variable x, and, those b and d respectively of their images by the function y as follows and ACCURATELY:

$\frac{\Delta y}{\Delta c}=\frac{b-d}{a-c}=\frac{d-b}{c-a}=\frac{b}{a}=\frac{d}{c}$

Verification

By the means of the following two parts, a) differential and b) integral, of the infinitesimal calculation and therein for the purpose of some historical revival and since the infinitesimal calculation is at the origin of the modern analysis or calculus:

a) Given that $y=\frac{b}{a}x$ or $y=\frac{d}{c}x$, as the vector line version of the matter, or, $y=\frac{b}{a}x+e$ or $y=\frac{d}{c}x+f$, as its associated affine straight line:

$y^{\prime }=\underset{x\to x_0}{\lim }\frac{y-y_0}{x-x_0}=\frac{b}{a}=\frac{d}{c}$

b)

$y={\displaystyle {\int }_0^x\frac{b}{a}\text{d}u}={\left[\frac{b}{a}u\right]}_0^x=\frac{b}{a}x+e$

$y={\displaystyle {\int }_0^x\frac{d}{c}\text{d}v}={\left[\frac{d}{c}v\right]}_0^x=\frac{d}{c}x+f$

And both (a) and (b), to conclude, dealing with theories of measure respectively associated to fractions and related to NJIKI’s suites in $\mathbb{Q}$, and to integral in ${ℝ}^2$.

In its meaning, the algebraic form $\frac{\alpha a+\beta d}{\alpha a+\beta c}$, in $\mathbb{Q}$, represents a certain weighted arithmetic MEAN VALUE of $\frac{b}{a}$ and $\frac{d}{c}$ affected with suitable weighting coefficients and whether $\frac{b}{a}\ne \frac{d}{c}$ or $\frac{b}{a}=\frac{d}{c}$ meaning in any case. And it does go the same way with $\frac{a+d}{a+c}$, in $\mathbb{Q}$, and particularly then in $\frac{\stackrel{\dot{}}{b}}{a}$ or $\frac{\stackrel{\dot{}}{d}}{c}$ of $\mathbb{Q}$, for the above both $\frac{\alpha a+\beta d}{\alpha a+\beta c}$ and $\frac{b+d}{a+c}$. More of facts, the algebraic form $\frac{b+d}{a+c}$ and from $\frac{b}{a}$ and $\frac{d}{c}$, also found in the hyperbolic geometry, developed in the 18^th century by LOBACHEVSKI [3] , and then by Poincare, can now be given much more MEANING and THAT from the euclidian geometry and more generally in the affine geometry.

Besides the above, it is obvious that the background and the results of the Article here are linked, for its title “the NJIKI’s fundamental THEOREM-DEFINITION on fractions in the mathematical set $\mathbb{Q}$ and by extension in $ℝ$ and $ℂ$” does speak itself and by showing: a) the LACK of some THEORETICAL KNOWLEDGE about $\mathbb{Q}$ and by extension in $ℝ$ and $ℂ$, as far as the said background is concerned, and, b) the coverage of that GAP, and associated with the practical applications in so many areas of the utilisation of fractions including the mathematics themselves and others namely DOCIMOMETRICS, PROBABLIMETRICS and ELECTION-METRICS, for example, as for the results. And not to forget as the result of results, the very wide transmission, around or over the WORLD, of the ABOVE BOTH, the improved theoretical knowledge about fractions carried up “by the NJIKI’s fundamental THEOREM-DEFINITION on fractions…” and its various practical applications, and the very wide transmission, around or over the WORLD, of the ABOVE BOTH ENABLED, through publication, by the American Journal of Computational Mathematics (AJCM) sponsored by the Scientific Research Publishing (SCIRP)!!! Warm congratulations and everlasting life to the AJCM/SCIRP then and for making “The NJIKI’s fundamental THEOREM-DEFINITION on fractions in the mathematical set $\mathbb{Q}$ and by extension in $ℝ$ and $ℂ$” worldwide known and shared and for the wellbeing all-round of the mankind.

