Ninjas Can't Divide By Zeroby WhoAteMyEggo
"Of course I divided by zero. I'm motherfuckin' Einstein."
私たちはできました：
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\title{\bf Announcement 354: What are $n = 2,1,0$ regular polygons inscribed in a disc? --
relations of $0$ and $\infty$ \\
(2017.2.8)}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
}
\date{\today}
\maketitle
{\bf Abstract: } In this announcement, as the representation of the point at infinity of the Riemann sphere by the zero $z = 0$, we will show its geometrical interpretation by $n$ regular polygons inscribed in a disc. This will show a delicate relation between $0$ and $\infty$. We will furthermore give surprising relations $0$ and $\infty$ which show a strong discontinuity at the point of infinity on the Riemann sphere.
\bigskip
\section{Introduction}
%\label{sect1}
By a {\bf natural extension} of the fractions
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we found the simple and beautiful result, for any complex number $b$
\begin{equation}
\frac{b}{0}=0,
\end{equation}
incidentally in \cite{s} by the Tikhonov regularization for the Hadamard product inversions for matrices and we discussed their properties and gave several physical interpretations on the general fractions in \cite{kmsy} for the case of real numbers. Meanwhile,
the result (1.2) is a very special case of very general fractional functions in \cite{cs}.
We thus should consider, for any complex number $b$, as (1.2);
that is, for the mapping
\begin{equation}
W = \frac{1}{z},
\end{equation}
the image of $z=0$ is $W=0$ ({\bf should be defined}). This fact seems to be a curious one in connection with our well-established popular image for the point at infinity on the Riemann sphere. Therefore, the division by zero will give great impacts to complex analysis and to our ideas for the space and universe.
However, for functions, we will need some modification {\bf by the idea of the division by zero calculus}:
For any formal Laurent expansion around $z=a$,
\begin{equation}
f(z) = \sum_{n=-\infty}^{\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}
\medskip
The typical example is that, as we can derive by the elementary way,
$$
\tan \frac{\pi}{2} =0.
$$
\medskip
We gave many examples with geometric meanings in \cite{mos}. See (\cite{cs,kmsy,msy,s,s16} for the related references.
In (\cite{mos}), many division by zero results in Euclidean spaces are given and the basic idea at the point at infinity should be changed. In (\cite{ms}), we gave beautiful geometrical interpretations of determinants from the viewpoint of the division by zero. The results show that the division by zero is our basic and elementary mathematics in our world.
For our ideas on the division by zero, see the survey style announcements stated in the references.
\section{$n=2,1,0$ regular polygons inscribed in a disc}
We consider $n$ regular polygons inscribed in a fixed disc with radius $a$. Then we note that their area $S_n$ and the lengths $L_n$ of the sum of the sides are given by
\begin{equation}
S_n= \frac{n a^2}{2} \sin \frac{2\pi}{n}
\end{equation}
and
\begin{equation}
L_n= 2n a \sin \frac{\pi}{n},
\end{equation}
respectively. For $n \ge 3$, the results are clear.
For $n =2$, we will consider two diameters that are the same. We can consider it as a generalized regular polygon inscribed in the disc as a degenerate case. Then, $S_2 =0$ and $L_2 =4a$, and the general formulas are valid.
Next, we will consider the case $n=1$. Then the corresponding regular polygon is a just diameter of the disc. Then, $S_1 =0 $ and $L_1 = 0$ that will mean that any regular polygon inscribed in the disc may not be formed and so its area and length of the side are zero.
Now we will consider the case $n=0$. Then, by the division by zero calculus, we obtain that
$S_0= \pi a^2$ and $L_0 = 2\pi a$. Note that they are the area and the length of the disc.
How to understand the results? Imagine contrary $n$ tending to infinity, then the corresponding regular polygons inscribed in the disc tend to the disc. Recall our new idea that the point at infinity is represented by $0$. Therefore, the results say that $n=0$ regular polygons are $n= \infty$ regular polygons inscribed in the disc in a sense and they are the disc. This is our interpretation of the theorem:
\medskip
{\bf Theorem.} {\it $n =0$ regular polygons inscribed in a disc are the whole disc.}
\medskip
\section{Our life figure}
As an interesting figure which shows an interesting relation between $0$ and infinity, we will consider a sector $\Delta_{\alpha}$ on the complex $z= x + iy$ plane
$$
\Delta_{\alpha} = \left\{ |\arg z| < \alpha; 0< \alpha < \frac{\pi}{2}\right\}.
$$
We will consider a disc inscribed in the sector $\Delta_{\alpha}$ whose center $(k,0)$ with radius $r$. Then, we have
\begin{equation}
r = k \sin \alpha.
