The Journey Begins
I do not feel like an alien in the universe. The more I examine the universe and study the details of its architecture, the more evidence I find that the universe in some sense must have known that we were coming.
– Freeman Dyson, Disturbing the Universe (1979)
T
he outer circle is a box full of air, going about its physics, chemistry and biology and doing nothing. But it is not without a craving to break out. The scientists as a rule fail to notice it, maybe because they are too caught up in the technical aspects of it, being under pressure to convince other technicians like themselves. So it usually falls on the popularizer of science to take notice. Since his job is to translate the thing back into normal language, it suddenly hits him that there is poetry there and no-one to appreciate it. Thus, in his book Cosmic Jackpot (2007), author Paul Davies was made to comment:
Even atheistic scientists will wax lyrical about the scale, the majesty, the harmony, the elegance, the sheer ingenuity of the universe of which they form so small and fragile a part … but [] do not necessarily interpret that as evidence for meaning or purpose in the universe.
Waiting for Godot
Fig 5.1. Poets have ever waxed lyrical on the joy of nature, so that the tragic note struck by Samuel Beckett in his play WAITING FOR GODOT (1953) serves as a timely reminder that the wait continues for the human being to finally arrive in full glory. The only prop in this absurdist play is a bare tree, which may well be symbolizing pi, waiting to be found out. The year tells us about the turning of the cube.
Artists and poets tend to take the opposite path, and being so much in awe of “nature” tend to worship it as pantheists. It’s not right either to shoot the messenger or worship him, the proper course being just to listen to the distress call. There is a malady out there and only we as human beings can put it right.
The tragedy is that it is all divided, has become part of a sweeping desolation known as The Divide. And this has persisted for aeons upon aeons, and the wait has been for Godot – something created in the image of God, and with three special gifts that link back to the divine. The craving of nature is best described as for the unity which brings all things together as one, to fix their divided individuality and revel in connectedness. This is what we mean by the world being rational. Reason connects things together, makes a symphony out of them. What is left behind is the cacophony of the separated notes.
For the most part this inner urge of the world is a passive thing. The music only appears in the mind of the poet who first saw that “host of golden daffodils” on the lakeside. Without the poet that music would never have been heard, and the longing of nature would only have remained a tragic note. So, life itself, even though still physics and chemistry, is only a tremendous outflow of the inner urge towards rational expression that exists in the outer circle. Anybody who has witnesses a sapling grow towards the sun has seen this urge firsthand.
So it is safe to say that life would never have stopped at any point in its evolution before giving rise to the pinnacle of Creation, the human being, in which there finds expression not only the physical and the rational urge to do things, but also the pure reason that leads finally to the goal of unity.
That which gave rise to first chemistry, then biology, was not the outer circle, but the inner one. All the subtlety of the universe is in the inner workings. And the ultimate goal is to join the dot with the infinite sphere beyond. It’s not that the joint circle does not exist, and the atoms are working to make it come into existence. The heavenly plane is implicit in things as they exist right now. As the 17th century German philosopher Gottfried Liebniz put it: the microcosm contains the macrocosm. In plain terms, the entire universe is inside each dot. And because it is, we have that amazing thing called calculus, which both Liebniz and his English arch rival Isaac Newton discovered independently at the same time.
Calculus and dividing zero by zero
Fig 5.2. Isaac Newton (1643-1727) and Gottfried Leibniz (1646-1716) were the culprits, ignoring the first commandment of mathematics not to divide by zero. But they hit gold, because what they mined in the process was the ideal circle.
It came about because they were bold enough to disobey the first commandment of mathematic: THOU SHALT NOT DIVIDE BY ZERO! They did something even worse, because they divided 0 by 0 – worse because utterly meaningless. But they happened to hit gold, because the two zeroes were standing for the inner and outer circles, and between them, entirely unsuspected, was the ideal circle. It was a piece of heaven that they had mined, and modern science continues to do the mining with the result of overflowing abundance of science and technology.
But the technology is only worthwhile if you appreciate the infinite subtlety of the inner circle, and through it the ideal circle. If the latest technological gadget only serves as temporary badge of honor, then I’m afraid you’re not in the frame of the story at all. The story of progress is all about fixing chaos, not self-image, nor creature comforts. Science is about the triumph of reason rather than technology. Calculus is a big part of the story because without it science would not have taken a single step.
The pressure of molecules and modern life
Fig 5.3. The jet-set life really amount to doing a whole lot of nothing, like molecules bumping into the walls of the box.
So the perfect circle is right there, even before the two circles come to join. It’s where all the real things are, the heavenly plane, the shop of ideal things which Socrates describes. It reaches out to us, just as do the prime numbers. Once we thought we could never glimpse this beauty, but we can. It’s through the symphony known as pi. This is what we have set out to do. And as we said before, beauty is not yet beautiful before it is appreciated.
If we know what the box is, we also know that the objective is to get out. To recap, the box has 8 corners, and it comes about by 2 and 3 mingling in the wrong way:
8 = 2+2x3
We will never get out if we allow the laws of physics to percolate over us, because that’s just another way of saying “do nothing”. Eating, drinking and being merry is just physics – that of the instincts. The more we do only this kind of “activity” we are just revving up the gas molecules in the box, which by Charles law increases temperature and pressure. It‘s not strange that the high flyers in the economic ladder are always complaining about pressure.
Instead, relax and think of the box as a room. If it is a room, it is meant to be for sitting down and doing things. Running around will just collect you bruises and bumps. Be assured that you will find desk and chair once you have come out of the frenzy. And once you have indeed settled down, there is going to be the guidelines right on top of the desk, without a doubt.
That’s in due course. But right now we are looking at how pi does the mending job of the messed up 2’s and 3’s. It does it through prime numbers, and first of all 8 of them are places in 8 corners of the box. 5 of them are called the major set [19, 2, 7, 5, 3], and 3 of them make up the minor set [13, 17, 11]. The task is to straighten out the mess. In other words, we will have to get the 2’s and 3’s mingling in the right and natural way, as follows:
7 = 2×2+3 (or 2+2+3)
12 = 2×2×3
The first gives 7 twice, so 14, and with the 12 implies 13. Remember from the last page that this is the blueprint for the reconstruction. One mistaken step is overcome by 3 correct ones. One original sin corrected by the proper use of the 3 royal gifts. 13 is therefore the symbol of regeneration, and recurs regularly in the journey ahead.
