原文作者：帕特丽夏罗斯曼，伦敦大学学院数学系荣誉研究员。
译文作者：小龙虾，哆嗒数学网翻译组成员。
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在1706年，一个名叫威廉?琼斯的不知名的数学老师第一次使用了一个符号来代表圆周率π，一个用数值可以接近却永远无法达到的理想概念。
任意圆周长与直径的恒定比值的历史和人们渴望测量的历史一样悠久，然而这个今天广为人知的比值π是起源于十八世纪早期。在这之前，这个比值用中古拉丁文晦涩地表示为：quantitas in quam cum multiflicetur diameter, provenietcircumferencia（这个量乘直径会得到周长）。
人们广泛认为是出生在瑞士的伟大数学家莱昂哈德?欧拉（1707-1783）将符号π引入普遍使用。事实上，在欧拉出生前一年的1706年，π第一次以它的现代含义出现在一个自学成才的数学老师威廉?琼斯的第二本书《新数学导论》中，这本书是基于他的教学笔记编写而成的。
在符号π出现前，像22/7和355/113的近似值被用来表示这个比值，这带来一种这个比值是个有理数的印象。尽管琼斯没有作证明，但是他相信π是个无理数，一个无限不循环小数，它不可能完全用数字形式表达。在《最新数学导论》中，他指出“…周长与直径的比值不可能由数字准确地表达”。因此需要用一个符号来表达这个可以接近却无法达到的理想概念。为此，琼斯认为只有一个纯的理想的符号才能满足需要。
在之前一个世纪，符号π被同时是教区长的数学家威廉?奥特雷德（1575-1660）用作另外的含义。在他的书《数学之钥》（在1631年第一次出版），他使用π代表给定圆的周长，所以他的π会随圆的直径的变化而变化，而不是现在代表一个常数。那时候圆的周长用'periphery'表示，因此用希腊对应字母“π”来表示。琼斯对π的使用是一个重要的奥特雷德没有实现的哲学进步，尽管奥特雷德引入了其他的数学符号，比如::表示比例以及'x'作为乘法的符号。
在奥特雷德去世的1660年，数学家约翰?科林斯（1625-1683）获得了奥特雷德数学图书馆中的一些书和论文，而琼斯也是通过约翰?科林斯获得了这些资料。
π的无理数特性直到1961年才被约翰?兰伯特（1728-1777）证明，然后在1882年费迪南?林德曼（1852-1939）证明了π是非代数的无理数，是一个超越数，即不能是任意次数的有理系数代数方程的解。有两个类型无理数的发现并没有贬低琼斯认识到周长与直径的比值不能用有理数表示的成就。
在第一次使用符号π之外，琼斯是非常令人感兴趣的，因为他与很多十八世纪的关键数学人物、科学人物与政治人物的联系。他还负责建设一个伟大的科学图书馆和数学档案馆，它们在他的赞助人麦克莱斯菲尔德家族的手中从当时一直保存了将近300年到现在。
尽管琼斯是带着数学成就去世的，但是他的出身是普通的。在大约1675年，他出生在安格尔西岛的一个小农场中。他唯一接受过的正式教育实在当地的慈善学校，在那里他展示出了数学才能，然后他被安排到伦敦的一个商人的帐房工作。后来，他航行到西印度群岛而且开始对航海感兴趣。后来他在一艘军舰上当数学老师。在1702年十月他参加了比戈战役，这场战役中英国人成功地拦截了由法国护送回西班牙西北部港口的西班牙舰队。胜利的水兵登上岸寻找金银，而根据廷茅斯男爵1807年的回忆录，对于琼斯来说最大的战利品是梦寐以求的文学珍品。
在琼斯回到英国后，他离开了海军然后开始在伦敦教数学，可能一开始在一个咖啡屋收取少量费用给人们上课。1702年，他出版了他的第一本书，《新实用航海艺术的纲要》。在这不久以后，他成为了菲利普约克的老师。后来菲利普约克（1690-1764）成为阿德威克第一任伯爵，他任大法官而且为介绍他的导师琼斯提供了无价的资源。
