№904

これは面白い、全く新しい知見です。　不等式、従来数学は　不等式の解は　図のように開区間ですが、ゼロ除算算法で、両端での値は　ー１ですから、不等式は閉区間が正解に　なります。現在の数学では、両端点での値は考えないと成っているので、この新しい解は、ゼロ除算算法が公認されるまで、　入試や学校では用いてはいけません。　しかし、新しい世界が出て居ることは　面白いのでは　ないでしょうか。何時から、新しい解が　公認されるでしょうか、楽しみです。　もちろん、私の世代では無理ですね。所謂極でも、有限確定値をとる　というのがゼロ除算算法の発見です。

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\begin{document}

\title{\bf Announcement 448:\\ Division by Zero;\\

Funny History and New World}

\author{再生核研究所}

\date{2018.08.20}

\maketitle

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{\bf Abstract: } Our division by zero research group wonder why our elementary results may still not be accepted by some wide world and very recently in our Announcements: 434 (2018.7.28),

437 (2018.7.30),

438(2018.8.6), \\

441(2018.8.9),

442(2018.8.10),

443(2018.8.11),

444(2018.8.14),

in Japanese, we stated their reasons and the importance of our elementary results. Here, we would like to state their essences. As some essential reasons, we found fundamental misunderstandings on the division by zero and so we would like to state the essences and the importance of our new results to human beings over mathematics.

We hope that:

close the mysterious and long history of division by zero that may be considered as a symbol of the stupidity of the human race and open the new world since Aristotle-Eulcid.

From the funny history of the division by zero, we will be able to realize that

human beings are full of prejudice and prejudice, and are narrow-minded, essentially.

\medskip

\section{Division by zero}

The division by zero with mysterious and long history was indeed trivial and clear as in the followings:

\medskip

By the concept of the Moore-Penrose generalized solution of the fundamental equation $az=b$, the division by zero was trivial and clear as $b/0=0$ in the {\bf generalized fraction} that is defined by the generalized solution of the equation $az=b$.

Note, in particular, that there exists a uniquely determined solution for any case of the equation $az=b$ containing the case $a=0$.

People, of course, consider as the division $b/a$ that it is the solution of the equation $ az =b$ and if $a=0$ then $0 \cdot z =0$ and so, for $b\ne0$ we can not consider the fraction $a/b$. We have been considered that the division by zero $b/0$ is impossible for mysteriously long years, since the document of zero in India in AD 628. In particular, note that Brahmagupta (598 -668 ?) established four arithmetic operations by introducing $0$ and at the same time he defined as $0/0=0$ in Brhmasphuasiddhnta. Our world history, however, stated that his definition $0/0=0$ is wrong over 1300 years, but, we will see that his definition is right and suitable. However, he did not give its reason and did not consider the importance case $1/0$ and the general fractions $b/0$. The division by zero was a symbol for {\bf impossibility} or to consider the division by zero was {\bf not permitted}. For this simple and clear conclusion, we did not definitely consider more on the division by zero. However, we see many and many formulas appearing the zero in denominators, one simple and typical example is in the function $w=1/z$ for $z=0$.

We did not consider the function at the origin $z=0$.

In this case, however, the serious interest happens in many physical problems and also in computer sciences, as we know.

When we can not find the solution of the fundamental equation $az=b$, it is fairly clear to consider the Moore-Penrose generalized solution in mathematics. Its basic idea and beautiful mathematics will be definite.

Therefore, we should consider the generalized fractions following the Moore-Penrose generalized inverse. Therefore, with its meaning and definition we should consider that $b/0=0$.

It will be very curious that we know very well the Moore-Penrose generalized inverse as a very fundamental and important concept, however, we did not consider the simplest case $ az =b$.

Its reason may be considered as follows: We will consider or imagine that the fraction $1/0$ may be like infinity or ideal one.

