# Science,English,and math 4th

I will start from now on

(兄の英語スレをよろしく‼)

*新しいレスの受付は終了しました*

- 投稿制限
- ハンドル名必須

【Thank you very much, Mayumichan】

I can start expressing myself in English again here because of Mayumichan who was a kind lady, so I`m grateful to her for it. 有難う、マユミちゃん😊

I`m going to start from the last response of the Science, math and English 3rd. Here we go.

【A marvelous inspiration from the Indian magician】

《Coming across a book on math》

A book on math decided the life of Ramanujan. Its title was basic summary on pure appliance of math written by an English teacher on math. Formulas for taking exam were written in the book.

It`s just that some 6000 theorems and formulae were...

【A marvelous inspiration from the Indian magician】

《Coming across a book on math》

It`s just that some 6000 theorems and formulae were written in the book with every title. It was dull and boring, but Ramanujan was absorbed in thought whether or not the theorems and formulas were right, so he made sure for himself.

It`s just that simple comments were added to every theorem and formula, but there were none of demonstrations for them, so he had to think of original ways so as to make sure those theorems and formulas were right, but it was often some hints led to new discovery of new theorems.

【A marvelous inspiration from the Indian magician】

《Coming across a book on math》

Ramanujan learned to write down on a notebook when discovering new theorems and formulas for himself.

After that those notebook were put in order and collected three notebooks, and have been possessed at library in Madras university, but results of theorems and formulae were written in the notebooks but there were none of demonstrations for them.

He learned math with self education, so one-third of description on the notebook were already discovered before, but rest of the two-thirds were new discoveries.

【A marvelous inspiration form the Indian magician】

《Coming across a book on math》

Its number are 3254 in total, and some of them are the ones without using the latest way no one can demonstrate.

77 years passed after his death, then all the theorems and formulae on the notebooks were demonstrated.

For example, pi. It`s expressed with addition of sum of numbers which continue infinitively. The author said he was going to express it in chapter 5, and this expression belongs to chapter 2.

The formula which Ramanujan thought of approaches the true value of the pi so quickly that we are...

【A marvelous inspiration from the Indian magician】

《Coming across a book on math》

The formula which Ramanujan thought of approaches a true value on the pie so quickly that we are forced to surprised at it.

When calculating first two numbers, its value is as same with that of the eighth decimal place on the true value of the pi.

To tell the truth, I`m not sure of the way of calculating, for I`ve never read the chapter 5, so I can`t have any comment so much, so I`m going to go on express as the book did.

Gottfried Leibniz is called a father on calculus of differential and integral.

【A marvelous inspiration from the Indian magician】

《Coming across a book on math》

He seems to be such a great mathematician that his ability on math was equal to Newton, though I have little knowledge on Newton, so I`m not sure how splendid both of them were on mathematical ability.

Leibniz also thought of formula on the pie. It seems to be well known and it was accurate to the third decimal place.

As a result, Ramanujan was better than Leibniz in relation to the way of calculation on the pi at least, but to my sorrow, I have little knowledge on the significance of the pi. I hate math.

【A marvelous inspiration from the Indian magician】

《Coming across a book on math》

It was not until 1987 after sixty years Ramanujan was dead that the accuracy on the formula of of pi by Ramanujan wasn’t demonstrated. After that the calculation on the pi has extended its number on digit by leaps and bounds.

No one seem to be able to think of the way like him, and when he was asked the resource of his imagination, he answered like the next.

I`m afraid that you can`t believe me, but the goddess who I pray for every day made me think of the way on the formula.

In addition he said an...

【A marvelous inspiration from the Indian magician】

《Coming across a book on math》

In addition he said an equation which doesn`t express any intention from the god is meaningless. Ramanujan discovered lots of theorems and formulas and they have influenced over lots of theories such as elementary particles, the cosmos, polymer chemistry, a study on cancer.

A famous physicist said studying Ramanujan has been to be significant, for his formulas aren’t only beautiful but have both something essential and something deep. It has turned out to be.

I have little knowledge on math, so my remark...

【A marvelous inspiration from the Indian magician】

《Coming across a book on math》

As you know, I have little knowledge on math, so my remark, though it`s just my poor translation from the book, is superficial.

《Half dozen of new theorem every morning》

In 1913 Ramanujan learned to extract some theorems from his notebook and to send first rank of mathematicians letters in England which was a suzerain for India, and one of the mathematicians was Godfrey Hardy who was a central person in English mathematical world then.

Hardy read the letter written unknown formula carefully about for ...

【A marvelous inspiration from the Indian magician】

《Half dozen of theorems every morning》

Hardy read the unknown theorems in letters carefully about for three hours with his colleague and came to an conclusion that the one who they dealt with was a genius without mistake.