在大约1706年，在琼斯发表了《新数学导论》时，他第一次得到了艾萨克牛顿的关注，他在其中解释了牛顿的微积分方法和其他数学新观念。在1708年，琼斯可以获得克林斯的图书馆和档案馆的丰富资料，包括许多牛顿在17世纪60年代写的信和论文。这些提高了公众对琼斯的兴趣对他的名声很有帮助。
出生相离半个世纪，克林斯和琼斯从来没有相见，然而由于图书馆和数学档案馆历史将这两个人永久的联系在一起。图书馆和数学档案馆由克林斯建立，琼斯继续管理，在他俩对收集书籍的热情下发展壮大。克林斯是贫困牧师的儿子，他在一个图书商那里当学徒。像琼斯一样他基本上也是自学，也走向海洋学习航海。在他回到伦敦后，他靠当老师和会计谋生。他拥有几个不断获利的岗位而且擅长理顺复杂的账目。
克林斯有个普通的志向就是开一个书店，但是他没有积累足够的资金。然而在1667年，他被选入皇家学会，成为不可缺少的成员，协助学会秘书亨利?奥尔登伯格处理数学事务。从那时开始，克林斯与牛顿以及很多顶尖的英国和国外数学家一样，代表学会起草数学笔记。
在1709年当琼斯申请基督医院数学学校校长时，他带了牛顿和埃蒙德哈雷的推荐信，尽管有这些，但他还是失败了。然而，琼斯之前的学生，现正从事法律事业的飞利浦约克他的导师推荐给托马斯帕克爵士（1667-1732），他是一个成功的律师并且在下一年将要成为下一人最高法院首席法官。琼斯加入了他的家庭，并成为他儿子乔治（1697-1764）的导师。这是他与帕克家庭常年交往的开始。
在那时，琼斯买下了克林斯的图书馆和档案馆，牛顿和德国数学家莱布尼茨正在辩论是谁先发明了微积分。在克林斯的数学论文中，琼斯发现了牛顿最早使用微积分的副本《分析》（1669），他在1711年出版了这本书。这本书之前仅仅是不公开的流传。从1703年担任皇家学会会长的牛顿不情愿让他的成果发表而且小心翼翼地保护自己的知识产权。然而，他把琼斯视为他的支持者。
在1712年，琼斯加入了皇家学会建立的确认微积分的最先发明者的委员会。琼斯把克林斯的论文和牛顿关于微积分的信件提供给了委员会，并且形成了一个有关争端的报告，这个报告《Commercium Epistolicum》在那一年发表，它的大部分内容都是基于克林斯的论文和牛顿关于微积分的信件撰写。尽管这个报告是匿名的，但它被牛顿编辑，所以很难认为是公正的。不出意料，它是站在牛顿一边的。（今天，大家认为牛顿和莱布尼茨都独立地发明了微积分，尽管莱布尼茨的标记法优于牛顿的而且是目前普遍使用的。）
到1712年，琼斯已经有稳固的数学成就了。在1718年，他的赞助人托马斯帕克爵士被成为大法官并且在1721年被封为麦克莱斯菲尔德伯爵。在那时，他已经用当时总计18350英镑购买了锡伯恩地产和城堡。锡伯恩城堡同样也成为了琼斯的家，在那时他几乎已经是一个家庭成员了。除了法律帕克对许多学科包括科学和数学有学术兴趣，而且他对科学和艺术还是一个慷慨的赞助商。他作为皇家天文学家在1721年“约会”哈雷彗星过程中有很大的影响。
但是在第一伯爵的人格中也有对立面。他似乎在拥有很强的能力和抱负的同时对财富也有危险强烈的欲望。他被指控贩卖大法官职务给最高竞买人，并且允许将让投资者的资金被滥用。在1725年帕克从大法官职位辞职，但是他仍被控告。他被罚缴纳30000英镑，并且被禁足在伦敦塔6周直到罚金缴齐。他的一些资产被变卖，他被枢密院除名。但是他并没有丧失锡伯恩，锡伯恩由麦克尔斯菲尔德家族拥有到现在。在1727年，他是牛顿葬礼送葬者之一，这恢复了一些他的尊严。
托马斯的儿子乔治帕克在1722年成为了沃灵福德的一个议员，并在锡伯恩度过了大量时间，在那里在琼斯的指导下，他丰富老了琼斯带来的图书馆和档案馆。乔治帕克对天文很有兴趣，在一个天文家朋友詹姆斯布拉德利（在1742年哈雷去世时成为第三皇家天文家）的帮助下，他在锡伯恩建立了一个天文台。
