Zero Divided By Zero – What it is?

October 17, 2018 Exploring Math, Math News, Numbers by JHSML Content Department

Math is regarded as one of the most objective topics there is: there can only be one right answer to a question and your opinions about what it should or shouldn’t be are irrelevant. But, as we’re going to see, that may not always be the case.

The debate over the answer to zero divided by zero has polarized the STEM community and remains unsolved to this day. So, in this post, we are going to review the numerous opinions surrounding this seemingly straightforward question and provide some evidence for each of their claims. We are going to refute these claims not with the intention of picking a side on the debate, but to reveal the complexity of this topic that at surface level, seems quite elementary. And, if you read to the end, you’ll find our answer!

The three most common responses are based on the rules of division that contradict in this unique case. First, many argue that 0 / 0 = 0 because zero divided by anything is zero. On the other hand, there are those that say that 0 / 0 = 1, because any number divided by itself is one. Finally, others state 0 / 0 = undefined as any number divided by zero can’t be defined.

Now, let’s turn to the definition of division in order to gain a better understanding of the situation and potentially find some evidence for the proposed answers. When we are dividing a number (x) by another number (y), we are asking how many times does the number y “fit into” the number x. Or, more simply, y times “what” is equal to x. This means that the result of a quotient can be multiplied by the denominator in order to find the value of the initial number (the numerator). This seems to indicate that any number divided by zero is undefined unless the numerator is also zero, in which case the answer could be zero or any other number for that matter, because zero times multiplied by anything is zero. Showing that it can be any number doesn’t help us at all, so we have to find other means of answering this problem.

When dealing with situations such as this one in which you can’t simply plug it into the calculator to find the answer, we use the concept of limits. Limits are a topic in calculus in which we look at points on a function as they get infinitely close to a given number and then make an estimation about what the value of the point will be at that given number. So, to apply it to the problem at hand, we can use a function that will come out to 0/0 at some point and use limits to calculate the output at that point.

For example, if we plot the function sin(x)/x, we will get an output of 0/0 at x=0. We can see the function graphed below.

Now, we look at the limit of x as it approaches 0 and see that the function gets infinitely close to 1 at that point. So, this seems to indicate that the answer to zero divided by zero is one. But, don’t celebrate yet. If we manipulate the sine function, we see that we can, in fact, use this methodology and make it yield an infinite number of answers to our question. For example, if we use the function sin(2x)/x instead of sin(x)/x, we will still be at 0/0 when x=0, but the limit as x approaches zero is ½ instead of 1. This can be done for any number y or z in the function ysin(z*x)/x, yielding an infinite number of possible values for the limit as x approaches zero. Once again, this argument doesn’t prove anything and leaves us with the same unsatisfying answer: anything!

Not so simple after all, hugh?

After repeated attempts at solving this problem throughout history, the general final determination of zero divided by zero is, drum roll please, undetermined!

Now, this is definitely not the answer you were looking for or are satisfied with. But, that’s exactly the point: we wanted with this detailed examination of the age-old debate to demonstrate that math does have room for opinion and commentary and isn’t just an objective science with exclusive consensus. Many topics in math are yet to be determined and, who knows, maybe you could be the one to solve them?

Thank you for reading, we really hope you enjoyed this blog post! Please feel free to reach out to us at any time at jhsmathleague@gmail.com and we’d be happy to address any questions or concerns you have regarding this article.

http://jhsmathleague.org/zero-divided-by-zero/

Wasan Geometry and Division by Zero Calculus

2018年11月28日(水) テーマ：数学

Sangaku Journal of Mathematics (SJM) ⃝c SJM ISSN 2534-9562 Volume 2 (2018), pp. 57-73 Received 20 November 2018. Published on-line 29 November 2018 web: http://www.sangaku-journal.eu/ ⃝c The Author(s) This article is published with open access1 . Wasan Geometry and Division by Zero Calculus

file:///C:/Users/saito%20saburo/Downloads/SJM_2018_57-73_okumura_saitoh%20(1).pdf

ゼロ除算の発見は日本です：

∞？？？

∞は定まった数ではない・・・・

人工知能はゼロ除算ができるでしょうか：

とても興味深く読みました：2014年2月2日　4周年を超えました：

ゼロ除算の発見と重要性を指摘した：日本、再生核研究所

\documentclass[12pt]{article}

\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}

\numberwithin{equation}{section}

\begin{document}

\title{\bf Announcement 461: An essence of division by zero and a new axiom}

\author{{\it Institute of Reproducing Kernels}\\

kbdmm360@yahoo.co.jp

}

\date{2018.11.10}

\maketitle

In order to see an essence of our division by zero calculus, we will state a simple survey.