Next year, Ramanujan was invited to Cambridge University, and learned to study with Hardy. Later Hardy said Ramanujan arrived with about a half dozen of theorems every morning, and Hardy guided him to add a demonstration to each of new theorems over and over again.

But Ramanujan had never received an orthodox education...

【A marvelous inspiration from the Indian magician】

《A half dozen of theorems every morning》

But Ramanujan had never received an orthodox education on math, so he hardly understood anything on the demonstration itself. Almost all of the theorems which Ramanuj submitted to was shown from the Goddess, and the ones which they looked at with his own eyes.

He was at a loss when asking to demonstrate the theorems.

For example, let’s suppose that there was a one who had never watched UFO, and asking to express what the UFO was. Then expressing the UFO seems to be hard. His situation resembles it.

【Marvelous inspiration from the Indian magician】

《Half dozens of theorems every morning》

Hardy gave up requesting demonstrations to Ramanujan soon, and when he was received the theorems, he regarded them as oracles from the god and adding demonstrations to the theorem was his job, he learned to be businesslike then.

The one which is worthy of mention among their joint research is a formula of approximate value on division number. What is the division number?

It’s...oh, I`m so sleepy that I can’t continue today any more. I have to go to bed, good night.

【Marvelous inspiration from the Indian magician】

《Half dozen of new theorems every morning》

The division number is a total number of the way on addition that a natural number is divided. It includes the number itself.

For example, 4 is expressed with 4, 3＋1、2＋2、2＋1＋1、1＋1＋1＋1, it is done with five ways, so the division number on 4 is five.

The bigger the original number is, the harder the calculation on division becomes, but the equation on an approximation value between Ramanujan and Hardy is proud of high precision.

Is it so marvelous? To tell the truth, I`m not sure of its significance.

【Marvelous magician from the Indian magician】

《A formula which has cleared a modern math》

But the communal operation between Ramanujan and Hardy didn’t continue for a long time.

Ramanujan wasn’t only a vegetarian but didn’t eat anything except for any dish cooked by Brahman. In addition, he was absorbed in cooperative operation with Hardy so much that he went on studying 30 hours without taking a rest and went on sleeping 20 hours after that.

He ruined his health due to the irregular life after three years when he went to England. He returned to India in 1919, but he was dead in the...

【Marvelous inspiration from the Indian magician】

《A formula which has cleared the modern math》

…he was dead in the next year. Then he was just the age of 32.

After returning to India, Ramanujan wrote down something significant on math on a notebook and its fragment happened to be discovered.

Mock theta function 擬テータ関数 and its formulas of which number were six thousand over were written down on the notebook. The discovery was so splendid that it was compared with the discovery on the tenth symphony by Beethoven.

To tell the truth, I have none of knowledge on Beethoven, besides...

【Marvelous inspiration from the Indian magician】

《A formula which has cleared the modern math》

To tell the truth, I`ve never have listened to any symphony on Beethoven, so I`m sure the simile on the author wasn’t suitable.

When the fragment on the notebook was discovered, lots of people thought it had common points with theta function developed by a German mathematician, so it was named like that, but what does it mean on the lots of formulas of the Mock theta function? Large part of them remain mysterious at present.

Theta function has played an important role on the superstring ...

【Marvelous inspiration from the Indian magician】

《A formula which has cleared the modern math》

The theta function has played an important role on superstring theory of modern physics.

As to the Mock theta function, we`ve hoped it`s related to the swelling energy on the universe and the grand unified theory in which trying to express all the power with unification, and lots of mathematicians and physicists have studied it enthusiastically at present.

As to the superstring theory, its easy way of thinking was on the internet, so I`m going to express it.

The theory says ultimate element...

【The theory on superstring】

The theory has said the ultimate element on material isn`t any particle but a string!

As to the scale on the superstring, it is said 10 ー35［m］, on the other hand, as to the scale on an atom is 10 — 10［m］, so the superstring is extremely minimum.

There are several hundreds of kind of elementary particles, and it is said we can express them with a single string.

When the string swings, waves of which number of swings are different happen, and each of waves is equal to each of the element particles, and the vacuum is filled with the strings.

Does it mean that...

【The theory on superstring】

Does it mean that each of elementary particles isn`t independent each other but each of them is connected as they were a single string? As a result, they are influenced each other?

As to its size, I understand it`s so minimum, but I`m not sure its unit, though even if I`ve heard, I`m afraid it won`t ring a bell.

I`m wondering how it is demonstrated, but lots of the people tried to do it at present.

I`m going to express the grand unified theory in relation to easy way of thinking.

【The grand unified theory】

There...

【The grand unified theory】

I`m going to express, being based on an expression in the internet, the elementary particles are expressed as a single string, everything is unified with a single one.