到1718年，琼斯将时间花费在锡伯恩和临近伦敦红狮广场的蒂博尔德的宫殿。在许多有影响力的数学家、天文家和自然哲学家中，他结识了罗杰科茨（1682-1716），他是剑桥第一个布卢米安天文学教授，他被很多人认为是那一年代牛顿之后最有才能的英国数学家。他被委托修订牛顿原理第二版的出版物。
当牛顿和科茨关系紧张时，琼斯便作他们的中间人。他显然有影响力而且相当的机智。在一封信中科茨对琼斯写道：“有件事情我自己不能很好地处理，需要您的协调…”。这件微妙的事情是对牛顿的一个方法改良的建议牛顿有难以相处的人格，必须小心对待。而琼斯可以做得很好。牛顿原理第二版在1713年出版，得到很大的赞扬。
牛顿在大多数时期像是高耸的巨人，科学界活在他的阴影下。琼斯和天文学学家、数学家约翰梅钦有广泛的通信。约翰梅钦从1718年开始在皇家学会担任秘书近30年。他也是学会调查微积分发明的委员会成员。他在格雷沙姆学院任天文学教授近40年，研究月球运动理论并且认为他自己是这一学科的专家。在写给琼斯的一封信中，他用富于幻想语言来抱怨牛顿的月球运动理论。
她（月球）通知我说他（牛顿）在她生命的整个过程中污辱她，公布说她因不规则和各种罪恶应感到内疚，继续说没有活着的人可以在任何时间发现她的位置。
他继续写道，他梅钦，知道月亮在什么地方而且他有能力获得“Lord Treasurer”提供发现海上经度的10000英镑，因为他的月球运动理论可以提高月亮航用表的准确度。
尽管梅钦没有获得那奖金，他的月球运动理论被描述为依照重力的月球运动规律并且在牛顿死后的1729年添加到了牛顿定理的英文版中。
梅钦也在周长与半径比值方面做了一系列工作，他的计算方法快速收敛。他的计算结果被印刷在琼斯1706年的书中“超过100个地方可以验证正确；由准确、文思敏捷、真正有天才的约翰梅钦先生计算..”梅钦使用其和收敛于π的无穷级数来计算。用数学术语意味着，无论有多少项求和这个和的值与π的值总是有差距尽管差距很小。梅钦使用的无穷级数里的项正负交替，所以和的值交替地小于和大于π。
琼斯也和海外人士保持联系。其中一位特别兴趣的是住在美洲的教友派信徒学者詹姆斯洛根（1674-1751）。洛根出生于爱尔兰，被教友派领导人和宾夕法尼亚州建立者威廉佩恩邀请作他的秘书。他把那里建设得很兴旺，最终买下了斯坦顿大农场，在那里他从50多岁退休并开始追寻他的兴趣包括数学和植物学。他拥有的图书馆有超过3万本书，是美国18世纪最出名的图书馆之一并且后来赠给费城。
在1732年，洛根写信给琼斯，信中内容与一个发明相关:“这里的一个年轻人…是非常有天赋的”。这个年轻人是托马斯戈弗雷（1704-1749），他是一个装玻璃工人，在1730年10月发明了一个可以在海上准确应用的仪表，因为这个仪表有一个单向透视玻璃太阳和地平线的反射图像排成一行。任意两个天体例如月亮和一个星星可以通过移动一个包含镜子的旋转臂排成一排，而且可以从量表中读出角度。这意味着船的移动不会干扰角度测量，因为物体和图像会同时移动。这是一个精巧的仪表。洛根认为可以用它确定海上经度。这个仪表就是现在我们知道的哈德利四分仪，尽管实际上是个八分仪。英国和美国都索要了这个发明的归属。英国天文学家约翰哈德利（1682-1744）在1730年的夏天制作了一个这样的仪表而且在接下来的五月把一个报告给了皇家学会。
洛根写了一个私人信件描述戈弗雷的发明给哈雷，然后皇家学会的会长称他为“尊敬的朋友”。这是一个友好的科学的沟通，而皇家学会照例没有阅读这个信件。洛根向琼斯询问这一遗漏。琼斯后来在1734年一月和学会提出这个议题，戈弗雷作为仪表的发明者的地位被确立，尽管不是第一发明者。
