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2020.05.31
カテゴリ： カテゴリ未分類
What is the history of the number 0? Is 0 really a number? Is it a good luck number?
Saburou Saitoh
·
just now
About ZERO, please look my book manuscript:
\newpage
\section{WHAT IS THE ZERO?}
The zero $0$ as the complex number or real number is given clearly by the axioms by the complex number field and real number field, respectively.
\index{Yamada field}
\medskip
{\bf Standard value}\index{standard value}
\medskip
The zero is a center and stand point (or bases, a standard value) of the coordinates - here we will consider our situation on the complex or real 2 dimensional spaces. By \index{stereographic projection}stereographic projection mapping or the Yamada field, the point at infinity $1/0$ is represented by zero. The origin of the coordinates and the point at infinity correspond with each other.
As the standard value, for the point $\omega_n = \exp \left(\frac{\pi}{n}i\right)$ on the unit circle $|z|=1$, for $n = 0$:
$$
\omega_0 = \exp \left(\frac{\pi}{0}i\right)=1, \quad \frac{\pi}{0} =0.
$$
For the mean value
$$
M_n = \frac{x_1 + x_2 +... + x_n}{n},
we have
$$
M_0 = 0 = \frac{0}{0}.
$$
\medskip
\medskip
For example, in very general partial differential equations, if the coefficients or terms are zero, we have some simple differential equations and the extreme case is all terms zero; that is, we have the trivial equation $0=0$; then its solution is zero. When we consider the converse, we see that the zero world is a fruitful one and it means some vanishing world. Recall \index{Yamane phenomena} the Yamane phenomena, the vanishing result is very simple zero, however, it is the result from some fruitful world. Sometimes, zero means void or nothing world, however, it will show some change as in the Yamane phenomena.
\medskip
{\bf From $0$ to $0$; $0$ means all and all are $0$}
\medskip
As we see from our life figure, \index{life figure}a story starts from the zero and ends to the zero. This will mean that $0$ means all and all are $0$, in a sense. The zero is a mother of all.
\medskip
{\bf Impossibility}\index{impossibility}
\medskip
As the solution of the simplest equation
\begin{equation}\label{what16.1}
ax =b
\end{equation}
we have $x=0$ for $a=0, b\ne 0$ as the standard value, or \index{Moore-Penrose generalized inverse}the Moore-Penrose generalized inverse. This will mean in a sense, the solution does not exist; to solve the equation \eqref{what16.1} is impossible.
We saw for different parallel lines or different parallel planes, their common point is the origin. Certainly they have the common point of the point at infinity and the point at infinity is represented by zero. However, we can understand also that they have no solutions, no common points, because the point at infinity is an ideal point.
We will consider the point P at the origin with starting at the time $t=0$ with velocity $V > 0$ and the point Q at the point $d >0$ with velocity $v >0$. Then, the time of coincidence P=Q is given by
$$
T = \frac{d}{V -v}.
$$
When $V=v$, we have, by the division by zero, $T=0$. This zero represents impossibility. We have many such situations.
We will consider the simple differential equation
\begin{equation}
m\frac{d^2x}{dt^2} =0, m\frac{d^2y}{dt^2} =-mg
\end{equation}
with the initial conditions, at $t =0$
$$
\frac{dx}{dt} = v_0 \cos \alpha , \quad \frac{dy}{dt} = v_0 \sin \alpha; \quad x=y=0.
$$
Then, the highest high $h$, arriving time $t$, the distance $d$ from the starting point at the origin to the point $y(2t) =0$ are given by
$$
h = \frac{v_0^2 \sin \alpha}{2g}, \quad d= \frac{v_0^2\sin 2\alpha}{g}
$$
and
$$
t= \frac{v_0 \sin \alpha}{g}.
$$
For the case $g=0$, we have $h=d =t=0$. We considered the case that they are infinity; however, our mathematics means zero, which shows impossibility.
These phenomena were looked in many cases on the universe; it seems that God does not like the infinity.
\medskip
As we stated already in the Bh\={a}skara's example -- sun and shadow \index{Bh\={a}skara's example}
\medskip
{\bf Zero represents void or nothing}
\medskip
On ZERO, the authors S. K. Sen and R. P. Agarwal \cite{sa16} published its history and many important properties. See also R. Kaplan \cite{kaplan}
\index{Sen, S. K.} \index{Agarwal, R. P.} and E. Sondheimer and A. Rogerson \cite{sr} on the very interesting books on zero and infinity. \index{Kaplan, R.} \index{Sondheimer, E.} \index{Rogerson, A.}
India has a great tradition on ZERO, VOID and INFINITY and they are familiar with those concepts.
Meanwhile, Europian (containing the USA) people do not like such basic ideas and they are not familiar with them.