There are four kinds of power in the natural world such as gravitation, electromagnetism, a strong power which works inside an atomic nucleus, and a weak power among the elementary particles.

Except for the gravitation, a theory which unifies other three powers is unified theory which is the theory for everything, and it`s called an ultimate theory which physicists have sought for.

The theory...

【The grand unified theory】

As to the theory which unifies the power, the law of universal gravitation by Newton is well-known. Other ones are the theory on magnetism by Maxwell and electroweak unified theory by Weinberg-Salam is noted.

The theory for everything which physics has sought for is unifying the four kinds of power which consist of the world, and an equation which expresses an origin on the world.

It is no exaggeration to say that the theory on physics has sought the theory for everything from various fields, and a way for thought on the theory is the superstring theory.

【What I`ve thought】

Though I`ve wanted to continue to express, but the battery on my iPad is about to run out.

When it is charged, we can express, but it is harmful for my iPad, so I hate it.

It will take more than a half day to charge completely, so please wait until then. I`m sorry for it.

By the way, I hate math, but learning something is fun, so studying math is necessary. Without studying math, I can`t understand physics, I`m afraid.

【The grand unified theory】

The theory said the world is made up with elementary particles and the elementary particles aren’t points but a thing like string.

As to the elementary particles, quarks like mesons, protons have been discovered. More than thirty years have passed since the theory was proposed, but lots of unclear parts remained.

If the theory is completed, we can understand not only the elementary particles themselves but the situation in which the universe is born and vanish. If trying to express it, unless the time and space is tenth dimension, there is a contradiction.

【The grand unified theory】

Tenth dimension? I can`t imagine it with my brain, so I`m going to return to my topic.

【Darkness on mathematicians who have thought of infinity】

The author said he used to heard a conversation between children in elementary school like the next.

The one asked the other. Do you know what the biggest number is?

The other answered. I know a trillion, but I don`t know bigger number than it.

The one said. Don`t you know it? It`s infinity.

The author said it`s a typical misunderstanding that we are apt to think of the infinity as a very big number, for we can`t ...

【Darkness on mathematicians who thought of infinity】

...for we can`t think of the infinity as the finite number like the trillion.

Other mathematician said like the next with regard to the Infinity.

If trying to substitute the infinity for a very big number, it resembles that there is the sky far away in the horizon, so substituting the sky for the sea which is far away. No one can admit it.

It was in the time of Ancient Greece when we the humankind learned to think on the infinity seriously, but philosophers who were typical mathematicians like Pythagoras, Plato, and Aristotle hated it.

【Darkness on mathematicians who thought of the infinity】

They thought everything in the world was finite, if having brought something infinite into a debate, it caused a confusion, so they hated the infinity.

To tell the truth, if trying to think of the infinity with a way of thinking something finite, there are lots of something mysterious and unreasonable, the author said like that.

For example, let`s suppose there were two group of natural numbers. The one is positive integral numbers alone, we call it A and the other is the positive integral numbers are squared alone, we call it B.

【Darkness on mathematicians who thought of the infinity】

Which have more elements, A or B?

Judging from the way of thinking in the finite world, needless to say it`s the A, for the natural numbers continue like 1,2,3 in the A…, on the other hand, the B is 1,4,9 …, the B has lots of space among the numbers, so the B is a part of the A. It`s an ordinary way of thinking.

But to my surprise, both groups of element on numbers are the same, for they are link to each other one by one like 1 and 1, 2 and 4, 3 and 9… we can classify them as a pair one by one. We call it one to one correspondence.

【Darkness on mathematicians who thought of the infinity】

Galileo who was an Italian pointed out that it`s strange, for it`s just a part, but both of numbers are the same. It`s difference between finite and infinite. It is said it`s a study in which we the humankind have placed essence on infinite for the first time.

Gauss who was a German said if handling the infinity as real numbers, something unreasonable is caused as Galileo said. When using the word of infinite, we should use it, as adverb, changing it into something big infinitively.

Even Galileo and Gauss were not sure what do do...

【Darkness on mathematicians who thought of the infinity】

Even Galileo and Gauss were unsure what to do in regard to the infinity, but it was Georg Cantor who settled down his research on the infinity outright for the first time and created an idea of a set 集合.

The set is gatherings, but in math it means something clear which we can judge whether or not it belongs to the group with some definition like the gathering on positive integral numbers, or the ones on hands when doing janken.

For example, gathering on something beautiful or the one on something delicious, we can`t judge an ...

【Darkness on mathematicians who have thought of the infinity】

...we can`t judge whether or not it includes a group, so we don`t call it the set in math.

