数学家思索数学问题时,用的是大脑的语言区域 吗?
果壳网2016-04-30 18:55:40阅读(0) 评论(0)
声明：本文由入驻搜狐公众平台的作者撰写，除搜狐官方账号外，观点仅代表作者本人，不代表搜狐立场。举报
长江后浪推前浪，前浪挂在高数上。在大学校园里，你总能见到为高等数学、数学分析和线性代数头疼不已的学生们。 “那些数学家们到底是用什么脑子来想数学问题的啊！”他们想着。
关于这个问题，科学家们也一直在争论：数学家在思索数学问题时动用的是和语言相关的区域，还是跟基本数字感相关的区域？
前阵子，法国巴黎-萨克雷大学的研究者就通过功能性磁共振成像（fMRI）对数学家的大脑进行了扫描。经过和非数学家进行对比，他们发现 人类大脑中负责高难度数学问题的区域，与负责基本数字感的区域基本相同。
换句话说， 数学家们并非是点亮了大脑的特殊区域才获得了不明觉厉的数学才能。他们思索高深抽象的数学概念时，跟我们为简单的运算题绞尽脑汁调用的是同样的大脑区域。相关结果[1]近日发表在了《美国科学院院刊》（PNAS）上。
语言还是数字：高级数学能力的起源 人类的大脑为什么能够处理高级的数学问题？至今人们也想不清这种甩其他动物几条街的卓越能力是如何进化而来的。长期以来，有研究者猜想这种能力与人类使用语言的能力息息相关，认为语言中的抽象化能力是我们处理高级数学问题的起点。不少数学家和物理学家质疑这种观点的可靠性。爱因斯坦就曾宣称：“词汇和语言，不管是写下来的还是说出口的，对我的思考过程似乎都没什么用处。”
略略略~
另一种假设主张，人类处理高级数学问题的能力也是从基本的数字，逻辑，分析等能力中演化出现的——在成年人中，处理数字概念所涉及的大脑区域与处理语言的部分几乎没有重叠。不过，也有数学家表示数字概念太过简单，不能代表高级数学能力。
没关系，有争议就有探究。在这次的研究中，玛丽·阿玛里克（Marie Amalric）和斯坦尼斯拉斯·德阿纳（Stanislas Dehaene）就召集了15名职业的数学家和15名学术地位与之同等的非数学专业研究者，对比他们在处理不同信息时脑部不同区域的活动差异。
玛丽·阿玛里克（左）和斯坦尼斯拉斯·德阿纳（右）试图利用fMRI找出数学家思考抽象数学问题时动用的脑部区域。图片来源：normalesup.org；institutfrancais.dk
数学家用哪里想数学问题在实验中，他们会听到一系列不同的命题：数学分析、代数、拓扑学和几何学领域的高级数学各18个，以及18个非数学领域（比如“在古希腊，还不起债的公民将沦为奴隶”）的命题。他们需要在听到命题4秒后判断这些命题是真的，假的，还是无意义的。在他们思考时，fMRI会记录下他们大脑各区域的活动。
在对非数学命题下判断时，数学组和非数学组正确率不相伯仲。而所谓术业有专攻，在判断具体数学命题时，数学组充分表现出了专业优势，正确率超过了60%，而非数学组的正确率则只有37%，跟随机瞎蒙蒙对的概率差不多。
来想一想（猜一猜）上述这些命题有没有意义，如果有，命题是真是假？图片来源：参考文献[1]
但谁正确率高并不是问题的核心。关键在于，fMRI成像的结果精确地描绘了大脑各区域在他们思考时的活跃情况。研究者发现，在处理判断与数学相关的命题时，双侧顶内沟区域（IPS），双侧颞下回区域（IT）以及前额叶皮层区域会被激活——而如果处理的问题与数学无关，它们则“无动于衷”，甚至还会表现出轻微的抑制现象。
那么，这些处理数学问题的部分和大脑中负责处理语言信息的部分究竟有多大关联呢？研究者们定位检测了传统理解中和语言相关的大脑区域，然后让被试看或听不同的句子。结果显示， 和与数学相关的句子相比，普通语义推理句对大脑中和语言处理相关区域的激活要更强。扫描对比也发现，在判断数学问题和进行语义推理时分别被激活的大脑区域，重叠部分非常微小。
高等数学和简单运算用同样的脑区有趣的是，这些脑区的活跃似乎与数学问题的难易无关——在后续实验中，研究者发现即使是再简单的数学问题，也可以激活这些脑区。而 只要与数学无关，无论问题多复杂都无法打动这些区域。这提示， 这些大脑区域非常专注于数学相关信息。
至此，处理高级数学问题的能力归功于语言的观点似乎无法支持了。那么，另一种猜想有怎么样呢？研究者发现，诸如数字和计算这种数学基本概念也同样激活IPS和IT区域——这些区域和数学家们之前处理高级数学问题时被激活的脑区重合了。
处理高级数学命题（红）与处理数字概念（绿）和计算（蓝）时显著更活跃的脑区基本重合。图片来源：参考文献[1]
等等，数学问题里难道不是本来就会提及数字和运算吗？嗯，为了避免这样的干扰，研究者在那72道数学命题中几乎没有提到任何数字。但结果依然指向一个可能：我们思考高级数学问题和处理基础数学概念时，用的是完全相同部分的大脑——至少在受过训练的人中是这样的。
所以呀，数学家们可不是靠什么“别的脑子”来想高深莫测的数学问题。 让数学家那些深邃思考得以形成的基础，也正是我们这些“普通人”用以发挥基础数学技能的神经网络。
当然，我们也无法排除一种可能性，即数学家在童年时受到的基础训练可能的确塑造了他们解决高级问题时所动用的神经回路。接下来，研究者也将继续探究，为什么那些为多数人“栽种”了基础算数能力的脑区，只将少数人带到了能从事高级数学研究的高度。
