独家｜趣味数学史：会计数也是文化天赋呢
2016-09-02 17:46:45
来源：搜狐读书 作者：【英】艾利克斯·贝洛斯
9月1日存在的这个开学周，大家都过得如何，有没有想起自己年少的求学时光？除非学霸，读书君觉得人人应该都有几门最痛恨的学科吧，简直是噩梦……如果要搞个投票，数学一定榜上有名。想当年，这一科考试难死过几多少年少女们……
可能是我们打开数学的方式不对？要求考试合格意味着往往错过真正有意思的内容。而宇宙之大，粒子之微，火箭之速，化工之巧，地球之变，生物之谜，日用之繁，实则无处不用数学。英国作家艾利克斯·贝洛斯在《数学世界漫游记》（译林出版社2015年6月版）觉得，数学其实很好玩，玩好了其实也很简单。它不光是一些枯燥的计算公式和无休止静的算数计算，从数数开始，它包含哲学、宗教、巫术、历史……
今天，我们有机会重新打开数学，从有关数学的无数花絮、灵感和弯路看起，从最基础的数数的进程与历史讲起，不一样的打开方式，看看你我还会不会成为数学学渣=。=
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亚马逊人只会数1-5？论会计数的重要性
虽然没有人知道确切的答案，但是，数字的存在不超过一万年的时间。这里，我所说的数字是关于数字的术语和符号。曾有理论认为，数字诞生于农业和通商，数字是通商中盘点货物所必须的工具，数字可以确保在买卖中不上当受骗。在我们的日常生活中，数字无处不在，很难想象如果没有数字的存在，人们如何解决生活中的各种问题。
我仍然觉得很奇怪，为什么在亚马逊地区的人的日常生活中没有出现大于5的数字。我问皮克，为什么印第安人会说“六条鱼”。比如说，某个人为六个人准备一顿饭，而且准备食物的人希望每个人都可以吃到一条鱼。
“这是不可能的，”皮克说。“在他们的生活中，根本就没‘我想为六个人准备鱼’这句话。”
手与数字的基情：怎样用手数到9 999 999 999！
当然，现代人们将十个数字当作一组，所以我们的数字系统由10个数字组成——0, 1, 2, 3, 4, 5, 6, 7, 8, 9。这组数字常被用作形成各种数字的符号，称作数字系统的基础，所以我们的数字系统是十进位的。
如果没有一个合理的数字系统，那么人们很难操纵和使用数字。一个好的计数系统需要拥有足够大的数字，当数到像100这样的数字时，人们不会感到喘不过气来，但与此同时，数字也不能过多，超过了人们记忆的能力也是不可取的。历史上，人们最常用的数字系统有五进制、十进制、二十进制，用这些进制的原因当然很显然。用这样的进制大概源于人类的身体构造。我们一只手上有5根手指，所以我们用一口气就能从1数到5。我们双手有十根手指，数到10时，我们也可以喘口气，之后将10根手指和10只脚趾加在一起，就变成20了。（有些计数系统是综合的。比如，林肯郡的牧羊人计算羊的头数的计算方法包含五进制、十进制和二十进制：第一个10个数字是单独存在的，之后的十位数字都是五个数字为一组。）用手指来计数的方式反应在很多数字词汇中，不少数词有双重含义。比如，俄语的数字5是piat，同时“张开的手”在俄语中是piast。同样，梵语中数字5是pantcha，和波斯语中“手”的单词pentcha有关。
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从人类开始计数的时候，人们便开始用手指帮助计算。将现代科技的发展归功于灵活多用的手指，这一点也不夸张。如果人类天生手臂和腿的末端长得像平平的树桩，也许现在我们的智力进化不会超越石器时代。笔和纸随处可得的时候，我们可以任意地将数字写下来，早在这个时期之前，我们需通过手指计数，即手语，进行交流。
在公元8世纪时，北逊布利亚的神学家“可敬的彼得”创造了一个计数系统，可以一直数到一百万。他的计数系统一部分来自算术，一部分来自爵士乐手。1和10位的数字由左手的手指来代表，百和千由右手的手指来表示。表示高阶数字时，将手在身体周围上上下下的移动——表示9000这个数字时，动作和教士的形象不符，彼得写道：“用左手抓住你的腰角，拇指朝向生殖器。”表示“一百万”的符号更能唤起人们的共鸣，表示成就和终结的一种自满的动作：双手紧握，手指交叠在一起。