2) A very short mathematical study between the relation of equality, marked ${=}_{1,1}$, or, ${=}_{\alpha ,\beta }$, and knowing that ${=}_{1,1}={=}_{\alpha ,\beta }$, within the $\mathbb{Q}$ equivalence classes or classes of equivalence $\frac{\stackrel{\dot{}}{b}}{a}$ or $\frac{\stackrel{\dot{}}{d}}{c}$, the additive operations or the operations of ADDITION ${+}_{1,1}$ and ${+}_{\alpha ,\beta }$, in $\mathbb{Q}$, and that so, from the generalization view Point: ${+}_{1,1}$, in $\mathbb{Q}$, generalizes ${=}_{1,1}$, or, ${=}_{\alpha ,\beta }$, in $\frac{\stackrel{\dot{}}{b}}{a}$ or $\frac{\stackrel{\dot{}}{d}}{c}$ of $\mathbb{Q}$, and, ${+}_{\alpha ,\beta }$, in $\mathbb{Q}$, generalizes ${+}_{1,1}$, in $\mathbb{Q}$, and then ${=}_{1,1}$, or, ${=}_{\alpha ,\beta }$, in $\frac{\stackrel{\dot{}}{b}}{a}$ or $\frac{\stackrel{\dot{}}{d}}{c}$ of $\mathbb{Q}$, by the property of transitivity of the relation of generalisation, in $\mathbb{Q}$.

To:

For:

Their various and numerous contributions, written or not in living voice, to my not dead-end paper and that is to say on the move.

(A) An INNOVATIVE mathematical study of the logical propositional expression $\frac{b}{a}=\frac{d}{c}=\frac{b+d}{a+c}$, in $\mathbb{Q}$:

Classically proved by setting $\frac{b}{a}=\frac{d}{c}=k$, and among other methods, leading successively to $k=\frac{b+d}{a+c}$ and then to $\frac{b}{a}=\frac{d}{c}=\frac{b+d}{a+c}$, the result, the above logical propositional expression can be viewed as the beginning of a NJIKI’s digital ACCURATE suite $s_k={\left(\frac{b}{a}\right)}_k=\frac{kb}{ka}$, with $\frac{b}{a}$ being irreducible and $k\in {\mathbb{Z}}^{*}$ while $\underset{k\to \infty }{\lim }{s}_k=\frac{b}{a}$ and meaning that s_k is constant or stationary, or, $s_h={\left(\frac{d}{c}\right)}_h=\frac{hd}{hc}$, with $\frac{d}{c}$ being irreducible and $h\in {\mathbb{Z}}^{*}$ while $\underset{h\to \infty }{\lim }{s}_h=\frac{d}{c}$ and meaning that s_h is constant or stationary, and the very NJIKI’s digital ACCURATE suite, s_k or s_h, DEFINING each an equivalence class or class of equivalence of $\mathbb{Q}$. On one way and in CALCULUS that is rigorously and accurately far better than the suites of Cauchy [28] , with reals, complexes or points of a metric space and more generally of an uniform space. And on the other way and in GENERAL ALGEBRA, in the double equality $\frac{b}{a}=\frac{d}{c}=\frac{b+d}{a+c}$, the second one, =, is proper and the first one, =, can be considered as an internal composition law marked * and of an ADDITIVE nature, logically and aesthetically far better than the ADDITION of DUNCE, and so as to make it be $\frac{b}{a}\ast \frac{d}{c}=\frac{b+d}{a+c}$. And that from $\frac{\stackrel{\dot{}}{b}}{a}\times \frac{\stackrel{\dot{}}{d}}{c}$ to $\frac{\stackrel{\dot{}}{d}}{a}$, or, from $\frac{\stackrel{\dot{}}{d}}{c}\times \frac{\stackrel{\dot{}}{d}}{c}$ to $\frac{\stackrel{\dot{}}{d}}{c}$, and not from $\mathbb{Q}\times \mathbb{Q}$ to $\mathbb{Q}$ where the function * is not even an application and because of the separation of its equivalence classes or classes of equivalence. And now, to formally summarize the above:

$\ast :\frac{\stackrel{\dot{}}{b}}{a}\times \frac{\stackrel{\dot{}}{b}}{a}\to \frac{\stackrel{\dot{}}{b}}{a}$

$\left(\frac{b}{a},\frac{d}{c}\right)\mapsto \ast \left(\frac{b}{a},\frac{d}{c}\right)=\frac{b}{a}\ast \frac{d}{c}=\frac{b+d}{a+c}$

or

$\ast :\frac{\stackrel{\dot{}}{d}}{c}\times \frac{\stackrel{\dot{}}{d}}{c}\to \frac{\stackrel{\dot{}}{d}}{c}$

$\left(\frac{b}{a},\frac{d}{c}\right)\mapsto \ast \left(\frac{b}{a},\frac{d}{c}\right)=\frac{b}{a}\ast \frac{d}{c}=\frac{b+d}{a+c}$

And all things showing or what can prove that, or, which can be shown or be proved from that:

* = = $\vee$ = = *, and where = is the 1^st one equality of the above double equality $\frac{b}{a}=\frac{d}{c}=\frac{b+d}{a+c}$ and being a particular case of ${+}_{1,1}$, and so so, with a very good tautology (= = $\vee$ = =) behind, besides.