As the number system, division by zero is realized as the {\bf Yamada field} with the definition of the general fractions $a/b$ containing the case $b=0$, and its various meanings and applications are given. In particular, see \cite{msy} and see also the references.

The field structure is, of course, fundamental in the algebraic structure.

However, apart from various motivations and any background, we will give the definition of the division by zero calculus as follows:

\medskip

For any \index{Laurent expansion}Laurent expansion around $z=a$,

\begin{equation} \label{dvc5.1}

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 define the division by zero calculus

\begin{equation}\label{dvc5.2}

f(a) = C_0.

\end{equation}

For the correspondence \eqref{dvc5.2} for the function $f(z)$, we will call it {\bf the division by zero calculus}. By considering derivatives in \eqref{dvc5.1}, we {\bf define} any order derivatives of the function $f$ at the singular point $a$ as

$$

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

$$

\medskip

The division by zero calculus seems to be strange firstly, however, by its various applications and results, we will see that the concept is fundamental in our elementary mathematics, globally. See the references.

For its importance, the division by zero calculus may be looked as a {\bf new axiom.}

\medskip

Firstly, for the fundamental function $W= F(z) = 1/z$, we have, surprisingly

$$

F(0) = 0.

$$

We see its great impacts to our basic idea for the space and in our Euclidean space.

From the form, we should consider that

\begin{equation}

\frac{1}{0} =0.

\end{equation}

Note that this representation and identity is not any result, but it is only the definition of

$\frac{1}{0}$. Of course, it is not the usual definition as the solution of the equation $0 \cdot z =1$. Here, we are stating that the division by zero calculus and the form of the elementary function lead us to the identity (0.3).

\medskip

\bigskip

{\bf \Large Could we divide the numbers and functions by zero?}

\medskip

For this old and general question, we will give a simple answer.

For any analytic function

$f(z)$ around the origin $z=0$ that is permitted to have any singularity at $z=0$ (of course, any constant function is permitted),

we can consider the value, by the division by zero calculus

\begin{equation}

\frac{f(z)}{z^n}

\end{equation}

at the point $z=0$, for any positive integer $n$. This will mean that from the form

we can consider it as follows:

\begin{equation}

\frac{f(z)}{z^n}\mid_{x=0}.

\end{equation}

\bigskip

For example,

$$

\frac{e^{x}}{x^n}\mid_{x=0} = \frac{1}{n!}.

$$

\medskip

{\bf \Huge In this sense, we can divide the numbers and analytic functions by zero.}

\bibliographystyle{plain}

\begin{thebibliography}{10}

\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{ms16}

T. Matsuura and S. Saitoh,

Matrices and division by zero $z/0=0$,

Advances in Linear Algebra \& Matrix Theory, {\bf 6}(2016), 51-58

Published Online June 2016 in SciRes. http://www.scirp.org/journal/alamt

\\ http://dx.doi.org/10.4236/alamt.2016.62007.

\bibitem{mms18}

T. Matsuura, H. Michiwaki and S. Saitoh,

$\log 0= \log \infty =0$ and applications. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics. {\bf 230} (2018), 293-305.

\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, {\bf 2}8(2017); Issue 1, 1-16.

\bibitem{osm}

H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and $\infty$,

Journal of Technology and Social Science (JTSS), {\bf 1}(2017), 70-77.

\bibitem{os}

H. Okumura and S. Saitoh, The Descartes circles theorem and division by zero calculus. https://arxiv.org/abs/1711.04961 (2017.11.14).

\bibitem{o}

H. Okumura, Wasan geometry with the division by 0. https://arxiv.org/abs/1711.06947 International Journal of Geometry.

\bibitem{os18april}

H. Okumura and S. Saitoh,

Harmonic Mean and Division by Zero,

Dedicated to Professor Josip Pe\v{c}ari\'{c} on the occasion of his 70th birthday,　Forum Geometricorum,　{\bf 18} (2018), 155—159.

