Divide by Zero Cookie Intolerance Posted by David Harley on July 6, 2015.

Divide by Zero Cookie Intolerance

Posted by David Harley on July 6, 2015.

Once upon a time – I guess it was in the late 1980s – a band I was in got into a debate over what we should call ourselves. I’d fairly recently switched from the building industry to the IT industry, and had become totally hooked on programming, which may explain why my own slightly Zen suggestion was ‘Division by Zero’. Yes, I know there is or was a Polish band with that name, but there wasn’t then… Anyway, I was outvoted.*

Still, decades on I was fascinated to read that The internet is obsessed with what happens when you ask Siri “what’s zero divided by zero?” It turns out that Apple’s virtual personal assistant thinks that’s a silly question and gives you an accurate enough but somewhat brusque response:

“Imagine that you have zero cookies and you split them evenly among zero friends. How many cookies does each person get? See? It doesn’t make sense. And Cookie Monster is sad that there are no cookies, and you are sad that you have no friends.”

Ouch! Well, maybe if you’re easily offended by an automated response, you’d better not ask the question. But the chances are that people have been interested in seeing/hearing the meme of the week rather than the real answer to the question, which is (literally) indeterminate, mathematically speaking. I’m not going to attempt to explain it because, frankly, my understanding of calculus doesn’t go much beyond being able to spell it.

However, 0/0 is actually a special case of division by zero. While Siri does offer a commonsense-ish explanation of why you can’t divide zero by itself, and you can certainly argue that in terms of everyday arithmetic, there isn’t much you can do with a number divided by nothing that you can’t do with a number that isn’t divided at all. However, there is actually a more formal reason why you ‘can’t’ divide by zero.

From a programmer’s viewpoint, though, there are practical issues. Calculators are normally programmed to return a ‘Divide by Zero’ error where a / is followed by a 0 followed by an =. In fact, many programming languages are specifically designed to prevent an illegal ‘Divide by Zero’ operation because the result of the operation is unreliable (to say the least). Sometimes, this will result in a compiled program grinding to a halt with an inscrutable error message along the lines of ‘ runtime error [numeric code] at [segment]:[offset]’ which can be very embarrassing for the programmer, especially one who may have no idea what he’s done to initiate an arithmetical operation. While I’m absolutely not a mathematician – and was never primarily a programmer – I did enough coding (some of it math-related) at that time to learn the hard way that sometimes you need to do some of your own exception handling in unexpected contexts. A program that crashes out with an error code is fine when you’re debugging it, not so fine when it does so out in the real world.

What does this have to do with security? Well, the accuracy and integrity of data certainly concern those of us working in security, and an unreliable application – well, any application may fail if the right combination of the wrong circumstances arises – can certainly give you a nasty shock with a high impact. Reportedly, in 1997 the USS Yorktown suffered a systems failure due to a divide-by-zero error in a Windows NT application that left it dead in the water for two and three-quarter hours. One contributor to the RISKS digest observed that:

Ideally, the next few generations of operating systems will end up being so incompatible with legacy systems that no country anywhere will be able to wage war.

David Harley

*I know you’ve only read this right to the end in the hope of finding out what we did call ourselves, so let me quench the fires of curiosity. We became the Flying Piglets. When I tell people that (not very often) they tend to say “Oh… you were quite famous then.” Evidently I need to improve my enunciation: the Flying Piglets (not to be confused with Three Flying Piglets) weren’t nearly as famous as the Flying Pickets.http://itsecurity.co.uk/2015/07/divide-by-zero-cookie-intolerance/

とても興味深く読みました：ゼロ除算の発見は再生核研究所

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\title{\bf  Announcement 412:  The 4th birthday of the division by zero $z/0=0$ \\

(2018.2.2)}

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

Kawauchi-cho, 5-1648-16,\\

Kiryu 376-0041, Japan\\

 }

\date{\today}

\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 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 references and the site: http://okmr.yamatoblog.net/

We wrote a global book manuscript \cite{s18} with 154 pages

 and stated in the preface and last section of the manuscript as follows:

\bigskip

{\bf Preface}

\medskip

 The division by zero has a long and mysterious story over the world (see, for example, H. G. Romig \cite{romig} and Google site with the division by zero) with its physical viewpoints since the document of zero in India on AD 628. In particular, note that Brahmagupta (598 -668 ?) established the 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.

