隐藏在24个数学公式背后的故事
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无
言
的
宇
宙
第一本「数学史话」
用诗意文字展示数学之美
兼具故事性、趣味性、科普性
全彩插图版
1
古代的定理
1.我们为什么信赖算术：世界上最简单的公式
2.抗拒新概念：零的发现
3.斜边的平方：毕达哥拉斯定理
4.圆的游戏：π的发现
5.从芝诺悖论起：至无穷的概念
6.杠杆作用的重要性：杠杆原理
约前2 0 0 0 年— 前1600年的一块写有楔形文字手稿的陶土书板叙述了一个代数-几何问题
在现代世界中，数学是一个高度一致的学科。世界上任何一个国家都会对同样的等式（例如a2＋b2＝c2）得出同样的认识与理解，无论在欧洲、亚洲、非洲或美洲，无不如此。
但过去的情况并非总是如此。回顾数学史，特别是古代世界的数学史，我们可以看到研究与学习数学的各种大不相同的途径和推理方式。在这一时期之内，数学逐步进化，脱离了它脱胎的学科—测绘、税收、建筑和天文学—变成了一门独立的科学。
打开算术之门的钥匙：
杰姆什德·阿尔卡什（1390—1450）的一份阿拉伯文手稿
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在埃及与美索不达米亚，算术和几何只不过是书吏的通才教育的一部分。从现今尚存的纸莎草纸文稿和楔形文字书板中看，数学当时似乎只是作为一套规则讲授的，其中几乎没有任何解释。
在中国，数学（或称数字艺术）的命运在很多个世纪中盈亏圆缺，兴衰交替。在中国唐代（618—907），数学是一门十分受人尊崇的学科，一切学者都必须学习；但另一方面，到了明朝（1368—1644），它却被归入“小学”之列！
在柏林工业大学数学系大楼外地面上镶嵌的π字图形
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最后，伊斯兰世界继承了希腊与印度两大不同的数学传统，并以伊斯兰数学家自己的新发现将之发扬光大，更把这些数学知识传播到了西欧，因此在数学史上占据了独特的地位。奇怪的是，现代数学的决定性转变仅仅发生在西欧……但这是将在后面讨论的课题。
2
探索时代的定理
7.口吃者的秘密：卡尔达诺公式
8.九重天上的秩序：开普勒的行星运行定律
9.书写永恒：费马最后定理
10.一片未曾探索过的大陆：微积分基本定理
11.关于苹果、传说……以及彗星：牛顿定律
12.伟大的探索者：欧拉定理
毛里茨·科内流斯·埃舍于1938 年创作的木刻《天与水，I》，这是前景与背景互换的一个例子
1548年8月10日，意大利城市米兰的圣玛利亚大教堂内人头攒动，挤满了好奇的旁观者。
他们赶来见证的并非教堂的宗教活动，而是一场“血”溅二十步的决斗，但决斗者使用的武器不是刀剑火器，而是数学，是智力。
决斗的一方是来自威尼斯的尼克罗·塔尔达利亚，他的对手是洛多维科·费拉里。后者出身农夫，后来成为吉罗拉莫·卡尔达诺的仆人。吉罗拉莫·卡尔达诺是米兰最著名的公民之一，他是内科医生、赌博高手以及多种智力活动的翘楚。
艾萨克·牛顿于1668 年制造的第一台反射式望远镜；图中背景是牛顿的《自然哲学的数学原理》手稿
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令人好奇的是，卡尔达诺本人根本就没露面。三年前他公布了一项数学公式，但那是塔尔达利亚告诉他的，并让他绝对保密，此事当时闹得沸沸扬扬。
然而，这一天他找了一个十分方便的借口躲出米兰，让他的仆人—这位仆人很可能在数学方面比他更为高明—为维护他的声誉出马应战。
手工着色的瑞士数学家欧拉雕刻像,约1770 年
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我们永远也不会知道那天在教堂里面究竟发生了什么。原计划的思想交锋显然变成了一次音量高低的较量。但根据间接的证据，我们觉得是费拉里获得了胜利。
在教堂中的这一战是数学史上最惨痛与离奇的争论的最后一幕—这场争论涉及自罗马帝国崩溃以来第一次在欧洲出现的全新数学发现的生存权。人们把这一公式与发现美洲大陆相比，因为这是有关世界的一项新鲜事实，任何古代书籍对此连一点暗示都没有。这一公式开创了数学上的一个探索时代，这一时代将改变世界数学的疆界，其深刻程度不亚于哥伦布的发现对于真实世界地理面貌的改变。
3
普罗米修斯时代的定理
13.新的代数：汉密尔顿与四元数
14.两颗流星：群论
15.鲸鱼几何与蚂蚁几何：非欧几何
16.我们信赖质数：质数定理
17.关于谱系的想法：傅立叶级数
18.上帝之眼中看到的光：麦克斯韦方程
捕捉对称： 约1550 — 1600 年，伊斯坦布尔的埃于普尤萨科浴室中的伊兹尼克陶瓷砖
如果你有幸前往爱尔兰的都柏林，你一定要乘巴士前往布鲁姆桥路，在皇家运河下车。也许你还没有意识到这一点，但你已经来到了史上最著名的数学涂鸦所在地。
如果你站在街道上，那座让这条街得名的石桥看上去很小，也看不出什么特色；但如果你走下街道，来到与运河的水道齐平的地方，而且走到桥的西端，你就会在大批大批现代人用喷漆涂鸦的潦草字迹中间发现一块饰板，上面写着这样的铭文：
