Who Did Derive First the Division by
Zero 1/0 and the Division by ZeroCalculus tan(π/2) = 0, log 0 = 0 as theOutputs of a Computer?
Saburou Saitoh
Institute of Reproducing Kernels
Kawauchi-cho, 5-1648-16, Kiryu 376-0041, JAPANsaburou.saitoh@gmail.com
March 11, 2019
Abstract:
In this short paper, we will introduce an essence of thedivision by zero calculus and the situation from the viewpoint of computersthat will contain a surprising news on the division by zero calculus.
Key Words: Division by zero, division by zero calculus, 0/0 = 1/0 =z/0 = tan(π/2) = log 0 = 0, infinity, discontinuous, Isabelle/HOL.
2010 Mathematics Subject Classification: 00A05, 00A09

1Introduction

In this short paper, we will introduce an essence of the division by zero
calculus and the situation in connection with computers based on [50].
The Institute of Reproducing Kernels is dealing with the theory of divisionby 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 Aristotele (BC384 - BC322)
and Euclid (BC 3 Century - ), and the division by zero is since Brahmagupta(598 - 668 ?).
1

For the details, see the references.
A simple and essential introduction of the division by zero is given by thedivision by zero calculus:
For any Laurent expansion around z = a,f (z) =−1∑

Cn (z − a) + C0 +nn=−∞∞∑Cn (z − a)n ,

(1.1)

n=1

we definef (a) = C0 ,(1.2)

as a value of the function f at the singular point z = a.
Here, we will note a naturality of the division by zero calculus.
Recall the Cauchy integral formula for analytic functions; for an analytic
function f (z) around z = a and for a smooth simple Jordan closed curve γ
enclosing one time the point a, we have
∫
f (z)
1dz.
f (a) =2πi γ z − aEven when the function f (z) has any singularity at the point a, we assume
that this formula is valid as the division by zero calculus. We define the valuef (a) by the Cauchy integral. For this background idea, we can consider thevalue of a function with some mean value of the function.
For the importance of this definition, the division by zero calculus may
be considered as a new axiom. This was discovered on May 8, 2014 from thevalue of the function exp(1/z) at the origin z = 0 ([2]).
In particular, for the function W = f (z) = 1/z, we have f (0) = 0. We
will write this result as
1= 0,0from the form. Here, the definition of 10 is given by this sense by meansof the division by zero calculus. Of course, 01 is not a usual fraction in theusual sense that 10 = X if and only if 1 = 0 × X; this means a contradiction.
See [21] for the details.
On February 16, 2019 H. Okumura introduced the surprising news in
Research Gate:
2

José Manuel Rodríguez Caballero
Added an answer
In the proof assistant Isabelle/HOL we have x/0 = 0 for each number x.
This is advantageous in order to simplify the proofs. You can download thisproof assistant here: https://isabelle.in.tum.de/.
J.M.R. Caballero kindly showed surprisingly several examples by the system that
πtan = 0,2log 0 = 0,1exp (x = 0) = 1,xand others. Precisely:
Dear Saitoh,
In Isabelle/HOL, we can define and redefine every function in different
ways. So, logarithm of zero depend upon our definition. The best definitionis the one which simplify the proofs the most. According to the experts, z/0= 0 is the best definition for division by zero.
tan(π/2) = 0
log 0 =
is undefined (but we can redefine it as 0)e0 = 1(but we can redefine it as 0)
00 = 1(but we can redefine it as 0).
In the attached file you will find some versions of logarithms and exponentials satisfying different properties. This file can be opened with thesoftware Isabelle/HOL from this webpage: https://isabelle.in.tum.de/Kind Regards,José M.
(2017.2.17.11:09).
At 2019.3.4.18:04 for my short question, we received:3

