Biostatistics and Biometrics Open Access Journal ISSN: 2573-2633

Mini Review Biostat Biometrics Open Acc J Volume 7 Issue 1 - May 2018 Copyright © All rights are reserved by Hiroshi Okumura DOI: 10.19080/BBOAJ.2018.07.555703 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 like toKeywords: show below.

Graph passes; Divide Circle. 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;

Mini Review Definition Let us consider a function satisfying F :R×→ R R WhatThe can same be definition seen from was the already definition given by Brahmagupta.

 a F( a, b) = ifb ≠ 0,  b . The definition (1) brings us several amazing phenomena. A F a,, b F c d= F ac , bd 0.  ( ) ( ) ( ) typical thing is that most things we have been thinking as infinity function yx=1, x = 0 implies y = 0, i.e., its graph passes through without paying attention are exactly For example, for the ab. the origin (0,0) . While, are not parallel, they This is a natural extension of the ordinary fraction axiii+ by += c0( i = 1, 2 ) 2 Fa( ,0) = Fa( ,0) = F( 2,2) Fa( ,0) = F( 2 a ,0) meet in the point 2 If the two lines bc−− bc a c ac 2 12 21,. 21 12 = F(2,1) Fa( ,0) = Fa( ,0) = 2 Fa( ,0) −− 1 ab12 ab 21 ab 12 ab 21 If they are parallel, i.e., ab−= ab 0, implies Fa( ,0) = 2 Fa( ,0) .Therefore Fa( ,0) = 0 for any real 12 21 that they meet in the origin The other typical thing is that the number a [1]. (0,0) . the definition (1) shows a 00= for any real number a. The fact shows that we have no choice but to define tan 900 = 0 a = 0 Hence, we define (1) two perpendicularly intersecting curves can be considered as they 0 touch. Because we get by the definition (1). Therefore For any real number a. if two curves intersect perpendicularly, the slopes of the two = tangents at the point of intersection are the same, i.e., we can still 00 0 considerLet us that consider the two a curves circle touchC at the point [5-8].r P Notice that we also get by the definition [1-4]. This O t be one of the simple discussion was made by Sin-Ei Takahashi, an emeritus of fixed radius with center 10 is not tangents of C from O and let θ l and t. Then professor at Yamagata University. As you see the definition of the on a half line with initial point (Figure 1). Let the inverse of 0, i.e., division by zero is very easy, but we have to notice that be the angle between 1 r 0× =×= 0 0 0. we can easily get OP = . 0 sinθ

Biostat Biometrics Open Acc J 7(1): BBOAJ.MS.ID.555703. (2018) 005 Biostatistics and Biometrics Open Access Journal

Hence the point P O t touches C in the case θ > 0, but it is orthogonal to C in the coincides with (Figure 3). The line case θ = 0. C because the tangent at the point of intersection has slope tan 900 = 0 by the But we can still consider that it touches l

definitionConclusion (1) if is a horizontal line. Mathematics is made upon the postulates. Any postulate Figure 1 should be taken into consideration, if it gives reasonable

conclusions 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 of this topics are already made [1-8], it seems that the research is entering next stage as experts consider the division by zero in eachReferences of the special fields of Science. 1. Figure 2 Kuroda M, Michiwaki H, Saitoh S, Yamane M (2014) New meanings of the division by zero and interpretations on 100=0 = 0 and 0=0 = 0. Int θ becomes smaller and smaller. 2. J Appl Math 27(2): 191-198. Then P O θ = 0, the We now consider the angle Matsuura T, Michiwaki H, Saitoh S (2017) log0=log1=0 and lines l and t coincide. Then one may think that the circle C has applications. Differential and Difference Equations with Applications. gets further away from (Figure2). In the case 3. Springer Proceedings in Mathematics & Statistics. Matsuura T, Saitoh S (2016) Matrices and Division by zero z/0 = 0. Adv moved away into point at infinity (this is not an exact expression 4. Linear Algebra & Matrix Theory 6: 51-58. though). But the definition (1) impliesrr OP = = = 0. Michiwaki H, Okumura H, Saitoh S (2017) Division by Zero z/0 = 0 in sin 00 0 5. Euclidean Spaces. Int J Math Comput 28(1) (2017) 1-16. Michiwaki H, Saitoh S, Yamada M (2016) Reality of the Division by Zero 6. z=0 = 0. Int J Appl Phys Math 6(1): 1-8. Okumura H, Saitoh S (2018) Remarks for the Twin Circles of Archimedes in a Skewed Arbelos by Okumura and Watanabe. Forum 7. Geom 18: 97-100. Okumura H, Saitoh S, Matsuura T (2017) Relations of 0 and 1. J Tech 8. Social Science 1(1): 70-77. Pinelas S, Saitoh S (2016) Division by zero calculus and differential Figure 3.θ = 0, 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. 06 DOI: 10.19080/BBOJ.2018.07.555703.