International Journal of Applied Physics and Mathematics

Reality of the Division by Zero z/0 = 0
Hiroshi Michiwaki1, Saburou Saitoh2, Masato Yamada2
1 NejiLaw Inc., 1-8-27-14F, Konan, Minato-Ku, Tokyo Japan.

2 Institute of Reproducing Kernels, Kawauchi-Cho, 5-1648-16, Kiryu 376-0041, Japan.

* Corresponding author. Tel.: +81277656755; e-mail: kbdmm360@yahoo.com.jp
Manuscript submitted November 1, 2015; accepted January 4, 2016.
doi: 10.17706/ijapm.2016.6.1.1-8

Abstract: In this paper, we will give some clear evidences of the reality of the division by zero z/0 = 0 with afundamental algebraic theorem, and physical and geometrical examples; that is, 1) a field structure
containing the division by zero, 2) by the gradient of the y axis on the (x, y) plane, 3) by the reflection 1/ zof z with respect to the unit circle with center at the origin on the complex z plane, and 4) by consideringrotation of a right circle cone having some very interesting phenomenon from some practical and physicalproblem.
Key words: Division by zero, 1/0 = 0, field, Y-field, computer science, reflection with respect to circle, point
at infinity, gradient, circle, curvature, EM radius, right circular cone.

1. Introduction
By a natural extension of the fractions𝑏(1.1)

𝑎for any complex numbers a and b, we found the surprising result, for any complex number b𝑏0= 0,(1.2)

incidentally in [1] by the Tikhonov regularization for the Hadamard product in versions for matrices andwe discussed their properties and gave several physical interpretations on the general fractions in [2] forthe case of real numbers. The result is a very special case for general fractional functions in [3].
The division by zero has a long and mysterious story over the world (see, for example, google site withthe division by zero) with its physical viewpoints since the document of zero in India on AD 628. However,Sin-Ei, Takahasi ([4]) (see also [2]) established a simple and decisive interpretation (1.2) by analyzing theextensions of fractions and by showing the complete characterization for the property (1.2):
Proposition 1 ([4], [5]). Let F be a function from C ×C to C such that𝐹 𝑏, 𝑎 𝐹 𝑐, 𝑑 = 𝐹 𝑏𝑐, 𝑎𝑑1

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for all𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝐶And𝐹 𝑏, 𝑎 =𝑏,𝑎𝑎, 𝑏 ∈ 𝐶, 𝑎 ≠ 0.

Then, we obtain, for any b ∈ C
𝐹 𝑏, 0 = 0.
We thus should consider, for any complex number b, as (1.2); that is, for the mapping
1𝑤=𝑧,(1.3)

the image of z = 0 is w = 0. 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 ([6]). Therefore, the division by zero will give great
impacts to complex analysis and to our idea for the world.
However, the division by zero (1.2) is now clear, indeed, for the introduction of (1.2), we have several
independent approaches as in:
A. by the generalization of the fractions by the Tikhonov regularization or by the Moore-Penrose
generalized inverse ([1], [2]),B. by the intuitive meaning of the fractions (division) by H. Michiwaki ([7]),C. by the unique extension of the fractions by S. Takahasi, as in Proposition 1([4], [5]),D.

by the extension of the fundamental function W = 1/z from C＼{0} onto C, that is, W = 1/z is a one toone and onto mapping from C＼{0} onto C＼{0} and the division by zero 1/0 = 0 gives a one to one
and onto mapping extension of the function W = 1/z from C onto C,

andE. by considering the values of functions with the mean values of functions.
Furthermore, we were able to find several meanings in the elementary geometry and physical meaningsof the division by zero.
In this paper, in order to show the importance of the division by zero, we will give:1) a field structure containing the division by zero ([8]),2) by the gradient of the y axis on the (x, y) plane ([9], [10]),3) by the reflection 1/ z of z with respect to the unit circle with center at the origin on the complex z
plane ([11]), and4) by considering rotation of a right circle cone having some very interesting phenomenon from some
practical and physical problem ([12]).
See also [13] for the relationship between fields and the division by zero, and the importance of thedivision by zero for computer science. However, they state that in the conclusion:
The theory of meadows depends upon the formal idea of a total inverse operator. We do not claim thatdivision by zero is possible in numerical calculations involving the rationals or reals. But we do claim thatzero totalized division is logically, algebraically and computationally useful: for some applications, allowing

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zero totalized division in formal calculations, based on equations and rewriting, is appropriate because it isconceptually and technically simpler than the conventional concept of partial division.
It seems that the relationship of the division by zero and field structures are abstract in their paper.
On the division by zero, see the survey style announcements 179,185,237,246,247,250 and 252 of theInstitute of Reproducing Kernels ([7]-[12], [14]). The division by zero is not only mathematical problems,
but also it will give great impacts to human beings and the idea on the universe. The Institute ofReproducing Kernels is presenting various opinions in Announcements (many in Japanese) on the universe.
It is clear that the items A, B, C and D are serious results in our mathematics.
At this moment, the following theorem may be looked as the fundamental theorem of the division byzero:
Theorem ([15]). Any analytic function takes a definite value at an isolated singular point with a naturalmeaning.
The essential meaning of this theorem may be understood in the sense of the mean values of analytic
functions.