7 and 12 are the immediate results, pressed into service straight away. Indeed, after dividing the 36 decimal place of pi by 19, these are the 2 factors that stand out, along with 2 other enormous factors, one of 13 digits and the other of 21:
12×7×8581666150511×229374624355487716489
The 7 wonders and the 12 labours
Fig 5.4. 7 and 12 take on the universe together, but 13 does it all on its own, and justifiably stands for the symbol of regeneration.
Now, since 7 comes about in 2 ways, it is the prime in charge, and the 19 is allowed to combine with the 12. It does so in 2 ways, giving 13 and 21, which is exactly the number of digits of the 2 large primes. This tells us that 7 is in complete control of all the digits of pi, and was marked out as the secret key for good reason.
(19 acts on a number in two ways. It can add one to it, or it can turn the number around. The reason behind this versatility must wait a while.)
The heritage of 7 also speaks for itself. 7 days in a week, 7 colors of the rainbow, 7 continents, all reflecting the 7 heavens. On the reverse side, the 7 deadly sins correspond to the 7 gates of hell.
As for the heritage of 12, the 12 in a dozen, the 12 constellations corresponding to the 12 months in a year, the 12 tribes of Israel and the 12 labors of Hercules.
We ask the bold question, why should 7 take all the pains to introduce 12 with such fanfare? How does it help to clear up the mess? We know that the 8 primes are going to do the real clearing up, and this by solving Rubik’s cube, which is by using 5 primes and 3 primes, and thus applying the “ 53” treatment. It’s a shorthand to describe the solution to Rubik’s cube, and you can see it right there in the major prime set:
[19, 2, 7, 5, 3]
Solving Rubik's cube
Fig 5.5. THE 5-3 TREATMENT! The world record for solving the cube stands at 5.55 seconds, which only underlines the 5-3. It starts with 12, meaning that the 12 edges of the cube start to turn. 7 means that 7 corners may be spun (2) across one fixed (19). There are 43,252,003,274,489,856,000 combinations to the thing. These digits end up as 29. So you will be interested to know that exactly 2 months and 9 days after Waiting for Godot opened in Paris (5 January 1953), it was Pi Day (14 March). Vladimir and Estragon were clearly waiting for pi in the sky.
As you can see, the solution comes right after 7. But take a look at what comes before. It’s 12, because 19 is read as 1. So you can see that 12 comes even before the cube has started turning. Before even this there is the key 7 turning, and take a close look at this. It does so using the philosophy of 2-by-2, described in the last page. This only means that it takes the first two (19, 2) and applies it to the last two (5, 3). As we saw in the last page, 2-by-2 describes how the whole world comes to fit into pi, so, it’s a fundamental thing. But here we are going even further – not only taking the world in, but trying to solve it.
So look closely. In the process of taking the first two, it gives 12. This is then applied to the second pair, 53, which is the turning of the Rubik. So 12 obviously means “start turning”. So generally 12 means START, or, more appropriate in the context of the human journey, the setting out.
It’s the obvious connotation to 12, and also the correct one. ONE-TWO, stepping out, setting out. It is not something that is obviously there, because go back and try to imagine what the void of Genesis was like, because it was just “2”, mind and matter. Only after God pronounced, “Let there be Light!” did 19 come before it to give rise to 12. It is the real beginning, as far as the ultimate salvation is concerned. 2-by-2 may have saved the world from the Great Flood, but it is ONE-TWO that is the real start to the human journey.
But ONE-TWO must be granted status of philosophy in its own right, because its contribution is just as momentous as that of 2-by-2. Remember the void of mind and matter, where not even 1 existed, because there was just no unity to the thing. After light came into the world, something else did in a simultaneous way. This thing is called TIME, which came to complement the void of SPACE. To set out on a journey is to set out in TIME. Physicists tell us that nothing can go faster than the speed of light, and it stands to reason because light and TIME came into existence simultaneously. The speed of light serves as the edge of the universe, and because of it we cannot go back in time either.
Let there be Light!
Fig 5.6. TIME stands still on a particle of LIGHT, because they came into existence at the same moment. The rest is lagging behind, but the light of reason allows us to catch up.
If ONE-TWO gave rise to TIME, it cannot be TIME itself. Instead, look at what comes immediately after – the 2-by-2. That is TIME! It cannot be anything else. That which once served to get the whole universe into pi, now serves as the gap towards that same goal. Life is but a journey through this gap, and in pi we identify the goal, in the sense of being the guide towards the real goal, which is, of course, Heaven.
The expression for TIME turns out to be 22, so at once you get the idea that 22/7 stands for the journey with the help of the secret key. The thing will be established bit by bit, but right now it is a inspired conjecture that 22 is the numerical symbol for TIME.
It must also be contrasted to the philosophy of doing nothing, in which the 2-by-2 has nowhere to go and ends up as simply 2, expressive of mind and matter, the dead-end of the materialist’s world.
Against the doing nothing, what does the turning of the cube actually achieve? Look again at the major prime set:
[19, 2, 7, 5, 3]
The beat of ONE-TWO picks up 7 and 5 straight away, which adds to 12, confirming that the START instruction has taken effect. Also, it inexorably joins with the 3 that comes next to give 123. This is a shorthand for the natural numbers. So, pi, at the onset, is determining the scope of the journey, which is all the numbers, and therefore everything.
There is also going to be a breaking out of the box too. Take another look at the intermediary 75. This implies 6, which is the symbol of the inner circles, as opposed to the outer.
Why 6 should symbolize the inner circle is pretty straightforward. It’s THE perfect number. This is said to happen when all the factors add to give the number again, so that:
6 = 1×2×3 and 1+2+3
The perfection of the number 6
Fig 5.7. One nudge and the triangle becomes a rectangle, 3 becomes 4, SPACE becomes TIME. This is why 6 takes credit of being the perfect number, and the basis of all structure.
Reflect on it further, because it is truly awe-inspiring! You will never see anything like it in any other number. It’s a number that comes back to itself, just as does 22/7, rational pi, but in a more direct fashion. It’s that rare quality of being a circle all by itself. Also, seeing the 2’s and 3’s involved, you will start to appreciate why this was exactly the tonic required to cure the malady of the mixed up 2’s and 3’s. The subtle involvement of 1 reminds, moreover, that unity is the goal.
