Their quotients in the calculator are both "error". Math books show they are both infinity.
Dear Eddie,
you should not consider the output of a calculation programme as mathematical "truth".
Mathematically spoken 1/0 and 0/0 are not the same: While 1/0 can be obtained as limes from 1/x with lim(x) --> 0 = infinity, the term 0/0 is not defined in mathematics - hence it cannot be stated that is also infinity. You could also say that the solution set S of the equation 0/0 = x is the empty set S = {}.
They are obviously not the same because '1/0' and '0/0' do not express the same numbers (1 versus 0). And this time I focus on the quotients, not their assumed results.
Actually, there is no reason that they are "obviously" not the same, as 0/1, 0/2,...0/n are all equal. The problem is that there isn't really any use for defining division by 0 the way one defines e.g., x^0 as 1 or 0! as 1, and good reason not to. a|b=c iff b=ca or "a divides b if and only if b= ca" is how division is defined in algebras. The only number for which b/a=c is equivalent to b=ca is 0, as 0/0=0 is equivalent to 0=0*0. As one can divide 0 by some number (but will always get 0) it is possible to define 0/0 without appealing to some special case or arbitrarily defining division by 0 in this case so that division in general or even the division of 0 in general will hold true. That can't be said for any integer n s.t. n/0= x provides something useful enough to make some division by zero equal to something by definition. So it is undefined. Put differently, one could create a consistent system which allowed division by 0 by defining it as equal to 0, but as there is no use for doing so and as this definition is inconsistent with how division is defined in general, why bother?
Besides the theoretical aspects mentioned above, there are two points worth mentioning:
1) floating point arithmetics (such as IEEE 754) usually defines the result of "1.0/0.0" as "positive infinity" and "0.0/0.0" as "not a number"
2) Sometimes it is convenient to define "0/0" in order to simply definitions; e.g. in the definition of b-spline basis functions $N_i^r(t)$ it is common convention to define/evaluate "0/0 := 0", so that the definition remains valid without having to formulate special rules.
0/0 is an indeterminate form and 1/0 is undefined, so they are not worth comparing. Math books do not say they are both infinity. This is commonly confused for the limit as x->0, where 1/x =infinity. Approaching is not the same as being the value. Andrew Messings explanation is one I recommend reading.
As for Torsten's comment related to IEEE floating point: Keep in mind IEEE standards defined this so they didn't need to throw a DIV/0 or NaN error. The reason why is because floating point needs to be flexible, and often times floating point numbers are used with scientific computing in mind. Situations where values close to 0 (not zero) occur all the time in that domain, and sometimes "limit-like" computations do occur. Torsten gives a good example.
EDIT: I fixed my first sentence. I stated something slightly off.
Daniel, math books do tell they are both infinity (with the symbol). However, it is true that some books qualify that 0/0 is meaningless although still symbolized by infinity, All of your answers, including those that were down voted, were very clarificatory and provides a picture of the situation on this two ratios that are so basic or elementary so they seem yet so complex. Thanks a lot. I hope you don;t mind if I up vote all of you. Perhaps it is time to propose a mathematical symbol for "meaningless" . Ed
Could you give an example of a text that does say both of these things are infinity? I am just curious. I have seen some people use a concept called complex infinity before (over complex numbers, not reals) to label numbers of that form (i.e., 1/0). But, 1/0 is not a real number as the quotient 1/0 cannot be defined.
Dan, both of them are given the infinity symbol, but which is called infinity and also called undefined. The book that said 0/0 is meaningless was written by Setek. I'll provide you the details later (I'll look for it). I believe your answer is very well in order which necessitates some explanations of the two ratios in our basic education mathematics and maybe in other schools, as you can see by the various ways, we in this forum answer the question. And that is the very reason why I asked the question. I myself need to be given information other than that I learned in basic ed and from the books I read. I thank all of you for the answers.
I often asked this question to college students and honestly, less than one percent of those I asked could answer 0/0.
Ed
In excel, 1/0=#DIV/0!
and 0/0= #DIV/0!
In the free online calculator http://www.calculateforfree.com/
1/0 = infinity and 0/0 = infinity
This Daniel, confuses us. It will be nice if mathematicians could give a categorical answer to 1/0 and 0/0 with the appropriate math symbols programmed in the calculator and the computers such as in excel.
Some would claim there is simply a difference between human-invented theory (e.g. as outlined in text books) and practice (e.g. electronic calculators), and this is not a joke!
Dear Eddie,
you should not consider the output of a calculation programme as mathematical "truth".
Mathematically spoken 1/0 and 0/0 are not the same: While 1/0 can be obtained as limes from 1/x with lim(x) --> 0 = infinity, the term 0/0 is not defined in mathematics - hence it cannot be stated that is also infinity. You could also say that the solution set S of the equation 0/0 = x is the empty set S = {}.
@Eddie: It is fundamental to understand that a computing device need not follow the same rules as the "actual thing". For example Excel will throw DIV/0 error, while some other systems (like IEEE floating point) may say "infinity" (in some cases). It is the nature of what is being done in practice a lot of the time. For example, some programming languages will allow you to overload operators to make them behave how you like. This can be completely independent of standard mathematics as you can always invent rules around them. You can think of these "tweaks" as including more axioms.
For example, it is trivial for me to write a method in a programming language that tells me "Uh oh" if A/B is where B=0. Your typical Calculus text will speak of 1/0 as an undefined term that can be approached using a limit (keep in mind that a limit isn't the same as the actual value unless it follows from certain properties), but 0/0 is indeterminate because it doesn't make sense in traditional algebraic systems.
This is essentially a more elaborated answer from what Marcel and Johannes said.
Hope this helps!
From a mathematics standpoint all the expressions a/0 do not make sense per se. In the complex numbers and every other standard set of numbers, dividing by 0, i.e. multiplying by the multiplicative inverse of 0, does not make sense at all, since there is no multiplicative inverse of 0 - there is a very easy proof for that. So the discussion, whether the two quotients are the same or not is a superficial one, since both quotients do not exist.
However, there are some mathematical applications, when divisions by 0 seem to show up and make sense, most notably in limit calculations. There, sometimes textbooks say something like "1/0 = infinity", and "0/0 = undefined expression". But in these very specialized applications these quotients are mere abbreviations for calculations of limits of sequences, and one should understand the limit calculations before memorizing the abbreviations. Otherwise, you can get it wrong: lim_{n->infinity} 1/(-1/n) takes the form 1/0, but the result is actually -infinity.