And for the recap of the major items: the operation of ADDITION ${+}_{\alpha ,\beta }$, in $\mathbb{Q}$, generalizes THOSE of the modular forms related to ${+}_{\alpha ,1}$ or ${+}_{1,\beta }$, in $\mathbb{Q}$, which THEMSELVES generalize THAT $\oplus$ or ${+}_{1,1}$, in $\mathbb{Q}$, which ITSELF generalizes THAT other = or *, in $\frac{\stackrel{\dot{}}{b}}{a}$ or $\frac{\stackrel{\dot{}}{d}}{c}$, belonging to $\mathbb{Q}$ and as one of its infinite equivalence classes or classes of equivalence corresponding to the ACCURATE suites of NJIKI s_k or s_h and among many numerous infinite others.

EXERCISE of APPLICATION in mathematics themselves or per se.

On the above pattern, make mathematically, in $\mathbb{Q}$ and from the analysis or calculus view point, and, in $\frac{\stackrel{\dot{}}{b}}{a}$ or $\frac{\stackrel{\dot{}}{d}}{c}$ and from the algebra one, an INNOVATIVE STUDY of the logical proposition $\frac{b}{a}=\frac{d}{c}=\frac{\alpha b+\beta d}{\alpha a+\beta c}$;

With = = ${\ast }_{\alpha ,\beta }\vee {\ast }_{\alpha ,\beta }$ = =

Notice that in $\frac{\stackrel{\dot{}}{b}}{a}$ or $\frac{\stackrel{\dot{}}{d}}{c}$, ${\ast }_{\alpha ,\beta }={\ast }_{1,1}=\ast$, meaning that there is no need of talking about any generalization between ${\ast }_{\alpha ,\beta }$ and ${\ast }_{1,1}$, in $\frac{\stackrel{\dot{}}{b}}{a}$ or $\frac{\stackrel{\dot{}}{d}}{c}$, and, as it is the case between ${+}_{\alpha ,\beta }$ and ${+}_{1,1}$, in $\mathbb{Q}$.

And what about the rational number $\frac{\alpha b+\beta d}{\alpha a+\beta c}\in \mathbb{Q}$ and the vector $\left(\begin{array}{c}\alpha a+\beta c\\ \alpha b+\beta d\end{array}\right)\in {\mathbb{Q}}_{jcn}$?

A partial mathematical study of the rational number $\frac{\alpha b+\beta d}{\alpha a+\beta c}\in \mathbb{Q}$ and the vector $\left(\begin{array}{c}\alpha a+\beta c\\ \alpha b+\beta d\end{array}\right)\in {\mathbb{Q}}_{jcn}$ and $Q_{jcn}$ being considered as that of the Njiki’s Z–module ( $Q_{jcn}$, +, .).

The rational number $\frac{\alpha b+\beta d}{\alpha a+\beta c}\in \mathbb{Q}$ is of the same class of equivalence or equivalence class than $\frac{b}{a}$ and $\frac{d}{c}$ in case that $\frac{b}{a}=\frac{d}{c}$ and what makes it be $\frac{b}{a}=\frac{d}{c}=\frac{\alpha b+\beta d}{\alpha a+\beta c}$ and which is a particular case of the general case $\frac{b}{a}{+}_{\alpha ,\beta }\frac{d}{c}=\frac{\alpha b+\beta d}{\alpha a+\beta c}$ and that is to say that even when $\frac{b}{a}\ne \frac{d}{c}$.

Now, in all cases or in any case, meaning whether $\frac{b}{a}=\frac{d}{c}$ or $\frac{b}{a}\ne \frac{d}{c}$, the rational number $\frac{\alpha b+\beta d}{\alpha a+\beta c}\in \mathbb{Q}$ and the vector $\left(\begin{array}{c}\alpha a+\beta c\\ \alpha b+\beta d\end{array}\right)$ do describe or characterize the same vector line and its associated affine straight line, and as their slope, in the field ( $\mathbb{Q},+,\times$), and vector director, in the NJIKI’s $\mathbb{Z}$-module ( ${\mathbb{Q}}_{jcn},+,.$), respectively.