Of course, a simple dose of 6 will not do the trick, just as 22/7 will not be ringmaster pi all on its own. It’s the symbolism that matters. There is a lot going on in the inner circle, but in the end it’s a perfect thing. And if we could ever see the whole thing at a glance it would probably be just as awe-inspiring as the vision of 6 above.
That 6 does indeed represent the equipment for this specific journey is seen by noticing that 53, the turning, implies 4, in turn implies 22. With the 7 before this gives 22/7 – the journey through TIME. We may read 53 alternatively as the 16th prime, and here again we see the symbol of the inner circle (6) bound to unity.
The 16th prime lodged between 5 and 3 gives 13 and 21, which again describes the 2 large factors. They can also give 11 and 19, and since the 19 joins with the original 16 gives 17, we have reproduced the minor prime set:
[13, 17, 11]
Now we are seriously in business. Having stepped out (12), we have met the inner circle (6), the source of all the provisions. Now with the last step, the two hands are gripped onto the cube, and it’s time to solve. Are we going to reach our destination? Well, 3 is how we describe the set above, and our final factor is 21 digits. That gives 321. This is also a symbol, and it is read as “The Arrival”, as opposed to the 123 of taking the universe on. It is the confirmation that unity is the assured goal.
`
The Platonic solids
Fig 5.8. The 5 regular solids formed the most prized knowledge of the ancient Greeks, and now turn out to be the most vivid illustration of the 5-3 treatment. The second is the mess-up of 2 and 3, and so results in the trap of the box. The 3rd shines hope on the situation by reversing the numbers. The fourth and fifth solids are clearly involved in the process of solving (“53-35”), while the first marks out the road for the journey as 33, explained in later pages.
6 was described as symbol of the inner circle, but there is an alternative reading, and in line with the cube of 8 corners. If the cube is an ideal solid of 8 corners, so too is the double pyramid the ideal solid of 6 corners. The pyramid, on a square rising to a vertex, symbolizes the yearning of the box to the perfection in the dot, the inner circle. But the pyramid we see on the ground is not the complete picture. There is another invisible and upside down pyramid which symbolizes the reach of heaven to the point below, which is also contained in the 6 cornered ideal figure. The star of David is the symbol of both these pyramids together, a shorthand way of saying “as above so below”, and pointing to the subtle workings of the inner circle.
As above, so below
Fig 5.9. AS ABOVE, SO BELOW; the star of David is all about linking heaven to earth, with a pyramid pointing to heaven and another coming down from it.http://thethirty-ninesteps.com/page_5-the_journey_begins.php
Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics\\
}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
}
\date{\today}
\maketitle
{\bf Abstract: } In this announcement, we shall introduce the zero division $z/0=0$. The result is a definite one and it is fundamental in mathematics.
\bigskip
\section{Introduction}
%\label{sect1}
By a natural extension of the fractions
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we, recently, found the surprising result, for any complex number $b$
\begin{equation}
\frac{b}{0}=0,
\end{equation}
incidentally in \cite{s} by the Tikhonov regularization for the Hadamard product inversions for matrices, and we discussed their properties and gave several physical interpretations on the general fractions in \cite{kmsy} for the case of real numbers. The result is a very special case for general fractional functions in \cite{cs}.
The division by zero has a long and mysterious story over the world (see, for example, google site with division by zero) with its physical viewpoints since the document of zero in India on AD 628, however,
Sin-Ei, Takahasi (\cite{taka}) (see also \cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing some full extensions of fractions and by showing the complete characterization for the property (1.2). His result will show that our mathematics says that the result (1.2) should be accepted as a natural one:
\bigskip
{\bf Proposition. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ such that
$$
F (b, a)F (c, d)= F (bc, ad)
$$
for all
$$
a, b, c, d \in {\bf C }
$$
and
$$
F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.
$$
Then, we obtain, for any $b \in {\bf C } $
$$
F (b, 0) = 0.
$$
}
\medskip
\section{What are the fractions $ b/a$?}
For many mathematicians, the division $b/a$ will be considered as the inverse of product;
that is, the fraction
\begin{equation}
\frac{b}{a}
\end{equation}
is defined as the solution of the equation
\begin{equation}
a\cdot x= b.
\end{equation}
The idea and the equation (2.2) show that the division by zero is impossible, with a strong conclusion. Meanwhile, the problem has been a long and old question:
As a typical example of the division by zero, we shall recall the fundamental law by Newton:
\begin{equation}
F = G \frac{m_1 m_2}{r^2}
\end{equation}
for two masses $m_1, m_2$ with a distance $r$ and for a constant $G$. Of course,
\begin{equation}
\lim_{r \to +0} F =\infty,
\end{equation}
however, in our fraction
\begin{equation}
F = G \frac{m_1 m_2}{0} = 0.
\end{equation}
\medskip
Now, we shall introduce an another approach. The division $b/a$ may be defined {\bf independently of the product}. Indeed, in Japan, the division $b/a$ ; $b$ {\bf raru} $a$ ({\bf jozan}) is defined as how many $a$ exists in $b$, this idea comes from subtraction $a$ repeatedly. (Meanwhile, product comes from addition).
In Japanese language for "division", there exists such a concept independently of product.
H. Michiwaki and his 6 years old girl said for the result $ 100/0=0$ that the result is clear, from the meaning of the fractions independently the concept of product and they said:
$100/0=0$ does not mean that $100= 0 \times 0$. Meanwhile, many mathematicians had a confusion for the result.
Her understanding is reasonable and may be acceptable:
$100/2=50 \quad$ will mean that we divide 100 by 2, then each will have 50.
$100/10=10 \quad$ will mean that we divide 100 by10, then each will have 10.
$100/0=0 \quad$ will mean that we do not divide 100, and then nobody will have at all and so 0.
Furthermore, she said then the rest is 100; that is, mathematically;
$$
100 = 0\cdot 0 + 100.
$$
Now, all the mathematicians may accept the division by zero $100/0=0$ with natural feelings as a trivial one?