2. Introduction of the Y-Field
We consider
C2 = C × C
and the direct decomposition
C2= (C＼{0})2+ ({0}× (C＼{0})) + ((C＼{0}) × {0}) + {0}2.
Then, we note
Theorem 1. For the set C2, we introduce the relation ～: for any(a, b), (c, d) ∈(C＼{0})2,𝑎, 𝑏 ~ 𝑐, 𝑑 ⇔ 𝑎𝑑 = 𝑏𝑐
and, for any (a, b), (c, d) ∉(C＼{0})2, in the above direct decomposition𝑎, 𝑏 ~ 𝑐, 𝑑 .
Then, the relation ∼ satisfies the equivalent relation.
Definition 1. For the quotient set by the relation ∼ of the set C2, we write it by A and for the class𝑎containing (a, b), we shall write it by 𝑏 .
Then, we obtain the main result:𝑎 𝑐Theorem 2. For any members , ∈ 𝐴, we introduce the product𝑏 𝑑𝑎 𝑐 𝑎𝑐∙ =𝑏 𝑑 𝑏𝑑
and the sum +3

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International Journal of Applied Physics and Mathematics𝑐,𝑑𝑎 𝑐𝑎+ =,𝑏 𝑑𝑏𝑎𝑑 + 𝑏𝑐,
𝑏𝑑

𝑖𝑓
𝑖𝑓
𝑖𝑓

𝑎 0=𝑏 1𝑐 0=𝑑 1𝑎 𝑐 0, ≠ ,𝑏 𝑑 1then, the product and the sum are well-defined and A becomes a field Y.
Indeed, we can see easily the followings: 1) Under the operation + ,Y becomes an abelian group and01= 0𝑌 is the unit element.
1Under the operation, Y＼{0Y }becomes an abelian group and 1 = 1Y is the unit element.
In Y , operations + and satisfy distributive law.
The results are very simple, however, we will have some confusions for the long history of the division byzero. The statement that for any (a, b), (c, d) ∈(C＼{0})2,𝑎, 𝑏 ~ 𝑐, 𝑑 ⇔ 𝑎𝑑 = 𝑏𝑐,
that is for non zero members, we introduce the usual fractions
like

0 1 0, ,1 0 0𝑎 𝑐,𝑏 𝑑and the identity0𝑎𝑏𝑐= . Other members
𝑑1are identified, and therefore, from 1 = 0, they are all zero. Hence, here, the division by zero 0 is

given by the definition as zero. As stated in the introduction, we can give the definition of the division by
1zero 0, in several ways. Here, following the structure of fields, the division by zero is introduced as zero.
1The division by zero 0 is not a mysterious one, but to look for its various meanings will be interesting
research topics.
Remark. We note that𝑎 𝑐 𝑎𝑑 + 𝑏𝑐+ =𝑏𝑑𝑏 𝑑is not well-defined.
Theorem 3. The two fields Y and C are homomorphic.
Indeed, consider the mapping f from Y to C:
𝑎𝑎𝑏 −1≠ 0Y𝑎𝑏𝑓: 𝑥 = | →𝑎𝑏0= 0Y .
𝑏Then, we can see easily the followings: 1) f(x+y) = f(x)+f(y), 2) f(x·y) =f(x)f(y), 3) f(1Y ) = 1, and 4) f is a
one to one and onto mapping from Y to C.
We define an inversion operation φY on Y asφY𝑎𝑏= .
𝑏𝑎

For the inverse element of x =a/b≠0Y, we will denote it by x-1.
Definition 2. We demote a binary operation / on Y as follows: For any x, y ∈ Y4

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𝑥 𝑦 = 𝑥 ∙ 𝜑Y 𝑦 =𝑥𝑦 −1 𝑦 ≠ 0Y ,0𝑦 = 0Y .

We will call the field Y with the operation φY0-divisible field. C becomesa 0-divisible field.
Theorem 3. By the homeophismf :Y → C, the inversion φ = f ◦φY ◦f -1 from C onto C is induced.
Indeed, in C, the operation φ = f ◦φY◦ f-1 is induced by the homomorphic f from the 0-divisible field C.
Then, for any z ∈ C,φ 𝑧 =𝑧 −1 𝑧 ≠ 0 ,0𝑧=0 .