(B) About the existence of an external multiplication operating over the set $\mathbb{Q}$

There does exist an external multiplication operating over the set $\mathbb{Q}$, with scalars in $\mathbb{Z}$ or $\mathbb{Q}$, marked $•$, and defined as follows:

$•:\mathbb{Z}\times \mathbb{Q}\to \mathbb{Q}$

$\left(\lambda ,\frac{b}{a}\right)\mapsto •\left(\lambda ,\frac{b}{a}\right)=\lambda •\frac{b}{a}=\frac{\lambda b}{\lambda a}=\frac{b}{a}$

or

$•:\mathbb{Q}\times \mathbb{Q}\to \mathbb{Q}$

$\left(\mu ,\frac{d}{c}\right)\mapsto •\left(\mu ,\frac{d}{c}\right)=\mu •\frac{d}{c}=\frac{\mu d}{\mu c}=\frac{d}{c}$

OBSERVATION

The external multiplication $•$ can be taken as the identity of $\mathbb{Q}$, $i{d}_{\mathbb{Q}}$, or as a vector homothety of $\mathbb{Q}$ of ratio 1, $h_1$. It also besides permits some very good definitions of both: 1) the Q equivalence classes or classes of equivalence, $\frac{\stackrel{\dot{}}{b}}{a}$, and 2) the NJIKI’s digital ACCURANTE suite, ${\left(\frac{b}{a}\right)}_{\lambda }$, and as follows:

1) $\frac{\stackrel{\dot{}}{b}}{a}=\left\{\frac{d}{c}\in \mathbb{Q}/\frac{d}{c}=\lambda \cdot \frac{b}{a}\wedge \lambda \in {\mathbb{Z}}^{*}\right\}$

2) ${\left(\frac{b}{a}\right)}_{\lambda }=\lambda \cdot \frac{b}{a}$ with $\lambda \in {\mathbb{Z}}^{*}$

It can easily be proved that:

1) The external multiplication $•$ is linked to ${+}_{1,1}$ as follows:

$\lambda •\frac{b}{a}=\underset{\lambda \text{times}\frac{b}{a}}{\underset{︸}{\frac{b}{a}{+}_{1,1}\cdots {+}_{1,1}\frac{b}{a}}}$ (by the definition of an operation of multiplication and by decomposition or development, in one way);

$=\underset{\lambda \text{times}b\text{and}\lambda \text{times}a}{\underset{︸}{\frac{b+\cdots +b}{a+\cdots +a}}}$ (by the definition of ${+}_{1,1}$ and its obvious associativity’s property, λ − 2 times);

$=\frac{\lambda b}{\lambda a}$ (by the definition of an operation of multiplication and by factorisation, on another way);

$=\frac{b}{a}$ (by the $\mathbb{Q}$ equivalence classes or classes of equivalence’s property or by the fractions simplification as well).

Or

$\mu •\frac{d}{c}=\underset{\mu \text{times}\frac{b}{a}}{\underset{︸}{\frac{d}{c}{+}_{1,1}\cdots {+}_{1,1}\frac{d}{c}}}=\frac{d+\cdots +d}{c+\cdots +c}=\frac{\mu d}{\mu c}=\frac{d}{c}$,

and within the $\mathbb{Q}$ equivalence classes or classes of equivalence the additive operation ${=}_{1,1}$ can also come in good handy in the place of ${+}_{1,1}$, assuming $\lambda$ or $\mu$ being (natural wholes) of $\mathbb{N}$, to make the enumeration constituting the heart of the proof easier, and by extension of $\mathbb{Z}$ and $\mathbb{Q}$, by some game of signs, − and +, between $\mathbb{Z}$ and $\mathbb{N}$, and, that of proportionality, by vector translation or change of reference bringing the Euclidian division in to the NJIKI’s divisibility, between $\mathbb{Q}$ and ( $\mathbb{Z}$ or $\mathbb{N}$).

2) The algebraic structure $\left(\mathbb{Q}\cup \left\{\frac{b}{0}∕b\in \mathbb{Z}\right\},{+}_{1,1},•\right)$, to be taken as an algebraic language translation of $\left({\mathbb{Q}}_{jcn},+,.\right)$, is a $\mathbb{Z}$-module and a $\mathbb{Q}$-vector plane said to be of NJIKI, and the null vector therein being $\frac{0}{0}$!!! And the one null vector $\frac{0}{0}$ being referred to as the NJIKI’s null vector whereas the elements or items of $\mathbb{Q}$ are here called non null vectors and as well as those of $\left\{\frac{b}{0}∕b\in {\mathbb{Z}}^{*}\right\}$.