在过了一些年的1736年琼斯写信给洛根，为没有及时回复道歉，他写道：
我的事务需要我全神贯注而且占据了我的思想以至于我有很少或者几乎没有时间考虑其他的事情甚至是数学。过去的这18年我缺少想法，我现在那些改进几乎是一个陌生的人。
但是在那个时间过后琼斯有关于数学学科的通信。可能是他不想鼓励洛根给他一些其他的发现。洛根是一个不知疲倦的通信者，他写的信比琼斯回复的信多很多。
当然琼斯脑海里是有其他东西的。像许多其他的研究科学的人，琼斯对经度问题感兴趣。他给皇家学会写信有关于当温度变化时时钟保持精确时间的课题。
他担任学会委员会成员并且在1749 年成为他的副会长。他的收入因工作清闲但报酬优厚的职位而大涨，这个职位是由他之前的学生建立的。他在阿德威克的影响下担任和平秘书，在乔治帕克的帮助下担任财政部副出纳员。然而他仍然在那时候经常发生的银行破产的作用下经历多次经济危机
琼斯在1731年完成了第二次婚姻，娶了比他小30岁的玛丽尼克斯，他们有三个孩子。在1747年他被选为育婴医院管理者，这时乔治帕克是副院长。是乔治让贺加斯为琼斯作画。尽管琼斯在这幅画中看起来令人注目，但是他被报道是一个矮小脸不长的威尔士人并且经常用粗暴和自由对待他的数学朋友。尽管如此，就像我们已经看到的，他知道在必要时如何变得机智而且展示盛意。
在他1749年74岁去世之后。皇家学会职员和图书馆管理员约翰罗伯特森说他去世时的情况比很多数学家好。他唯一存活的儿子，也叫威廉，那时只有三岁。他为人知的名字是奥连塔尔琼斯，他是一个出色语言学家和文献学者而且他精通印度法律而且他被正式封爵。
在1750年，乔治帕克撰写了一篇论文，这篇论文被皇家学会阅读而且被命名为评论太阳和月亮年。乔治是采用阳历最重要的支持者而且在1752年将新年从3月25日改到1月1日。有些人可能认为日历的修订是威廉琼斯科学遗产的一部分。在同一年，帕克被选为皇家学会会长，他直到去世都担任这一职务。
按照琼斯的意愿，他把学术书籍给乔治帕克作为他接受了帕克很多帮助的证明与鸣谢。帕克从琼斯继承的科学书籍和档案馆里的论文保存在锡伯恩的图书馆中。得到这些资料受到了严格的控制，尽管需要承认的是他们代表了他们在私人手中的最重要的书籍。在2000年剑桥大学图书馆在遗产彩票基金一笔基金的帮助下花费6370000英镑购买了档案馆的书信和论文。在2005年麦克莱斯菲尔德图书馆最终在索斯比以世界第六大销售额卖掉。
在琼斯的一生中，他将赞助商留住的能力十分重要而且他为他们服务得很好。从历史的角度来看，琼斯为麦克莱斯菲尔德做出贡献远大于他从赞助商的获取，正是这样，他为世界留下了智力遗产。
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\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 326: The division by zero z/0=0 - its impact to human beings through education and research\\
(2016.10.17)}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
}
\date{\today}
\maketitle
{\bf Abstract: } In this announcement, for its importance we would like to state the
situation on the division by zero and propose basic new challenges to education and research on our wrong world history.