\medskip
For simplicity, we shall consider the numbers on non-negative real numbers. We wish to define the division (or fraction) $b/a$ following the usual procedure for its calculation, however, we have to take care for the division by zero:
The first principle, for example, for $100/2 $ we shall consider it as follows:
$$
100-2-2-2-,...,-2.
$$
How may times can we subtract $2$? At this case, it is 50 times and so, the fraction is $50$.
The second case, for example, for $3/2$ we shall consider it as follows:
$$
3 - 2 = 1
$$
and the rest (remainder) is $1$, and for the rest $1$, we multiple $10$,
then we consider similarly as follows:
$$
10-2-2-2-2-2=0.
$$
Therefore $10/2=5$ and so we define as follows:
$$
\frac{3}{2} =1 + 0.5 = 1.5.
$$
By these procedures, for $a \ne 0$ we can define the fraction $b/a$, usually. Here we do not need the concept of product. Except the zero division, all the results for fractions are valid and accepted.
Now, we shall consider the zero division, for example, $100/0$. Since
$$
100 - 0 = 100,
$$
that is, by the subtraction $100 - 0$, 100 does not decrease, so we can not say we subtract any from $100$. Therefore, the subtract number should be understood as zero; that is,
$$
\frac{100}{0} = 0.
$$
We can understand this: the division by $0$ means that it does not divide $100$ and so, the result is $0$.
Similarly, we can see that
$$
\frac{0}{0} =0.
$$
As a conclusion, we should define the zero divison as, for any $b$
$$
\frac{b}{0} =0.
$$
See \cite{kmsy} for the details.
\medskip
\section{In complex analysis}
We thus should consider, for any complex number $b$, as (1.2);
that is, for the mapping
\begin{equation}
w = \frac{1}{z},
\end{equation}
the image of $z=0$ is $w=0$. This fact seems to be a curious one in connection with our well-established popular image for the point at infinity on the Riemann sphere.
However, we shall recall the elementary function
\begin{equation}
W(z) = \exp \frac{1}{z}
\end{equation}
$$
= 1 + \frac{1}{1! z} + \frac{1}{2! z^2} + \frac{1}{3! z^3} + \cdot \cdot \cdot .
$$
The function has an essential singularity around the origin. When we consider (1.2), meanwhile, surprisingly enough, we have:
\begin{equation}
W(0) = 1.
\end{equation}
{\bf The point at infinity is not a number} and so we will not be able to consider the function (3.2) at the zero point $z = 0$, meanwhile, we can consider the value $1$ as in (3.3) at the zero point $z = 0$. How do we consider these situations?
In the famous standard textbook on Complex Analysis, L. V. Ahlfors (\cite{ahlfors}) introduced the point at infinity as a number and the Riemann sphere model as well known, however, our interpretation will be suitable as a number. We will not be able to accept the point at infinity as a number.
As a typical result, we can derive the surprising result: {\it At an isolated singular point of an analytic function, it takes a definite value }{\bf with a natural meaning.} As the important applications for this result, the extension formula of functions with analytic parameters may be obtained and singular integrals may be interpretated with the division by zero, naturally (\cite{msty}).
\bigskip
\section{Conclusion}
The division by zero $b/0=0$ is possible and the result is naturally determined, uniquely.
The result does not contradict with the present mathematics - however, in complex analysis, we need only to change a little presentation for the pole; not essentially, because we did not consider the division by zero, essentially.
The common understanding that the division by zero is impossible should be changed with many text books and mathematical science books. The definition of the fractions may be introduced by {\it the method of Michiwaki} in the elementary school, even.
Should we teach the beautiful fact, widely?:
For the elementary graph of the fundamental function
$$
y = f(x) = \frac{1}{x},
$$
$$
f(0) = 0.
$$
The result is applicable widely and will give a new understanding for the universe ({\bf Announcement 166}).
\medskip
If the division by zero $b/0=0$ is not introduced, then it seems that mathematics is incomplete in a sense, and by the intoduction of the division by zero, mathematics will become complete in a sense and perfectly beautiful.
\bigskip
section{Remarks}
For the procedure of the developing of the division by zero and for some general ideas on the division by zero, we presented the following announcements in Japanese:
\medskip
{\bf Announcement 148} (2014.2.12): $100/0=0, 0/0=0$ -- by a natural extension of fractions -- A wish of the God
\medskip
{\bf Announcement 154} (2014.4.22): A new world: division by zero, a curious world, a new idea
\medskip
{\bf Announcement 157} (2014.5.8): We wish to know the idea of the God for the division by zero; why the infinity and zero point are coincident?
\medskip
{\bf Announcement 161} (2014.5.30): Learning from the division by zero, sprits of mathematics and of looking for the truth
\medskip
{\bf Announcement 163} (2014.6.17): The division by zero, an extremely pleasant mathematics - shall we look for the pleasant division by zero: a proposal for a fun club looking for the division by zero.
\medskip
{\bf Announcement 166} (2014.6.29): New general ideas for the universe from the viewpoint of the division by zero
\medskip
{\bf Announcement 171} (2014.7.30): The meanings of product and division -- The division by zero is trivial from the own sense of the division independently of the concept of product
\medskip
{\bf Announcement 176} (2014.8.9): Should be changed the education of the division by zero
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\bibitem{cs}
L. P. Castro and S.Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
\bibitem{kmsy}
S. Koshiba, H. Michiwaki, S. Saitoh and M. Yamane,
An interpretation of the division by zero z/0=0 without the concept of product
(note).
\bibitem{kmsy}
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
Int. J. Appl. Math. Vol. 27, No 2 (2014), pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{msty}
H. Michiwaki, S. Saitoh, M. Takagi and M. Yamada,
A new concept for the point at infinity and the division by zero z/0=0
(note).
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. Vol.4 No.2 (2014), 87-95. http://www.scirp.org/journal/ALAMT/
\bibitem{taka}
S.-E. Takahasi,
{On the identities $100/0=0$ and $ 0/0=0$}
(note).
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operators on the real and complex fields. (submitted)
\end{thebibliography}
\end{document}
アインシュタインも解決できなかった「ゼロで割る」問題
http://matome.naver.jp/odai/2135710882669605901
Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.
https://notevenpast.org/dividing-nothing/
私は数学を信じない。 アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。
I do not feel like an alien in the universe. The more I examine the universe and study the details of its architecture, the more evidence I find that the universe in some sense must have known that we were coming.