\newpage
1 view · Answer requested by Richard Giraldo
Saburou Saitoh
·
just now
About ZERO, please look my book manuscript:
\newpage
\section{WHAT IS THE ZERO?}
The zero $0$ as the complex number or real number is given clearly by the axioms by the complex number field and real number field, respectively.
\index{Yamada field}
\medskip
{\bf Standard value}\index{standard value}
\medskip
The zero is a center and stand point (or bases, a standard value) of the coordinates - here we will consider our situation on the complex or real 2 dimensional spaces. By \index{stereographic projection}stereographic projection mapping or the Yamada field, the point at infinity $1/0$ is represented by zero. The origin of the coordinates and the point at infinity correspond with each other.
As the standard value, for the point $\omega_n = \exp \left(\frac{\pi}{n}i\right)$ on the unit circle $|z|=1$, for $n = 0$:
$$
\omega_0 = \exp \left(\frac{\pi}{0}i\right)=1, \quad \frac{\pi}{0} =0.
$$
For the mean value
$$
M_n = \frac{x_1 + x_2 +... + x_n}{n},
we have
$$
M_0 = 0 = \frac{0}{0}.
$$
\medskip
\medskip
For example, in very general partial differential equations, if the coefficients or terms are zero, we have some simple differential equations and the extreme case is all terms zero; that is, we have the trivial equation $0=0$; then its solution is zero. When we consider the converse, we see that the zero world is a fruitful one and it means some vanishing world. Recall \index{Yamane phenomena} the Yamane phenomena, the vanishing result is very simple zero, however, it is the result from some fruitful world. Sometimes, zero means void or nothing world, however, it will show some change as in the Yamane phenomena.
\medskip
{\bf From $0$ to $0$; $0$ means all and all are $0$}
\medskip
As we see from our life figure, \index{life figure}a story starts from the zero and ends to the zero. This will mean that $0$ means all and all are $0$, in a sense. The zero is a mother of all.
\medskip
{\bf Impossibility}\index{impossibility}
\medskip
As the solution of the simplest equation
\begin{equation}\label{what16.1}
ax =b
\end{equation}
we have $x=0$ for $a=0, b\ne 0$ as the standard value, or \index{Moore-Penrose generalized inverse}the Moore-Penrose generalized inverse. This will mean in a sense, the solution does not exist; to solve the equation \eqref{what16.1} is impossible.
We saw for different parallel lines or different parallel planes, their common point is the origin. Certainly they have the common point of the point at infinity and the point at infinity is represented by zero. However, we can understand also that they have no solutions, no common points, because the point at infinity is an ideal point.
We will consider the point P at the origin with starting at the time $t=0$ with velocity $V > 0$ and the point Q at the point $d >0$ with velocity $v >0$. Then, the time of coincidence P=Q is given by
$$
T = \frac{d}{V -v}.
$$
When $V=v$, we have, by the division by zero, $T=0$. This zero represents impossibility. We have many such situations.
We will consider the simple differential equation
\begin{equation}
m\frac{d^2x}{dt^2} =0, m\frac{d^2y}{dt^2} =-mg
\end{equation}
with the initial conditions, at $t =0$
$$
\frac{dx}{dt} = v_0 \cos \alpha , \quad \frac{dy}{dt} = v_0 \sin \alpha; \quad x=y=0.
$$
Then, the highest high $h$, arriving time $t$, the distance $d$ from the starting point at the origin to the point $y(2t) =0$ are given by
$$
h = \frac{v_0^2 \sin \alpha}{2g}, \quad d= \frac{v_0^2\sin 2\alpha}{g}
$$
and
$$
t= \frac{v_0 \sin \alpha}{g}.
$$
For the case $g=0$, we have $h=d =t=0$. We considered the case that they are infinity; however, our mathematics means zero, which shows impossibility.
These phenomena were looked in many cases on the universe; it seems that God does not like the infinity.
\medskip
As we stated already in the Bh\={a}skara's example -- sun and shadow \index{Bh\={a}skara's example}
\medskip
{\bf Zero represents void or nothing}
\medskip
On ZERO, the authors S. K. Sen and R. P. Agarwal \cite{sa16} published its history and many important properties. See also R. Kaplan \cite{kaplan}
\index{Sen, S. K.} \index{Agarwal, R. P.} and E. Sondheimer and A. Rogerson \cite{sr} on the very interesting books on zero and infinity. \index{Kaplan, R.} \index{Sondheimer, E.} \index{Rogerson, A.}
India has a great tradition on ZERO, VOID and INFINITY and they are familiar with those concepts.
Meanwhile, Europian (containing the USA) people do not like such basic ideas and they are not familiar with them.
\newpage
1 view · Answer requested by Richard Giraldo
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2020.05.31 13:28:45
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