\end{equation}
Then, note that as $k$ tends to zero, $r$ tends to zero, meanwhile $k$ tends to $+\infty$, $r$ tends to $+\infty$. However, by our division by zero calculus, we see that immediately that
\begin{equation}
[r ]_{r =\infty}= 0.
\end{equation}
For this fact, note the following:
The behavior of the space around the point at infinity may be considered by that of the origin by the linear transform $W = 1/z$. We thus see that
\begin{equation}
\lim_{z \to \infty} z = \infty,
\end{equation}
however,
\begin{equation}
[z]_{z =\infty} =0,
\end{equation}
by the division by zero. Here, $[z]_{z =\infty}$ denotes the value of the function $W=z$ at the topological point at the infinity in one point compactification by Aleksandrov. The difference of (3.3) and (3.4) is very important as we see clearly by the function $W = 1/z$ and the behavior at the origin. The limiting value to the origin and the value at the origin are different. For surprising results, we will state the property in the real space as follows:
\begin{equation}
\lim_{x\to +\infty} x =+\infty , \quad \lim_{x\to -\infty} x = -\infty,
\end{equation}
however,
\begin{equation}
[x]_{ +\infty } =0, \quad [x]_{ -\infty } =0.
\end{equation}
Of course, two points $+\infty$ and $ -\infty$ are the same point as the point at infinity. However, $\pm$ will be convenient in order to show the approach directions. In \cite{mos}, we gave many examples for this property.
On the sector, we see that from the origin as the point 0, the inscribed discs are increasing endlessly, however their final disc reduces to the origin suddenly -
it seems that the whole process looks like our life in the viewpoint of our initial and final.
\section{H. Okumura's example}
The suprising example by H. Okumura which was sent to the Institute on 2016.12.1.9:40 will show a new phenomenon at the point at infinity.
On the sector $\Delta_{\alpha}$, we shall change the angle and we consider a fixed circle $C_a, a > 0$ with radius $a$ inscribed in the sectors. We see that when the circle tends to $+\infty$, the angles $\alpha$ tend to zero. How will be the case $\alpha = 0$? Then, we will not be able to see the position of the circle. Surprisingly enough, then the circle $C_a$ is the circle with center at the origin $0$. This result is derived from the division by zero calculus for the formula
\begin{equation}
k =\frac{a}{\sin \alpha}.
\end{equation}
The two lines $ \arg z = \alpha$ and $ \arg z = -\alpha$ were tangential lines of the circle
$C_a$ and now they are the positive real line. The gradient of the positive real line is of course zero. Note here that the gradient of the positive imaginary line is zero by the division by zero calculus that means $\tan \frac{\pi}{2} =0$. Therefore, we can understand that the positive real line is still a tangential line of the circle $C_a$.
This will show some great relation between zero and infinity. We can see some mysterious property around the point at infinity.
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{cs}
L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
\bibitem{kmsy}
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
Int. J. Appl. Math. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{msy}
H. Michiwaki, S. Saitoh, and M.Yamada,
Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. {\bf 6}(2015), 1--8. http://www.ijapm.org/show-63-504-1.html
\bibitem{ms}
T. Matsuura and S. Saitoh,
Matrices and division by zero $z/0=0$, Advances in Linear Algebra
\& Matrix Theory, 6 (2016), 51-58. http://dx.doi.org/10.4236/alamt.2016.62007 http://www.scirp.org/journal/alamt
\bibitem{mos}
H. Michiwaki, H. Okumura, and S. Saitoh,
Division by Zero $z/0 = 0$ in Euclidean Spaces.
International Journal of Mathematics and Computation Vol. 28(2017); Issue 1, 2017), 1-16.
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. {\bf 4} (2014), no. 2, 87--95. http://www.scirp.org/journal/ALAMT/
\bibitem{s16}
S. Saitoh, A reproducing kernel theory with some general applications,
Qian,T./Rodino,L.(eds.): Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, {\bf 177}(2016), 151-182 (Springer).
\bibitem{ann179}
Announcement 179 (2014.8.30): Division by zero is clear as z/0=0 and it is fundamental in mathematics.