The set like the positive integral numbers of squared can link to other set on natural numbers one by one, so we call it a countable set.

The element on number of the countable set is as same as the set on natural numbers, so Cantor adopted a way of saying that they are the same cardinality. The author said the cardinality 濃度 in Japanese, but it`s different from the one like salty water. It`s different from concentration.

For example...

【Darkness on mathematicians who thought of the infinity】

For example, there are two kinds of set like 1.2.3 and a.b.c. Each element on number is three, so two sets on the cardinality is the same.

In short, the cardinality is its number. Why wasn’t adopted a Japanese, 個数 when translating？ if the set is large infinitively, when counting and saying 何個, it struck us as incongruous, the author said.

A set is same the in the natural numbers on cardinality means that each element on the set is linked to that of the set on natural numbers each other. All of them can make a pair.

There is an...

【Darkness on mathematician who thought of the infinity】

When there is an infinitive set and lining up its elements on a line number at a place of integral number like 1.2.3...one by one, there are the ones which we can`t put there exactly, so we are forced to put them on other space except for the natural numbers like 0.5, or route 2.

Then the situation on the number line of the infinite set is different from the one of natural numbers. It seems that the one on the infinitive set is more crowded, so the cardinality was translated into 濃度 in Japanese, the author said.

A set which we can...

【Darkness in which mathematicians have thought the infinity】

《We`ve seen but we don`t believe it》

The set like the squared natural numbers which we can count on the cardinality is named aleph 0. Its element on the number of squared and the one on natural number is the same.

Aleph is the first alphabet in Hebrew.

There is the whole integral numbers including negative integral numbers and rational numbers which we can express with fractional numbers of which denominator and numerator can be expressed with integral numbers. Canton showed a degree on crowded of the two set are also aleph 1.

【Darkness on mathematicians who thought of the infinity】

《We`ve seen but don’t believe it》

Oh! I made a mistake. Cantor showed the two of set were aleph 0.

By the way, I adopted the word of cardinality as the degree of crowded on the set, but it’s hard to understand, so I make it rule to say the degree of crowded from now on.

In the next, Cantor showed the set on irrational number is more crowded than aleph 0, and indicated it’s aleph 1.

In fact, both the number of points included on a number line, the ones on a plane and the ones in the space are the same, the author said like that.

【Darkness on mathematicians who have thought of infinity】

《I’ve seen but I don’t believe》

Cantor also reached the same conclusion, and he was very surprised at it and wrote to his friend and he said he’d seen it but he didn’t believe it. His phrase of this is very well known.

There are simple drawing on a number line and coordinates on the book. Three points like 0.12, 0.3456, and 0.789012 both on the number line and coordinates.

0,12 on the number line is able to show on the coordinates. We can show it at the place 0.1 and 0.2 on the coordinates. If line up the number after the...

【Darkness on mathematicians who thought of the infinity】

《I’ve seen it but don’t believe it》

It means that 0•1 links to the x axis, the horizontal one, and 0•2 links to the y axis, the vertical axis.

When lining up odd number after the decimal point and linking to the x axis and even numbers to the y axis. Then the number of 0.3456 on the number line links 0.35 and 0.46 on the coordinates.

Those links between the one on the number line and other on the plain is made up. Though both of the points on the number line and the plain are infinitive, but both number of the elements are the same.

【Darkness to mathematicians who thought of the infinity】

《I’ve seen it, but don’t believe it》

I’m not good at expressing something, so I’m afraid whether or not I can make myself understand, but it can’t be helped. I can’t do anything more than that. Please pardon me for the poor expression.

The author continued like the next.

He’d had a recognition that points gather and they become a line, and the line gathers and they become the plain before, so he couldn’t believe that the number of points on a straight line and those on a plain were the same easily, but it was clear that he...

【Darkness to mathematicians who have thought of the infinity】

《I’ve seen it but don’t believe it》

...but it was clear that he couldn’t refute. Then he was aware that common knowledge in the finite world isn’t accepted in the infinite world.

Besides, the degree on crowded isn’t limited to aleph 0 and aleph 1 alone. It’s limitless. Infinity isn’t a very big number but the general term on numberless numbers which have boundless expansion we can’t imagine at all in the finite world.

The mathematician said he saw it but didn’t believe it. I’ve also sympathized with him. Math is mysterious.

【Darkness to mathematicians who have thought of the infinity】

《Confrontation between the great mathematician and his disciple》

As the concept on the set was born the infinity became an object for an argument scientifically for the first time then.

Cantor was a pioneer who opened the gate for a mysterious land like the Garden of Eden where theologians alone were permitted to go into. He opened the gate of the mystery land on the infinity for other mathematicians.

But his great achievement by the marvelous genius was so advanced that it wasn’t evaluated respectably when he was alive.