（编辑：Calo）
一个AI【思考题】既然数学家的大脑并没有什么特异功能，你的大脑也没有受过严重的损伤，为什么你就是会挂在高数上呢？
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http://mt.sohu.com/20160430/n447197001.shtml
Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics\\
}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
\date{\today}
\maketitle
{\bf Abstract: } In this announcement, we shall introduce the zero division $z/0=0$. The result is a definite one and it is fundamental in mathematics.
\bigskip
\section{Introduction}
%\label{sect1}
By a natural extension of the fractions
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we, recently, found the surprising result, for any complex number $b$
\begin{equation}
\frac{b}{0}=0,
\end{equation}
incidentally in \cite{s} by the Tikhonov regularization for the Hadamard product inversions for matrices, and we discussed their properties and gave several physical interpretations on the general fractions in \cite{kmsy} for the case of real numbers. The result is a very special case for general fractional functions in \cite{cs}.
The division by zero has a long and mysterious story over the world (see, for example, google site with division by zero) with its physical viewpoints since the document of zero in India on AD 628, however,
Sin-Ei, Takahasi (\cite{taka}) (see also \cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing some full extensions of fractions and by showing the complete characterization for the property (1.2). His result will show that our mathematics says that the result (1.2) should be accepted as a natural one:
\bigskip
{\bf Proposition. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ such that
$$
F (b, a)F (c, d)= F (bc, ad)
$$
for all
$$
a, b, c, d \in {\bf C }
$$
and
$$
F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.
$$
Then, we obtain, for any $b \in {\bf C } $
$$
F (b, 0) = 0.
$$
}
\medskip
\section{What are the fractions $ b/a$?}
For many mathematicians, the division $b/a$ will be considered as the inverse of product;
that is, the fraction
\begin{equation}
\frac{b}{a}
\end{equation}
is defined as the solution of the equation
\begin{equation}
a\cdot x= b.