For the fundamental function $W =1/ z $ we did not consider any value at the origin $z = 0$. Many and many people consider its value by the limiting like $+\infty $ and $- \infty$ or the

point at infinity as $\infty$. However, their basic idea comes from {\bf continuity} with the common sense or

based on the basic idea of Aristotle. --

For the related Greece philosophy, see \cite{a,b,c}. However, as the division by zero we have to consider its value of

the function $W =1 /z$ as zero at $z = 0$. We will see that this new definition is valid widely in

mathematics and mathematical sciences, see (\cite{mos,osm}) for example. Therefore, the division by zero will give great impacts to calculus, Euclidian geometry, analytic geometry, complex analysis and the theory of differential equations in an undergraduate level and furthermore to our basic ideas for the space and universe.

For the extended complex plane, we consider its stereographic projection mapping as the Riemann sphere and the point at infinity is realized as the north pole in the Alexsandroff's one point compactification.

The Riemann sphere model gives a beautiful and complete realization of the extended complex plane through the stereographic projection mapping and the mapping has beautiful properties like isogonal (equiangular) and circle to circle correspondence (circle transformation). Therefore, the Riemann sphere is a very classical concept \cite{ahlfors}.

\medskip

Now, with the division by zero we have to admit the strong discontinuity at the point at infinity. To accept this strong discontinuity seems to be very difficult, and therefore we showed many and many examples for giving the evidences over $800$ items.

\medskip

We back to our general fractions $1/0=0/0=z/0=0$ for its importances.

\medskip

H. Michiwaki and his 6 years old daughter Eko Michiwaki stated that in about three weeks after the discovery of the division by zero that

division by zero is trivial and clear from the concept of repeated subtraction and they showed the detailed interpretation of the general fractions. Their method is a basic one and it will give a good introduction of division and their calculation method of divisions.

We can say that division by zero, say $100/0$ means that we do not divide $100$ and so the number of the divided ones is zero.

\medskip

Furthermore,

recall the uniqueness theorem by S. Takahasi on the division by zero:

\medskip

{\bf Proposition 1.1 }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ satisfying

$$

F (b, a)F (c, d)= F (bc, ad)

$$

for all

$$

a, b, c, d \in {\bf C }

$$

and

$$

F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.

$$

Then, we obtain, for any $b \in {\bf C } $

$$

F (b, 0) = 0.

$$

}

Note that the complete proof of this proposition is simply given by 2 or 3 lines.

In the long mysterious history of the division by zero, this proposition seems to be decisive.

Indeed, Takahasi's assumption for the product property should be accepted for any generalization of fraction (division). Without the product property, we will not be able to consider any reasonable fraction (division).

Following Proposition 1.1, we should {\bf define}

$$

F (b, 0) = \frac{b}{0} =0,

$$

and consider, for any complex number $b$, as $0$;

that is, for the mapping

\begin{equation}

W = f(z) = \frac{1}{z},

\end{equation}

the image of $z=0$ is $W=0$ ({\bf should be defined from the form}).

\medskip

Furthermore,

the simple field structure containing division by zero was established by M. Yamada.

\medskip

In addition, for the fundamental function $f(z) = 1/z$, note that

the function is odd function

$$

f(z) = - f(-z)

$$

and if the function may be extended as an odd function at the origin $z=0$, then the identity $f(0) = 1/0 =0$ has to be satisfied. Further, if the equation

$$

\frac{1}{z} =0

$$

has a solution, then the solution has to be $z=0$.

\medskip

\section{Division by zero calculus}

As the number system containing the division by zero, the Yamada field structure is complete.

However, for applications of the division by zero to {\bf functions}, we need the concept of the division by zero calculus for the sake of uniquely determinations of the results and for other reasons.

For example, for the typical linear mapping

\begin{equation}

W = \frac{z - i}{z + i},

\end{equation}

it gives a conformal mapping on $\{{\bf C} \setminus \{-i\}\}$ onto $\{{\bf C} \setminus \{1\}\}$ in one to one and from \begin{equation}

W = 1 + \frac{-2i}{ z - (-i)},

\end{equation}

we see that $-i$ corresponds to $1$ and so the function maps the whole $\{{\bf C} \}$ onto $\{{\bf C} \}$ in one to one.