“就在威廉·若宛·汉密尔顿爵士2于1843年10月16日走过这里的时候，他天才的灵光一闪，发现了四元数乘法的基本定理i2＝j2＝k2＝ijk＝-1，并把它刻在这座桥的石头上。”
| 让我们现在暂时假定你就是一头鲸鱼。在深邃的大洋里光线不是很有用，因为水中很暗。所以你主要靠声音来感受外界、与外界交流。在你的世界中，两点之间的最短距离将是声波走过的路径。对于你来说，这就相当于一条直线 |
说实话，没什么人知道汉密尔顿是不是真的把他的公式刻到了布鲁姆桥上。这个故事起源于许多年后他写给他儿子阿奇博尔德的一封信。
跟许多家族故事一样，这件事也很可能是经过加工的。然而汉密尔顿因为发现了四元数而兴奋莫名，这一点倒应该是确凿无疑的。他认为四元数是他一生事业的巅峰。
19世纪之前只有一种代数和一种几何。在数学家们的头脑中，发明任何不同的东西的想法几乎根本就不存在。19世纪改变了这一切。无论对数学还是对整个外部世界，这都是一个革命的时代。
4
我们这个时代的定理
19.光电效应：量子与相对论
20.从劣质雪茄到威斯敏斯特大教堂：狄拉克公式
21.王国缔造者：陈省身-高斯-博内公式
22.有一点儿无限：连续统假说
23.混沌理论：洛伦兹方程
24.驯虎：布莱克-斯科尔斯方程
用铁屑揭示两块条形磁铁产生的磁场磁力线
最能代表20世纪时代精神的科学家非爱因斯坦莫属。
他是顽童与预言家的结合体。他时而在一幅著名的照片中伸出舌头，时而又在另一幅中以厌世嫉俗的目光凝视着我们。他的头发乱作一团，他对社会传统缺乏关心：这正是科学的流行图像。他是古往今来第一位摇滚歌星式的科学家。
他是认识了光的量子化性质的第一位物理学家；他是意识到物质与能量间等价关系的第一人；他的名字是相对论的同义词。而他同时也超越了科学的界限。他用他的名声推动和平主义的发展，至少在纳粹德国的兴起使他无法保持这一立场之前一直如此。
爱因斯坦做出的发现是他的性格、他所生活的时代以及他的智力这三者的共同结晶。
量子粒子的一种数码解释
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令人感到新奇的是，爱因斯坦在他早期的职业生涯中并不是一个数学爱好者。他过去的数学老师赫尔曼·闵可夫斯基曾经写道：“他在大学里是一条懒狗。他从来就没正眼看过数学一次。”
但爱因斯坦的态度随着时间发生了彻底的变化。1908年，闵可夫斯基以数学方法重新书写了狭义相对论，帮助这一理论赢得了人们的承认。如果爱因斯坦不能理解非欧几何，他永远也不可能写下他的广义相对论。
有时候，后入教的皈依者会成为最好的传道者。虽然爱因斯坦是一位不情不愿的数学家，但说他增加了数学这一学科的光彩，这一点他当之无愧。
对圆环面形宇宙的艺术式数码解释
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（来源：http://mt.sohu.com/20161023/n471063118.shtml）
ゼロ除算（1÷0・0÷0）はどうでしょうか：
\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}
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J. A. Bergstra, Y. Hirshfeld and J. V. Tucker,
Meadows and the equational specification of division (arXiv:0901.0823v1[math.RA] 7 Jan 2009).
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L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
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M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
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H. Michiwaki, H. Okumura, and S. Saitoh,
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T. S. Reis and J.A.D.W. Anderson,
Transdifferential and Transintegral Calculus,
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\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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