In IEEE floating point calculations every possible arithmetic operation has a defined result. There, 1/0 = +inf, 0/0 = sNaN (silent not a number). Clear: the computer has to produce some result. However, floating point numbers are algebraically very weird. They are not associative, inverses are in general not unique, not all operations are invertible, etc. So they do not have the properties a mathematician would require from "true numbers".
Daniel, here's the book
Fundamentals of Mathematics. Fifth Edition
MacMillan Publishing Company
New York
by William G. Setek, Jr. 1989
I am learning so much from this discussion. Thanks.
In calculus :1/0=infinity, and 0/0 is meaningless or undetermined. the all meaningless
are: inf-inf, 1^inf, inf^0,0^inf
The answer depends on what we mean by the expressions 1/0 and 0/0. The usual definition of a/b as a rational number, where a and b are integers, is that a/b is the unique real number r such that a = b*r. Using this definition for the expressions 1/0 and 0/0, we ask: Is there a unique real number, r, such that 1/0 = r or 0/0 = r? In both cases, the answer is no. For the expression, 1/0, there obviously is no real number r such that 1 = 0*r, and for the expression 0/0 there is no unique real number r such that 0 = 0*r. Hence neither 1/0 nor 0/0 is well defined over the reals using the definition of 1/0 and 0/0 as rational expressions of the form a/b. Alternatively, we can express 0/0 as a limit: Let 0/0 = lim(x-->x*) f(x)/g(x), where lim(x-->x*)f(x) = 0 = lim(x-->x*)g(x). We now have the problem that we can make 0/0 be any number by our choice of f(x) and g(x). For example, let f(x) = rx and let g(x)= x, then lim(x-->x*)f(x)/g(x) = lim(x-->x*) rx/x = lim(x-->x*) r = r. Thus we have the problem that if we try to define 0/0 using limits, we can make 0/0 be any value, r. For the expression 1/0, of course, we do get a unique limit, namely infinity, which is also a way of saying that 1/0 is larger than any real number r.
Robert: You are quite right -- I was thinking of magnitudes, but I should have said that in the limit as x goes to 0 through positive numbers then 1/0 is larger than any positive real number and as x goes to 0 through negative numbers then 1/0 is less than any negative real number, hence there are two limits.
Dear Dr. AL-Rammahi:
In calculus, 1/0 is not equal to infinity. The limit of 1/x as x approaches infinity is 0. 1/0 is undefined. Only 0/0 can be easily made to equal 0 without special definitions (as division OF 0 is defined, and will always be 0 hence division of 0 by 0 requires no special treatment; however, it is generally still considered undefined). Of course, if we extend infinitely along the number line to the left, we generally use infinity to symbolize the result. This does not make 1/0 anything other than undefined.
Dear #Andrew Messing.
Who said anything about "The limit of 1/x as x approaches infinity is 0"? The question was about 1/0, so if you want to talk about limits you will have to consider lim x->0 (1/x).
Dear Andrew and Fernando, you have been providing this forum essential information that benefits all of us. I hope you can extend your sharing and also understanding to infinity. Peace. Ed
0/0 is often encountered in applications as a limit. For example, for any constant a, f_a (x) = sin(a x)/x is 0/0 for x = 0 if the numerator and denominator are computed separately, but f_a (x) is a smooth function for which f_a(x) -> a as x -> 0.
1/0 = +infinity or-infinity
0/0 could be anything (it is a limit as in the definition of Derivative of a function at a point)... Finite or infinite or does not exist but not infinite.
Yes, 0/0 has no meaning in and of itself, but when it is encountered, it is always appropriate to investigate the numerator and denominator for values of the independent variables in the denominator nearby those that make the denominator vanish. In many applications relevant to physical science and engineering, the numerator/ denominator ratio will be found to tend smoothly to a fixed value in the limits of those variables at which the denominator vanishes. Of course this is not always the case, but the limiting form is always the first thing that one should check.
The question deals with contradictory “potential infinitude” and “actual infinitude” concepts have being accompanying we human with a long-drawn-out and ceaseless “potential infinity--actual infinity” debate since antiquity. But It is impossible to understand and solve such a question scientifically in present science theory because of the suspended defects in the “Infinitude” Related Foundation of Mathematics disclosed by all kinds of suspended infinitude paradoxes (such as Zeno’s Paradox).
“Are the quotients of 1/0 and 0/0 the same?” is a very good but tough question because it touches the basic stones of 2500-year old present traditional infinity system: (1) infinity conception, (2) infinity related number forms as well as (3) infinity related number treating theory and techniques (limit theory). This question has been challenging we human for more than 2500 years, it should be solved by we human.
In applied mathematics, the question worries no one, people just operate and do as required. But the poor philosophical explanations have been troubling us human ever since the concept of “infinity” came to being. It proves that we can not run away from philosophical ones.
Can we use the “brackets-placing rule"(１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．．) applied in the divergent proof of Harmonious Series (1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．．) to produce infinite numbers bigger than 1/2 or 1 or 2 or 3 or 4…?
Miguel，the situation in the “brackets-placing rule"(１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．．) applied in the divergent proof of Harmonious Series (1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．．) to produce infinite numbers bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… is totally different from that of Grandi's series:1-1+1-1+1-1...
Actually our operation in Harmonious Series Paradox is: to change an infinitely decreasing Harmonious Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant or any infinitely increasing series with the property of Un--->infinity--------this means that not matter what kind of runner (even a runner with the speed of modern jet plane) held the race with the tortoise in Zeno’s Paradox, the runner will never catch up with it.
Miguel, I am sorry that I don’t quite understand your idea above, would you please express in another way？
Dear Miguel,
“Achilles and Tortoise” Zeno’s Paradox has disclosed the "real infinite--potential infinite" fundamental defects in present traditional infinite related science theory system. In this theory system, the critical defects are unsolvable because we are not allowed to forget either "real infinite" or "potential infinite" in such paradox (exactly same situation happen in Harmonious Series Paradox). That is why this paradox family has been troubling us human for more than 2500 years.