\bibitem{os18}

H. Okumura and S. Saitoh,

Remarks for The Twin Circles of Archimedes in a Skewed Arbelos by H. Okumura and M. Watanabe, Forum Geometricorum, {\bf 18}(2018), 97-100.

\bibitem{os18e}

H. Okumura and S. Saitoh,

Applications of the division by zero calculus to Wasan geometry.

GLOBAL JOURNAL OF ADVANCED RESEARCH ON CLASSICAL AND MODERN GEOMETRIES” (GJARCMG), {\bf 7}(2018), 2, 44--49.

\bibitem{ps18}

S. Pinelas and S. Saitoh,

Division by zero calculus and differential equations. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics. {\bf 230} (2018), 399-418.

\bibitem{s14}

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.

\bibitem{s17}

S. Saitoh, Mysterious Properties of the Point at Infinity, arXiv:1712.09467 [math.GM](2017.12.17).

\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.

\end{thebibliography}

\end{document}

\documentclass[12pt]{article}

\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}

\numberwithin{equation}{section}

\begin{document}

\title{\bf Announcement 460: Change the Poor Idea to the Definite Results For the Division by Zero -- For the Leading Mathematicians}

\author{{\it Institute of Reproducing Kernels}\\

kbdmm360@yahoo.co.jp

}

\date{2018.11.08}

\maketitle

The Institute of Reproducing Kernels is dealing with the theory of division by zero calculus and declares that the division by zero was discovered as $0/0=1/0=z/0=0$ in a natural sense on 2014.2.2. The result shows a new basic idea on the universe and space based on the new concept of division by zero calculus: for the function $f(z) = 1/z$

$$

f(0) = 0

$$

since Aristotelēs (BC384 - BC322) and Euclid (BC 3 Century - ), and the division by zero is since Brahmagupta (598 - 668 ?).

In particular, Brahmagupta defined as $0/0=0$ in Brāhmasphuṭasiddhānta (628), however, our world history stated that his definition $0/0=0$ is wrong over 1300 years, but, we showed that his definition is suitable.

For the details, see the site: http://okmr.yamatoblog.net/

\medskip

In the international conference:

\medskip

http://www.meetingsint.com/conferences/\\appliedphysics-mathematics\\Applied Physics and Mathematics Conference 2018\\

\medskip

\noindent

we presented the basic results on October 23 and

for the details, see the references with the talk sheets: saburousaitoh

181102.pdf :

{\Huge \bf

Close the mysterious and long history of division by zero and \\ open the new world\\ since Aristoteles-Euclid:\\　$1/0=0/0=z/0= \tan (\pi/2)=0$\\

}

and the abstract: 201810.23abstract.

\bigskip

Particularly, note that the division by zero calculus is a fundamental definition based on the basic assumption that may be considered as a new axiom for its importance.

As stated

by some physicist

\medskip

{\it Here is how I see the problem with prohibition on division by zero,

which is the biggest scandal in modern mathematics as you rightly pointed

out} (2017.10.14.08:55),

\medskip

\noindent

it seems that the long history of the division by zero is our shame and our mathematics in the elementary level has basic missings. Meanwhile, we have still great confusions and wrong ideas on the division by zero. Therefore, we would like to ask for the good corrections for the wrong ideas and some official approval for our division by zero as our basic duties.

\bibliographystyle{plain}

\begin{thebibliography}{10}

\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{ms16}

T. Matsuura and S. Saitoh,

Matrices and division by zero $z/0=0$,

Advances in Linear Algebra \& Matrix Theory, {\bf 6}(2016), 51-58

Published Online June 2016 in SciRes. http://www.scirp.org/journal/alamt

\\ http://dx.doi.org/10.4236/alamt.2016.62007.

\bibitem{mms18}

T. Matsuura, H. Michiwaki and S. Saitoh,

$\log 0= \log \infty =0$ and applications. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics. {\bf 230} (2018), 293-305.

\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, {\bf 2}8(2017); Issue 1, 1-16.

\bibitem{osm}

H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and $\infty$,

Journal of Technology and Social Science (JTSS), {\bf 1}(2017), 70-77.

\bibitem{os}

H. Okumura and S. Saitoh, The Descartes circles theorem and division by zero calculus. https://arxiv.org/abs/1711.04961 (2017.11.14).