\end{equation}
The idea and the equation (2.2) show that the division by zero is impossible, with a strong conclusion. Meanwhile, the problem has been a long and old question:
As a typical example of the division by zero, we shall recall the fundamental law by Newton:
\begin{equation}
F = G \frac{m_1 m_2}{r^2}
\end{equation}
for two masses $m_1, m_2$ with a distance $r$ and for a constant $G$. Of course,
\begin{equation}
\lim_{r \to +0} F =\infty,
\end{equation}
however, in our fraction
\begin{equation}
F = G \frac{m_1 m_2}{0} = 0.
\end{equation}
\medskip
Now, we shall introduce an another approach. The division $b/a$ may be defined {\bf independently of the product}. Indeed, in Japan, the division $b/a$ ; $b$ {\bf raru} $a$ ({\bf jozan}) is defined as how many $a$ exists in $b$, this idea comes from subtraction $a$ repeatedly. (Meanwhile, product comes from addition).
In Japanese language for "division", there exists such a concept independently of product.
H. Michiwaki and his 6 years old girl said for the result $ 100/0=0$ that the result is clear, from the meaning of the fractions independently the concept of product and they said:
$100/0=0$ does not mean that $100= 0 \times 0$. Meanwhile, many mathematicians had a confusion for the result.
Her understanding is reasonable and may be acceptable:
$100/2=50 \quad$ will mean that we divide 100 by 2, then each will have 50.
$100/10=10 \quad$ will mean that we divide 100 by10, then each will have 10.
$100/0=0 \quad$ will mean that we do not divide 100, and then nobody will have at all and so 0.
Furthermore, she said then the rest is 100; that is, mathematically;
$$
100 = 0\cdot 0 + 100.
$$
Now, all the mathematicians may accept the division by zero $100/0=0$ with natural feelings as a trivial one?
\medskip
For simplicity, we shall consider the numbers on non-negative real numbers. We wish to define the division (or fraction) $b/a$ following the usual procedure for its calculation, however, we have to take care for the division by zero:
The first principle, for example, for $100/2 $ we shall consider it as follows:
$$
100-2-2-2-,...,-2.
$$
How may times can we subtract $2$? At this case, it is 50 times and so, the fraction is $50$.
The second case, for example, for $3/2$ we shall consider it as follows:
$$
3 - 2 = 1
$$
and the rest (remainder) is $1$, and for the rest $1$, we multiple $10$,
then we consider similarly as follows:
$$
10-2-2-2-2-2=0.
$$
Therefore $10/2=5$ and so we define as follows:
$$
\frac{3}{2} =1 + 0.5 = 1.5.
$$
By these procedures, for $a \ne 0$ we can define the fraction $b/a$, usually. Here we do not need the concept of product. Except the zero division, all the results for fractions are valid and accepted.
Now, we shall consider the zero division, for example, $100/0$. Since
$$
100 - 0 = 100,
$$
that is, by the subtraction $100 - 0$, 100 does not decrease, so we can not say we subtract any from $100$. Therefore, the subtract number should be understood as zero; that is,
$$
\frac{100}{0} = 0.
$$
We can understand this: the division by $0$ means that it does not divide $100$ and so, the result is $0$.
Similarly, we can see that
$$
\frac{0}{0} =0.
$$
As a conclusion, we should define the zero divison as, for any $b$
$$
\frac{b}{0} =0.
$$
See \cite{kmsy} for the details.
\medskip
\section{In complex analysis}
We thus should consider, for any complex number $b$, as (1.2);
that is, for the mapping
\begin{equation}
w = \frac{1}{z},
\end{equation}
the image of $z=0$ is $w=0$. This fact seems to be a curious one in connection with our well-established popular image for the point at infinity on the Riemann sphere.
However, we shall recall the elementary function
\begin{equation}
W(z) = \exp \frac{1}{z}
\end{equation}
$$
= 1 + \frac{1}{1! z} + \frac{1}{2! z^2} + \frac{1}{3! z^3} + \cdot \cdot \cdot .
$$
The function has an essential singularity around the origin. When we consider (1.2), meanwhile, surprisingly enough, we have:
\begin{equation}
W(0) = 1.