【Darkness to mathematicians who have thought of the infinity】

《Confrontation between a great mathematician and his disciple》

Far from it! The idea which Cantor thought of was often criticized and attacked from other mathematicians, especially the one who criticized him severely was Leopoldo Kronecker who trained Cantor.

Kronecker was a great mathematician in Germany and appeared in the book several times, the author said like that, but to my sorrow, I don’t remember him.

He used to treat Cantor with affection before and when finding his job at an university, he lent a hand then, but ...

【Darkness to mathematicians who thought of the infinity】

《Confrontation between a great mathematician and his disciple》

...but when Cantor started to study irrational number and the infinity, Kronecker criticized Cantor who used to be his favorite pupil. He called his disciple the one who was harmful to others. He began to oppose his pupil.

He thought numbers which weren’t able to express with integral number and the ones which weren’t finite weren’t worthy of thinking. He asserted that the one which wasn’t expressed with fraction of integral number and irregular number continued after...

【Darkness to mathematicians who thought of the infinity】

《Confrontation between a great mathematician and his disciple》

Kronecker asserted that the one which wasn’t able to express with fraction of integral number and a number continued after the decimal point irregularly was nonsense.

Students in junior high school learn the irrational number at present, but the great mathematician who led the world on math didn’t recognize it 150 years before. We are forced to be surprised, but handling something endless and the one which we can’t see its end mathematically is hard and courageous.

【Darkness to mathematicians who thought of the infinity】

《After being sick spiritually and in his later years...》

Cantor was grieved over being lacking of understanding and obstinate personal remarks from his Kronecker who used to be his teacher.

There was one more thing from which Cantor suffered in the latter half of his life as if he had been pursed and attacked. It was a demonstration on hypothesis of continuum 連続体.

It means that there aren’t any other degree on crowded number between aleph 0 and aleph 1.

The continuum is a set on whole real numbers including rational and ...

【Darkness to mathematicians who thought of the infinity】

《As being sick spiritually in his late years...》

Continuum is a set of the whole real number including rational number and irrational number, and a number line is filled with the real number.

The hypothesis on the continuum is a hypothesis that an infinite set which has an element that it is more than natural number and is less than real number doesn’t exist.

No one can prove nor show any evidence against the hypothesis. It has been demonstrated clearly at present, but Cantor believed he could and challenged over and over again...

【Darkness to mathematicians who thought of the infinity】

《As being sick spiritually in his late years...》

...but needless to say he failed to do it, it’s natural, for no one can demonstrate it, but Cantor lost his confidence in his own ability as mathematician because of the setback.

Repeated criticism from Kronecker and the failure on demonstration of the hypothesis on the continuum. Two of the agony spread dark shadow over his heart and he fell sick spiritually at last.

In addition he suddenly started to be absorbed in a study on English history and English literature in his late years.

【Darkness to mathematicians who thought of the infinity】

《As being sick spiritually in his late years...》

His theme was demonstrating a theory that the true author on dramas written by Shakespeare was Francis Bacon.

A German mathematician said no one can drive out of us from the paradise which Cantor opened for us, but the infinite world where Cantor reached with wings made of intellect and imagination may not have been any paradise but may have been a hell where devils live for him.

【A perfectionist who demonstrated an imperfect theorem】

《If saying I’m a liar, is it true?》

Do you...

【A perfectionist who demonstrated an imperfect theorem】

《If saying I’m a liar, is is true?》

Do you know a paradox on self-reliance?

There is a premise or theory which look right, but a conclusion which we can’t accept easily is caused from the premise and theory, we call the conclusion the paradox.

As an example on the paradox of self-reliance, an well known is a remark that I’m a liar. It seems an ordinary one, but thinking of it carefully, the remark is a contradiction.

If saying I’m a liar, it means that the remark that I’m a liar itself is a lie, so in short I’m honest, but ...

【A perfectionist who demonstrated an imperfect theorem】

《If saying I’m a liar, is it true?》

...but the premise started from the remark that I’m a liar, but there is the conclusion that I’m honest. It’s a contradiction. Then let’s suppose that the remark was a lie. Then what’s happened?

I’m honest, so the remark that I’m a liar is also true, so I’m a liar. As a result, it’s also a contradiction.

In short, as to the remark that I’m a liar, neither we can say it’s true nor a lie. In general, if a sentence has a construction that it’s false, we can’t judge whether it’s true or not.

【A perfectionist who demonstrated an imperfect theorem】

《If saying I’m a liar, is it true?》

We call it the paradox on self-reference.

A rule saying there are no rules without exceptions, or a bill on the wall saying to prohibit to put any bill on the war are well known as the paradox on self-reliance.

By the way how about math?