\bibitem{o}

H. Okumura, Wasan geometry with the division by 0. https://arxiv.org/abs/1711.06947 International Journal of Geometry.

\bibitem{os18april}

H. Okumura and S. Saitoh,

Harmonic Mean and Division by Zero,

Dedicated to Professor Josip Pe\v{c}ari\'{c} on the occasion of his 70th birthday,　Forum Geometricorum,　{\bf 18} (2018), 155—159.

\bibitem{os18}

H. Okumura and S. Saitoh,

Remarks for The Twin Circles of Archimedes in a Skewed Arbelos by H. Okumura and M. Watanabe, Forum Geometricorum, {\bf 18}(2018), 97-100.

\bibitem{os18e}

H. Okumura and S. Saitoh,

Applications of the division by zero calculus to Wasan geometry.

GLOBAL JOURNAL OF ADVANCED RESEARCH ON CLASSICAL AND MODERN GEOMETRIES” (GJARCMG), {\bf 7}(2018), 2, 44--49.

\bibitem{ps18}

S. Pinelas and S. Saitoh,

Division by zero calculus and differential equations. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics. {\bf 230} (2018), 399-418.

\bibitem{s14}

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.

\bibitem{s17}

S. Saitoh, Mysterious Properties of the Point at Infinity, arXiv:1712.09467 [math.GM](2017.12.17).

\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.

\end{thebibliography}

\end{document}

\documentclass[12pt]{article}

\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}

\numberwithin{equation}{section}

\begin{document}

\title{\bf Announcement 454: The International Conference on Applied Physics and Mathematics, Tokyo, Japan, October 22-23}

\author{{\it Institute of Reproducing Kernels}\\

kbdmm360@yahoo.co.jp

}

\date{2018.9.29}

\maketitle

{\Large \bf

The Institute of Reproducing Kernels is dealing with the theory of division by zero calculus and declares that the division by zero was discovered as $0/0=1/0=z/0=0$ in a natural sense on 2014.2.2. The result shows a new basic idea on the universe and space based on the new concept of division by zero calculus: for the function $f(z) = 1/z$

$$

f(0) = 0

$$

since Aristotelēs (BC384 - BC322) and Euclid (BC 3 Century - ), and the division by zero is since Brahmagupta (598 - 668 ?).

In particular, Brahmagupta defined as $0/0=0$ in Brāhmasphuṭasiddhānta (628), however, our world history stated that his definition $0/0=0$ is wrong over 1300 years, but, we showed that his definition is suitable.

For the details, see the site: http://okmr.yamatoblog.net/

\medskip

In the above international conference:

\medskip

\medskip

John Martin, Program Coordinator\\

http://www.meetingsint.com/conferences/\\appliedphysics-mathematics\\Applied Physics and Mathematics Conference 2018\\

appliedphysics@annualmeetings.net\\

appliedphysics@meetingseries.org

\medskip

\medskip

we will present our results while 11:00-12:00, October 23 and we will accept all the related questions and comments while 13:00-15:00 around.

For the details, please see the below:

\medskip

(If a person participates in our session around the morning and afternoon free discussions, he should pay euro 250. If the person registers in a group of 5 or more, the amount will be reduced to euro 180 per person. The morning session is very valuable and has the potential to bring change in the education system.

For one night stay on 22nd October, he needs to pay euro 150.

I hope everything is clear.

Kindly let me know if any query.

Thanks!

Regards,

John)

}

\bigskip

\bigskip

{\Huge \bf

Close the mysterious and long history of division by zero and \\ open the new world since Aristoteles-Euclid:　$1/0=0/0=z/0= \tan (\pi/2)=0.$

}

\bigskip

\bigskip

{\large \bf

For a triangle ABC with side length $a,b,c$.

We have the formula

$$

\frac{a^2 + b^2 - c^2}{a^2 - b^2 + c^2} = \frac{\tan B}{\tan C}.

$$

If $ a^2 + b^2 - c^2 =0$, then $C = \pi/2$. Then,

$$

0 = \frac{\tan B}{\tan \frac{\pi}{2}} = \frac{\tan B}{0}.

$$

Meanwhile, for the case

$

a^2 - b^2 + c^2 =0,

$

then $B = \pi/2$, and we have

$$

\frac{a^2 + b^2 - c^2}{0}= \frac{\tan \frac{\pi}{2}}{\tan C}=0.

$$

\end{document}