\end{equation}
{\bf The point at infinity is not a number} and so we will not be able to consider the function (3.2) at the zero point $z = 0$, meanwhile, we can consider the value $1$ as in (3.3) at the zero point $z = 0$. How do we consider these situations?
In the famous standard textbook on Complex Analysis, L. V. Ahlfors (\cite{ahlfors}) introduced the point at infinity as a number and the Riemann sphere model as well known, however, our interpretation will be suitable as a number. We will not be able to accept the point at infinity as a number.
As a typical result, we can derive the surprising result: {\it At an isolated singular point of an analytic function, it takes a definite value }{\bf with a natural meaning.} As the important applications for this result, the extension formula of functions with analytic parameters may be obtained and singular integrals may be interpretated with the division by zero, naturally (\cite{msty}).
\bigskip
\section{Conclusion}
The division by zero $b/0=0$ is possible and the result is naturally determined, uniquely.
The result does not contradict with the present mathematics - however, in complex analysis, we need only to change a little presentation for the pole; not essentially, because we did not consider the division by zero, essentially.
The common understanding that the division by zero is impossible should be changed with many text books and mathematical science books. The definition of the fractions may be introduced by {\it the method of Michiwaki} in the elementary school, even.
Should we teach the beautiful fact, widely?:
For the elementary graph of the fundamental function
$$
y = f(x) = \frac{1}{x},
$$
$$
f(0) = 0.
$$
The result is applicable widely and will give a new understanding for the universe ({\bf Announcement 166}).
\medskip
If the division by zero $b/0=0$ is not introduced, then it seems that mathematics is incomplete in a sense, and by the intoduction of the division by zero, mathematics will become complete in a sense and perfectly beautiful.
\bigskip
section{Remarks}
For the procedure of the developing of the division by zero and for some general ideas on the division by zero, we presented the following announcements in Japanese:
\medskip
{\bf Announcement 148} (2014.2.12): $100/0=0, 0/0=0$ -- by a natural extension of fractions -- A wish of the God
\medskip
{\bf Announcement 154} (2014.4.22): A new world: division by zero, a curious world, a new idea
\medskip
{\bf Announcement 157} (2014.5.8): We wish to know the idea of the God for the division by zero; why the infinity and zero point are coincident?
\medskip
{\bf Announcement 161} (2014.5.30): Learning from the division by zero, sprits of mathematics and of looking for the truth
\medskip
{\bf Announcement 163} (2014.6.17): The division by zero, an extremely pleasant mathematics - shall we look for the pleasant division by zero: a proposal for a fun club looking for the division by zero.
\medskip
{\bf Announcement 166} (2014.6.29): New general ideas for the universe from the viewpoint of the division by zero
\medskip
{\bf Announcement 171} (2014.7.30): The meanings of product and division -- The division by zero is trivial from the own sense of the division independently of the concept of product
\medskip
{\bf Announcement 176} (2014.8.9): Should be changed the education of the division by zero
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\bibitem{cs}
L. P. Castro and S.Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
\bibitem{kmsy}
S. Koshiba, H. Michiwaki, S. Saitoh and M. Yamane,
An interpretation of the division by zero z/0=0 without the concept of product
(note).
\bibitem{kmsy}
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
Int. J. Appl. Math. Vol. 27, No 2 (2014), pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{msty}
H. Michiwaki, S. Saitoh, M. Takagi and M. Yamada,
A new concept for the point at infinity and the division by zero z/0=0
(note).
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. Vol.4 No.2 (2014), 87-95. http://www.scirp.org/journal/ALAMT/
\bibitem{taka}
S.-E. Takahasi,
{On the identities $100/0=0$ and $ 0/0=0$}
(note).