– Freeman Dyson, Disturbing the Universe (1979)
T
he outer circle is a box full of air, going about its physics, chemistry and biology and doing nothing. But it is not without a craving to break out. The scientists as a rule fail to notice it, maybe because they are too caught up in the technical aspects of it, being under pressure to convince other technicians like themselves. So it usually falls on the popularizer of science to take notice. Since his job is to translate the thing back into normal language, it suddenly hits him that there is poetry there and no-one to appreciate it. Thus, in his book Cosmic Jackpot (2007), author Paul Davies was made to comment:
Even atheistic scientists will wax lyrical about the scale, the majesty, the harmony, the elegance, the sheer ingenuity of the universe of which they form so small and fragile a part … but [] do not necessarily interpret that as evidence for meaning or purpose in the universe.
Waiting for Godot
Fig 5.1. Poets have ever waxed lyrical on the joy of nature, so that the tragic note struck by Samuel Beckett in his play WAITING FOR GODOT (1953) serves as a timely reminder that the wait continues for the human being to finally arrive in full glory. The only prop in this absurdist play is a bare tree, which may well be symbolizing pi, waiting to be found out. The year tells us about the turning of the cube.
Artists and poets tend to take the opposite path, and being so much in awe of “nature” tend to worship it as pantheists. It’s not right either to shoot the messenger or worship him, the proper course being just to listen to the distress call. There is a malady out there and only we as human beings can put it right.
The tragedy is that it is all divided, has become part of a sweeping desolation known as The Divide. And this has persisted for aeons upon aeons, and the wait has been for Godot – something created in the image of God, and with three special gifts that link back to the divine. The craving of nature is best described as for the unity which brings all things together as one, to fix their divided individuality and revel in connectedness. This is what we mean by the world being rational. Reason connects things together, makes a symphony out of them. What is left behind is the cacophony of the separated notes.
For the most part this inner urge of the world is a passive thing. The music only appears in the mind of the poet who first saw that “host of golden daffodils” on the lakeside. Without the poet that music would never have been heard, and the longing of nature would only have remained a tragic note. So, life itself, even though still physics and chemistry, is only a tremendous outflow of the inner urge towards rational expression that exists in the outer circle. Anybody who has witnesses a sapling grow towards the sun has seen this urge firsthand.
So it is safe to say that life would never have stopped at any point in its evolution before giving rise to the pinnacle of Creation, the human being, in which there finds expression not only the physical and the rational urge to do things, but also the pure reason that leads finally to the goal of unity.
That which gave rise to first chemistry, then biology, was not the outer circle, but the inner one. All the subtlety of the universe is in the inner workings. And the ultimate goal is to join the dot with the infinite sphere beyond. It’s not that the joint circle does not exist, and the atoms are working to make it come into existence. The heavenly plane is implicit in things as they exist right now. As the 17th century German philosopher Gottfried Liebniz put it: the microcosm contains the macrocosm. In plain terms, the entire universe is inside each dot. And because it is, we have that amazing thing called calculus, which both Liebniz and his English arch rival Isaac Newton discovered independently at the same time.
Calculus and dividing zero by zero
Fig 5.2. Isaac Newton (1643-1727) and Gottfried Leibniz (1646-1716) were the culprits, ignoring the first commandment of mathematics not to divide by zero. But they hit gold, because what they mined in the process was the ideal circle.
It came about because they were bold enough to disobey the first commandment of mathematic: THOU SHALT NOT DIVIDE BY ZERO! They did something even worse, because they divided 0 by 0 – worse because utterly meaningless. But they happened to hit gold, because the two zeroes were standing for the inner and outer circles, and between them, entirely unsuspected, was the ideal circle. It was a piece of heaven that they had mined, and modern science continues to do the mining with the result of overflowing abundance of science and technology.
But the technology is only worthwhile if you appreciate the infinite subtlety of the inner circle, and through it the ideal circle. If the latest technological gadget only serves as temporary badge of honor, then I’m afraid you’re not in the frame of the story at all. The story of progress is all about fixing chaos, not self-image, nor creature comforts. Science is about the triumph of reason rather than technology. Calculus is a big part of the story because without it science would not have taken a single step.
The pressure of molecules and modern life
Fig 5.3. The jet-set life really amount to doing a whole lot of nothing, like molecules bumping into the walls of the box.
So the perfect circle is right there, even before the two circles come to join. It’s where all the real things are, the heavenly plane, the shop of ideal things which Socrates describes. It reaches out to us, just as do the prime numbers. Once we thought we could never glimpse this beauty, but we can. It’s through the symphony known as pi. This is what we have set out to do. And as we said before, beauty is not yet beautiful before it is appreciated.
If we know what the box is, we also know that the objective is to get out. To recap, the box has 8 corners, and it comes about by 2 and 3 mingling in the wrong way:
8 = 2+2x3
We will never get out if we allow the laws of physics to percolate over us, because that’s just another way of saying “do nothing”. Eating, drinking and being merry is just physics – that of the instincts. The more we do only this kind of “activity” we are just revving up the gas molecules in the box, which by Charles law increases temperature and pressure. It‘s not strange that the high flyers in the economic ladder are always complaining about pressure.
Instead, relax and think of the box as a room. If it is a room, it is meant to be for sitting down and doing things. Running around will just collect you bruises and bumps. Be assured that you will find desk and chair once you have come out of the frenzy. And once you have indeed settled down, there is going to be the guidelines right on top of the desk, without a doubt.
That’s in due course. But right now we are looking at how pi does the mending job of the messed up 2’s and 3’s. It does it through prime numbers, and first of all 8 of them are places in 8 corners of the box. 5 of them are called the major set [19, 2, 7, 5, 3], and 3 of them make up the minor set [13, 17, 11]. The task is to straighten out the mess. In other words, we will have to get the 2’s and 3’s mingling in the right and natural way, as follows:
7 = 2×2+3 (or 2+2+3)
12 = 2×2×3
The first gives 7 twice, so 14, and with the 12 implies 13. Remember from the last page that this is the blueprint for the reconstruction. One mistaken step is overcome by 3 correct ones. One original sin corrected by the proper use of the 3 royal gifts. 13 is therefore the symbol of regeneration, and recurs regularly in the journey ahead.