Many people even suggested a simple calculation of 1+1/2+1/4+1/8+…=2 or a real race between Mr. Smith and a strong tortoise to solve this paradox in history, but it is like an apple to an oyster as we know.
Could we have a short break for following dish?
When we study ”the meaning of zero" and the location of zero in “number spectrum” in our mathematics, an unbalanced defect can be easily discovered: “zero" appears on one side of the “number spectrum” as a kind of mathematical language telling people a situation of “ nothing, not-being,…”; but on the other side of the “number spectrum” we lack of another kind of mathematical language telling people an opposite situation to “zero”------“ something, being,…”.
We need a new number symbol with opposite meaning to zero locating at the opposite side of zero in the “number spectrum” to make up the structural incompleteness of “number spectrum” and to complete the existence of “zero”.
Sincerely yours,
Geng
Geng, from a different perspective, the number line in fact, presents to us two other sides of zero, one side is less than being (negative) and the other side, more than being (positive).
Ed
Wasan Geometry and Division by Zero Calculus
2018年11月28日(水) テーマ：数学
Sangaku Journal of Mathematics (SJM) ⃝c SJM ISSN 2534-9562 Volume 2 (2018), pp. 57-73 Received 20 November 2018. Published on-line 29 November 2018 web: http://www.sangaku-journal.eu/ ⃝c The Author(s) This article is published with open access1 . Wasan Geometry and Division by Zero Calculus
file:///C:/Users/saito%20saburo/Downloads/SJM_2018_57-73_okumura_saitoh%20(1).pdf
ゼロ除算の発見は日本です：
∞？？？
∞は定まった数ではない・・・・
人工知能はゼロ除算ができるでしょうか：
とても興味深く読みました：2014年2月2日 4周年を超えました：
ゼロ除算の発見と重要性を指摘した：日本、再生核研究所
神の数式で ゼロ除算を用いると どうなるのでしょうか という質問が 寄せられています。
神の数式：
神の数式が解析関数でかけて居れば、 特異点でローラン展開して、正則部の第1項を取れば、 何時でも有限値を得るので、 形式的に無限が出ても 実は問題なく 意味を有します。
物理学者如何でしょうか。
計算機は 正しい答え 0/0=0 を出したのに計算機は何時、1/0=0 ができるようになるでしょうか。
📷
カテゴリ：カテゴリ未分類
そこで、計算機は何時、1/0=0 ができるようになるでしょうか。 楽しみにしています。 もうできる進化した 計算機をお持ちの方は おられないですね。
これは凄い、面白い事件では？ 計算機が人間を超えている 例では？
面白いことを発見しました。 計算機は 正しい答え 0/0=0
を出したのに、 この方は 間違いだと 言っている、思っているようです。
0/0=0 は １３００年も前に 算術の発見者によって与えられたにも関わらず、世界史は間違いだと とんでもないことを言ってきた。 世界史の恥。 実は a/0=0 が 何時も成り立っていた。 しかし、ここで 分数の意味を きちんと定義する必要がある。 計算機は、その意味さえ知っているようですね。 計算機、人間より賢くなっている 様が 出て居て 実に 面白い。
https://steemkr.com/utopian-io/@faisalamin/bug-zero-divide-by-zero-answers-is-zero
２０１８．１０．１１．１１：２３
https://plaza.rakuten.co.jp/reproducingkerne/diary/201810110003/
計算機は 正しい答え 0/0=0 を出したのに
カテゴリ：カテゴリ未分類
面白いことを発見しました。 計算機は 正しい答え 0/0=0
を出したのに、 この方は 間違いだと 言っている、思っているようです。
0/0=0 は １３００年も前に 算術の発見者によって与えられたにも関わらず、世界史は間違いだと とんでもないことを言ってきた。 実は a/0=0 が 何時も成り立っていた。しかし、ここで 分数の意味を きちんと定義する必要がある。 計算機は、その意味さえ知っているようですね。 計算機、人間より賢くなっている様が 出て居て 実に面白い。
https://steemkr.com/utopian-io/@faisalamin/bug-zero-divide-by-zero-answers-is-zero
２０１８．１０．１１．１１：２３
ゼロ除算、ゼロで割る問題、分からない、正しいのかなど、 良く理解できない人が 未だに 多いようです。そこで、簡潔な一般的な 解説を思い付きました。 もちろん、学会などでも述べていますが、 予断で 良く聞けないようです。まず、分数、a/b は a 割る b のことで、これは 方程式 b x=a の解のことです。ところが、 b がゼロならば、 どんな xでも 0 x =0 ですから、a がゼロでなければ、解は存在せず、 従って 100/0 など、ゼロ除算は考えられない、できないとなってしまいます。 普通の意味では ゼロ除算は 不可能であるという、世界の常識、定説です。できない、不可能であると言われれば、いろいろ考えたくなるのが、人間らしい創造の精神です。 基本方程式 b x=a が b がゼロならば解けない、解が存在しないので、困るのですが、このようなとき、従来の結果が成り立つような意味で、解が考えられないかと、数学者は良く考えて来ました。 何と、 そのような方程式は 何時でも唯一つに 一般化された意味で解をもつと考える 方法があります。 Moore-Penrose 一般化逆の考え方です。 どんな行列の 逆行列を唯一つに定める 一般的な 素晴らしい、自然な考えです。その考えだと、 b がゼロの時、解はゼロが出るので、 a/0=0 と定義するのは 当然です。 すなわち、この意味で 方程式の解を考えて 分数を考えれば、ゼロ除算は ゼロとして定まる ということです。ただ一つに定まるのですから、 この考えは 自然で、その意味を知りたいと 考えるのは、当然ではないでしょうか？初等数学全般に影響を与える ユークリッド以来の新世界が 現れてきます。
ゼロ除算の誤解は深刻：
最近、3つの事が在りました。
私の簡単な講演、相当な数学者が信じられないような誤解をして、全然理解できなく、目が回っているいるような印象を受けたこと、 相当ゼロ除算の研究をされている方が、基本を誤解されていたこと、1/0 の定義を誤解されていた。 相当な才能の持ち主が、連続性や順序に拘って、４年以上もゼロ除算の研究を避けていたこと。
これらのことは、人間如何に予断と偏見にハマった存在であるかを教えている。 まずは ゼロ除算は不可能であるの 思いが強すぎで、初めからダメ、考えない、無視の気持ちが、強い。 ゼロ除算を従来の 掛け算の逆と考えると、不可能であるが 証明されてしまうので、割り算の意味を拡張しないと、考えられない。それで、 1/0,0/0,z/0 などの意味を発見する必要がある。 それらの意味は、普通の意味ではないことの 初めの考えを飛ばして ダメ、ダメの感情が 突っ走ている。 非ユークリッド幾何学の出現や天動説が地動説に変わった世界史の事件のような 形相と言える。
２０１８．９．２２．６：４１ ゼロ除算の４つの誤解：
１． ゼロでは割れない、ゼロ除算は 不可能である との考え方に拘って、思考停止している。 普通、不可能であるは、考え方や意味を拡張して 可能にできないかと考えるのが 数学の伝統であるが、それができない。
２． 可能にする考え方が 紹介されても ゼロ除算の意味を誤解して、繰り返し間違えている。可能にする理論を 素直に理解しない、 強い従来の考えに縛られている。拘っている。
３． ゼロ除算を関数に適用すると 強力な不連続性を示すが、連続性のアリストテレス以来の 連続性の考えに囚われていて 強力な不連続性を受け入れられない。数学では、不連続性の概念を明確に持っているのに、不連続性の凄い現象に、ゼロ除算の場合には 理解できない。
４． 深刻な誤解は、ゼロ除算は本質的に定義であり、仮定に基づいているので 疑いの気持ちがぬぐえず、ダメ、怪しいと誤解している。数学が公理系に基づいた理論体系のように、ゼロ除算は 新しい仮定に基づいていること。 定義に基づいていることの認識が良く理解できず、誤解している。
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.