\bibitem{ann185}
Announcement 185 (2014.10.22): The importance of the division by zero $z/0=0$.
\bibitem{ann237}
Announcement 237 (2015.6.18): A reality of the division by zero $z/0=0$ by geometrical optics.
\bibitem{ann246}
Announcement 246 (2015.9.17): An interpretation of the division by zero $1/0=0$ by the gradients of lines.
\bibitem{ann247}
Announcement 247 (2015.9.22): The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.
\bibitem{ann250}
Announcement 250 (2015.10.20): What are numbers? - the Yamada field containing the division by zero $z/0=0$.
\bibitem{ann252}
Announcement 252 (2015.11.1): Circles and
curvature - an interpretation by Mr.
Hiroshi Michiwaki of the division by
zero $r/0 = 0$.
\bibitem{ann281}
Announcement 281 (2016.2.1): The importance of the division by zero $z/0=0$.
\bibitem{ann282}
Announcement 282 (2016.2.2): The Division by Zero $z/0=0$ on the Second Birthday.
\bibitem{ann293}
Announcement 293 (2016.3.27): Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0.
\bibitem{ann300}
Announcement 300 (2016.05.22): New challenges on the division by zero z/0=0.
\bibitem{ann326}
Announcement 326 (2016.10.17): The division by zero z/0=0 - its impact to human beings through education and research.
\bibitem{ann352}
Announcement 352 (2017.2.2): On the third birthday of the division by zero z/0=0.
\end{thebibliography}
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\title{\bf Announcement 326: The division by zero z/0=0 - its impact to human beings through education and research\\
(2016.10.17)}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
}
\date{\today}
\maketitle
{\bf Abstract: } In this announcement, for its importance we would like to state the
situation on the division by zero and propose basic new challenges to education and research on our wrong world history.
\bigskip
\section{Introduction}
%\label{sect1}
By a {\bf natural extension} of the fractions
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we found the simple and beautiful result, for any complex number $b$
\begin{equation}
\frac{b}{0}=0,
\end{equation}
incidentally in \cite{s} by the Tikhonov regularization for the Hadamard product inversions for matrices and we discussed their properties and gave several physical interpretations on the general fractions in \cite{kmsy} for the case of real numbers.
The division by zero has a long and mysterious story over the world (see, for example, Google site with the division by zero) with its physical viewpoints since the document of zero in India on AD 628, however,
Sin-Ei Takahasi (\cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing the extensions of fractions and by showing the complete characterization for the property (1.2):
\bigskip
{\bf Proposition 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.
We should define $F(b,0)= b/0 =0$, in general.
\medskip
We thus should consider, for any complex number $b$, as (1.2);
that is, for the mapping
\begin{equation}
W = \frac{1}{z},
\end{equation}
the image of $z=0$ is $W=0$ ({\bf should be defined}). This fact seems to be a curious one in connection with our well-established popular image for the point at infinity on the Riemann sphere. Therefore, the division by zero will give great impact to complex analysis and to our ideas for the space and universe.
However, the division by zero (1.2) is now clear, indeed, for the introduction of (1.2), we have several independent approaches as in:
\medskip
1) by the generalization of the fractions by the Tikhonov regularization and by the Moore-Penrose generalized inverse,
\medskip
2) by the intuitive meaning of the fractions (division) by H. Michiwaki - repeated subtraction method,
\medskip
3) by the unique extension of the fractions by S. Takahasi, as in the above,
\medskip
4) by the extension of the fundamental function $W = 1/z$ from ${\bf C} \setminus \{0\}$ into ${\bf C}$ such that $W =1/z$ is a one to one and onto mapping from $ {\bf C} \setminus \{0\} $ onto ${\bf C} \setminus \{0\}$ and the division by zero $1/0=0$ is a one to one and onto mapping extension of the function $W =1/z $ from ${\bf C}$ onto ${\bf C}$,
\medskip
and
\medskip
5) by considering the values of functions with the mean values of functions.
\medskip
Furthermore, in (\cite{msy}) we gave the results in order to show the reality of the division by zero in our world:
\medskip
\medskip
A) a field structure containing the division by zero --- the Yamada field ${\bf Y}$,
\medskip
B) by the gradient of the $y$ axis on the $(x,y)$ plane --- $\tan \frac{\pi}{2} =0$,
\medskip
C) by the reflection $W =1/\overline{z}$ of $W= z$ with respect to the unit circle with center at the origin on the complex $z$ plane --- the reflection point of zero is zero, not the point at infinity.