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operators on the real and complex fields. (submitted)
\end{thebibliography}
\end{document}
アインシュタインも解決できなかった「ゼロで割る」問題
http://matome.naver.jp/odai/2135710882669605901
Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.
https://notevenpast.org/dividing-nothing/
私は数学を信じない。 アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。
1423793753.460.341866474681。
Einstein's Only Mistake: Division by Zero
http://refully.blogspot.jp/2012/05/einsteins-only-mistake-division-by-zero.html
果壳网2016-04-30 18:55:40阅读(0) 评论(0)
声明：本文由入驻搜狐公众平台的作者撰写，除搜狐官方账号外，观点仅代表作者本人，不代表搜狐立场。举报
长江后浪推前浪，前浪挂在高数上。在大学校园里，你总能见到为高等数学、数学分析和线性代数头疼不已的学生们。 “那些数学家们到底是用什么脑子来想数学问题的啊！”他们想着。
关于这个问题，科学家们也一直在争论：数学家在思索数学问题时动用的是和语言相关的区域，还是跟基本数字感相关的区域？
前阵子，法国巴黎-萨克雷大学的研究者就通过功能性磁共振成像（fMRI）对数学家的大脑进行了扫描。经过和非数学家进行对比，他们发现 人类大脑中负责高难度数学问题的区域，与负责基本数字感的区域基本相同。
换句话说， 数学家们并非是点亮了大脑的特殊区域才获得了不明觉厉的数学才能。他们思索高深抽象的数学概念时，跟我们为简单的运算题绞尽脑汁调用的是同样的大脑区域。相关结果[1]近日发表在了《美国科学院院刊》（PNAS）上。
语言还是数字：高级数学能力的起源 人类的大脑为什么能够处理高级的数学问题？至今人们也想不清这种甩其他动物几条街的卓越能力是如何进化而来的。长期以来，有研究者猜想这种能力与人类使用语言的能力息息相关，认为语言中的抽象化能力是我们处理高级数学问题的起点。不少数学家和物理学家质疑这种观点的可靠性。爱因斯坦就曾宣称：“词汇和语言，不管是写下来的还是说出口的，对我的思考过程似乎都没什么用处。”
略略略~
另一种假设主张，人类处理高级数学问题的能力也是从基本的数字，逻辑，分析等能力中演化出现的——在成年人中，处理数字概念所涉及的大脑区域与处理语言的部分几乎没有重叠。不过，也有数学家表示数字概念太过简单，不能代表高级数学能力。
没关系，有争议就有探究。在这次的研究中，玛丽·阿玛里克（Marie Amalric）和斯坦尼斯拉斯·德阿纳（Stanislas Dehaene）就召集了15名职业的数学家和15名学术地位与之同等的非数学专业研究者，对比他们在处理不同信息时脑部不同区域的活动差异。
玛丽·阿玛里克（左）和斯坦尼斯拉斯·德阿纳（右）试图利用fMRI找出数学家思考抽象数学问题时动用的脑部区域。图片来源：normalesup.org；institutfrancais.dk
数学家用哪里想数学问题在实验中，他们会听到一系列不同的命题：数学分析、代数、拓扑学和几何学领域的高级数学各18个，以及18个非数学领域（比如“在古希腊，还不起债的公民将沦为奴隶”）的命题。他们需要在听到命题4秒后判断这些命题是真的，假的，还是无意义的。在他们思考时，fMRI会记录下他们大脑各区域的活动。
在对非数学命题下判断时，数学组和非数学组正确率不相伯仲。而所谓术业有专攻，在判断具体数学命题时，数学组充分表现出了专业优势，正确率超过了60%，而非数学组的正确率则只有37%，跟随机瞎蒙蒙对的概率差不多。
来想一想（猜一猜）上述这些命题有没有意义，如果有，命题是真是假？图片来源：参考文献[1]
但谁正确率高并不是问题的核心。关键在于，fMRI成像的结果精确地描绘了大脑各区域在他们思考时的活跃情况。研究者发现，在处理判断与数学相关的命题时，双侧顶内沟区域（IPS），双侧颞下回区域（IT）以及前额叶皮层区域会被激活——而如果处理的问题与数学无关，它们则“无动于衷”，甚至还会表现出轻微的抑制现象。
那么，这些处理数学问题的部分和大脑中负责处理语言信息的部分究竟有多大关联呢？研究者们定位检测了传统理解中和语言相关的大脑区域，然后让被试看或听不同的句子。结果显示， 和与数学相关的句子相比，普通语义推理句对大脑中和语言处理相关区域的激活要更强。扫描对比也发现，在判断数学问题和进行语义推理时分别被激活的大脑区域，重叠部分非常微小。
高等数学和简单运算用同样的脑区有趣的是，这些脑区的活跃似乎与数学问题的难易无关——在后续实验中，研究者发现即使是再简单的数学问题，也可以激活这些脑区。而 只要与数学无关，无论问题多复杂都无法打动这些区域。这提示， 这些大脑区域非常专注于数学相关信息。
至此，处理高级数学问题的能力归功于语言的观点似乎无法支持了。那么，另一种猜想有怎么样呢？研究者发现，诸如数字和计算这种数学基本概念也同样激活IPS和IT区域——这些区域和数学家们之前处理高级数学问题时被激活的脑区重合了。
处理高级数学命题（红）与处理数字概念（绿）和计算（蓝）时显著更活跃的脑区基本重合。图片来源：参考文献[1]