Eπi =-1 （1748）（Leonhard Euler）
E = mc 2 （1905）（Albert Einstein）
1/0=0/0=0 （2014年2月2日再生核研究所）
ゼロ除算（division by zero）1/0=0/0=z/0= tan (pi/2)=0
https://ameblo.jp/syoshinoris/entry-12420397278.html
1+1=2 （ ）
a2+b2=c2 （Pythagoras）
1/0=0/0=0（2014年2月2日再生核研究所）
[2981] viXra:1902.0058 [pdf] submitted on 2019-02-03 22:47:53
We Can Divide the Numbers and Analytic Functions by Zero\\ with a Natural Sense.
Authors: Saburou Saitoh
http://vixra.org/abs/1902.0058
To divide by zero is to multiply by zero.
Hence
0/0 = 0 times 0=0,
1/0 = 1 times 0=0.
Also
z/0 = z times 0=0.
This is my thought.
In present classical infinite related mathematics, the unavoidable confusion of “potential infinite” and “actual infinite” have made people unable to know (impossible to know) what 0 is and what the quotients of 1/0 and 0/0 are.
Since infinite concept came to our science, people have been studying and cognizing systematically the infinite related things. The “potential infinite” and “actual infinite” concepts based “classical infinite idea” was constructed and the “classical infinite idea” based present classical infinite related science theory system was build: classical infinite idea, classical mathematical analysis, classical number system, classical limit theory, classical set theory and classical one to one correspondence theory. These six fields constitute an integrated qualitative and quantitative cognizing system for infinite things in human science. But, we have to acknowledge after careful studies to the infinite related science history that people have never been able to explain scientifically what “potential infinite” and “actual infinite” concepts are, what roles they play and how they act in above six fields although they have been unavoidable ever since. And even more, people have been unable to know the inevitable ligaments among these two fundamental concepts and above six fields. These fatal defects unable us to answer self-justificationly following four conundrums in present classical infinite related science theory system:
(1) What are “potential infinite” and “actual infinite”, what are the essential differences between them? What is the relationship between present classical infinite idea and them, what roles they play and how they act in infinite idea, why? Do we have “potential infinite idea--potential infinite theory system” only basing on “potential infinite” concept or “actual infinite idea--actual infinite theory system” only basing on “actual infinite” concept or “potential infinite and actual infinite idea-- potential infinite and actual infinite theory system” which has close relationship with both “potential infinite” and “actual infinite” concepts or “non potential infinite and non actual infinite idea--non potential infinite and non actual infinite theory system” which is nothing to do with both concepts of “potential infinite” and “actual infinite”, why and how?
(2) What is the relationship between present classical mathematical analysis and the concepts of “potential infinite” and “actual infinite”, what roles they play and how they act in mathematical analysis, why? Do we have “potential infinite mathematical analysis” only basing on “potential infinite” concept or “actual infinite mathematical analysis” only basing on “actual infinite” concept or “potential infinite and actual infinite mathematical analysis” which has close relationship with both “potential infinite” and “actual infinite” concepts or “non potential infinite and non actual infinite mathematical analysis” which is nothing to do with both concepts of “potential infinite” and “actual infinite”, why and how?
(3) What is the relationship between present classical number system and the concepts of “potential infinite” and “actual infinite”, what roles they play and how they act in number system, why? Do we have “potential infinite number system” only basing on “potential infinite” concept or “actual infinite number system” only basing on “actual infinite” concept or “potential infinite and actual infinite number system” which has close relationship with both “potential infinite” and “actual infinite” concepts or “non potential infinite and non actual infinite number system” which is nothing to do with both concepts of “potential infinite” and “actual infinite”, why and how?
(4) What is the relationship between present classical limit theory and the concepts of “potential infinite” and “actual infinite”, what roles they play and how they act in limit theory, why? Do we have “potential infinite limit theory” only basing on “potential infinite” concept or “actual infinite limit theory” only basing on “actual infinite” concept or “potential infinite and actual infinite limit theory” which has close relationship with both “potential infinite” and “actual infinite” concepts or “non potential infinite and non actual infinite limit theory” which is nothing to do with both concepts of “potential infinite” and “actual infinite”, why and how?
The fundamental defects in “potential infinite--actual infinite” based present classical infinite theory system necessarily lead to fatal defects in the foundations of above four infinite related mathematics fields, greatly reduce the scientificity of our cognizing results to “infinite things”.
In present classical infinite related mathematics, the unavoidable confusion of “potential infinite” and “actual infinite” have made people unable to know (impossible to know) what 0 is and what the quotients of 1/0 and 0/0 are.
The question is how to define z/0.