There is a proposition which is the matter we can judge whether it’s true or not objectively in math. Is there any proposition which we can’t judge whether or not it’s true?

There are two of things alone in math. The one which is demonstrated it’s true, and...

【A perfectionist who demonstrated an imperfect theorem】

《If saying I’m a liar, is it true?》

...or the one which is demonstrated it’s false. Lots of people recognize it in common.

Needless to say, there are some propositions which we can’t judge whether or not it’s true, but it’s because of us the humankind’s inability, so they will be classified either of truth or falsehood. Plenty of people seems to have thought like that, don’t they?

But there is a person who made it clear that there is a proposition in which neither we can demonstrate whether or not it’s true in math. He is Kurt Godel.

【A perfectionist who demonstrated an imperfect theorem】

《If saying I’m a liar, is it true?》

Kurt Godel was a Czech.

Mathematic world used to be in a chaotic state at a time from the latter in the 19th century to the beginning in the 20th century.

Geometry which was born an interest from a figure and developed into technical skill for civil engineering and navigating ships.

Algebra which started to seek for something unknown and developed into theories on equations.

Differential and integral calculus which was necessary so as to ask for a value on an area of a figure and to solve...

【A perfectionist who demonstrated an imperfect theorem】

《If saying I’m a liar, is it true?》

Differential and integral calculus which is necessary in order to ask for the value on an area and to solve physical phenomena.

Statistics which started to rule over a nation.

The thorny on probability which was born in the pursuit of benefits when gambling.

They were born one after another with no connection, and grew up respectively as if each field had lived together in a building named math.

Then a movement of trying to rebuild the mathematic world with the concept of set established by Cantor

【A perfectionist who demonstrated an imperfect theorem】

《The paradox on a barber》

The theory on set is the origin from modern math, the feeling was heightened up then, and Bertrand Russel who was much talked as the greatest logician since Aristotle was aware that what is called the paradox by Russel on the set. Its example on the next paradox of a barber is well known.

There is a single barber shop in a town, and the barber shop is managed by a single barber. The barber was imposed a rule on himself.

If there are the ones who don’t shave for themselves in the town, he will shave for them.

【A perfectionist who demonstrated an imperfect theorem】

《A paradox on a barber》

If there are the ones who shave for themselves in the town, the barber don’t shave for them.

By the way, who shaved for the barber?

As he said he won’t shave for the ones who shave for themselves in the town, if he shaves for himself, his remark is inconsistent.

On the other hand, as he said he will shave for the ones who don’t shave for themselves in the town, his action is contradictory to his remark.

The barber imposed the rule on himself, but the rule prevents himself from shaving, but he can’t choose...

【A perfectionist who demonstrated an imperfect theorem】

《The paradox on a barber》

...but he can’t choose not to shave for himself.

So Russel wrote Principia Mathematica made up with three volumes with Alfred North Whitehead who was his teacher so as to avoid the paradox.

The book is based on the theory on set and tried to unify the whole math which we the humankind acquired until then and to demonstrate it with signs alone. It was written with the magnificent concept.

Almost all the part of the first volume was used so as to define 1 alone. Needless to say, after that its pace increased.

【A perfectionist who demonstrated an imperfect theorem】

《The paradox on a barber》

Then high level mathematical concept appeared one after another, but its last part came to an end with a description that from this place we can seek a value with the same way. It was unfinished and the description seemed to be irresponsible and apathetic, the author said like that.

《if number of stamps are below five, its gift...》

Though having described that demonstrating with signs alone simply, it was one of great feats by Russell and his teacher. Its base is a truth table 真偽表.

For example, let’s...

【A perfectionist who demonstrated an imperfect theorem】

《If the number of stamps are below five, its gift...》

Let’s suppose there was a card on which has several collected stamps, and there is a promise that if the stamps are more than five, its owner gets a present.

The condition of more than collected five stamps is P and other condition of being able to get the present is Q.

I’m going to think over whether or not P and Q are true, and if P stands up, then Q does, then what is its relation between true and false?

1 If the collected stamps are more than five, the owner can get a present.

【A perfectionist who demonstrated an imperfect theorem】

《If the collected stamps are below five, its present...》

1 When the Q is true as the proposition which means the promise shows, the collected stamps are more than five, so P is true too, as a result P＝ Q stands up, so P ＝Q is true.

2 Though the collected stamps are more than five, the owner doesn’t get the present. Then P is true, but Q is against the proposition, so P ＝Q doesn’t stand up, so P ＝ Q isn’t true.

3 As the collected stamps are below five, the owner doesn’t get the present, but it is showed in the promise, so P ＝ Q is true

【A perfectionist who demonstrated an imperfect theorem】

《If the collected stamps are below five, the present...》

3 So, P ＝ Q stands up.

These proposition struck no one as incongruous until now, but as to the last 4, everyone is hard to accept it at first.