 The division by zero $1/0=0/0=z/0$ itself will be quite clear and trivial with several natural extensions of the fractions against the mysterously long history, as we can see from the concepts of the Moore-Penrose generalized inverses or the Tikhonov regularization method to the fundamental equation $az=b$, whose solution leads to the definition $z =b/a$.

  However, the result (definition) will show that

      for the elementary mapping

\begin{equation}

W = \frac{1}{z},

\end{equation}

the image of $z=0$ is $W=0$ ({\bf should be defined from the form}). 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 (\cite{ahlfors}). �As the representation of the point at infinity of the Riemann sphere by the

zero $z =  0$, we will see some delicate relations between $0$ and $\infty$ which show a strong

discontinuity at the point of infinity on the Riemann sphere. We did not consider any value of the elementary function $W =1/ z $ at the origin $z = 0$, because we did not consider the division by zero

$1/ 0$ in a good way. 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 will 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, Euclidean geometry, analytic geometry, differential equations,  complex analysis in the undergraduate level and to our basic ideas for the space and universe.

We have to arrange globally our modern mathematics in our undergraduate level. Our common sense on the division by zero will be wrong, with our basic idea on the space and the universe since Aristotle and Euclid. We would like to show clearly these facts in this book. The content is in the undergraduate level.

\bigskip

\bigskip

{\bf Conclusion}

\medskip

 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.

This book is an elementary mathematics  on our division by zero as the first publication of  books for the topics. The contents  have wide connections to various fields beyond mathematics. The author expects the readers write some philosophy, papers and essays on the division by zero from this simple source book.

The division by zero theory may be developed and expanded greatly as in the author's conjecture whose break theory was recently given surprisingly and deeply by  Professor Qi'an Guan \cite{guan} since 30 years proposed  in \cite{s88} (the original is in \cite {s79}).

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.

\bibliographystyle{plain}

\begin{thebibliography}{10}

\bibitem{ahlfors}

L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.

\bibitem{cs}

L. P.  Castro and S. Saitoh,  Fractional functions and their representations,  Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.

\bibitem{guan}

Q.  Guan,  A proof of Saitoh's conjecture for conjugate Hardy H2 kernels, arXiv:1712.04207.

\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{ms18}

T. Matsuura and S. Saitoh,

Division by zero calculus and singular integrals. (Submitted for publication)

\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.

\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, 2017), 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{os18}

H. Okumura and S. Saitoh,

Applications of the division by zero calculus to Wasan geometry.

(Submitted for publication).

\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.

\bibitem{romig}

H. G. Romig, Discussions: Early History of Division by Zero,

American Mathematical Monthly, Vol. {\bf 3}1, No. 8. (Oct., 1924), pp. 387-389.

\bibitem{s79}

S. Saitoh, The Bergman norm and the Szeg$\ddot{o}$ norm, Trans. Amer. Math. Soc. {\bf 249} (1979), no. 2, 261--279.

\bibitem{s88}

 S. Saitoh, Theory of reproducing kernels and its applications. Pitman Research Notes in Mathematics Series, {\bf 189}. Longman Scientific \& Technical, Harlow; copublished in the United States with John Wiley \& Sons, Inc., New York, 1988. x+157 pp. ISBN: 0-582-03564-3

\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. (Springer) .

\bibitem{s17}

S. Saitoh, Mysterious Properties of the Point at Infinity、

arXiv:1712.09467 [math.GM](2017.12.17).

\bibitem{s18}

S. Saitoh, Division by zero calculus (154 pages: draft): （http://okmr.yamatoblog.net/）

\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{a}

https://philosophy.kent.edu/OPA2/sites/default/files/012001.pdf

\bibitem{b}

http://publish.uwo.ca/~jbell/The 20Continuous.pdf

\bibitem{c}

http://www.mathpages.com/home/kmath526/kmath526.htm

\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.

\bibitem{ann354}

Announcement 354(2017.2.8):　What are $n = 2,1,0$ regular polygons inscribed in a disc? -- relations of $0$ and infinity.

\bibitem{362}

Announcement 362(2017.5.5): Discovery of the division by zero as  $0/0=1/0=z/0=0$

 \bibitem{380}

Announcement 380 (2017.8.21):  What is the zero?

\bibitem{388}

Announcement 388(2017.10.29):   Information and ideas on zero and division by zero (a project).

 \bibitem{409}

Announcement 409 (2018.1.29.): 　Various Publication Projects on the Division by Zero.

\bibitem{410}

Announcement 410 (2018.1 30.):  What is mathematics? -- beyond logic; for great challengers on the division by zero.

\end{thebibliography}

\end{document}