7 and 12 are the immediate results, pressed into service straight away. Indeed, after dividing the 36 decimal place of pi by 19, these are the 2 factors that stand out, along with 2 other enormous factors, one of 13 digits and the other of 21:
12×7×8581666150511×229374624355487716489
The 7 wonders and the 12 labours
Fig 5.4. 7 and 12 take on the universe together, but 13 does it all on its own, and justifiably stands for the symbol of regeneration.
Now, since 7 comes about in 2 ways, it is the prime in charge, and the 19 is allowed to combine with the 12. It does so in 2 ways, giving 13 and 21, which is exactly the number of digits of the 2 large primes. This tells us that 7 is in complete control of all the digits of pi, and was marked out as the secret key for good reason.
(19 acts on a number in two ways. It can add one to it, or it can turn the number around. The reason behind this versatility must wait a while.)
The heritage of 7 also speaks for itself. 7 days in a week, 7 colors of the rainbow, 7 continents, all reflecting the 7 heavens. On the reverse side, the 7 deadly sins correspond to the 7 gates of hell.
As for the heritage of 12, the 12 in a dozen, the 12 constellations corresponding to the 12 months in a year, the 12 tribes of Israel and the 12 labors of Hercules.
We ask the bold question, why should 7 take all the pains to introduce 12 with such fanfare? How does it help to clear up the mess? We know that the 8 primes are going to do the real clearing up, and this by solving Rubik’s cube, which is by using 5 primes and 3 primes, and thus applying the “ 53” treatment. It’s a shorthand to describe the solution to Rubik’s cube, and you can see it right there in the major prime set:
[19, 2, 7, 5, 3]
Solving Rubik's cube
Fig 5.5. THE 5-3 TREATMENT! The world record for solving the cube stands at 5.55 seconds, which only underlines the 5-3. It starts with 12, meaning that the 12 edges of the cube start to turn. 7 means that 7 corners may be spun (2) across one fixed (19). There are 43,252,003,274,489,856,000 combinations to the thing. These digits end up as 29. So you will be interested to know that exactly 2 months and 9 days after Waiting for Godot opened in Paris (5 January 1953), it was Pi Day (14 March). Vladimir and Estragon were clearly waiting for pi in the sky.
As you can see, the solution comes right after 7. But take a look at what comes before. It’s 12, because 19 is read as 1. So you can see that 12 comes even before the cube has started turning. Before even this there is the key 7 turning, and take a close look at this. It does so using the philosophy of 2-by-2, described in the last page. This only means that it takes the first two (19, 2) and applies it to the last two (5, 3). As we saw in the last page, 2-by-2 describes how the whole world comes to fit into pi, so, it’s a fundamental thing. But here we are going even further – not only taking the world in, but trying to solve it.
So look closely. In the process of taking the first two, it gives 12. This is then applied to the second pair, 53, which is the turning of the Rubik. So 12 obviously means “start turning”. So generally 12 means START, or, more appropriate in the context of the human journey, the setting out.
It’s the obvious connotation to 12, and also the correct one. ONE-TWO, stepping out, setting out. It is not something that is obviously there, because go back and try to imagine what the void of Genesis was like, because it was just “2”, mind and matter. Only after God pronounced, “Let there be Light!” did 19 come before it to give rise to 12. It is the real beginning, as far as the ultimate salvation is concerned. 2-by-2 may have saved the world from the Great Flood, but it is ONE-TWO that is the real start to the human journey.
But ONE-TWO must be granted status of philosophy in its own right, because its contribution is just as momentous as that of 2-by-2. Remember the void of mind and matter, where not even 1 existed, because there was just no unity to the thing. After light came into the world, something else did in a simultaneous way. This thing is called TIME, which came to complement the void of SPACE. To set out on a journey is to set out in TIME. Physicists tell us that nothing can go faster than the speed of light, and it stands to reason because light and TIME came into existence simultaneously. The speed of light serves as the edge of the universe, and because of it we cannot go back in time either.
Let there be Light!
Fig 5.6. TIME stands still on a particle of LIGHT, because they came into existence at the same moment. The rest is lagging behind, but the light of reason allows us to catch up.
If ONE-TWO gave rise to TIME, it cannot be TIME itself. Instead, look at what comes immediately after – the 2-by-2. That is TIME! It cannot be anything else. That which once served to get the whole universe into pi, now serves as the gap towards that same goal. Life is but a journey through this gap, and in pi we identify the goal, in the sense of being the guide towards the real goal, which is, of course, Heaven.
The expression for TIME turns out to be 22, so at once you get the idea that 22/7 stands for the journey with the help of the secret key. The thing will be established bit by bit, but right now it is a inspired conjecture that 22 is the numerical symbol for TIME.
It must also be contrasted to the philosophy of doing nothing, in which the 2-by-2 has nowhere to go and ends up as simply 2, expressive of mind and matter, the dead-end of the materialist’s world.
Against the doing nothing, what does the turning of the cube actually achieve? Look again at the major prime set:
[19, 2, 7, 5, 3]
The beat of ONE-TWO picks up 7 and 5 straight away, which adds to 12, confirming that the START instruction has taken effect. Also, it inexorably joins with the 3 that comes next to give 123. This is a shorthand for the natural numbers. So, pi, at the onset, is determining the scope of the journey, which is all the numbers, and therefore everything.
There is also going to be a breaking out of the box too. Take another look at the intermediary 75. This implies 6, which is the symbol of the inner circles, as opposed to the outer.
Why 6 should symbolize the inner circle is pretty straightforward. It’s THE perfect number. This is said to happen when all the factors add to give the number again, so that:
6 = 1×2×3 and 1+2+3
The perfection of the number 6
Fig 5.7. One nudge and the triangle becomes a rectangle, 3 becomes 4, SPACE becomes TIME. This is why 6 takes credit of being the perfect number, and the basis of all structure.
Reflect on it further, because it is truly awe-inspiring! You will never see anything like it in any other number. It’s a number that comes back to itself, just as does 22/7, rational pi, but in a more direct fashion. It’s that rare quality of being a circle all by itself. Also, seeing the 2’s and 3’s involved, you will start to appreciate why this was exactly the tonic required to cure the malady of the mixed up 2’s and 3’s. The subtle involvement of 1 reminds, moreover, that unity is the goal.