The mathematical set $\mathbb{Q}\cup \left\{\frac{b}{0}∕b\in \mathbb{Z}\right\}$, marked ${\mathbb{Q}}_{nve}$, is named or called the NJIKI’s vector extension of $\mathbb{Q}$. Its extension in $ℝ$ and $ℂ$ are respectively marked ${ℝ}_{nve}$ and ${ℂ}_{nve}$. And to summarize: ${\mathbb{Q}}_{nve}=\mathbb{Q}\cup \left\{\frac{b}{0}/b\in \mathbb{Z}\right\}$, ${ℝ}_{nve}=ℝ\cup \left\{\frac{b}{0}/b\in ℝ\right\}$ and ${ℂ}_{nve}=ℂ\cup \left\{\frac{b}{0}/b\in ℂ\right\}$. Besides, the common canonical basis of ( ${\mathbb{Q}}_{nve},{+}_{1,1},•$), ( ${ℝ}_{nve},{+}_{1,1},•$), ( ${ℂ}_{nve},{+}_{1,1},•$) is $B=\left(\frac{1}{0},\frac{0}{1}\right)$. co-tangentially, or, $B=\left(\frac{0}{1},\frac{1}{0}\right)$ tangentially. Making it so be as follows: $\forall \frac{b}{a}\in {\mathbb{Q}}_{nve}$ $\vee$ $\forall \frac{b}{a}\in {ℝ}_{nve}$ $\vee$ $\forall \frac{b}{a}\in {ℂ}_{nve}$, $\frac{b}{a}=b•\frac{1}{0}{+}_{1,1}a•\frac{0}{1}$ $\vee$ $\frac{d}{c}=d•\frac{1}{0}{+}_{1,1}c•\frac{0}{1}$ $\vee$ $\frac{f}{e}=f•\frac{1}{0}{+}_{1,1}e•\frac{0}{1}$.

3) $\alpha \frac{b}{a}{+}_{1,1}\beta \frac{d}{c}=\frac{\alpha b+\beta d}{\alpha a+\beta c}$, by linear combination. And as remark, that same result, $\frac{\alpha b+\beta d}{\alpha a+\beta c}$, obtained by linear combination, can also be achieved by a single one additive operation ${+}_{\alpha ,\beta }$, in $\mathbb{Q}$, and as follows $\frac{b}{a}{+}_{\alpha ,\beta }\frac{d}{c}=\frac{\alpha b+\beta d}{\alpha a+\beta c}$. Meaning that the additive operation ${+}_{\alpha ,\beta }$, in $\mathbb{Q}$, can be taken as bearing a linear combination affected with coefficients $\alpha$ and $\beta$. Another major remark is that, by definition, the linear combination $\alpha •\frac{b}{a}{+}_{1,1}\beta •\frac{d}{c}$ has 3 other versions, options, alternatives or possibilities of calculation and that so because of the 2 equalities $\frac{\alpha b}{\alpha a}=\frac{b}{a}$ and $\frac{\beta d}{\beta c}=\frac{d}{c}$ leading to 2 × 2 ways of computing $\alpha •\frac{b}{a}{+}_{1,1}\beta •\frac{d}{c}$, and those 3 other versions are: a) $\alpha •\frac{b}{a}{+}_{1,1}\beta •\frac{d}{c}=\frac{\alpha b+d}{\alpha a+c}$, b) $\alpha •\frac{b}{a}{+}_{1,1}\beta •\frac{d}{c}=\frac{b+\beta d}{a+\beta c}$ and c) $\alpha •\frac{b}{a}{+}_{1,1}\beta •\frac{d}{c}=\frac{b+d}{a+c}$, respectively corresponding to ${+}_{\alpha ,1}$, ${+}_{1,\beta }$ and ${+}_{1,1}$. And, diagrammatically:

$\alpha •\frac{b}{a}{+}_{1,1}\beta •\frac{d}{c}=$ $\begin{array}{l}\frac{\alpha b+\beta d}{\alpha a+\beta c}\left(0\right)\equiv \left(\frac{b}{a}{+}_{\alpha ,\beta }\frac{d}{c}\right)\hfill \\ \frac{\alpha b+d}{\alpha a+c}\left(1\right)\equiv \left(\frac{b}{a}{+}_{\alpha ,1}\frac{d}{c}\right)\hfill \\ \frac{b+\beta d}{a+\beta c}\left(2\right)\equiv \left(\frac{b}{a}{+}_{1,\beta }\frac{d}{c}\right)\hfill \\ \frac{b+d}{a+c}\left(3\right)\equiv \left(\frac{b}{a}{+}_{1,1}\frac{d}{c}\right)\hfill \end{array}$

And the middle ones therein and that is to say (1) and (2) being close to some modular forms.