仅仅一百年前，还没有一本有关算术的手册可以记录指算的图解。现今，它几乎已经成为一种失传的艺术。但是，在世界的很多地方，人们仍然在使用指算。在印度，商人们不想让旁观者知道交易的内容，便会藏在斗篷或布之后，触碰手指的关节交易。在中国，一个聪明绝顶的人用一种（看似相当复杂的）技巧让你能从1数到接近十亿——9,999,999,999。每个手指都被想象成有9个点——每个手指的褶皱线上有3个点，正如图上所画的一样。最后边的小指代表数字1到9，向右边第二根手指代表数字10到90，中指代表数字100到900，以此类推，每个手指都代表下面的10个数字。因此，一个人的手指能够数完地球上所有的人数，就像有种说法，一手掌握世间万物。
也许，最令人好奇的就是巴布亚地区的于普诺（Yupno），这里的每个人都有一个属于自己的短小曲调，就像名字一样，或者说一段信号曲。他们拥有一个计数系统，包括鼻孔、眼睛、乳头、肚脐和高潮部位，数字31是“左边的肾”，数字32是“右边的肾”，数字33是“阴茎”。你可以想象，数字33在三种宗教中的重要性（基督死去时的年纪，国王大卫当政的年限，穆斯林祈祷珠串上的珠子的数目），特别有趣的是于普诺人用阴茎代表数字33，其实他们对此感到很害羞。当他们提到数字33时，委婉地用“男人之物”来指代。研究者还不知道是否女性也有同样的指代，因为他们没有完整的数字系统，而且拒绝回答各种问题。这个群落的数字上限是数字34，他们把它称作“一个死人”。
怎样用身体其他部分来计数？脑洞太大了，还有点污……
怎样用身体其他部分来计数？脑洞太大了，还有点污……
在不同的文化中，不仅仅用手指和脚趾计数，人们还用身体其他的部位帮助算术。19世纪末，英国的人类学家们探险到达了托雷斯海峡的岛屿上，水道分隔开了澳大利亚和新几内亚。在那，他们发现了一个群落，将“右手的小指”当做数字1，“右手的无名指”当做数字2，以此类推，直到“右手腕”当做数字6，“右胳膊肘”当做数字7，再到“右肩”、“右胸骨”、“左手臂”、“手”、“脚”、“双腿”，最后“右脚小指”当做数字33。随后的探险和研究中，科学家们继续发现了这个地区的许多群落，同样使用“身体计算”的算术系统。
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十进制是否最好？瑞典国王：太粗野了！
在西方，十进制的数字系统已经沿用千百年了。尽管，十进制和人类的身体很和谐，但是仍然有人提出，是否十进制就是最合理的计算进制法？事实上，有人指出，它来源于人类身体，所以不是个好选择。瑞典的国王查尔斯十二世废除了十进制，他认为十进制是“粗野和头脑简单的人”用手指乱摸来摸去的。他认为，在现代的斯堪的纳维亚，一进制“更方便、更有用”。
所以，在1716年，他下令科学家伊曼纽•史威登堡设计一个新的计算系统，即64进制的计算系统。他最终完成了这个了不起的任务，灵感来源于立方体，计算方法为4×4×4。查尔斯国王发动战争（战争输了），北方大战，他认为军队里使用的计算，比如测量火药盒的容积时，用立方体数字为进制单位，则更容易。伏尔泰写道：“他灵机一动只能说明他很喜欢特别且麻烦困难的东西。”64进制需要64个独一无二的数字名称（或者数字符号）——一种荒唐、不合理且又不方便的系统。因此，史威登堡简化了这个数字系统，变成八进制，提出了一个新的计数法。然而，1718年，当史威登堡即将公开新的计算系统时，国王被子弹击中——随即他的八进制的美梦——死得彻彻底底。
但是，查尔斯十二世这么做，的确很有道理。为什么人们总是要坚持使用十进制呢？难道因为它和我们手指、脚趾加起来的数目一样？比如，如果人类长得像迪斯尼中的动画人物，一只手上只有三根手指和一个大拇指，那么几乎可以肯定我们将会活在一个八进制的世界里：分数的最高分为八，汇编最优的八个图表，规定八分钱为一角钱。数学不会因为换了一个数字系统而有任何改变。生性好斗的瑞典人有理由质问，到底哪种进制更适合科学研究需要——而不是根据我们的解剖学，选择数字系统。
文化不同会影响数学好坏？中国孩子数学更好之谜
乔治•奥威尔的《1984》一书中，主人公温斯顿•史密斯写道：“自由就是自由，就像二加二等于四一样简单。”奥威尔的这段话对前苏联的自由和数学两个方面都作了评论。二加二总是等于四。没有人能告诉你还有别的答案。数学上的真理不受到文化和意识形态的影响。
另一个方面，我们对数学的理解经常受到文化的冲击。比如，选用十进制不是以数学理论为前提，而是来源于生物和生理学，即人类手指和脚趾的数目。语言对数学的理解影响，同样很大。比如，在西方，我们对如何选用表示数字的词汇一直很保守。