We, however, would like to state that the division by zero z/0 = 0 is essentially, just the definition, and wecan derive all the properties of the division by zero, essentially, from the definition. Furthermore, by theidea of this session, we can introduce the fundamental concept of the divisions (fractions) in any field.
We should use the 0-divisible field Y for the complex numbers field C as complex numbers, by this simplemodification. Then, we can see the new world that the point at infinity is represented by zero and the zero
division z/0 is possible. Note that, however, the point ∞ at infinity makes sense still as the limit
1lim = ∞.

𝑧→0 𝑧

The ∞ is not a definite number and is considered as the limit.

3. The Gradient of y-Axis Is ZeroConsider the lines y = ax with gradients a through the origin (0, 0) on the (x, y) plane. Consider the twolimits that a (>0) tends to +∞ and a (<0) tends to −∞, respectively. As their limits, we see that the limiting
lines are y axis, in a sense. Note that the gradient of the y axis is zero, and is not infinity. This example shows
as in the graph of the function y = f(x) = 1/x at x = 0 as f(0) = 0, that was introduced by the division by zero
1/0 = 0 mathematically ([1], [2], [5], [10]).
For this result, Professor H. Begehr kindly referred to the gradient of the y axis in the above: If thegradient of the imaginary axis is 0 this would mean tan(π/2) = 0, right? Of course this would be aconsequence of 1/0 = 0! We had sent the e-mail, soon as follows:
For the gradient of y axis, we can define it as zero, in a very natural way and in the intuitive sense; ofcourse, we can give its definition precisely.
However, as you stated, we can derive it formally by the division by zero
1/0 = 0; this deduction will be very interested in itself, because, the formal result 1/0 = 0 coincidents
with the natural sense.
Surprisingly enough, this would mean tan(π/2) = 0, right? This is right in our sense; we gave thedefinition of the values for analytic functions at an isolated singular point as in the theorem in theintroduction.
As the fundamental results, we would like to state that1) The gradient of the y axis is zero,and2)𝜋tan = 0.
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4. Reflection Points
For simplicity, we will consider the unit circle |z| = 1 on the complex z =x + iy plane. Then, we have thereflection formulaz* 1z(4.1)

for any point z, as well-known ([6]). For the reflection point 1/ z , there is no problem for the points z≠0, ∞.
As the classical result, the reflection of zero is the point at infinity and conversely, for the point at infinity
we have the zero point. The reflection is a one to one and onto mapping between the inside and the outsideof the unit circle.
Are these correspondences, however, suitable? Does there exist the point at ∞, really? Is the point atinfinity corresponding to the zero point? Is the point at ∞ reasonable from the practical point of view?
Indeed, where can we find the point at infinity? Of course, we know pleasantly the point at infinity on theRiemann sphere, however, on the complex z-plane it seems that we can not find the corresponding point.
When we approach to the origin on a radial line, it seems that the correspondence reflection points
approach to the point at infinity with the direction (on the radial line).
On the concept of the division by zero, there is no point at infinity ∞ as the numbers. For any point z suchthat |z| >1, there exists the unique point z* by (4.1). Meanwhile, for any point z such that |z| <1 except z = 0,
there exits the unique point z*by (4.1). Here, note that for z = 0, by the division by zero, z*= 0. Furthermore,
we can see thatlim𝑧→0 𝑧 ∗ = ∞,(4.2)

however, for z = 0 itself, by the division by zero, we have z*= 0. This will mean a strong discontinuity of the1functions w = 𝑧 and (4.1) at the origin z = 0; that is a typical property of the division by zero. This strong
discontinuity may be looked in the above reflection property, physically.
Should we exclude the point at infinity, from the numbers? We were able to look the strong
discontinuity of the division by zero in the reflection with respect to circles, physically (geometrical optics).
The division by zero gives a one to one and onto mapping of the reflection (4.1) from the whole complex
plane onto the whole complex plane.
The infinity ∞ may be considered as in (4.2) as the usual sense of limits, however, the infinity ∞ is not adefinite number.