Of course, a simple dose of 6 will not do the trick, just as 22/7 will not be ringmaster pi all on its own. It’s the symbolism that matters. There is a lot going on in the inner circle, but in the end it’s a perfect thing. And if we could ever see the whole thing at a glance it would probably be just as awe-inspiring as the vision of 6 above.
That 6 does indeed represent the equipment for this specific journey is seen by noticing that 53, the turning, implies 4, in turn implies 22. With the 7 before this gives 22/7 – the journey through TIME. We may read 53 alternatively as the 16th prime, and here again we see the symbol of the inner circle (6) bound to unity.
The 16th prime lodged between 5 and 3 gives 13 and 21, which again describes the 2 large factors. They can also give 11 and 19, and since the 19 joins with the original 16 gives 17, we have reproduced the minor prime set:
[13, 17, 11]
Now we are seriously in business. Having stepped out (12), we have met the inner circle (6), the source of all the provisions. Now with the last step, the two hands are gripped onto the cube, and it’s time to solve. Are we going to reach our destination? Well, 3 is how we describe the set above, and our final factor is 21 digits. That gives 321. This is also a symbol, and it is read as “The Arrival”, as opposed to the 123 of taking the universe on. It is the confirmation that unity is the assured goal.
`
The Platonic solids
Fig 5.8. The 5 regular solids formed the most prized knowledge of the ancient Greeks, and now turn out to be the most vivid illustration of the 5-3 treatment. The second is the mess-up of 2 and 3, and so results in the trap of the box. The 3rd shines hope on the situation by reversing the numbers. The fourth and fifth solids are clearly involved in the process of solving (“53-35”), while the first marks out the road for the journey as 33, explained in later pages.
6 was described as symbol of the inner circle, but there is an alternative reading, and in line with the cube of 8 corners. If the cube is an ideal solid of 8 corners, so too is the double pyramid the ideal solid of 6 corners. The pyramid, on a square rising to a vertex, symbolizes the yearning of the box to the perfection in the dot, the inner circle. But the pyramid we see on the ground is not the complete picture. There is another invisible and upside down pyramid which symbolizes the reach of heaven to the point below, which is also contained in the 6 cornered ideal figure. The star of David is the symbol of both these pyramids together, a shorthand way of saying “as above so below”, and pointing to the subtle workings of the inner circle.
As above, so below
Fig 5.9. AS ABOVE, SO BELOW; the star of David is all about linking heaven to earth, with a pyramid pointing to heaven and another coming down from it.http://thethirty-ninesteps.com/page_5-the_journey_begins.php
Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics\\
}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
}
\date{\today}
\maketitle
{\bf Abstract: } In this announcement, we shall introduce the zero division $z/0=0$. The result is a definite one and it is fundamental in mathematics.
\bigskip
\section{Introduction}
%\label{sect1}
By a natural extension of the fractions
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we, recently, found the surprising result, for any complex number $b$
\begin{equation}
\frac{b}{0}=0,
\end{equation}
incidentally in \cite{s} by the Tikhonov regularization for the Hadamard product inversions for matrices, and we discussed their properties and gave several physical interpretations on the general fractions in \cite{kmsy} for the case of real numbers. The result is a very special case for general fractional functions in \cite{cs}.
The division by zero has a long and mysterious story over the world (see, for example, google site with division by zero) with its physical viewpoints since the document of zero in India on AD 628, however,
Sin-Ei, Takahasi (\cite{taka}) (see also \cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing some full extensions of fractions and by showing the complete characterization for the property (1.2). His result will show that our mathematics says that the result (1.2) should be accepted as a natural one:
\bigskip
{\bf Proposition. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ such that
$$
F (b, a)F (c, d)= F (bc, ad)
$$
for all
$$
a, b, c, d \in {\bf C }
$$
and
$$
F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.
$$
Then, we obtain, for any $b \in {\bf C } $
$$
F (b, 0) = 0.
$$
}
\medskip
\section{What are the fractions $ b/a$?}
For many mathematicians, the division $b/a$ will be considered as the inverse of product;
that is, the fraction
\begin{equation}
\frac{b}{a}
\end{equation}
is defined as the solution of the equation
\begin{equation}
a\cdot x= b.
\end{equation}
The idea and the equation (2.2) show that the division by zero is impossible, with a strong conclusion. Meanwhile, the problem has been a long and old question:
As a typical example of the division by zero, we shall recall the fundamental law by Newton:
\begin{equation}
F = G \frac{m_1 m_2}{r^2}
\end{equation}
for two masses $m_1, m_2$ with a distance $r$ and for a constant $G$. Of course,
\begin{equation}
\lim_{r \to +0} F =\infty,
\end{equation}
however, in our fraction
\begin{equation}
F = G \frac{m_1 m_2}{0} = 0.
\end{equation}
\medskip
Now, we shall introduce an another approach. The division $b/a$ may be defined {\bf independently of the product}. Indeed, in Japan, the division $b/a$ ; $b$ {\bf raru} $a$ ({\bf jozan}) is defined as how many $a$ exists in $b$, this idea comes from subtraction $a$ repeatedly. (Meanwhile, product comes from addition).
In Japanese language for "division", there exists such a concept independently of product.
H. Michiwaki and his 6 years old girl said for the result $ 100/0=0$ that the result is clear, from the meaning of the fractions independently the concept of product and they said:
$100/0=0$ does not mean that $100= 0 \times 0$. Meanwhile, many mathematicians had a confusion for the result.
Her understanding is reasonable and may be acceptable:
$100/2=50 \quad$ will mean that we divide 100 by 2, then each will have 50.
$100/10=10 \quad$ will mean that we divide 100 by10, then each will have 10.
$100/0=0 \quad$ will mean that we do not divide 100, and then nobody will have at all and so 0.
Furthermore, she said then the rest is 100; that is, mathematically;
$$
100 = 0\cdot 0 + 100.
$$
Now, all the mathematicians may accept the division by zero $100/0=0$ with natural feelings as a trivial one?
