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In this paper, a new viewpoint of the division by zero z/0 = 0 in matrices is introduced and the results will show that the division by zero is our elementary and fundamental mathematics. New and practical meanings for many mathematical and physical formulas for the denominator zero cases may be given. Furthermore, a new space idea for the point at infinity for the Eucleadian plane is also introduced.

By a natural extension of the fractions

for any complex numbers a and b, we found the simple and beautiful result, for any complex number b

incidentally in [

The division by zero has a long and mysterious story over the world (see, for example, Google site with the division by zero) with its physical viewpoints since the document of zero in India on AD 628; however, Sin-Ei Takahasi ( [

Proposition 1. Let F be a function from

for all

and

Then, we obtain, for any

Note that the complete proof of Proposition 1 is simply done with 2 or 3 lines.

We thus should consider, for any complex number b, as (2); that is, for the mapping

the image of

However, the division by zero (2) is now clear; indeed, for the introduction of (2), we have several independent approaches as in:

1) by the generalization of the fractions by the Tikhonov regularization or by the Moore-Penrose generalized inverse,

2) by the intuitive meaning of the fractions (division) by H. Michiwaki,

3) by the unique extension of the fractions by S. Takahasi, as in the above,

4) by the extension of the fundamental function

5) by considering the values of functions with the mean values of functions.

Furthermore, in ( [

A) a field structure containing the division by zero―the Yamada field

B) by the gradient of the y axis on the

C) by the reflection

D) by considering rotation of a right circular cone having some very interesting phenomenon from some practical and physical problem.

See J. A. Bergstra, Y. Hirshfeld and J. V. Tucker ( [

Meanwhile, J. P. Barukcic and I. Barukcic ( [

Furthermore, T. S. Reis and J.A.D.W. Anderson ( [

Meanwhile, we should refer to up-to-date information:

Riemann Hypothesis Addendum―Breakthrough

Kurt Arbenz: https://www.researchgate.net/publication/272022137 Riemann Hypothesis Addendum―Break- through.

Here, we recall Albert Einstein’s words on mathematics: Blackholes are where God divided by zero. I don’t believe in mathematics. George Gamow (1904-1968) Russian-born American nuclear physicist and cosmologist remarked that “it is well known to students of high school algebra” that division by zero is not valid; and Einstein admitted it as the biggest blunder of his life (Gamow, G., My World Line (Viking, New York). p 44, 1970).

For the definitions

and

are valid and the sum is given by

However, the sum law

is, in general, not valid, however, if

In this paper, we will discuss the division by zero in matrices and we will be able to see that the division by zero is our elementary and fundamental mathematics. We will introduce a new space for the Euclidean plane. Indeed, for the point at infinity on the Riemann sphere, we will introduce a new idea and fact.

We will recall the elementary Cramer’s law. We write lines by

The common point

By the division by zero, we can understand that if

even when the two lines are the same.

The division by zero, in particular, means, surprisingly, that the point at infinity is represented by zero, that is, the coincidence of the point at infinity and the origin. Precisely, the point at infinity (topological point) is represented by

This fact may be understood that the point at infinity is reflected to the origin. In this sense, the origin will have double natures of the native origin and reflection of the point at infinity. The latter has a strong discontinuity.

We will be able to see the whole Euclidean plane by the stereographic projection into the Riemann sphere―We think that in the Euclidean plane, there does not exist the point at infinity.

However, we can consider it as a limit like ¥. Recall the definition of

The behavior of the space around the point at infinity may be considered by that around the origin by the linear transform

however,

by the division by zero. Here,

however,

Of course, two points

In order to see some realization of the properties of (12) and (13), we will consider the triangle with the basic edge (side) a and high h. Then, the area S of the triangle is given by

By fixing the high h and the line containing the side a, we will consider the limiting

However, we will see that

just like the division by zero, because, when

The strong discontinuity of the division by zero is appeared as the broken of the triangles. These phenomena may be looked in many situations as the universe one. We can consider similar problems for many types volumes. However, the simplest cases are disc and sphere (ball) with radius 1/R. When

The results in Section 4 may be interpreted beautifully by analytic geometry and matrix theory.

We write lines by

The area S of the triangle surrounded by these lines is given by

where

and

For a function

the radius R of the circle

If

When we apply the division by zero to functions, we can consider, in general, many ways.

For example, for the function

However, from the identity―the Laurent expansion around

we have

For analytic functions we can give uniquely determined values at isolated singular points by the values by means of the Laurent expansions as in (22), however, the values by means of the Laurent expansions are not always reasonable. We will need to consider many interpretations for reasonable values. In many formulas in mathematics and physics, we can see that the division by zero is valid. See [

The center of the circle (19) is given by

Therefore, the center of a general line

may be considered as the origin

We consider the functions

The distance d of the centers of the circles

If

Then,

Meanwhile, the identity

We write four planes by

The volume V of the tetrahedron surrounded by these planes is given by

where

and

For the torsion formula

if

We will give a typical physical example of the division by zero.

We will consider a spring with two spring constants

by Hooke’s law. We know, in particular, if

and by the division by zero,

that is very reasonable. In particular, by Hooke’s law, we see that

The corresponding result for the case of Ohmu’s law is similar and valid.

We will consider the fundamental ordinary differential equation

satisfying the initial conditions

Then we have the solution

with

Then, for

In this case, by continuity we can obtain the same result:

However, in the next example, we will need essentially the concept of the division by zero.

We will consider the typical ordinary differential equation

satisfying the initial conditions

Then we have the solution

Then, if

Furthermore, if

We can find many and many such examples.

By the division by zero

In the sense of the division by zero, the number of infinity should be excluded; however, in limits we can consider the infinity in the common sense. We should distinguish the infinity in the senses of a point at infinity in one point compactification and of the infinity in some limit. The point at infinity in one point compactification is represented by the number

The first author is supported in part by the Grant-in-Aid for the Scientific Research (C) (2) (No. 26400192). Saitoh wishes to express his deep thanks Professor Haydar Akca for his kind invitation of the paper ( [

Tsutomu Matsuura,Saburou Saitoh, (2016) Matrices and Division by Zero z/0 = 0. Advances in Linear Algebra & Matrix Theory,06,51-58. doi: 10.4236/alamt.2016.62007