It is as it was programmed by the HOL team.
Jose M.
On Mar 4, 2019, Saburou Saitoh wrote:
Dear José M.
I have the short question.
For your outputs for the division by zero calculus, for the input, is it somedirect or do you need some program???
With best regards, Sincerely yours,
Saburou Saitoh 2019.3.4.18:00
As we stated in [14], the important point in the division by zero problem
is on its definition (meaning of division.), because in the usual sense,
we can not consider the division by zero.
L. C. Paulson stated that I would guess that Isabelle has used this convention 1/0 = 0 since the 1980s and introduced his book [7] referred to thisfact. However, in his group the importance of this fact seems to be entirely
ignored at this moment as we see from the book.
The result 1/0 = 0 has a long tradition of Isabelle, however, the result
has not been accepted by the world.
Indeed, S. K. Sen and R. P. Agarwal [22] referred to the paper [2] inconnection with division by zero, however, their understandings on the paper
seem to be not suitable (not right) and their ideas on the division by zeroseem to be traditional, indeed, they stated as a conclusion of the introductionof the book that:“Thou shalt not divide by zero” remains valid eternally.
However, in [20] we stated simply based on the division by zero calculusthat
We Can Divide the Numbers and Analytic Functions by Zerowith a Natural Sense.
In these situations, the results of J.M.R. Caballero will be very interested.
For some precise information, we would like to ask for the question that
Who did derive first the division by zero 1/0 and the division byzero calculus tan(π/2) = 0, log 0 = 0 as the outputs of a computer?
If it is possible, we would like to know the related details.
Furthermore, we got the important informations from Caballero’s e-mails
to some person who was saying 0/0 = 1 for many years:
42019.3.6.15:23: To accept that x/0 = 0 produces no contradiction, using
the rules of Isabelle/HOL. You could download the software and try to provethat 1 = 0 (this will be impossible). http://isabelle.in.tum.de/There are millions of dollars invested in Isabelle/HOL, where x/0 = 0(this is not a joke): https://www.cl.cam.ac.uk/ lp15/Grants/Alexandria/
Prof. Saitoh derived that x/0 = 0 from pure geometric intuition and hewas right.
2019.3.8.22:23: The deduction that z/0 = 0, for any z, is based in Saitoh’sgeometric intuition and it is currently applied in proof assistant technology,
which are useful in industry and in the military.
Acknowledgement
The author wishes to express his deep thanks Hiroshi Okumura and José
Manuel Rodríguez Caballero for many exciting e-mails.

References[1] C. B. Boyer, An early reference to division by zero, The Journal of theAmerican Mathematical Monthly, 50 (1943), (8), 487- 491. Retrieved
March 6, 2018, from the JSTOR database.
[2] M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane, New meanings of the division by zero and interpretations on 100/0 = 0 andon 0/0 = 0, Int. J. Appl. Math. 27 (2014), no 2, pp. 191-198, DOI:
10.12732/ijam.v27i2.9.
[3] T. Matsuura and S. Saitoh, Matrices and division by zero z/0 = 0,Advances in Linear Algebra & Matrix Theory, 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.
[4] T. Matsuura, H. Michiwaki and S. Saitoh, log 0 = log ∞ = 0 andapplications. Differential and Difference Equations with Applications.
Springer Proceedings in Mathematics & Statistics. 230 (2018), 293-305.

5[5] H. Michiwaki, S. Saitoh and M.Yamada, Reality of the division by zero
z/0 = 0, IJAPM International J. of Applied Physics and Math. 6(2015),
1–8. http://www.ijapm.org/show-63-504-1.html[6] H. Michiwaki, H. Okumura and S. Saitoh, Division by Zero z/0 = 0 in
Euclidean Spaces, International Journal of Mathematics and Computation, 28(2017); Issue 1, 1-16.
[7] T. Nipkow, L. C. Paulson and M. Wenzel, Isabelle/HOL A Proof Assistant for Higher-Order Logic, Lecture Notes in Computer Science,Springer E E002 E E.
[8] H. Okumura, S. Saitoh and T. Matsuura, Relations of 0 and ∞, Journalof Technology and Social Science (JTSS), 1(2017), 70-77.
[9] H. Okumura and S. Saitoh, The Descartes circles theorem and divisionby zero calculus. https://arxiv.org/abs/1711.04961 (2017.11.14).
[10] H. Okumura, Wasan geometry with the division by 0.
https://arxiv.org/abs/1711.06947 International Journal of Geometry, 7(2018), No. 1, 17-20.
[11] H. Okumura and S. Saitoh, Harmonic Mean and Division by Zero,
Dedicated to Professor Josip Pečarić on the occasion of his 70th birthday, Forum Geometricorum, 18 (2018), 155—159.
[12] H. Okumura and S. Saitoh, Remarks for The Twin Circles of Archimedes
in a Skewed Arbelos by H. Okumura and M. Watanabe, Forum Geometricorum, 18(2018), 97-100.
[13] 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), 7(2018), 2, 44–49.
[14] H. Okumura and S. Saitoh, Wasan Geometry and Division by Zero Calculus, Sangaku Journal of Mathematics (SJM), 2 (2018), 57–73.
[15] S. Pinelas and S. Saitoh, Division by zero calculus and differential equations. Differential and Difference Equations with Applications. Springer
Proceedings in Mathematics & Statistics. 230 (2018), 399-418.
6

