Biostatistics and Biometrics
Open Access Journal
ISSN: 2573-2633

Mini Review

Biostat Biometrics Open Acc J

Volume 7 Issue 1 - May 2018
DOI: 10.19080/BBOAJ.2018.07.555703

Copyright © All rights are reserved by Hiroshi Okumura

Is It Really Impossible To Divide By Zero?

Hiroshi Okumura
Ph.D, Maebashi Institute of Technology (retired), Thailand
Submission: March 26, 2018; Published: May 22, 2018

*Corresponding author: Hiroshi Okumura, Ph.D. Maebashi Institute of Technology (retired), Mueang District KhonKaen 40000, Thailand,Tel:; Email:
Abstract
It has been said that to define to divide by zero is impossible. But recent attempts show that it can be done very easily, which we would liketo show below.

Keywords: Divide by zero; Impossible; Satisfying; Natural extension; Ordinary fraction; Real number; Amazing phenomena; Origin;
Perpendicularly; Intersecting curves; Tangents; Intersection; Two curves; Infinity; Orthogonal; Slope; Horizontal line; Fixed radius; Initial point;
Graph passes; Circle.

Mini Review
Definition
Let us consider a function F : R × R → R satisfyinga, b)ifb ≠ 0, F ( a=b.

 F ( a, b ) F ( c, d ) = F ( ac, bd )This is a natural extension of the ordinary fraction a b .
While,=F ( a, 0 )

2=F ( a, 0 )2= F ( 2,1=) F ( a, 0 )F ( 2, 2 =) F ( a, 0 )F ( 2 a, 0 )

2=F ( a, 0 ) 2 F ( a, 0 )1

implies F ( a, 0 ) = 2 F ( a, 0 ) .Therefore F ( a, 0 ) = 0 for any real
number a [1]. The fact shows that we have no choice but to define
a 0 = 0 for any real number a. Hence, we define
a= 0 		(1)0

For any real number a.

Notice that we also get 0 0 = 0 by the definition [1-4]. This
simple discussion was made by Sin-Ei Takahashi, an emeritus
professor at Yamagata University. As you see the definition of thedivision by zero is very easy, but we have to notice that 1 0 is not
the inverse of 0, i.e.,
10 × = 0 × 0 = 0.
0

Biostat Biometrics Open Acc J 7(1): BBOAJ.MS.ID.555703. (2018)

The same definition was already given by Brahmagupta.

What can be seen from the definition

The definition (1) brings us several amazing phenomena. A
typical thing is that most things we have been thinking as infinity
without paying attention are exactly 0. For example, for thefunction y = 1 x , x = 0 implies y = 0, i.e., its graph passes throughthe origin ( 0, 0 ) .
If the two lines ai x + bi y + ci = 0 ( i = 1, 2 ) are not parallel, theymeet in the point b1c2 − b2 c1 a2 c1 − a1c2 ,.
 a1b2 − a2b1 a1b2 − a2b1 0, the definition (1) shows
If they are parallel, i.e., a1b2 − a2b1 =that they meet in the origin ( 0, 0 ) . The other typical thing is that thetwo perpendicularly intersecting curves can be considered as theytouch. Because we get tan 900 = 0 by the definition (1). Therefore
if two curves intersect perpendicularly, the slopes of the twotangents at the point of intersection are the same, i.e., we can still
consider that the two curves touch at the point [5-8].

Let us consider a circle C of fixed radius r with center Pon a half line with initial point O (Figure 1). Let t be one of thetangents of C from O and let θ be the angle between l and t. Then
we can easily get

OP =r.
sin θ005

Biostatistics and Biometrics Open Access JournalHence the point P coincides with O (Figure 3). The line
C in the case θ > 0, but it is orthogonal to C in thetouchestcase θ = 0. But we can still consider that it touches C because thetangent at the point of intersection has slope tan 900 = 0 by thedefinition (1) if l is a horizontal line.

Conclusion
Figure 1

Mathematics is made upon the postulates. Any postulate
should be taken into consideration, if it gives reasonableconclusions with new insights. The research of the division by
zero was established by Saburou Saitoh [8], an emeritus professor
at Gunma University, four years ago. Since the foundation ofthis topics are already made [1-8], it seems that the research isentering next stage as experts consider the division by zero ineach of the special fields of Science.

References
Figure 2

We now consider the angle θ becomes smaller and smaller.
Then P gets further away from O (Figure2). In the case θ = 0, thelines l and t coincide. Then one may think that the circle C has
moved away into point at infinity (this is not an exact expressionthough). But the definition (1) implies

rr
OP== = 0.
sin 00 0

1. Kuroda M, Michiwaki H, Saitoh S, Yamane M (2014) New meanings ofthe division by zero and interpretations on 100=0 = 0 and 0=0 = 0. Int
J Appl Math 27(2): 191-198.
2. Matsuura T, Michiwaki H, Saitoh S (2017) log0=log1=0 andapplications. Differential and Difference Equations with Applications.
Springer Proceedings in Mathematics & Statistics.
3. Matsuura T, Saitoh S (2016) Matrices and Division by zero z/0 = 0. Adv
Linear Algebra & Matrix Theory 6: 51-58.

4. Michiwaki H, Okumura H, Saitoh S (2017) Division by Zero z/0 = 0 in
Euclidean Spaces. Int J Math Comput 28(1) (2017) 1-16.

5. Michiwaki H, Saitoh S, Yamada M (2016) Reality of the Division by Zeroz=0 = 0. Int J Appl Phys Math 6(1): 1-8.

6. Okumura H, Saitoh S (2018) Remarks for the Twin Circles ofArchimedes in a Skewed Arbelos by Okumura and Watanabe. ForumGeom 18: 97-100.
7. Okumura H, Saitoh S, Matsuura T (2017) Relations of 0 and 1. J TechSocial Science 1(1): 70-77.
Figure 3. θ = 0,This work is licensed under Creative
Commons Attribution 4.0 Licens
DOI: 10.19080/BBOJ.2018.07.555703

8. Pinelas S, Saitoh S (2016) Division by zero calculus and differential
equations, Differential and Difference Equations with Applications.
Springer Proceedings in Mathematics & Statistics.

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How to cite this article: Hiroshi O. Is It Really Impossible To Divide By Zero?. Biostat Biometrics Open Acc J. 2018; 7(1): 555703.
DOI: 10.19080/BBOJ.2018.07.555703.