Meanwhile, note that for

\begin{equation}

W = (z - i) \cdot \frac{1}{z + i},

\end{equation}

if we enter $z= -i$ in the way

\begin{equation}

[(z - i)]_{z =-i} \cdot \left[ \frac{1}{z + i}\right]_{z =-i} = (-2i) \cdot 0= 0,

\end{equation}

we have another value.

\medskip

In many cases, the above two results will have practical meanings and so, we will need to consider many ways for the application of the division by zero and we will need to check the results obtained, in some practical viewpoints. We referred to this delicate problem with many examples.

Therefore, we will introduce the division by zero calculus that give important values for functions. For any Laurent expansion around $z=a$,

\begin{equation}

f(z) = \sum_{n=-\infty}^{-1} C_n (z - a)^n + C_0 + \sum_{n=1}^{\infty} C_n (z - a)^n,

\end{equation}

we obtain the identity, by the division by zero

\begin{equation}

f(a) = C_0.

\end{equation}

Note that here, there is no problem on any convergence of the expansion (2.5) at the point $z = a$, because all the terms $(z - a)^n$ are zero at $z=a$ for $n \ne 0$.

\medskip

For the correspondence (2.6) for the function $f(z)$, we will call it {\bf the division by zero calculus}. By considering the formal derivatives in (2.5), we {\bf can define any order derivatives of the function} $f$ at the singular point $a$; that is,

$$

f^{(n)}(a) = n! C_n.

$$

\medskip

{\bf Apart from the motivation, we define the division by zero calculus by (2.6).}

With this assumption, we can obtain many new results and new ideas. However, for this assumption we have to check the results obtained whether they are reasonable or not. By this idea, we can avoid any logical problems. -- In this point, the division by zero calculus may be considered as an axiom.

\medskip

This paragraph is very important. Our division by zero is just definition and the division by zero is an assumption. Only with the assumption and definition of the division by zero calculus, we can create and enjoy our new mathematics. Therefore, the division by zero calculus may be considered as a new axiom.

Of course, its strong motivations were given. We did not consider any value {\bf at the singular point} $a$ for the Laurent expansion (2.5). Therefore, our division by zero is a new mathematics entirely and isolated singular points are a new world for our mathematics.

We had been considered properties of analytic functions {\bf around their isolated singular points.}

The typical example of the division zero calculus is $\tan (\pi/2) = 0$ and the result gives great impacts to analysis and geometry.

See the references for the materials.

\medskip

For an identity, when we multiply zero, we obtain the zero identity that is a trivial.

We will consider the division by zero to an equation.

For example, for the simple example for the line equation on the $x, y$ plane

$$

ax + by + c=0

$$

we have, formally

$$

x + \frac{by + c}{a} =0,

$$

and so, by the division by zero, we have, for $a=0$, the reasonable result

$$

x = 0.

$$

However, from

$$

\frac{ax + by}{c} + 1 =0,

$$

for $c=0$, we have the contradiction, by the division by zero

$$

1 =0.

$$

For this case, we can consider that

$$

\frac{ax + by}{c} + \frac{c}{c} =0,

$$

that is always valid. {\bf In this sense, we can divide an equation by zero.}

\section{Conclusion}

Apparently, the common sense on the division by zero with a long and mysterious history is wrong and our basic idea on the space around the point at infinity is also wrong since Euclid. On the gradient or on derivatives we have a great missing since $\tan (\pi/2) = 0$. Our mathematics is also wrong in elementary mathematics on the division by zero.

We have to arrange globally our modern mathematics with our division by zero in our undergraduate level.

We have to change our basic ideas for our space and world.

We have to change globally our textbooks and scientific books on the division by zero.

From the mysterious history of the division by zero, we will be able to study what are human beings and about our narrow-minded.

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\end{document}