4 Though the collected stamp are below five, the owner gets the present. Then the owner must have been indignant at it, and it would said it shouldn’t have gathered the stamp so hard, but the first proposition said nothing in relation to the time when the stamps were below five. It doesn’t say that the owner can’t get the present.

【A perfectionist who demonstrated an imperfect theorem】

《If the collected stamps are below five, the present...》

4 As a result, P ＝ Q stands up, the author said like that, and continued.

It seems to be quibbling, but sometimes presents are left so much that the sponsor delivered presents for the guests of which number on stamps were below five, but the first promise doesn’t prohibit it.

If looking at the both of 3 and 4, the promise said nothing when the stamps were below five, so even if the owner of the stamps whether or not got the present, the proposition, it means the promise, is true

【A perfectionist who demonstrated an imperfect theorem】

《Semantics and syntax》

Even if it’s true mathematically, if there are lots of quibbles like that, I’m afraid that a riot may have been caused, though the presents were delivered, so it seems to be all right.

A way of thinking over the meaning of each condition and judging whether or not the proposition is true. It’s called a semantics 意味論的方法.

But if adopting the semantics, a meaning of words which we usually everyday life apt to come into a conclusion when judging something. Sometimes it’s unclear and shows us equivocal situation...

【A perfectionist who demonstrated an imperfect theorem】

《Semantics and syntax》

It’s sometimes unclear and shows us an equivocal situation we can’t judge whether or not it’s true, but as math aims at being perfect, so it isn’t very good.

Creating signs which we don’t use everyday life newly and showing a proposition with the sighs alone, the idea was born. The truth table is its first step. It seems to be a kind of a chart. What kind of chart? I’m sure I have to express it, but I feel sleepy, so I’m going to give it up tonight. Good night, everyone.

【A perfectionist who demonstrated an imperfect theorem】

《Semantics and syntax》

If using the truth table, we can judge various propositions mechanically. Soon, relying on symbols alone, a way of making ahead with a demonstration was contrived as if we had calculated. We call it a syntax.

In the syntax, without thinking of its meaning at all, we demonstrate formally according to the fixed rule. Once having fixed its rule, as the demonstration proceeds automatically we should be careful of an axiom, it means a premise when starting.

If the axiom was in the wrong, an conclusion which we get...

【A perfectionist who demonstrated an imperfect theorem】

《Semantics and syntax》

If the axiom includes something wrong, a conclusion from which we get will be in the wrong automatically.

The syntax on demonstration for math has been arranged by an English, George Boole, and a German Frege Gottlob and has been completed through the Principia Mathematica by Russell and Whitehead.

《The truth table》

I expressed that if the collected stamps are more than five, its owner can get a present before, so I’m going to show it with the truth table.

Then the truth table has twenty square. A vertical ...

【A perfectionist who demonstrated an imperfect theorem】

《The truth table》

I’m afraid it’s just a recitation of my expression before. There should have been a better way, but it doesn’t occur to me. I’m sorry for it.

The vertical row has five. A place of the upper part on the rightmost is a blank column, and from the second place, 1,2,3, and 4.

1 is the condition that the collected stamp is more than five, the owner can get a present.

2 is the one that the collected stamp is more than five, but the owner can’t get the present.

3 is the one that the collected stamp is below five, so the...

【A perfectionist who demonstrated an imperfect theorem】

《The truth table》

3 is the one that there is below five on the collected stamp, and the owner couldn’t get the present.

4 is the one that there is below five on the collected stamp, but the owner could get the present.

Each of those four conditions connects with two elements and we can reach a conclusion. Whether it’s the truth or not.

One of the elements is P that the collect stamp is more than five, and the second is Q that being able to get the present. The third element its conclusion.

For example, both of the P and Q are the...

【A perfectionist who demonstrated an imperfect theorem】

《The truth table》

For example, both P and Q are the truth, so the conclusion is the truth. It means that the owner has more than five collected stamps, so it can get the present. Then three of the columns which 1 links to P, Q, and conclusion are all truth.

When making ahead with other connection, we can get each conclusion which I expressed before.

Thus we can understand its conclusion mechanically. It’s the truth table. As you know, I’m not good at expressing something, so I’m afraid whether or not I could make myself understood.

【A perfectionist who demonstrated an imperfect theorem】

《The imperfect theorem by Godel》

The demonstration on the imperfect theorem by Godel seems to be very hard. I can express what the author said but it doesn’t always mean that I can understand what it means.

The author said he was going to omit its process and to introduce us the conclusion which Godel demonstrated in the imperfect theorem. Needless to say, I agree with the author.