5. Circles and Curvature — An Interpretation of the Division by Zero r/0 = 0
We consider a solid body called right circular cone whose bottom is a disc with radius r2. We cut the bodywith a disc of radius r1(0 < r1 < r2) that is parallel to the bottom disc. We denote the distance by d between
the both discs and R the distance between the top point of the cone and the bottom circle on the surface ofthe cone. Then, R is calculated by Eko Michiwachi(8 years old daughter of Mr. H. Michiwaki) as follows:
𝑅=𝑟2
𝑑 2 − 𝑟2 − 𝑟1 2 ,𝑟2 − 𝑟1

that is called EM radius, because by the rotation of the cone on the plane, the bottom circle writes the circle

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International Journal of Applied Physics and Mathematicsof radius R. We denote by K = K(R) = 1/R the curvature of the circle with radius R. We fix the distance d.
Now note that:
𝑟1 → 𝑟2 ⇒ 𝑅 → ∞.
This will be natural in the sense that when r1= r2, the circle with radius R becomes a line.
However, the division by zero will mean that when r1= r2, the above EM radius formula makes sense and
R = 0. What does it mean? Here, note that, however, then the curvature K = K(0) = 0 by the division by zero;that is, the circle with radius R becomes a line, similarly. The curvature of a point (circle of radius zero) iszero.

Acknowledgments
The authors wish to express their deep thanks Professor Haydar Akca for his kind invitation of a paper
based on some recent results for the division by zero. Saitoh wishes to express his sincere thanks
Professors H. G. W. Begehr, J. A. Bergstra, Hiroshi Okumura, F. Stenger, Lukasz T. Stepien, Sin-Ei, Takahashiand Dr. Masako Takagi for their valuable information and suggestions.

References[1] Saitoh, S., (2014). Generalized inversions of Hadamard and tensorproducts for matrices. Advances in
Linear Algebra & Matrix Theory, 4(2), 87-95.
[2] Kuroda, M., Michiwaki, H., Saitoh, S., & Yamane, M. (2014). New meanings of the division by zero andinterpretations on 100/0 = 0 and on 0/0 = 0. Int. J. Appl. Math, 27(2), 191-198.
[3] Castro, L. P., & Saitoh, S. (2013). Fractional functions and their representations. Complex Anal. Oper.
Theory, 7(4), 1049-1063.
[4] Takahasi, S.-E. (2014). On the identities 100/0 = 0 and 0/0 = 0.
[5] Takahasi, S.-E., Tsukada, M., & Kobayashi, Y. (2015). Classification of continuous fractional binary
operations on the real and complex fields. Tokyo Journal of Mathematics, 8(2).
[6] Ahlfors, L. V. (1966). Complex Analysis. McGraw-Hill Book Company.
[7] Division by zero is clear as z/0=0 and it is fundamental in mathematics. Announcement 179.
[8] What are numbers? — The Yamada field containing the division by zero z/0 = 0. Announcement 250.
[9] An interpretation of the division by zero 1/0 = 0 by the gradients of lines. Announcement 246.
[10] The gradient of y-axis is zero and tan(π/2) = 0 by the division by zero 1/0 = 0. Announcement 247.
[11] A reality of the division by zero z/0 = 0 by geometrical optics. Announcement 237.
[12] Circles and curvature — An interpretation by Mr. Hiroshi Michiwaki of the division by zero r/0 = 0.
Announcement 252.
[13] Bergstra, J. A., Hirshfeld Y., & Tucker, J. V. (2009). Meadows and the Equational Specification of Division.
[14] The importance of the division by zero z/0 = 0. Announcement 185.
[15] Michiwaki, H., Saitoh, S., & Takagi, M. A new concept for the point at infinity and the division by zeroz/0=0.
Hiroshi Michiwaki was born at Gunma Prefecture, Japan on July 17, 1977. He was dropped out ofelementary school 4 or 5 years. He established NejiLawInc., and was the president in 2009. His ideas are
great as in the followings: Tokyo Venture Technology Award, Grand Prize winner in 2011; Good DesignAward, Gold Prize winner in 2011. He had been entrusted advance development of the strategic key

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technology from Ministry of Economy, Trade and Industry (METI) in 2012, and it's completed in 2015;
Good Design Award invitation judge in 2012. Japan Innovators Award, Excellent Award winner in 2015.
Saburou Saitoh was born at Tochigi Prefecture, Japan on March 4, 1944. He completed the postgraduate
courses of master and Ph.D. at Tokyo Institute of Technology. He got academic positions at Shibaura
Institute of Technology (1971-1976) and Gunma University (1976-2009). He was appointed as emeritus
professor of Gunma University in 2009. After that, he got a five years researcher position atCIDMA,University of Aveiro (2009-2014). He published, with his collaborators, over 150 papers and 7
books (indexed in the new zbMATH interface). Moreover, together with professor Yoshihiro Sawano, he isplanning to publish in Springer a fundamental monograph entitled Theory of Reproducing Kernels andApplications.
Masato Yamada was born at Maebashi, Japan on February 18, 1969, and he is a part-time lecturer ofGunma University, Gunma Prefectual Woman's University, National Institute Technology, Gunma College,
Maebashi Institute of Technology and so on. He obtained his doctor's degree from division of electronics
and computing, Graduate School of Engineering, Gunma University. He is interested in mathematicaleducation and educational technology.

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