EXAMPLE

$c•\frac{b}{a}{+}_{1,1}a•\frac{d}{c}=$ $\begin{array}{l}\frac{cb+ad}{ca+ac}\left(0\right)\equiv \left(\frac{b}{a}{+}_{c,a}\frac{d}{c}\right)=\frac{1}{2}\left(\frac{b}{a}+\frac{d}{c}\right)\hfill \\ \frac{cb+d}{ca+c}\left(1\right)\equiv \left(\frac{b}{a}{+}_{c,1}\frac{d}{c}\right)\hfill \\ \frac{b+ad}{a+ac}\left(2\right)\equiv \left(\frac{b}{a}{+}_{1,a}\frac{d}{c}\right)\hfill \\ \frac{b+d}{a+c}\left(3\right)\equiv \left(\frac{b}{a}{+}_{1,1}\frac{d}{c}\right)\hfill \end{array}$

EXERCISE 1

Make a short mathematical study, from the above example, of the six (6) equalities between (0), (1), (2) and (3), and 2 by 2. And that so, both in the field ( $\mathbb{Q},+,\times$) and the NJIKI’s $\mathbb{Q}$-vector plane $\left(\mathbb{Q}\cup \left\{\frac{b}{0}/b\in \mathbb{Z}\right\},{+}_{1,1},•\right)$.

Conclusion: The linear combination $\alpha •\frac{b}{a}{+}_{1,1}\beta •\frac{d}{c}$ within the NJIKI’s $\mathbb{Q}$-vector plane $\left(\mathbb{Q}\cup \left\{\frac{b}{0}/b\in \mathbb{Z}\right\},{+}_{1,1},•\right)$, in terms of the set theory, can then be considered as follows: $\begin{array}{c}\alpha •\frac{b}{a}{+}_{1,1}\beta •\frac{d}{c}\in \left\{\alpha •\frac{b}{a}{+}_{1,1}\beta •\frac{d}{c}\right\}=\left\{\frac{\alpha b+\beta d}{\alpha a+\beta c},\frac{\alpha b+d}{\alpha a+c,}\frac{b+\beta d}{a+\beta c},\frac{b+d}{a+c}\right\}\\ =\left\{\frac{b}{a}{+}_{\alpha ,\beta }\frac{d}{c},\frac{b}{a}{+}_{\alpha ,1}\frac{d}{c},\frac{b}{a}{+}_{1,\beta }\frac{d}{c},\frac{b}{a}{+}_{1,1}\frac{d}{c}\right\}\end{array}$.

And it is called or named, for the purpose of the cause, “the NJIKI’s 4 elements of 1 number-set or set-number” including mixed up 2 modular forms, ${+}_{\alpha ,1}$ and ${+}_{1,\beta }$, 1 addition and also found in the hyperbolic geometry, ${+}_{1,1}$, another 1 addition, ${+}_{\alpha ,\beta }$, specially found in DOCIMOMETRICS, for some detailed description about it, and its items are equals 2 by 2 under certain circumstances and particularly and in all cases under those of the $\mathbb{Q}$ equivalence classes or classes of equivalence.

And, to make it, the story, turn into an image it is as if “the NJIKI’s 4 elements of 1 number-set or set-number”, by its linear combination $\alpha •\frac{b}{a}{+}_{1,1}\beta •\frac{d}{c}$, in ( ${\mathbb{Q}}_{nve},{+}_{1,1},•$), were gathering into the same toolbox or dissection kit, etc., the 4 addictive operations, in $\mathbb{Q}$, ${+}_{\alpha ,\beta }$, ${+}_{\alpha ,1}$, ${+}_{1,\beta }$, and ${+}_{1,1}$. And that last one ${+}_{1,1}$, generalizing the 4 other equal addictive operations ${=}_{\alpha ,\beta }$, ${=}_{\alpha ,1}$, ${=}_{1,\beta }$, and ${=}_{1,1}$ within the $\mathbb{Q}$ equivalence classes or classes of equivalence, being as a generator at the theoretical basis of the construction of all others including even the number-set or set-number itself, $\alpha •\frac{b}{a}{+}_{1,1}\beta •\frac{d}{c}$, in ( ${\mathbb{Q}}_{nve},{+}_{1,1},•$),. And that so, in addition to both its natural and granulometric character and its use, by default, when it comes to get involved into applications.