And my answer is to define as follows:
The definition should be made by binary operation, and
z/0 = z times 0.
The question is very simple, which has been asked since antiquity again and again troubling people without any scientific results because it touches the very foundation of present classical infinite theory---------what “infinite, potential infinite and actual infinite” are? But, it is not that easy to answer.
The facts we are facing to are: in present classical infinite theory system, there is neither special quantitative cognizing tool for “potential infinitesimal, potential infinite many, potential infinities” mathematics things------ because we are unable to know (impossible to know) what “potential infinite” is, and nor special quantitative cognizing tool for “actual infinitesimal, actual infinite many, actual infinity” mathematics things ------ because we are unable to know (impossible to know) what “actual infinite” is. It is this fatal defect that unable people to conduct clear, scientific and self-justification quantitative cognizing activities to any infinite things--------they can be either “potential infinite” with all the necessary qualities and manifestations of “potential infinite” or “actual infinite” with all the necessary qualities and manifestations of “actual infinite”, just depending on one’s favorite.
How will be our results on the division by zero?
With best regards,
Sincerely yours, Saburou Saitoh
2019.2.16.18:45
Close the mysterious and long history of division by zero and open the new world since Aristotelēs-Euclid: 1/0=0/0=z/0= \tan (\pi/2)=0.
Sangaku Journal of Mathematics (SJM) c ⃝SJMISSN 2534-9562 Volume 2 (2018), pp. 57-73 Received 20 November 2018. Published on-line 29 November 2018 web: http://www.sangaku-journal.eu/ c ⃝The Author(s) This article is published with open access1.
Wasan Geometry and Division by Zero Calculus
∗Hiroshi Okumura and ∗∗Saburou Saitoh
Saburou Saitoh
[5] viXra:1902.0240 submitted on 2019-02-13 22:57:25, (0 unique-IP downloads)
Zero and Infinity; Their Interrelation by Means of Division by Zero
Authors: Saburou Saitoh Category: General Mathematics
[4] viXra:1902.0223 submitted on 2019-02-12 18:39:18, (2 unique-IP downloads)
Horn Torus Models for the Riemann Sphere and Division by Zero
Authors: Wolfgang W. D\"aumler, Hiroshi Okumura, Vyacheslav V. Puha, Saburou Saitoh Category: Functions and Analysis
[3] viXra:1902.0204 submitted on 2019-02-11 18:46:02, (6 unique-IP downloads)
A Meaning and Interpretation of Minus Areas of Figures by Means of Division by Zero
Authors: Saburou Saitoh Category: General Mathematics
[2] viXra:1902.0187 submitted on 2019-02-10 22:40:35, (3 unique-IP downloads)
The Simple and Typical Physical Examples of the Division by Zero 1/0=0 by Ctes\'ibio (BC. 286-222) and e. Torricelli (1608 1646)
Authors: Saburou Saitoh Category: General Mathematics
[1] viXra:1902.0058 submitted on 2019-02-03 22:47:53, (26 unique-IP downloads)
We Can Divide the Numbers and Analytic Functions by Zero\\ with a Natural Sense.
Authors: Saburou Saitoh Category: General Mathematics
With deep thanks.
I would like to take a conntact with you by e-mail:
計算機がゼロ除算ができるという、情報が入った。そこで、 関係者に我々の研究を下記のように 送った： Isabelle は 相当前に 便宜的という理由で ゼロ除算を用いていたが、仲間内では 評価されていない との 情報を得ていた。著書原稿にも そのことを触れている。 重要性に 気づいてきた 可能性がある。 ２０１９．２．１６．２０：００
斎藤先生
RGの以下のQ&Aスレッドで Isabelleというソフトを使って x/0=0 for all x を証明したと言っていますね。
https://www.researchgate.net/post/Are_the_quotients_of_1_0_and_0_0_the_same#view=5c67b4f3aa1f0937e5503789
奥村博拝 How will be our results on the division by zero?
With best regards, Sincerely yours,
Saburou Saitoh 2019.2.16.18:35 Close the mysterious and long history of division by zero and open the new world since Aristotelēs-Euclid: 1/0=0/0=z/0= \tan (\pi/2)=0.
Sangaku Journal of Mathematics (SJM) c ⃝SJMISSN 2534-9562 Volume 2 (2018), pp. 57-73 Received 20 November 2018. Published on-line 29 November 2018 web: http://www.sangaku-journal.eu/ c ⃝The Author(s) This article is published with open access1.
Wasan Geometry and Division by Zero Calculus
∗Hiroshi Okumura and ∗∗Saburou Saitoh
Saburou Saitoh
[5] viXra:1902.0240 submitted on 2019-02-13 22:57:25, (0 unique-IP downloads)
Zero and Infinity; Their Interrelation by Means of Division by Zero
Authors: Saburou Saitoh Category: General Mathematics
[4] viXra:1902.0223 submitted on 2019-02-12 18:39:18, (2 unique-IP downloads)
Horn Torus Models for the Riemann Sphere and Division by Zero
Authors: Wolfgang W. D\"aumler, Hiroshi Okumura, Vyacheslav V. Puha, Saburou Saitoh Category: Functions and Analysis
[3] viXra:1902.0204 submitted on 2019-02-11 18:46:02, (6 unique-IP downloads)
A Meaning and Interpretation of Minus Areas of Figures by Means of Division by Zero
Authors: Saburou Saitoh Category: General Mathematics
[2] viXra:1902.0187 submitted on 2019-02-10 22:40:35, (3 unique-IP downloads)
The Simple and Typical Physical Examples of the Division by Zero 1/0=0 by Ctes\'ibio (BC. 286-222) and e. Torricelli (1608 1646)
Authors: Saburou Saitoh Category: General Mathematics
[1] viXra:1902.0058 submitted on 2019-02-03 22:47:53, (26 unique-IP downloads)
We Can Divide the Numbers and Analytic Functions by Zero\\ with a Natural Sense.
Authors: Saburou Saitoh Category: General Mathematics
計算機が ゼロ除算が できたと言ってきた。
凄いことでは？ 関心のある方は 解明してほしい。
この方と 連絡が 取れました。
in 8 hours José Manuel Rodríguez Caballero Added an answer In the proof assistant Isabelle/HOL we have x/0 = 0 for each number x. This is advantageous in order to simplify the proofs. You can download this proof assistant here:
Close the mysterious and long history of division by zero and open the new world since Aristotelēs-Euclid: 1/0=0/0=z/0= \tan (\pi/2)=0.
