International Journal of Applied Mathematics————————————————————–
Volume 27 No. 2 2014, 191-198
ISSN: 1311-1728 (printed version); ISSN: 1314-8060 (on-line version)
doi: http://dx.doi.org/10.12732/ijam.v27i2.9

NEW MEANINGS OF THE DIVISION BY ZERO AND
INTERPRETATIONS ON 100/0 = 0 AND ON 0/0 = 0
Masao Kuroda1 , Hiroshi Michiwaki2 ,Saburou Saitoh3 § , Masami Yamane4
1 Gunma University Foundation for Science and Technology

1-5-1 Tenjin-cho, Kiryu 376-8515, JAPAN
2,3,4 Institute of Reproducing Kernels
Kawauchi-cho, 5-1648-16, Kiryu 376-0041, JAPAN

Abstract: In this paper, we shall give simple and natural, however, thesurprising identities 100/0 = 0 and 0/0 = 0 by a natural extension of fractions
with the concept of Tikhonov regularization using the theory of reproducing
kernels. We shall give interpretations of the results.
AMS Subject Classification: 42A38, 30C40, 46E22, 32A36
Key Words: general fractional function, fraction, Moore-Penrose generalized
inverse, Tikhonov regularization, reproducing kernel, singular point, computer,algorithm, Newton law, big bang, Laurant expansion, singularity of analyticfunction, Picard’s great theorem, Doppler effect, capillary pressure, LaplaceYoung equation
1. Introduction
In order to state the essential idea in elementary way, we shall consider thefractions on the real numbers. Then, by a natural extension of the fractionb(1.1)
afor any numbers a and b, we shall, in particular, show the surprising results
Received:

March 7, 2014

§ Correspondence authorc 2014 Academic Publications

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M. Kuroda, H. Michiwaki, S. Saitoh, M. Yamane
1000= 0, = 0.
00(1.2)

However, in our recent paper [3] or the concept of the Moore-Penrose generalized inverse, the above results are clear - indeed, the results (1.2) themselves
were found in a very general vector version in Saitoh ([7]), however, many mathematicians do not admit the results. So, we would like to show the results in theelementary means and we shall give several interpretations of the results. Then,the results may be considered as very natural ones. Furthermore, the results
will create many models of problems in analysis and physics on the universe.

2. Definition of General Fractions
We shall give the definition first: for any real numbers a and b, the general
fractionsb(2.1)
aare defined by the following way:
For any fixed λ > 0, the minimum member of the Tikhonov function in x
λx2 + (ax − b)2 ;(2.2)

that is,ab
λ + a2is the fraction in the sense of Tikhonov. By taking the limitxλ (a, b) =lim xλ (a, b) =λ→+0b,a(2.3)(2.4)

we shall define the general fractions (2.1).
Note that, for a 6= 0, the definition (2.1) is the same as the ordinary sense,
however, when a = 0, we obtain the desired results, since xλ (0, b) = 0, always.
For the general theory of the Tikhonov regularization and many applications, see the cited references.
Mathematicians will expect if there exists some realization of our mathematics in the real world models.
So, we wonder: does there exist some real examples supporting the above
results ([8])?
Recently, Takahashi ([9]) established a simple and natural interpretationof (1.2) by analyzing any extensions of fractions and by showing the completeNEW MEANINGS OF THE DIVISION BY ZERO AND...

193

characterization for such property (1.2). Furthermore, he examined several
fundamental properties of the general fractions. His result will show that ourmathematics says that the results (1.2) should be accepted as natural ones.
However, the results will be curious for our feelings and so, the above question
will be vital still as a very important problem.
Dr. M. Dalla Riva proved simply and directly the following:
Theorem. Let F be a function from R × R to R such that
F (a, b)F (c, d) = F (ac, bd)
for alla, b, c, d ∈ R
andF (a, b) =a,ba, b ∈ R, b 6= 0.

Then, we obtain, for any a ∈ R
F (a, 0) = 0.

Proof. We have
F (a, 0) = F (a, 0)1 = F (a, 0)

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

Thus F (a, 0) = 2F (a, 0) which implies the desired result F (a, 0) = 0 for alla ∈ R.
Several mathematicians pointed out to the authors that the notations of100/0 and 0/0 are not good for the sake of the generalized sense, however,there does not exist other natural and good meaning for them. Why should weneed and use any new notations for involving the notations? We should use theusual notations, we think so.

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M. Kuroda, H. Michiwaki, S. Saitoh, M. Yamane
3. An Interpretation of 01 = 0

We will consider that the result 10 = 0 is a curious one, because
1= +∞x→+0 xlim

and meanwhile,lim

1x→−0 x= −∞.

However, recall the graph of the function
1.
xThen, we see the beautiful point 0 as the center of the graph or 0 is the meanvalue of +∞ and −∞, and the mean value is the basic property in mathematics
and in the universe.
However, when we image the graph as a big bang at the origin, then we will
be able to enjoy the graph and then we can expect the half some world fromthe negative real axis.
y=4. A Physical Interpretation of 00 = 0
We shall give a simple physical model showing the result 00 = 0. We shallconsider a disc with x2 + y 2 ≤ a2 rolling uniformly with a positive constantangle velocity ω with the center at the origin. Then we see, at the only origin,ω = 0 and at all other points, ω is a constant. That is, the velocity and theradius r are zero at the origin. This will mean that, in the general formulav = rω,or, inω=
we have at the origin,

v
r

0= 0.
0We shall not be able to obtain the result fromvlim ω = lim ,r→+0r→+0 r

because it is the constant.