\bigskip
\section{Introduction}
%\label{sect1}
By a {\bf natural extension} of the fractions
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we found the simple and beautiful 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 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 (\cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing the extensions of fractions and by showing the complete characterization for the property (1.2):
\bigskip
{\bf Proposition 1. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ satisfying
$$
F (b, a)F (c, d)= F (bc, ad)
$$
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 } $
$$
F (b, 0) = 0.
$$
}
Note that the complete proof of this proposition is simply given by 2 or 3 lines.
We should define $F(b,0)= b/0 =0$, in general.
\medskip
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 $W=0$ ({\bf should be defined}). 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. Therefore, the division by zero will give great impact to complex analysis and to our ideas for the space and universe.
However, the division by zero (1.2) is now clear, indeed, for the introduction of (1.2), we have several independent approaches as in:
\medskip
1) by the generalization of the fractions by the Tikhonov regularization and by the Moore-Penrose generalized inverse,
\medskip
2) by the intuitive meaning of the fractions (division) by H. Michiwaki - repeated subtraction method,
\medskip
3) by the unique extension of the fractions by S. Takahasi, as in the above,
\medskip
4) by the extension of the fundamental function $W = 1/z$ from ${\bf C} \setminus \{0\}$ into ${\bf C}$ such that $W =1/z$ is a one to one and onto mapping from $ {\bf C} \setminus \{0\} $ onto ${\bf C} \setminus \{0\}$ and the division by zero $1/0=0$ is a one to one and onto mapping extension of the function $W =1/z $ from ${\bf C}$ onto ${\bf C}$,
\medskip
and
\medskip
5) by considering the values of functions with the mean values of functions.
\medskip
Furthermore, in (\cite{msy}) we gave the results in order to show the reality of the division by zero in our world:
\medskip
\medskip
A) a field structure containing the division by zero --- the Yamada field ${\bf Y}$,
\medskip
B) by the gradient of the $y$ axis on the $(x,y)$ plane --- $\tan \frac{\pi}{2} =0$,
\medskip
C) by the reflection $W =1/\overline{z}$ of $W= z$ with respect to the unit circle with center at the origin on the complex $z$ plane --- the reflection point of zero is zero, not the point at infinity.
\medskip
and
\medskip
D) by considering rotation of a right circular cone having some very interesting
phenomenon from some practical and physical problem.
\medskip
In (\cite{mos}), many division by zero results in Euclidean spaces are given and the basic idea at the point at infinity should be changed. In (\cite{ms}), we gave beautiful geometrical interpretations of determinants from the viewpoint of the division by zero. The results show that the division by zero is our basic and elementary mathematics in our world.
\medskip
See J. A. Bergstra, Y. Hirshfeld and J. V. Tucker \cite{bht} for the relationship between fields and the division by zero, and the importance of the division by zero for computer science. It seems that the relationship of the division by zero and field structures are abstract in their paper.
Meanwhile, J. P. Barukcic and I. Barukcic (\cite{bb}) discussed recently the relation between the divisions $0/0$, $1/0$ and special relative theory of Einstein. However, their logic seems to be curious and their results contradict with ours.
Furthermore, T. S. Reis and J.A.D.W. Anderson (\cite{ra,ra2}) extend the system of the real numbers by introducing an ideal number for the division by zero $0/0$.