Sangaku Journal of Mathematics (SJM) c ⃝SJMISSN 2534-9562 Volume 2 (2018), pp. 57-73 Received 20 November 2018. Published on-line 29 November 2018 web: http://www.sangaku-journal.eu/ c ⃝The Author(s) This article is published with open access1.
Wasan Geometry and Division by Zero Calculus
∗Hiroshi Okumura and ∗∗Saburou Saitoh
Yes, they are all zero. Beautiful!!
José Manuel Rodríguez Caballero
Added an answer
In the proof assistant Isabelle/HOL we have x/0 = 0 for each number x. This is advantageous in order to simplify the proofs. You can download this proof assistant here: https://isabelle.in.tum.de/
Close the mysterious and long history of division by zero and open the new world since Aristotelēs-Euclid: 1/0=0/0=z/0= \tan (\pi/2)=0.
Sangaku Journal of Mathematics (SJM) c ⃝SJMISSN 2534-9562 Volume 2 (2018), pp. 57-73 Received 20 November 2018. Published on-line 29 November 2018 web: http://www.sangaku-journal.eu/ c ⃝The Author(s) This article is published with open access1.
Wasan Geometry and Division by Zero Calculus
∗Hiroshi Okumura and ∗∗Saburou Saitoh
José Manuel Rodríguez Caballero
Added an answer
In the proof assistant Isabelle/HOL we have x/0 = 0 for each number x. This is advantageous in order to simplify the proofs. You can download this proof assistant here: https://isabelle.in.tum.de/
Close the mysterious and long history of division by zero and open the new world since Aristotelēs-Euclid: 1/0=0/0=z/0= \tan (\pi/2)=0.
Sangaku Journal of Mathematics (SJM) c ⃝SJMISSN 2534-9562 Volume 2 (2018), pp. 57-73 Received 20 November 2018. Published on-line 29 November 2018 web: http://www.sangaku-journal.eu/ c ⃝The Author(s) This article is published with open access1.
Wasan Geometry and Division by Zero Calculus
∗Hiroshi Okumura and ∗∗Saburou Saitoh
It is the time to accept the definition z/0=0 by S. Saitoh, since z/0=0 has been derived by Isabelle/HOL. Then the definition of the inverse should be updated as follows:
$a$ and $b$ are the inverse to each other if and only if $a=1/b$ holds.
By the new definition, 0 is the inverse of 0.
Please read the comments:
The typical good comment for the first draft is given by some physicist as follows:
Here is how I see the problem with prohibition on division by zero,
which is the biggest scandal in modern mathematics as you rightly pointed
out (2017.10.14.08:55)
A typical wrong idea will be given as follows:
mathematical life is very good without division by zero (2018.2.8.21:43).
It is nice to know that you will present your result at the Tokyo Institute of Technology. Please remember to mention Isabelle/HOL, which is a software in which x/0 = 0. This software is the result of many years of research and a millions of dollars were invested in it. If x/0 = 0 was false, all these money was for nothing. Right now, there is a team of mathematicians formalizing all the mathematics in Isabelle/HOL, where x/0 = 0 for all x, so this mathematical relation is the future of mathematics. https://www.cl.cam.ac.uk/~lp15/Grants/Alexandria/
José Manuel Rodríguez Caballero
Added an answer
In the proof assistant Isabelle/HOL we have x/0 = 0 for each number x. This is advantageous in order to simplify the proofs. You can download this proof assistant here: https://isabelle.in.tum.de/
Close the mysterious and long history of division by zero and open the new world since Aristotelēs-Euclid: 1/0=0/0=z/0= \tan (\pi/2)=0.
Sangaku Journal of Mathematics (SJM) c ⃝SJMISSN 2534-9562 Volume 2 (2018), pp. 57-73 Received 20 November 2018. Published on-line 29 November 2018 web: http://www.sangaku-journal.eu/ c ⃝The Author(s) This article is published with open access1.
Wasan Geometry and Division by Zero Calculus
∗Hiroshi Okumura and ∗∗Saburou Saitoh
２０１９．３．１４．１１：３０
Of course division by zero in any calculator or computing program will lead to undefined answer. The difference is 0/0 arises from a limit of a function for which sometimes you can calculate a finite answer, for example, using L'Hospital rule.
1/0 is a limit of a function thus
limx→0+ (1/x)= +∞ and limx→0- (1/x)= -∞
While 0/0 is indeterminate.
The deduction that z/0 = 0, for any z, is based in Saitoh's geometric intuition and it is currently applied in proof assistant technology, which are useful in industry and in the military.
Is It Really Impossible To Divide By Zero?
https://juniperpublishers.com/bboaj/pdf/BBOAJ.MS.ID.555703.pdf
Dear the leading person:
How will be the below information?
The biggest scandal:
The typical good comment for the first draft is given by some physicist as follows:
Here is how I see the problem with prohibition on division by zero,
which is the biggest scandal in modern mathematics as you rightly pointed out (2017.10.14.08:55)
A typical wrong idea will be given as follows:
mathematical life is very good without division by zero (2018.2.8.21:43).
It is nice to know that you will present your result at the Tokyo Institute of Technology. Please remember to mention Isabelle/HOL, which is a software in which x/0 = 0. This software is the result of many years of research and a millions of dollars were invested in it. If x/0 = 0 was false, all these money was for nothing. Right now, there is a team of mathematicians formalizing all the mathematics in Isabelle/HOL, where x/0 = 0 for all x, so this mathematical relation is the future of mathematics. https://www.cl.cam.ac.uk/~lp15/Grants/Alexandria/ José Manuel Rodríguez Caballero
Added an answer
In the proof assistant Isabelle/HOL we have x/0 = 0 for each number x. This is advantageous in order to simplify the proofs. You can download this proof assistant here: https://isabelle.in.tum.de/
Nevertheless, you can use that x/0 = 0, following the rules from Isabelle/HOL and you will obtain no contradiction. Indeed, you can check this fact just downloading Isabelle/HOL: https://isabelle.in.tum.de/ and copying the following code theory DivByZeroSatoih imports Complex_Main begin theorem T: ‹x/0 + 2000 = 2000› for x :: complex by simp end
2019/03/30 18:42 (11 時間前)
Close the mysterious and long history of division by zero and open the new world since Aristotelēs-Euclid: 1/0=0/0=z/0= \tan (\pi/2)=0.