在所有欧洲国家的语言中，表示数字的词汇几乎不是都遵循一个规律。在英语中，我们会说数字21,22,23，而我们国人（编者注：作者为英国人，这里说的数字发音和语言结构均为英语）却不这么说，会改用数字11、12、13。数字11和12具有独特构造，尽管13由3和10构成，数字3排列在10之前——不像数字23中，数字3排列在数字20之后。在英语词汇中，数字10到20之间，数字构造看似比较混乱。
然而，在中国、日本和韩国，代表数字的词汇遵循一定规律。数字11写成10和1相加，数字12写成10和2，之后是10和3、10和4，直到数字19写成10和9。数字20是两个10，数字21是两个10再加1。所有这些数字的发音就和写下来的一样。所以如何？因此，不同的数字词汇对年纪较小的孩子影响区别很大。实验反复证明，亚洲儿童比欧洲儿童更易学会数数。在一项研究中，研究对象是中国和美国4至5岁的孩子，两个国籍的儿童在学习数数时，对12以前的数字掌握程度相当，但是学习更难的数字时，中国儿童的发展水平比美国儿童提前大约一年时间。一个有规律的数字系统，使人更容易理解。一个简单的加法，比如：25+32，如果把数字表达成两个10加5，再加三个10加2，这样计算的步骤更简便，更易得出正确答案：五个10加7。
并不是每个西语的数字词汇都是无规律的。比如，威尔士语中的数字表达和中文的数词规律差不多。威尔士语中，数字11写成un deg un（1,10,1）；数字21写成 un deg dau（1，10，2），等等。牛津大学的安娜•多克和德里斯•劳埃德测试了威尔士地区的一个村庄中说威尔士语和英语的儿童，分析他们的数学能力的差异性。也许，亚洲儿童的数学能力比美国儿童好源于文化因素，如：亚洲儿童花费更多的时间在数学练习上，对数学学习更加认真，但是如果儿童们住在同一种环境中，文化的因素就可以被排除了。多克和劳埃德发现，说威尔士语和英语的儿童的数学能力差不多，但是，说威尔士语的儿童的确在某些数学领域上更优秀——比如，读、比较、熟练地计算两位数字等方面。
速记数字，广东人可以秒掉全世界的人？
西方语言和亚洲语言相比，西方语言中数字的位置没什么，11到19之间的数词也没什么规律，这些都让西方语言处于劣势。对于西方人来说，需要花更长的时间读出数字，这可以算是一个数词缺陷。
在《数感》中，斯坦尼斯拉斯•德阿内写下一连串数字：4、6、5、3、9、7、6，让我们花20秒时间记住它们。英语国家人士有50%的几率正确地记住7个数字。相比之下，讲中文的人有50%的几率正确地记住9个数字。斯坦尼斯拉斯说，这是因为人类大脑一次性记住的数字，在两秒钟的一个记忆循环中，人们可以一次性读出多少个数字。
在中文中，数字1到9都只包含一个音节，十分简洁：yi,er,san,si,wu,liu,qi,ba,jiu。读出每个音节只需四分之一秒，所以讲中文的人在两秒钟之内可以读出9个数字。相比之下，需要三分之一秒念每个英文数词（由于笨重冗长的英文数字7，有两个音节，有拖长音的英文数字3），所以我们在两秒钟之内，只能记住7个数字。同时，讲广东话的人保持最高的记忆记录，他们的数字发音比普通话更简短，在两秒钟之内，讲广东话的人可以记住10个数字。
（感谢译林出版社授权并提供文字）
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作者: [英]艾利克斯·贝洛斯
出版社: 译林出版社
原作名: Alex’s Adventures in Numberland
译者: 孟天
出版年: 2015-7 http://book.sohu.com/20160902/n467387816.shtml
出版社: 译林出版社
原作名: Alex’s Adventures in Numberland
译者: 孟天
出版年: 2015-7 http://book.sohu.com/20160902/n467387816.shtml
読んでためになりました：
\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}
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}
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.
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
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