[16] H. G. Romig, Discussions: Early History of Division by Zero, American
Mathematical Monthly, 31, No. 8. (Oct., 1924), 387-389.
[17] S. Saitoh, Generalized inversions of Hadamard and tensor products formatrices, Advances in Linear Algebra & Matrix Theory. 4 (2014), no.
2, 87–95. http://www.scirp.org/journal/ALAMT/[18] 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, 177(2016), 151-182.
[19] S. Saitoh, Mysterious Properties ofarXiv:1712.09467 [math.GM](2017.12.17).

thePointatInfinity,[20] S. Saitoh, We Can Divide the Numbers and Analytic Functions byZero with a Natural Sense, viXra:1902.0058 submitted on 2019-02-03
22:47:53.
[21] S. Saitoh, Zero and Infinity; Their Interrelation by Means of Divisionby Zero, viXra:1902.0240 submitted on 2019-02-13 22:57:25.
[22] S.K.S. Sen and R. P. Agarwal, ZERO A Landmark Discovery, theDreadful Volid, and the Unitimate Mind, ELSEVIER (2016).
[23] S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operations on the real and complex fields, Tokyo
Journal of Mathematics, 38(2015), no. 2, 369-380.
[24] Announcement 179 (2014.8.30): Division by zero is clear as z/0=0 and
it is fundamental in mathematics.
[25] Announcement 185 (2014.10.22): The importance of the division by zero
z/0 = 0.
[26] Announcement 237 (2015.6.18): A reality of the division by zero z/0 = 0
by geometrical optics.
[27] Announcement 246 (2015.9.17): An interpretation of the division byzero 1/0 = 0 by the gradients of lines.

7[28] Announcement 247 (2015.9.22): The gradient of y-axis is zero andtan(π/2) = 0 by the division by zero 1/0 = 0.
[29] Announcement 250 (2015.10.20): What are numbers? - the Yamada
field containing the division by zero z/0 = 0.
[30] Announcement 252 (2015.11.1): Circles and curvature - an interpretation by Mr. Hiroshi Michiwaki of the division by zero r/0 = 0.
[31] Announcement 281 (2016.2.1): The importance of the division by zeroz/0 = 0.
[32] Announcement 282 (2016.2.2): The Division by Zero z/0 = 0 on theSecond Birthday.
[33] Announcement 293 (2016.3.27): Parallel lines on the Euclidean planefrom the viewpoint of division by zero 1/0=0.
[34] Announcement 300 (2016.05.22): New challenges on the division by zero
z/0=0.
[35] Announcement 326 (2016.10.17): The division by zero z/0=0 - its impact
to human beings through education and research.
[36] Announcement 352(2017.2.2): On the third birthday of the division by
zero z/0=0.
[37] Announcement 354(2017.2.8): What are n = 2, 1, 0 regular polygons
inscribed in a disc? – relations of 0 and infinity.
[38] Announcement 362(2017.5.5): Discovery of the division by zero as 0/0 =1/0 = z/0 = 0.
[39] Announcement 380 (2017.8.21): What is the zero?
[40] Announcement 388(2017.10.29): Information and ideas on zero and division by zero (a project).
[41] Announcement 409 (2018.1.29):
Division by Zero.

Various Publication Projects on the[42] Announcement 410 (2018.1 30): What is mathematics? – beyond logic;
for great challengers on the division by zero.
8[43] Announcement 412(2018.2.2): The 4th birthday of the division by zero
z/0 = 0.
[44] Announcement 433(2018.7.16): Puha’s Horn Torus Model for the Riemann Sphere From the Viewpoint of Division by Zero.
[45] Announcement 448(2018.8.20): Division by Zero; Funny History andNew World.
[46] Announcement 454(2018.9.29): The International Conference on Applied Physics and Mathematics, Tokyo, Japan, October 22-23.
[47] Announcement 460(2018.11.06): Change the Poor Idea to the Definite
Results For the Division by Zero - For the Leading Mathematicians.
[48] Announcement 461(2018.11.10): An essence of division by zero and anew axiom.
[49] Announcement 471(2019.2.2): The 5th birthday of the division by zero
z/0 = 0.
[50] Announcement 478(2019.3.4): Who did derive first the division by zero
1/0 and the division by zero calculus tan(π/2) = 0, log 0 = 0 as theoutputs of a computer?

9