\medskip
and
\medskip
D) by considering rotation of a right circular cone having some very interesting
phenomenon from some practical and physical problem.
\medskip
In (\cite{mos}), many division by zero results in Euclidean spaces are given and the basic idea at the point at infinity should be changed. In (\cite{ms}), we gave beautiful geometrical interpretations of determinants from the viewpoint of the division by zero. The results show that the division by zero is our basic and elementary mathematics in our world.
\medskip
See J. A. Bergstra, Y. Hirshfeld and J. V. Tucker \cite{bht} for the relationship between fields and the division by zero, and the importance of the division by zero for computer science. It seems that the relationship of the division by zero and field structures are abstract in their paper.
Meanwhile, J. P. Barukcic and I. Barukcic (\cite{bb}) discussed recently the relation between the divisions $0/0$, $1/0$ and special relative theory of Einstein. However, their logic seems to be curious and their results contradict with ours.
Furthermore, T. S. Reis and J.A.D.W. Anderson (\cite{ra,ra2}) extend the system of the real numbers by introducing an ideal number for the division by zero $0/0$.
Meanwhile, we should refer to up-to-date information:
{\it Riemann Hypothesis Addendum - Breakthrough
Kurt Arbenz
https://www.researchgate.net/publication/272022137 Riemann Hypothesis Addendum - Breakthrough.}
\medskip
Here, we recall Albert Einstein's words on mathematics:
Blackholes are where God divided by zero.
I don't believe in mathematics.
George Gamow (1904-1968) Russian-born American nuclear physicist and cosmologist remarked that "it is well known to students of high school algebra" that division by zero is not valid; and Einstein admitted it as {\bf the biggest blunder of his life} [1]：
1. Gamow, G., My World Line (Viking, New York). p 44, 1970.
Apparently, the division by zero is a great missing in our mathematics and the result (1.2) is definitely determined as our basic mathematics, as we see from Proposition 1. Note its very general assumptions and many fundamental evidences in our world in (\cite{kmsy,msy,mos}). The results will give great impact on Euclidean spaces, analytic geometry, calculus, differential equations, complex analysis and physical problems.
The mysterious history of the division by zero over one thousand years is a great shame of mathematicians and human race on the world history, like the Ptolemaic system (geocentric theory). The division by zero will become a typical symbol of foolish human race with long and unceasing struggles. Future people will realize this fact as a definite common sense.
We should check and fill our mathematics, globally and beautifully, from the viewpoint of the division by zero. Our mathematics will be more perfect and beautiful, and will give great impact to our basic ideas on the universe.
For our ideas on the division by zero, see the survey style announcements.
\section{Basic Materials of Mathematics}
(1): First, we should declare that the divison by zero is possible in the natural and uniquley determined sense and its importance.
(2): In the elementary school, we should introduce the concept of division by the idea of repeated subtraction method by H. Michiwaki whoes method is applied in computer algorithmu and in old days for calculation of division. This method will give a simple and clear method for calculation of division and students will be happy to apply this simple method at the first stage. At this time, they will be able to understand that the division by zero is clear and trivial as $a/0=0$ for any $a$. Note that Michiwaki knows how to apply his method to the complex number field.
(3): For the introduction of the elemetary function $y= 1/x$, we should give the definition of the function at the origin $x=0$ as $y = 0$ by the division by zero idea and we should apply this definition for the occasions of its appearences, step by step, following the curriculum and the results of the division by zero.
(4): For the idea of the Euclidean space (plane), we should introduce, at the first stage, the concept of steleographic projection and the concept of the point at infinity -
one point compactification. Then, we will be able to see the whole Euclidean plane, however, by the division by zero, the point at infinity is represented by zero. We can teach the very important fact with many geometric and analytic geometry methods. These topics will give great pleasant feelings to many students.
Interesting topics are: parallel lines, what is a line? - a line contains the origin as an isolated
point for the case that the native line does not through the origin. All the lines pass the origin, our space is not the Eulcildean space and is not Aristoteles for the strong discontinuity at the point at infinity (at the origin). - Here note that an orthogonal coordinates should be fixed first for our all arguments.
(5): The inversion of the origin with respect to a circle with center the origin is the origin itself, not the point at infinity - the very classical result is wrong. We can also prove this elementary result by many elementary ways.