Meanwhile, we should refer to up-to-date information:
{\it Riemann Hypothesis Addendum - Breakthrough
Kurt Arbenz
https://www.researchgate.net/publication/272022137 Riemann Hypothesis Addendum - Breakthrough.}
\medskip
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 {\bf the biggest blunder of his life} [1]：
1. Gamow, G., My World Line (Viking, New York). p 44, 1970.
Apparently, the division by zero is a great missing in our mathematics and the result (1.2) is definitely determined as our basic mathematics, as we see from Proposition 1. Note its very general assumptions and many fundamental evidences in our world in (\cite{kmsy,msy,mos}). The results will give great impact on Euclidean spaces, analytic geometry, calculus, differential equations, complex analysis and physical problems.
The mysterious history of the division by zero over one thousand years is a great shame of mathematicians and human race on the world history, like the Ptolemaic system (geocentric theory). The division by zero will become a typical symbol of foolish human race with long and unceasing struggles. Future people will realize this fact as a definite common sense.
We should check and fill our mathematics, globally and beautifully, from the viewpoint of the division by zero. Our mathematics will be more perfect and beautiful, and will give great impact to our basic ideas on the universe.
For our ideas on the division by zero, see the survey style announcements.
\section{Basic Materials of Mathematics}
(1): First, we should declare that the divison by zero is possible in the natural and uniquley determined sense and its importance.
(2): In the elementary school, we should introduce the concept of division by the idea of repeated subtraction method by H. Michiwaki whoes method is applied in computer algorithmu and in old days for calculation of division. This method will give a simple and clear method for calculation of division and students will be happy to apply this simple method at the first stage. At this time, they will be able to understand that the division by zero is clear and trivial as $a/0=0$ for any $a$. Note that Michiwaki knows how to apply his method to the complex number field.
(3): For the introduction of the elemetary function $y= 1/x$, we should give the definition of the function at the origin $x=0$ as $y = 0$ by the division by zero idea and we should apply this definition for the occasions of its appearences, step by step, following the curriculum and the results of the division by zero.
(4): For the idea of the Euclidean space (plane), we should introduce, at the first stage, the concept of steleographic projection and the concept of the point at infinity -
one point compactification. Then, we will be able to see the whole Euclidean plane, however, by the division by zero, the point at infinity is represented by zero. We can teach the very important fact with many geometric and analytic geometry methods. These topics will give great pleasant feelings to many students.
Interesting topics are: parallel lines, what is a line? - a line contains the origin as an isolated
point for the case that the native line does not through the origin. All the lines pass the origin, our space is not the Eulcildean space and is not Aristoteles for the strong discontinuity at the point at infinity (at the origin). - Here note that an orthogonal coordinates should be fixed first for our all arguments.
(5): The inversion of the origin with respect to a circle with center the origin is the origin itself, not the point at infinity - the very classical result is wrong. We can also prove this elementary result by many elementary ways.
(6): We should change the concept of gradients; on the usual orthogonal coordinates $(x,y)$,
the gradient of the $y$ axis is zero; this is given and proved by the fundamental result
$\tan (\pi/2) =0$. The result is trivial in the definition of the Yamada field. This result is derived also from the {\bf division by zero calculus}:
\medskip
For any formal Laurent expansion around $z=a$,
\begin{equation}
f(z) = \sum_{n=-\infty}^{\infty} C_n (z - a)^n,
\end{equation}
we obtain the identity, by the division by zero
\begin{equation}
f(a) = C_0.
\end{equation}
\medskip
This fundamental result leads to the important new definition:
From the viewpoint of the division by zero, when there exists the limit, at $ x$
\begin{equation}
f^\prime(x) = \lim_{h\to 0} \frac{f(x + h) - f(x)}{h} =\infty
\end{equation}
or
\begin{equation}
f^\prime(x) = -\infty,
\end{equation}
both cases, we can write them as follows:
\begin{equation}
f^\prime(x) = 0.