Sangaku Journal of Mathematics (SJM) c ⃝SJMISSN 2534-9562 Volume 2 (2018), pp. 57-73 Received 20 November 2018. Published on-line 29 November 2018 web: http://www.sangaku-journal.eu/ c ⃝The Author(s) This article is published with open access1.
Wasan Geometry and Division by Zero Calculus
∗Hiroshi Okumura and ∗∗Saburou Saitoh
２０１９．３．１４．１１：３０
今受け取ったメールです。 何十年もゼロ除算の研究をされてきた人が、積極的に我々の理論の正当性を認めてきた。
Re: 1/0=0/0=0 example JAMES ANDERSON [email protected] apr, 2 at 15:03 All,
Saitoh’s claim is wider than 1/0 = 0. It is x/0 = 0 for all real x. Real numbers are a field. The axioms of fields define the multiplicative inverse for every number except zero. Saitoh generalises this inverse to give 0^(-1) = 0. The axioms give the freedom to do this. The really important thing is that the result is zero - a number for which the field axioms hold. So Saitoh’s generalised system is still a field. This makes it attractive for algebraic reasons but, in my view, it is unattractive when dealing with calculus.
There is no milage in declaring Saitoh wrong. The only objections one can make are to usefulness. That is why Saitoh publishes so many notes on the usefulness of his system. I do the same with my system, but my method is to establish usefulness by extending many areas of mathematics and establishing new mathematical results.
That said, there is value in examining the logical basis of the various proposed number systems. We might find errors in them and we certainly can find areas of overlap and difference. These areas inform the choice of number system for different applications. This analysis helps determine where each number system will be useful.
James Anderson
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 471: The 5th birthday of the division by zero $z/0=0$ \\
(2019.2.2)}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
{\bf [email protected]}\\
}
\date{\today}
\maketitle
The Institute of Reproducing Kernels is dealing with the theory of division by zero calculus and declares that the division by zero was discovered as $0/0=1/0=z/0=0$ in a natural sense on 2014.2.2. The result shows a new basic idea on the universe and space since Aristotelēs (BC384 - BC322) and Euclid (BC 3 Century - ), and the division by zero is since Brahmagupta (598 - 668 ?).
For the details, see the references and the site: http://okmr.yamatoblog.net/
We wrote a global book manuscript \cite{s18} with 235 pages
and stated in the preface and last section of the manuscript as follows:
\bigskip
{\bf Preface}
\medskip
The division by zero has the long and mysterious history over the world (see, for example, \index{H. G. Romig} \cite{boyer, romig} and Google site with the division by zero) with its physical viewpoint since the document of zero in India in AD 628. In particular, note that \index{Brahmagupta} Brāhmasphuṭasiddhānta (598 -668 ?) established four arithmetic operations by introducing $0$ and at the same time he defined as $0/0=0$ in
Brāhmasphuṭasiddhānta. We have been, however, considering that his definition $0/0=0$ is wrong over 1300 years, but, we will see that his definition is right and suitable.
The division by zero $1/0=0/0=z/0$ itself will be quite clear and trivial with several natural extensions of fractions against the mysteriously long history, as we can see from the concept of the Moore-Penrose generalized inverse \index{Moore-Penrose} \index{Tikhonov regularization} to the fundamental equation $az=b$, whose solution leads to the definition of $z =b/a$.
However, the result (definition) will show that
for the elementary mapping
$$
W = \frac{1}{z},
$$
the image of $z=0$ is $W=0$ ({\bf should be defined from the form}). 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 \index{Riemann sphere} (\cite{ahlfors}). As the representation of the \index{point at infinity} point at infinity of the \index{Riemann sphere} Riemann sphere by the
zero $z = 0$, we will see some delicate relations between $0$ and $\infty$ which show a strong \index{discontinuity}
discontinuity at the point of infinity on the Riemann sphere. We did not consider any value of the elementary function $W =1/ z $ at the origin $z = 0$, because we did not consider the division by zero
$1/ 0$ in a good way. Many and many people consider its value by limiting like $+\infty $ and $- \infty$ or the
point at infinity as $\infty$. However, their basic idea comes from {\bf continuity} with the common sense or
based on the basic idea of Aristotelēs %Aristotle\index{Aristotle}.
--
For the related Greek philosophy, see \cite{a,b,c}. However, as the division by zero we will consider the value of
the function $W =1 /z$ as zero at $z = 0$. We will see that this new definition is valid widely in
mathematics and mathematical sciences, see (\cite{mos,osm}) for example. Therefore, the division by zero will give great impacts to calculus, Euclidean geometry, analytic geometry, differential equations, complex analysis at the undergraduate level and to our basic idea for the space and universe.
We have to arrange globally our modern mathematics at our undergraduate level. Our common sense on the division by zero will be wrong, with our basic idea on the space and universe since Aristotelēs and Euclid. We would like to show clearly these facts in this book. The content is at the undergraduate level.
Close the mysterious and long history of division by zero that may be considered as a symbol of the stupidity of the human race and open the new world since Aristotel{$\bar{\rm e}$}s-Eulcid.
\bigskip
\bigskip
{\bf Conclusion}
\medskip
Apparently, the common sense on the division by zero with a long and mysterious history is wrong and our basic idea on the space around the point at infinity is also wrong since Euclid. On the gradient or on derivatives we have a great missing since $\tan (\pi/2) = 0$. Our mathematics is also wrong in elementary mathematics on the division by zero.
This book is elementary on our division by zero as the first publication of books for the topics. The contents have wide connections to various fields beyond mathematics. The author expects the readers to write some philosophy, papers and essays on the division by zero from this simple source book.
The division by zero theory may be developed and expanded greatly as in the author's conjecture whose break theory was recently given surprisingly and deeply by Professor \index{Qi'an Guan}Qi'an Guan \cite{guan} since 30 years proposed in \cite{s88} (the original is in \cite {s79}).
We have to arrange globally our modern mathematics with our division by zero in our undergraduate level.
We have to change our basic ideas for our space and world.