等等，数学问题里难道不是本来就会提及数字和运算吗？嗯，为了避免这样的干扰，研究者在那72道数学命题中几乎没有提到任何数字。但结果依然指向一个可能：我们思考高级数学问题和处理基础数学概念时，用的是完全相同部分的大脑——至少在受过训练的人中是这样的。
所以呀，数学家们可不是靠什么“别的脑子”来想高深莫测的数学问题。 让数学家那些深邃思考得以形成的基础，也正是我们这些“普通人”用以发挥基础数学技能的神经网络。
当然，我们也无法排除一种可能性，即数学家在童年时受到的基础训练可能的确塑造了他们解决高级问题时所动用的神经回路。接下来，研究者也将继续探究，为什么那些为多数人“栽种”了基础算数能力的脑区，只将少数人带到了能从事高级数学研究的高度。
（编辑：Calo）
一个AI【思考题】既然数学家的大脑并没有什么特异功能，你的大脑也没有受过严重的损伤，为什么你就是会挂在高数上呢？
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http://mt.sohu.com/20160430/n447197001.shtml
Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics\\
}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
\date{\today}
\maketitle
{\bf Abstract: } In this announcement, we shall introduce the zero division $z/0=0$. The result is a definite one and it is fundamental in mathematics.
\bigskip
\section{Introduction}
%\label{sect1}
By a natural extension of the fractions
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we, recently, found the surprising result, for any complex number $b$
\begin{equation}
\frac{b}{0}=0,
\end{equation}
incidentally in \cite{s} by the Tikhonov regularization for the Hadamard product inversions for matrices, and we discussed their properties and gave several physical interpretations on the general fractions in \cite{kmsy} for the case of real numbers. The result is a very special case for general fractional functions in \cite{cs}.
The division by zero has a long and mysterious story over the world (see, for example, google site with division by zero) with its physical viewpoints since the document of zero in India on AD 628, however,
Sin-Ei, Takahasi (\cite{taka}) (see also \cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing some full extensions of fractions and by showing the complete characterization for the property (1.2). His result will show that our mathematics says that the result (1.2) should be accepted as a natural one:
\bigskip
{\bf Proposition. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ such that
$$
F (b, a)F (c, d)= F (bc, ad)
$$
for all
$$
a, b, c, d \in {\bf C }
$$
and
$$
F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.
$$
Then, we obtain, for any $b \in {\bf C } $
$$
F (b, 0) = 0.
$$
}
\medskip
\section{What are the fractions $ b/a$?}
For many mathematicians, the division $b/a$ will be considered as the inverse of product;
that is, the fraction
\begin{equation}
\frac{b}{a}
\end{equation}
is defined as the solution of the equation
\begin{equation}
a\cdot x= b.