EXERCISE 2

Show the following said to be the two mathematics’ coachings or the two double inequalities of NJIKI:

1) $\forall \left(\frac{b}{a},\frac{d}{c}\right)\in {\mathbb{Q}}^{+2}\wedge \forall \left(\alpha ,\beta \right)\in {\mathbb{Q}}^{\ast +2}$, $\frac{b}{a}\le \frac{d}{c}\Rightarrow \frac{b}{a}\le \alpha •\frac{b}{a}{+}_{1,1}\beta •\frac{d}{c}\le \frac{d}{c}$

2) $\forall \left(\frac{b}{a},\frac{d}{c}\right)\in {\mathbb{Q}}^{+2}\wedge \forall \left(\alpha ,\beta \right)\in {\mathbb{Q}}^{+*2}$, $\frac{b}{a}<\frac{d}{c}\Rightarrow \frac{b}{a}<\alpha \cdot \frac{b}{a}{+}_{1,1}\beta \cdot \frac{d}{c}<\frac{d}{c}$

Note:

1) a fraction is definitely both a number, said to be relative according to NJIKI and as explained below, in Part (C), and not rational as usual, and, a vector.

2) $\forall \left(\frac{b}{a},\frac{d}{c}\right)\in {\mathbb{Q}}^{+2}$, $\frac{b}{a}<\frac{b}{a}+\frac{d}{c}\wedge \frac{d}{c}<\frac{b}{a}+\frac{d}{c}$

EXERCISE 3, or, the QUIZ that makes it talk about Mathematics as the 1^st Natural Sciences among all those of that category.

Say it practically physically how to interpret the theoretical weighting coefficients λ and μ in the 1 number-set or set-number $\left\{\lambda \cdot \frac{b}{a}{+}_{1,1}\mu \cdot \frac{d}{c}\right\}$. Justify your answer (and the which answer having a very big interest in the PART (C) below).

ANSWER PROPOSAL

They, λ and μ in the 1 number-set or set-number $\left\{\lambda \cdot \frac{b}{a}{+}_{1,1}\mu \cdot \frac{d}{c}\right\}$, practically physically can, and for the sake of various and numerous applications, be referred to as the adjustment or conversion coefficients into the same physical units or scales between those common to a and b, and, those other common to c and d. And by so doing, for the justification now, in order to enable the 2 factorisations between both the b and d level, and, the a and c one, being by the same unit or scale term that can then be simplified or not.

EXERCISE 4: Fractions and Theories of Representation

For the tip about it, this EXERCISE 4 is a conclusive exercise consisting in a comparative practical study between the classical or traditional operation of addition + and that carried out by the NJIKI’s 4 elements of 1 number-set or set-number $\alpha \cdot \frac{b}{a}{+}_{1,1}\beta \cdot \frac{d}{c}\in \left\{\alpha \cdot \frac{b}{a}{+}_{1,1}\beta \cdot \frac{d}{c}\right\}$, in $\mathbb{Q}$. Knowing that from $\frac{b}{a}=\frac{\lambda b}{\lambda a}$ and $\frac{d}{c}=\frac{\mu d}{\mu c}$, 1) $\frac{b}{a}+\frac{d}{c}=\frac{bc+ad}{ac}$, 2) $\frac{\lambda b}{\lambda a}+\frac{\mu d}{\mu c}=\frac{bc+ad}{ac}$, 3) $\frac{\lambda b}{\lambda a}+\frac{d}{c}=\frac{bc+ad}{ac}$, and 4) $\frac{b}{a}+\frac{\mu d}{\mu c}=\frac{bc+ad}{ac}$; and what shows the independence of $\frac{b}{a}+\frac{d}{c}$ upon $\lambda$ and $\mu$. And the final step of it being into how $\frac{b}{a}+\frac{d}{c}$ and $\alpha \cdot \frac{b}{a}{+}_{1,1}\beta \cdot \frac{d}{c}$ can be viewed from their splitting into proportional segments or sectors on or in a straight line or a disc representation in a plane, for example. By answering to the question whether $\frac{b}{a}+\frac{d}{c}$ can make sense by extracting $\frac{b}{a}$ and $\frac{d}{c}$ from two different physical items, of the same nature, and of course, or not. And idem with $\alpha \cdot \frac{b}{a}{+}_{1,1}\beta \cdot \frac{d}{c}$ . The ultimate aim of such an exercise being to set out the existence of constraints or conditions or not in the practical use of $\frac{b}{a}+\frac{d}{c}$ or of $\alpha \cdot \frac{b}{a}{+}_{1,1}\beta \cdot \frac{d}{c}$. And beyond over that, the establishment of the existence of the eventual limits of mathematics, and the mathematics remaining much more formal than meaningful.