\end{equation}
\medskip
For the elementary ordinary differential equation
\begin{equation}
y^\prime = \frac{dy}{dx} =\frac{1}{x}, \quad x > 0,
\end{equation}
how will be the case at the point $x = 0$? From its general solution, with a general constant $C$
\begin{equation}
y = \log x + C,
\end{equation}
we see that, by the division by zero,
\begin{equation}
y^\prime (0)= \left[ \frac{1}{x}\right]_{x=0} = 0,
\end{equation}
that will mean that the division by zero (1.2) is very natural.
In addition, note that the function $y = \log x$ has infinite order derivatives and all the values are zero at the origin, in the sense of the division by zero.
However, for the derivative of the function $y = \log x$, we have to fix the sense at the origin, clearly, because the function is not differentiable, but it has a singularity at the origin. For $x >0$, there is no problem for (2.6) and (2.7). At $x = 0$, we see that we can not consider the limit in the sense (2.3). However, for $x >0$ we have (2.6) and
\begin{equation}
\lim_{x \to +0} \left(\log x \right)^\prime = +\infty.
\end{equation}
In the usual sense, the limit is $+\infty$, but in the present case, in the sense of the division by zero, we have:
\begin{equation}
\left[ \left(\log x \right)^\prime \right]_{x=0}= 0
\end{equation}
and we will be able to understand its sense graphycally.
By the new interpretation for the derivative, we can arrange many formulas for derivatives, by the division by zero. We can modify many formulas and statements in calculus and we can apply our concept to the differential equation theory and the universe in connetion with derivatives.
(7): We shall introduce the typical division by zero calculus.
For the integral
\begin{equation}
\int x(x^{2}+1)^{a}dx=\frac{(x^{2}+1)^{a+1}}{2(a+1)}\quad(a\ne-1),
\end{equation}
we obtain, by the division by zero,
\begin{equation}
\int x(x^{2}+1)^{-1}dx=\frac{\log(x^{2}+1)}{2}.
\end{equation}
We will consider the fundamental ordinary differential equations
\begin{equation}
x^{\prime \prime}(t) =g -kx^{\prime}(t)
\end{equation}
with the initial conditions
\begin{equation}
x(0) = -h, x^{\prime}(0) =0.
\end{equation}
Then we have the solution
\begin{equation}
x(t) = \frac{g}{k}t + \frac{g(e^{-kt}- 1)}{k^2} - h.
\end{equation}
Then, for $k=0$, we obtain, immediately, by the division by zero
\begin{equation}
x(t) = \frac{1}{2}g t^2 -h.
\end{equation}
In those examples, we were able to give valuable functions for denominator zero cases. The division by zero calculus may be applied to many cases as a new fundamental calculus over l'Hôpital's rule.
(8): When we apply the division by zero to functions, we can consider, in general, many ways. For example,
for the function $z/(z-1)$, when we insert $z=1$ in numerator and denominator, we have
\begin{equation}
\left[\frac{z}{z-1}\right]_{z = 1} = \frac{1}{0} =0.
\end{equation}
However,
from the identity --
the Laurent expansion around $z=1$,
\begin{equation}
\frac{z}{z-1} = \frac{1}{z-1} + 1,
\end{equation}
we have
\begin{equation}
\left[\frac{z}{z-1}\right]_{z = 1} = 1.
\end{equation}
For analytic functions we can give uniquely determined values at isolated singular points by the values by means of the Laurent expansions as the division by zero calculus, 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, however, we can see that the division by zero calculus is reasonably valid. See \cite{kmsy,msy}.
\section{Albert Einstein's biggest blunder}
The division by zero is directly related to the Einstein's theory and various
physical problems
containing the division by zero. Now we should check the theory and the problems by the concept of the RIGHT and DEFINITE division by zero. Now is the best time since 100 years from Albert Einstein. It seems that the background knowledge is timely fruitful.