\medskip
For simplicity, we shall consider the numbers on non-negative real numbers. We wish to define the division (or fraction) $b/a$ following the usual procedure for its calculation, however, we have to take care for the division by zero:
The first principle, for example, for $100/2 $ we shall consider it as follows:
$$
100-2-2-2-,...,-2.
$$
How may times can we subtract $2$? At this case, it is 50 times and so, the fraction is $50$.
The second case, for example, for $3/2$ we shall consider it as follows:
$$
3 - 2 = 1
$$
and the rest (remainder) is $1$, and for the rest $1$, we multiple $10$,
then we consider similarly as follows:
$$
10-2-2-2-2-2=0.
$$
Therefore $10/2=5$ and so we define as follows:
$$
\frac{3}{2} =1 + 0.5 = 1.5.
$$
By these procedures, for $a \ne 0$ we can define the fraction $b/a$, usually. Here we do not need the concept of product. Except the zero division, all the results for fractions are valid and accepted.
Now, we shall consider the zero division, for example, $100/0$. Since
$$
100 - 0 = 100,
$$
that is, by the subtraction $100 - 0$, 100 does not decrease, so we can not say we subtract any from $100$. Therefore, the subtract number should be understood as zero; that is,
$$
\frac{100}{0} = 0.
$$
We can understand this: the division by $0$ means that it does not divide $100$ and so, the result is $0$.
Similarly, we can see that
$$
\frac{0}{0} =0.
$$
As a conclusion, we should define the zero divison as, for any $b$
$$
\frac{b}{0} =0.
$$
See \cite{kmsy} for the details.
\medskip
\section{In complex analysis}
We thus should consider, for any complex number $b$, as (1.2);
that is, for the mapping
\begin{equation}
w = \frac{1}{z},
\end{equation}
the image of $z=0$ is $w=0$. This fact seems to be a curious one in connection with our well-established popular image for the point at infinity on the Riemann sphere.
However, we shall recall the elementary function
\begin{equation}
W(z) = \exp \frac{1}{z}
\end{equation}
$$
= 1 + \frac{1}{1! z} + \frac{1}{2! z^2} + \frac{1}{3! z^3} + \cdot \cdot \cdot .
$$
The function has an essential singularity around the origin. When we consider (1.2), meanwhile, surprisingly enough, we have:
\begin{equation}
W(0) = 1.
\end{equation}
{\bf The point at infinity is not a number} and so we will not be able to consider the function (3.2) at the zero point $z = 0$, meanwhile, we can consider the value $1$ as in (3.3) at the zero point $z = 0$. How do we consider these situations?
In the famous standard textbook on Complex Analysis, L. V. Ahlfors (\cite{ahlfors}) introduced the point at infinity as a number and the Riemann sphere model as well known, however, our interpretation will be suitable as a number. We will not be able to accept the point at infinity as a number.
As a typical result, we can derive the surprising result: {\it At an isolated singular point of an analytic function, it takes a definite value }{\bf with a natural meaning.} As the important applications for this result, the extension formula of functions with analytic parameters may be obtained and singular integrals may be interpretated with the division by zero, naturally (\cite{msty}).
\bigskip
\section{Conclusion}
The division by zero $b/0=0$ is possible and the result is naturally determined, uniquely.
The result does not contradict with the present mathematics - however, in complex analysis, we need only to change a little presentation for the pole; not essentially, because we did not consider the division by zero, essentially.
The common understanding that the division by zero is impossible should be changed with many text books and mathematical science books. The definition of the fractions may be introduced by {\it the method of Michiwaki} in the elementary school, even.
Should we teach the beautiful fact, widely?:
For the elementary graph of the fundamental function
$$
y = f(x) = \frac{1}{x},
$$
$$
f(0) = 0.
$$
The result is applicable widely and will give a new understanding for the universe ({\bf Announcement 166}).
\medskip
If the division by zero $b/0=0$ is not introduced, then it seems that mathematics is incomplete in a sense, and by the intoduction of the division by zero, mathematics will become complete in a sense and perfectly beautiful.
\bigskip
section{Remarks}
For the procedure of the developing of the division by zero and for some general ideas on the division by zero, we presented the following announcements in Japanese:
\medskip
{\bf Announcement 148} (2014.2.12): $100/0=0, 0/0=0$ -- by a natural extension of fractions -- A wish of the God
\medskip
{\bf Announcement 154} (2014.4.22): A new world: division by zero, a curious world, a new idea
\medskip
{\bf Announcement 157} (2014.5.8): We wish to know the idea of the God for the division by zero; why the infinity and zero point are coincident?
\medskip
{\bf Announcement 161} (2014.5.30): Learning from the division by zero, sprits of mathematics and of looking for the truth
\medskip
{\bf Announcement 163} (2014.6.17): The division by zero, an extremely pleasant mathematics - shall we look for the pleasant division by zero: a proposal for a fun club looking for the division by zero.
\medskip
{\bf Announcement 166} (2014.6.29): New general ideas for the universe from the viewpoint of the division by zero
\medskip
{\bf Announcement 171} (2014.7.30): The meanings of product and division -- The division by zero is trivial from the own sense of the division independently of the concept of product
\medskip
{\bf Announcement 176} (2014.8.9): Should be changed the education of the division by zero
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\bibitem{cs}
L. P. Castro and S.Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
\bibitem{kmsy}
S. Koshiba, H. Michiwaki, S. Saitoh and M. Yamane,
An interpretation of the division by zero z/0=0 without the concept of product
(note).
\bibitem{kmsy}
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
Int. J. Appl. Math. Vol. 27, No 2 (2014), pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{msty}
H. Michiwaki, S. Saitoh, M. Takagi and M. Yamada,
A new concept for the point at infinity and the division by zero z/0=0
(note).
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. Vol.4 No.2 (2014), 87-95. http://www.scirp.org/journal/ALAMT/
\bibitem{taka}
S.-E. Takahasi,
{On the identities $100/0=0$ and $ 0/0=0$}
(note).
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operators on the real and complex fields. (submitted)
\end{thebibliography}
\end{document}
アインシュタインも解決できなかった「ゼロで割る」問題
http://matome.naver.jp/odai/2135710882669605901
Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.
https://notevenpast.org/dividing-nothing/
私は数学を信じない。 アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。
AD
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