And now, practice or exercise with $\frac{2}{5}+\frac{1}{3}$ and $\lambda \cdot \frac{2}{5}{+}_{1,1}\mu \cdot \frac{1}{3}$ in the following 3 arbitrarily chosen cases: 1) $\lambda =\mu$, by default, 2) $\lambda =10\mu$ and 3) $\lambda =3\mu$. Else $\lambda \cdot \frac{2}{5}{+}_{1,1}\mu \cdot \frac{1}{3}$ remains once at least or twice parametric and in $\lambda$ or $\mu$. So then, and that being said, shift to the generalization of the above with: 1) $\frac{b}{a}+\frac{d}{c}=\frac{bc+ad}{ac}$ opening way to 3 scales of ac, and especially in the case that $ac>bc+ad$, namely a, c and ac, and, 2) $\lambda \cdot \frac{b}{a}{+}_{1,1}\mu \cdot \frac{d}{c}$, setting $\lambda =k\mu$ (1) and where $\lambda$ and $\mu$ are respective proportional units of $\frac{b}{a}$ and $\frac{d}{c}$ in a certain physical system of units. And by setting either $\lambda =1$ or (e) $\mu =1$ (2), one moves towards modular forms. Meaning that the modular forms hold their origin into the NJIKI’s 4 elements of 1 number-set or set-number which themselves derive from the NJIKI’s Fundamental Theorem-Definition… In other words or terms, the beginning of the major items of the modular forms is at the prolongation of the NJIKI’s Fundamental Theorem-Definition…, and that through the NJIKI’s 4 elements of 1 number-set or set-number. And what makes so then another much more merit to the ongoing paper or article, to salute or celebrate and because it got jumbled over and straight to the study of the modular forms depending upon it, in relation to the natural and methodic order of bringing into construction the mathematical stuffs. And the same way of reasoning can be apply as concerning the hyperbolic geometry.

(C) The special idea behind the NJIKI’s proposal about the logical exchange, interchange, reverse or permutation of names by predicates between the current “relative” integers of $\mathbb{Z}$ and the “rational” fractions or numbers of $\mathbb{Q}$, and, special idea of reverse or permutation of names to make it be reviewed by other mathematicians around the world and as the opening of the debate.

The “relative” integers of $\mathbb{Z}$, according to NJIKI, will henceforth become the “rational” integers of $\mathbb{Z}$ still, and, the “rational” fractions or numbers of $\mathbb{Q}$ the “relative” fractions or numbers of $\mathbb{Q}$. And all things of “relative”, linked to $\mathbb{Q}$, due to its equivalence classes or classes of equivalence, $\frac{b}{a}=\frac{kb}{ka}$ with $k\in {\mathbb{Z}}^{*}$, and which can be said to be of a multiplicative type, giving relative choice of possibilities in the use of fractions of the same equivalence class or class of equivalence , and fractions of the same equivalence class or class of equivalence called its representatives and indeed its relative representatives constituting the heart of the announced debate in the title, and according to the problem cases to go through, as in the arithmetic ones. For example, to make $\frac{b}{a}+\frac{d}{c}$, one has to use $\frac{bc}{ac}$ and $\frac{ad}{ac}$ and then two other fractions, relative to the first ones $\frac{b}{a}$ and $\frac{d}{c}$, and they themselves resulting from an infinite system of units as follows: $\frac{b}{a}=b\cdot a^{-1}=\frac{b}{c^{-1}}\cdot a^{-1}{c}^{-1}=bc\cdot {\left(ac\right)}^{-1}=\frac{bc}{ac}$ and $\frac{d}{c}=d\cdot c^{-1}=ad\cdot \frac{c^{-1}}{a}=ad\cdot a^{-1}{c}^{-1}=ad\cdot {\left(ac\right)}^{-1}=\frac{ad}{ac}$.

And finally, by factorization $\frac{bc}{ac}+\frac{ad}{ac}=\frac{1}{ac}\left(bc+ad\right)=\frac{bc+ad}{ac}=\frac{b}{a}+\frac{d}{c}$, or, $bc\cdot {\left(ac\right)}^{-1}+ad{\left(ac\right)}^{-1}=\left(bc+ad\right){\left(ac\right)}^{-1}=\frac{bc+ad}{ac}=\frac{b}{a}+\frac{d}{c}$.