\end{equation}
The idea and the equation (2.2) show that the division by zero is impossible, with a strong conclusion. Meanwhile, the problem has been a long and old question:
As a typical example of the division by zero, we shall recall the fundamental law by Newton:
\begin{equation}
F = G \frac{m_1 m_2}{r^2}
\end{equation}
for two masses $m_1, m_2$ with a distance $r$ and for a constant $G$. Of course,
\begin{equation}
\lim_{r \to +0} F =\infty,
\end{equation}
however, in our fraction
\begin{equation}
F = G \frac{m_1 m_2}{0} = 0.
\end{equation}
\medskip
Now, we shall introduce an another approach. The division $b/a$ may be defined {\bf independently of the product}. Indeed, in Japan, the division $b/a$ ; $b$ {\bf raru} $a$ ({\bf jozan}) is defined as how many $a$ exists in $b$, this idea comes from subtraction $a$ repeatedly. (Meanwhile, product comes from addition).
In Japanese language for "division", there exists such a concept independently of product.
H. Michiwaki and his 6 years old girl said for the result $ 100/0=0$ that the result is clear, from the meaning of the fractions independently the concept of product and they said:
$100/0=0$ does not mean that $100= 0 \times 0$. Meanwhile, many mathematicians had a confusion for the result.
Her understanding is reasonable and may be acceptable:
$100/2=50 \quad$ will mean that we divide 100 by 2, then each will have 50.
$100/10=10 \quad$ will mean that we divide 100 by10, then each will have 10.
$100/0=0 \quad$ will mean that we do not divide 100, and then nobody will have at all and so 0.
Furthermore, she said then the rest is 100; that is, mathematically;
$$
100 = 0\cdot 0 + 100.
$$
Now, all the mathematicians may accept the division by zero $100/0=0$ with natural feelings as a trivial one?
\medskip
For simplicity, we shall consider the numbers on non-negative real numbers. We wish to define the division (or fraction) $b/a$ following the usual procedure for its calculation, however, we have to take care for the division by zero:
The first principle, for example, for $100/2 $ we shall consider it as follows:
$$
100-2-2-2-,...,-2.
$$
How may times can we subtract $2$? At this case, it is 50 times and so, the fraction is $50$.
The second case, for example, for $3/2$ we shall consider it as follows:
$$
3 - 2 = 1
$$
and the rest (remainder) is $1$, and for the rest $1$, we multiple $10$,
then we consider similarly as follows:
$$
10-2-2-2-2-2=0.
$$
Therefore $10/2=5$ and so we define as follows:
$$
\frac{3}{2} =1 + 0.5 = 1.5.
$$
By these procedures, for $a \ne 0$ we can define the fraction $b/a$, usually. Here we do not need the concept of product. Except the zero division, all the results for fractions are valid and accepted.
Now, we shall consider the zero division, for example, $100/0$. Since
$$
100 - 0 = 100,
$$
that is, by the subtraction $100 - 0$, 100 does not decrease, so we can not say we subtract any from $100$. Therefore, the subtract number should be understood as zero; that is,
$$
\frac{100}{0} = 0.
$$
We can understand this: the division by $0$ means that it does not divide $100$ and so, the result is $0$.
Similarly, we can see that
$$
\frac{0}{0} =0.
$$
As a conclusion, we should define the zero divison as, for any $b$
$$
\frac{b}{0} =0.
$$
See \cite{kmsy} for the details.
\medskip
\section{In complex analysis}
We thus should consider, for any complex number $b$, as (1.2);
that is, for the mapping
\begin{equation}
w = \frac{1}{z},
\end{equation}
the image of $z=0$ is $w=0$. This fact seems to be a curious one in connection with our well-established popular image for the point at infinity on the Riemann sphere.
However, we shall recall the elementary function
\begin{equation}
W(z) = \exp \frac{1}{z}
\end{equation}
$$
= 1 + \frac{1}{1! z} + \frac{1}{2! z^2} + \frac{1}{3! z^3} + \cdot \cdot \cdot .
$$
The function has an essential singularity around the origin. When we consider (1.2), meanwhile, surprisingly enough, we have:
\begin{equation}
W(0) = 1.
\end{equation}
{\bf The point at infinity is not a number} and so we will not be able to consider the function (3.2) at the zero point $z = 0$, meanwhile, we can consider the value $1$ as in (3.3) at the zero point $z = 0$. How do we consider these situations?