Note that the Big Bang also may be related to the division by zero like the blackholes.
\section{Computer systems}
The above Professors listed are wishing the contributions in order to avoid the division by zero trouble in computers. Now, we should arrange new computer systems in order not to meet the division by zero trouble in computer systems.
By the division by zero calculus, we will be able to overcome troubles in Maple for specialization problems.
\section{General ideas on the universe}
The division by zero may be related to religion, philosophy and the ideas on the universe, and it will creat a new world. Look the new world introduced.
\bigskip
We are standing on a new generation and in front of the new world, as in the discovery of the Americas. Should we push the research and education on the division by zero?
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{bb}
J. P. Barukcic and I. Barukcic, Anti Aristotle—The Division of Zero by Zero. Journal of Applied Mathematics and Physics, {\bf 4}(2016), 749-761.
doi: 10.4236/jamp.2016.44085.
\bibitem{bht}
J. A. Bergstra, Y. Hirshfeld and J. V. Tucker,
Meadows and the equational specification of division (arXiv:0901.0823v1[math.RA] 7 Jan 2009).
\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}
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. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{ms}
T. Matsuura and S. Saitoh,
Matrices and division by zero $z/0=0$, Advances in Linear Algebra
\& Matrix Theory, 6, 51-58. http://dx.doi.org/10.4236/alamt.2016.62007 http://www.scirp.org/journal/alamt
\bibitem{msy}
H. Michiwaki, S. Saitoh, and M.Yamada,
Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. {\bf 6}(2015), 1--8. http://www.ijapm.org/show-63-504-1.html
\bibitem{mos}
H. Michiwaki, H. Okumura, and S. Saitoh,
Division by Zero $z/0 = 0$ in Euclidean Spaces.
International Journal of Mathematics and Computation
(in press).
\bibitem{ra}
T. S. Reis and J.A.D.W. Anderson,
Transdifferential and Transintegral Calculus,
Proceedings of the World Congress on Engineering and Computer Science 2014 Vol I
WCECS 2014, 22-24 October, 2014, San Francisco, USA
\bibitem{ra2}
T. S. Reis and J.A.D.W. Anderson,
Transreal Calculus,
IAENG International J. of Applied Math., {\bf 45}(2015): IJAM 45 1 06.
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. {\bf 4} (2014), no. 2, 87--95. http://www.scirp.org/journal/ALAMT/
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operations on the real and complex fields, Tokyo Journal of Mathematics, {\bf 38}(2015), no. 2, 369-380.
\bibitem{ann179}
Announcement 179 (2014.8.30): Division by zero is clear as z/0=0 and it is fundamental in mathematics.
\bibitem{ann185}
Announcement 185 (2014.10.22): The importance of the division by zero $z/0=0$.
\bibitem{ann237}
Announcement 237 (2015.6.18): A reality of the division by zero $z/0=0$ by geometrical optics.
\bibitem{ann246}
Announcement 246 (2015.9.17): An interpretation of the division by zero $1/0=0$ by the gradients of lines.
\bibitem{ann247}
Announcement 247 (2015.9.22): The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.
\bibitem{ann250}
Announcement 250 (2015.10.20): What are numbers? - the Yamada field containing the division by zero $z/0=0$.
\bibitem{ann252}
Announcement 252 (2015.11.1): Circles and
curvature - an interpretation by Mr.
Hiroshi Michiwaki of the division by
zero $r/0 = 0$.
\bibitem{ann281}
Announcement 281 (2016.2.1): The importance of the division by zero $z/0=0$.
\bibitem{ann282}
Announcement 282 (2016.2.2): The Division by Zero $z/0=0$ on the Second Birthday.
\bibitem{ann293}
Announcement 293 (2016.3.27): Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0.
\bibitem{ann300}
Announcement 300 (2016.05.22): New challenges on the division by zero z/0=0.
\end{thebibliography}
\end{document}
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