(6): We should change the concept of gradients; on the usual orthogonal coordinates $(x,y)$,
the gradient of the $y$ axis is zero; this is given and proved by the fundamental result
$\tan (\pi/2) =0$. The result is trivial in the definition of the Yamada field. This result is derived also from the {\bf division by zero calculus}:
\medskip
For any formal Laurent expansion around $z=a$,
\begin{equation}
f(z) = \sum_{n=-\infty}^{\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}
\medskip
This fundamental result leads to the important new definition:
From the viewpoint of the division by zero, when there exists the limit, at $ x$
\begin{equation}
f^\prime(x) = \lim_{h\to 0} \frac{f(x + h) - f(x)}{h} =\infty
\end{equation}
or
\begin{equation}
f^\prime(x) = -\infty,
\end{equation}
both cases, we can write them as follows:
\begin{equation}
f^\prime(x) = 0.
\end{equation}
\medskip
For the elementary ordinary differential equation
\begin{equation}
y^\prime = \frac{dy}{dx} =\frac{1}{x}, \quad x > 0,
\end{equation}
how will be the case at the point $x = 0$? From its general solution, with a general constant $C$
\begin{equation}
y = \log x + C,
\end{equation}
we see that, by the division by zero,
\begin{equation}
y^\prime (0)= \left[ \frac{1}{x}\right]_{x=0} = 0,
\end{equation}
that will mean that the division by zero (1.2) is very natural.
In addition, note that the function $y = \log x$ has infinite order derivatives and all the values are zero at the origin, in the sense of the division by zero.
However, for the derivative of the function $y = \log x$, we have to fix the sense at the origin, clearly, because the function is not differentiable, but it has a singularity at the origin. For $x >0$, there is no problem for (2.6) and (2.7). At $x = 0$, we see that we can not consider the limit in the sense (2.3). However, for $x >0$ we have (2.6) and
\begin{equation}
\lim_{x \to +0} \left(\log x \right)^\prime = +\infty.
\end{equation}
In the usual sense, the limit is $+\infty$, but in the present case, in the sense of the division by zero, we have:
\begin{equation}
\left[ \left(\log x \right)^\prime \right]_{x=0}= 0
\end{equation}
and we will be able to understand its sense graphycally.
By the new interpretation for the derivative, we can arrange many formulas for derivatives, by the division by zero. We can modify many formulas and statements in calculus and we can apply our concept to the differential equation theory and the universe in connetion with derivatives.
(7): We shall introduce the typical division by zero calculus.
For the integral
\begin{equation}
\int x(x^{2}+1)^{a}dx=\frac{(x^{2}+1)^{a+1}}{2(a+1)}\quad(a\ne-1),
\end{equation}
we obtain, by the division by zero,
\begin{equation}
\int x(x^{2}+1)^{-1}dx=\frac{\log(x^{2}+1)}{2}.
\end{equation}
We will consider the fundamental ordinary differential equations
\begin{equation}
x^{\prime \prime}(t) =g -kx^{\prime}(t)
\end{equation}
with the initial conditions
\begin{equation}
x(0) = -h, x^{\prime}(0) =0.
\end{equation}
Then we have the solution
\begin{equation}
x(t) = \frac{g}{k}t + \frac{g(e^{-kt}- 1)}{k^2} - h.
\end{equation}
Then, for $k=0$, we obtain, immediately, by the division by zero
\begin{equation}
x(t) = \frac{1}{2}g t^2 -h.
\end{equation}
In those examples, we were able to give valuable functions for denominator zero cases. The division by zero calculus may be applied to many cases as a new fundamental calculus over l'Hôpital's rule.
(8): When we apply the division by zero to functions, we can consider, in general, many ways. For example,
for the function $z/(z-1)$, when we insert $z=1$ in numerator and denominator, we have
\begin{equation}
\left[\frac{z}{z-1}\right]_{z = 1} = \frac{1}{0} =0.
\end{equation}
However,
from the identity --
the Laurent expansion around $z=1$,
\begin{equation}
\frac{z}{z-1} = \frac{1}{z-1} + 1,
\end{equation}
we have
\begin{equation}
\left[\frac{z}{z-1}\right]_{z = 1} = 1.