In the famous standard textbook on Complex Analysis, L. V. Ahlfors (\cite{ahlfors}) introduced the point at infinity as a number and the Riemann sphere model as well known, however, our interpretation will be suitable as a number. We will not be able to accept the point at infinity as a number.
As a typical result, we can derive the surprising result: {\it At an isolated singular point of an analytic function, it takes a definite value }{\bf with a natural meaning.} As the important applications for this result, the extension formula of functions with analytic parameters may be obtained and singular integrals may be interpretated with the division by zero, naturally (\cite{msty}).
\bigskip
\section{Conclusion}
The division by zero $b/0=0$ is possible and the result is naturally determined, uniquely.
The result does not contradict with the present mathematics - however, in complex analysis, we need only to change a little presentation for the pole; not essentially, because we did not consider the division by zero, essentially.
The common understanding that the division by zero is impossible should be changed with many text books and mathematical science books. The definition of the fractions may be introduced by {\it the method of Michiwaki} in the elementary school, even.
Should we teach the beautiful fact, widely?:
For the elementary graph of the fundamental function
$$
y = f(x) = \frac{1}{x},
$$
$$
f(0) = 0.
$$
The result is applicable widely and will give a new understanding for the universe ({\bf Announcement 166}).
\medskip
If the division by zero $b/0=0$ is not introduced, then it seems that mathematics is incomplete in a sense, and by the intoduction of the division by zero, mathematics will become complete in a sense and perfectly beautiful.
\bigskip
section{Remarks}
For the procedure of the developing of the division by zero and for some general ideas on the division by zero, we presented the following announcements in Japanese:
\medskip
{\bf Announcement 148} (2014.2.12): $100/0=0, 0/0=0$ -- by a natural extension of fractions -- A wish of the God
\medskip
{\bf Announcement 154} (2014.4.22): A new world: division by zero, a curious world, a new idea
\medskip
{\bf Announcement 157} (2014.5.8): We wish to know the idea of the God for the division by zero; why the infinity and zero point are coincident?
\medskip
{\bf Announcement 161} (2014.5.30): Learning from the division by zero, sprits of mathematics and of looking for the truth
\medskip
{\bf Announcement 163} (2014.6.17): The division by zero, an extremely pleasant mathematics - shall we look for the pleasant division by zero: a proposal for a fun club looking for the division by zero.
\medskip
{\bf Announcement 166} (2014.6.29): New general ideas for the universe from the viewpoint of the division by zero
\medskip
{\bf Announcement 171} (2014.7.30): The meanings of product and division -- The division by zero is trivial from the own sense of the division independently of the concept of product
\medskip
{\bf Announcement 176} (2014.8.9): Should be changed the education of the division by zero
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\bibitem{cs}
L. P. Castro and S.Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
\bibitem{kmsy}
S. Koshiba, H. Michiwaki, S. Saitoh and M. Yamane,
An interpretation of the division by zero z/0=0 without the concept of product
(note).
\bibitem{kmsy}
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
Int. J. Appl. Math. Vol. 27, No 2 (2014), pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{msty}
H. Michiwaki, S. Saitoh, M. Takagi and M. Yamada,
A new concept for the point at infinity and the division by zero z/0=0
(note).
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. Vol.4 No.2 (2014), 87-95. http://www.scirp.org/journal/ALAMT/
\bibitem{taka}
S.-E. Takahasi,
{On the identities $100/0=0$ and $ 0/0=0$}
(note).
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operators on the real and complex fields. (submitted)
\end{thebibliography}
\end{document}
アインシュタインも解決できなかった「ゼロで割る」問題
http://matome.naver.jp/odai/2135710882669605901
Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.
https://notevenpast.org/dividing-nothing/
私は数学を信じない。 アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。
1423793753.460.341866474681。
Einstein's Only Mistake: Division by Zero
http://refully.blogspot.jp/2012/05/einsteins-only-mistake-division-by-zero.html
AD
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