We have to change globally our textbooks and scientific books on the division by zero.
\bigskip
Our division by zero research group wonders why our elementary results may still not be accepted by some wide world.
\medskip
%We hope that:
%close the mysterious and long history of division by zero that may be considered as a symbol of the stupidity of the human race and open the new world since Aristotle-Eulcid.
% \medskip
From the funny history of the division by zero, we will be able to realize that
\medskip
human beings are full of prejudice and prejudice, and are narrow-minded, essentially.
\medskip
It seems that the long history of the division by zero is our shame and our mathematics in the elementary level has basic missings. Meanwhile, we have still great confusions and wrong ideas on the division by zero. Therefore, we would like to ask for the good corrections for the wrong ideas and some official approval for our division by zero as our basic duties.
\bibliographystyle{plain}
\begin{thebibliography}{10}
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\bibitem{boyer}
C. B. Boyer, An early reference to division by zero, The Journal of the American Mathematical Monthly, {\bf 50} (1943), (8), 487- 491. Retrieved March 6, 2018, from the JSTOR database.
\bibitem{cs}
L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
\bibitem{dops}
W. W. D\"aumler, H. Okumura, V. V. Puha and S. Saitoh,
Horn Torus Models for the Riemann Sphere and Division by Zero. (manuscript).
\bibitem{guan}
Q. Guan, A proof of Saitoh's conjecture for conjugate Hardy H2 kernels, arXiv:1712.04207.
\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 $0/0=0$,
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\bibitem{ms16}
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Advances in Linear Algebra \& Matrix Theory, {\bf 6}(2016), 51-58
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\\ http://dx.doi.org/10.4236/alamt.2016.62007.
\bibitem{mms18}
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\bibitem{msy}
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\bibitem{mos}
H. Michiwaki, H. Okumura and S. Saitoh,
Division by Zero $z/0 = 0$ in Euclidean Spaces,
International Journal of Mathematics and Computation, {\bf 2}8(2017); Issue 1, 1-16.
\bibitem{osm}
H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and $\infty$,
Journal of Technology and Social Science (JTSS), {\bf 1}(2017), 70-77.
\bibitem{os}
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\bibitem{o}
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\bibitem{os18april}
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Harmonic Mean and Division by Zero,
Dedicated to Professor Josip Pe\v{c}ari\'{c} on the occasion of his 70th birthday, Forum Geometricorum, {\bf 18} (2018), 155—159.
\bibitem{os18}
H. Okumura and S. Saitoh,
Remarks for The Twin Circles of Archimedes in a Skewed Arbelos by H. Okumura and M. Watanabe, Forum Geometricorum, {\bf 18}(2018), 97-100.
\bibitem{os18e}
H. Okumura and S. Saitoh,
Applications of the division by zero calculus to Wasan geometry.
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\bibitem{os1811}
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Wasan Geometry and Division by Zero Calculus,
Sangaku Journal of Mathematics (SJM), {\bf 2 }(2018), 57--73.
\bibitem{ps18}
S. Pinelas and S. Saitoh,
Division by zero calculus and differential equations. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics. {\bf 230} (2018), 399-418.
\bibitem{romig}
H. G. Romig, Discussions: Early History of Division by Zero,
American Mathematical Monthly, {\bf 3}1, No. 8. (Oct., 1924), 387-389.
\bibitem{s79}
S. Saitoh, The Bergman norm and the Szeg\"{o} norm, Trans. Amer. Math. Soc., {\bf 249} (1979), no. 2, 261-279.
\bibitem{s88}
S. Saitoh, Theory of reproducing kernels and its applications. Pitman Research Notes in Mathematics Series, {\bf 189}. Longman Scientific \&Technical, Harlow; copublished in the United States with John Wiley \& Sons, Inc., New York, (1988). x+157 pp. ISBN: 0-582-03564-3.
\bibitem{s14}
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{s16}
S. Saitoh, A reproducing kernel theory with some general applications,
Qian,T./Rodino,L.(eds.): Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, {\bf 177}(2016), 151-182.
\bibitem{s17}
S. Saitoh, Mysterious Properties of the Point at Infinity, arXiv:1712.09467 [math.GM](2017.12.17).
\bibitem{s18}
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\bibitem{ttk}
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\bibitem{a}
https://philosophy.kent.edu/OPA2/sites/default/files/012001.pdf
\bibitem{b}
http://publish.uwo.ca/~jbell/The 20Continuous.pdf
\bibitem{c}
http://www.mathpages.com/home/kmath526/kmath526.htm
\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.
\bibitem{ann326}
Announcement 326 (2016.10.17): The division by zero z/0=0 - its impact to human beings through education and research.
\bibitem{ann352}
Announcement 352(2017.2.2): On the third birthday of the division by zero z/0=0.
\bibitem{ann354}
Announcement 354(2017.2.8): What are $n = 2,1,0$ regular polygons inscribed in a disc? -- relations of $0$ and infinity.
\bibitem{362}
Announcement 362(2017.5.5): Discovery of the division by zero as $0/0=1/0=z/0=0$
\bibitem{380}
Announcement 380 (2017.8.21): What is the zero?
\bibitem{388}
Announcement 388(2017.10.29): Information and ideas on zero and division by zero (a project).
\bibitem{409}
Announcement 409 (2018.1.29.): Various Publication Projects on the Division by Zero.
\bibitem{410}
Announcement 410 (2018.1 30.): What is mathematics? -- beyond logic; for great challengers on the division by zero.
\bibitem{412}
Announcement 412(2018.2.2.): The 4th birthday of the division by zero $z/0=0$.
\bibitem{433}
Announcement 433(2018.7.16.): Puha's Horn Torus Model for the Riemann Sphere From the Viewpoint of Division by Zero.
\bibitem{448}
Announcement 448(2018.8.20): Division by Zero;
Funny History and New World.
\bibitem{454}
Announcement 454(2018.9.29): The International Conference on Applied Physics and Mathematics, Tokyo, Japan, October 22-23.
\bibitem{460}
Announcement 460(2018.11.06): Change the Poor Idea to the Definite Results For the Division by Zero - For the Leading Mathematicians.
\bibitem{461}
Announcement 461(2018.11.10): An essence of division by zero and a new axiom.
\end{thebibliography}
\end{document}
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