\end{equation}
For analytic functions we can give uniquely determined values at isolated singular points by the values by means of the Laurent expansions as the division by zero calculus, however, the values by means of the Laurent expansions are not always reasonable. We will need to consider many interpretations for reasonable values. In many formulas in mathematics and physics, however, we can see that the division by zero calculus is reasonably valid. See \cite{kmsy,msy}.
\section{Albert Einstein's biggest blunder}
The division by zero is directly related to the Einstein's theory and various
physical problems
containing the division by zero. Now we should check the theory and the problems by the concept of the RIGHT and DEFINITE division by zero. Now is the best time since 100 years from Albert Einstein. It seems that the background knowledge is timely fruitful.
Note that the Big Bang also may be related to the division by zero like the blackholes.
\section{Computer systems}
The above Professors listed are wishing the contributions in order to avoid the division by zero trouble in computers. Now, we should arrange new computer systems in order not to meet the division by zero trouble in computer systems.
By the division by zero calculus, we will be able to overcome troubles in Maple for specialization problems.
\section{General ideas on the universe}
The division by zero may be related to religion, philosophy and the ideas on the universe, and it will creat a new world. Look the new world introduced.
\bigskip
We are standing on a new generation and in front of the new world, as in the discovery of the Americas. Should we push the research and education on the division by zero?
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{bb}
J. P. Barukcic and I. Barukcic, Anti Aristotle—The Division of Zero by Zero. Journal of Applied Mathematics and Physics, {\bf 4}(2016), 749-761.
doi: 10.4236/jamp.2016.44085.
\bibitem{bht}
J. A. Bergstra, Y. Hirshfeld and J. V. Tucker,
Meadows and the equational specification of division (arXiv:0901.0823v1[math.RA] 7 Jan 2009).
\bibitem{cs}
L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
\bibitem{kmsy}
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
Int. J. Appl. Math. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{ms}
T. Matsuura and S. Saitoh,
Matrices and division by zero $z/0=0$, Advances in Linear Algebra
\& Matrix Theory, 6, 51-58. http://dx.doi.org/10.4236/alamt.2016.62007 http://www.scirp.org/journal/alamt
\bibitem{msy}
H. Michiwaki, S. Saitoh, and M.Yamada,
Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. {\bf 6}(2015), 1--8. http://www.ijapm.org/show-63-504-1.html
\bibitem{mos}
H. Michiwaki, H. Okumura, and S. Saitoh,
Division by Zero $z/0 = 0$ in Euclidean Spaces.
International Journal of Mathematics and Computation
(in press).
\bibitem{ra}
T. S. Reis and J.A.D.W. Anderson,
Transdifferential and Transintegral Calculus,
Proceedings of the World Congress on Engineering and Computer Science 2014 Vol I
WCECS 2014, 22-24 October, 2014, San Francisco, USA
\bibitem{ra2}
T. S. Reis and J.A.D.W. Anderson,
Transreal Calculus,
IAENG International J. of Applied Math., {\bf 45}(2015): IJAM 45 1 06.
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. {\bf 4} (2014), no. 2, 87--95. http://www.scirp.org/journal/ALAMT/
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operations on the real and complex fields, Tokyo Journal of Mathematics, {\bf 38}(2015), no. 2, 369-380.
\bibitem{ann179}
Announcement 179 (2014.8.30): Division by zero is clear as z/0=0 and it is fundamental in mathematics.
\bibitem{ann185}
Announcement 185 (2014.10.22): The importance of the division by zero $z/0=0$.
\bibitem{ann237}
Announcement 237 (2015.6.18): A reality of the division by zero $z/0=0$ by geometrical optics.
\bibitem{ann246}
Announcement 246 (2015.9.17): An interpretation of the division by zero $1/0=0$ by the gradients of lines.
\bibitem{ann247}
Announcement 247 (2015.9.22): The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.
\bibitem{ann250}
Announcement 250 (2015.10.20): What are numbers? - the Yamada field containing the division by zero $z/0=0$.
\bibitem{ann252}
Announcement 252 (2015.11.1): Circles and
curvature - an interpretation by Mr.
Hiroshi Michiwaki of the division by
zero $r/0 = 0$.
\bibitem{ann281}
Announcement 281 (2016.2.1): The importance of the division by zero $z/0=0$.
\bibitem{ann282}
Announcement 282 (2016.2.2): The Division by Zero $z/0=0$ on the Second Birthday.
\bibitem{ann293}
Announcement 293 (2016.3.27): Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0.
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