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195

5. By the Newton’s Law
We shall recall the fundamental law by Newton:
F =c

m1 m2
r2(5.1)for two masses m1 , m2 with a distance r and a constant c. Of course,
lim F = ∞,r→+0(5.2)

however, as in our fraction
F =0=cm1 m2.
0(5.3)

Of course, here, we can consider the above interpretation for the mathematical formula (5.1) as the new interpretation (5.3). In the ideal case, when
the two masses are at point, the force F will not be positive, it will be reduced
to zero.
For the Coulomb’s law, see similar formulas.
Furthermore, as well-known, the bright at a point at the distance r fromthe origin is given by the formulaP
B = k 2,r(5.4)

where k is a constant and P is the amount of the light. Of course, we have, atthe infinity:
B = 0.
(5.5)
Then, meanwhile, may we consider as
B=0

(5.6)

at the origin r = 0? Then we can obtain our formulakP= 0,0

as in our new formula.
We found several very interesting results from several physical formulas thatour general fractions are very reasonable physically. We see that such generalinterpretations exist in our beings around.

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M. Kuroda, H. Michiwaki, S. Saitoh, M. Yamane
6. Analytic Function Theory, Singularity

We shall recall the elementary function
W (z) = exp=1+1z(6.1)

111+++ · · ·.
1!z 2!z 23!z 3This function has an essential singularity around the origin. However, surprisingly enough, we have
W (0) = 1.

(6.2)

From this fact, we will have a great world for singularity points problems ofanalytic functions, widely and deeply. Recall Picard’s great theorem for theserious meanings.

7. An Interpretation of 0 × 0 = 100 from 100/0 = 0The expression 100/0 = 0 will represent some division by the zero in a sense,
not the usual one, and so, we will be able to consider some product in the sense0 × 0 = 100.
We shall show such interpretation.
We shall consider same two masses m, however, their constant velocities v
for the origin are the same in the real line, in the symmetry way: We consider
the moving energy product E 2 ,11mv 2 × m(−v)2 = E 2 .
22(7.1)

We shall consider at the origin and we assume the two masses stop at the origin(possible in some case). Then, we can consider, formally0 × 0 = E2.

(7.2)

The moving energies turn to other energies, however, we can obtain some interpretation as in the above.

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197

8. Capillary Pressure in a Narrow Capillary Tube
In a narrow capillary tube saturated with fluid such as water, the capillarypressure is simply expressed as follows:
Pc =2σ,r(8.1)

where P c is capillary pressure (suction pressure), σ is surface tension, and ris radius. If r is zero, there is no pressure. However P c shows infinity, in thecommon meaning.
This simple equation is based on the Laplace-Young equation
1
1P =σ+,(8.2)
R1 R2
where R1 and R2 are two principal radii of curvature at any point on the surface
of a droplet or a bubble and in the case of spherical form, R1 = R2 = R. Fora spherical bubble the pressure difference across the bubble film is zero as thepressure is the same on both sides of the film. The Laplace-Young equation
reduces to1
1+= 0.
(8.3)
R1 R2
On the other hand, when the diameter of a bubble is decreased and becomes
0 (R = 0), the bubbles collapse, and enormous energy is generated. The accumulated free energy in the bubble is released instantaneously.
M. Yamane referred to the Doppler effects that will have many related
problems and should examine them physically. We hope our new formula may
contribute to many models of physical problems.
Meanwhile, Professor Igarashi ([4]) showed the natural interpretation for ourgeneral fractions with computer languages by natural algorithms. In particular,
he showed that the general fractions are natural ones from the viewpoint ofcomputer algorithms.
For the above problems, the statements are not so simple, and so we expect
their publication by separate covers.

Acknowledgments
The authors wish to express deep thanks to the following professors and Drs,
in particular, for their valuable discussions and communications. Indeed, the198

M. Kuroda, H. Michiwaki, S. Saitoh, M. Yamane

topics are very fundamental, and so we were requested to have many information
details on the topics among the related mathematicians ([8]):H. G. W. Begehr, Yoshihide Igarashi, Kazuko Kato, Shin-ichi Nakagiri,
Mitsuharu Otani, Matteo Dalla Riva, Lukasz T. Stepien, Masako Takagi, SinEi Takahashi, and Dmitry Vorotnikov.
S. Saitoh is supported in part by the Grant-in-Aid for the Scientific Research(C)(2)(No. 24540113).

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Mathematics, 161, Springer, Basel (2012), 87-101.
[2] L.P. Castro, S. Saitoh, Y. Sawano and A.S. Silva, Discrete linear differentialequations, Analysis, 32, No. 3 (2012), 181-191.
[3] L.P. Castro, S. Saitoh, Fractional functions and their representations,
Complex Analysis and Operator Theory, 7 (2013), 1049-1063, doi:
10.1007/s11785-011-0154-1.
[4] Y. Igarashi, Definitions of fractions by simple programs, Manuscript.
[5] S. Saitoh, Integral Transforms, Reproducing Kernels and their Applications, Pitman Research Notes in Mathematics Series, Volume 369, Addison Wesley-Longman, London (1997).
[6] S. Saitoh, Theory of reproducing kernels: Applications to approximate
solutions of bounded linear operator functions on Hilbert spaces, AMS
Translations, Ser. 2, 230 (2010), 107-134.
[7] S. Saitoh, Generalized inversions of Hadamard and tensor products formatrices, Advances in Linear Algebra & Matrix Theory, To Appear:
http://www.scirp.org/journal/alamt.
[8] S. Saitoh, A natural interpretation on 100/0= 0 and 0/0=0, and an open
question, Manuscript.
[9] S. Takahashi, On the identities 100/0=0 and 0/0 =0, Submitted.