Considering the formal syntax in which we treat natural numbers, a proposition exists in which no one can demonstrate, though when handling other things...

【A perfectionist who demonstrated an imperfect theorem】

《The imperfect theorem by Godel》

...though when handling other things, the proposition may have been existed.

The author said a reader who is quick to pick up on things may be aware of it, it’s the same structure as the paradox on self reference which the author showed us, I’m a liar, so neither we can affirm nor deny it, though to my sorrow, I can’t be conscious of it.

As to the imperfect theorem by Godel, its title has been out of control on account of shocking its title, the title alone has been active, separating from its ....

【A perfectionist who demonstrated the imperfect theorem】

《The imperfect theorem by Godel》

...separating from its original meaning, some people said that it was demonstrated that the math was in the wrong, and others said we could see the limit on us the humankind. It has been done frequently with a plausible way, but it’s a misunderstanding altogether.

The imperfection theorem means that there is a proposition which neither we can affirm nor deny, but it doesn’t mean that math has failed nor a thing which we thought to be right was in the wrong.

《The later years on the perfectionist》

【The perfectionist who demonstrated the imperfect theorem】

《Later years of the perfectionist》

Godel showed an imperfection on a formal demonstration with the syntax, and after that, it prompted a remarkable progress on not only math, logic but computational science, for a computer understands an order and repeating judgement is formal, especially it is said that it has influenced on Alan Turing who changed the part of on the base of computer into a theory and was called a father of artificial father.

To my sorrow, it’s too hard to understand for me.

Godel defected to America with his...

【The perfectionist who demonstrated the imperfect theorem】

《Later years of the perfectionist》

Godel defected to America with his wife so as to avoid a persecution on the Jews from Nazi in his mid thirties. Their guarantor was Einstein.

There seemed to be an oral examination on American constitution in order to get a citizenship in America.

On the day of the exam, Godel said like the next others around him.

When studying the constitution, it turns out that America has a possibility in itself that it may change into a despotic nation lawfully.

Einstein was forced to panic.

Godel was...

【The perfectionist who demonstrated the imperfect theorem】

《Later years of the perfectionist》

Godel was a perfectionist and a nervous temperament, especially in his later years, its tendency became strong.

Except meals which his wife cooked, he tried not to eat anything, and he was afraid of being assassinated by poisonous gas that he left all the windows open even if in winter.

At last when his wife was in hospital, he starved to death because he didn’t eat anything. Then his weight was no more than 29.5 kg.

【Beauty in math exists in internal pleasure】

《If math isn’t beautiful...》

【Beauty in math exists in its internal pleasure】

《If math isn’t beautiful...》

Tchaikovsky said if math isn’t beautiful, math itself wasn’t born perhaps. Except for something beautiful, is there other thing which the greatest geniuses have been in math attracted?

Do you think whether or not math is beautiful? Do you think you are sure of it? Or you don’t think so? The author said he’s thought math is beautiful, but I don’t think it at all.

When looking up the word on the beauty in 広辞苑, it says it stimulates perception, sensation, and feeling and internal pleasure is caused. What is ...?

【Beauty in math exists in internal pleasure】

《If math isn’t beautiful...》

What is caused internal pleasure from math?

The author said his theory is that it depends greatly on the four next natures of math.

1 symmetry

2 rationality

3 elements of surprise

4 being concise

《Symmetry》

If Tokyo tower and Mt.Fuji are left-right asymmetric, lots of the people have never been fascinated like that.

In math, as to a figure, when folding from a point of line and lay two of them, two of the figures fit perfectly, we call it a line symmetry.

When being in a point in its center, and it turns...

【Something beautiful in math exists in internal pleasure】

《Symmetry》

When being a point in the center and rotating 180 degree, if the figure fits perfectly with the same position before it rotated, we call the figure a point symmetry.

In a numerical formula, if exchanging an order of the number, it reaches the same value, we call it a symmetric expression.

When being able to recognize something symmetric in the figure or in the numeral expression, I find it beautiful and it natural. The author said like that.

《Rationality》

Have you ever heard of saying that when a swallow flies low...

【Something beautiful in math exists in internal pleasure】

《Rationality》

Have you ever heard a saying that when a swallow flies low, it will rain?

Foreseeing a weather from a natural phenomenon close to us beforehand like that is called a weather lore.

It seems it’s caused from an experience, but it has a ground.

When a low pressure zone which make it rain approaches, atmosphere which contains lots of moisture flows into near the ground, so insects which are prey to the swallow can’t fly high because their wings are heavy with humid atmosphere.

As a result, the swallows which try to...

【Something beautiful in math exists in internal pleasure】

《Rationality》

As a result, the swallow which tries to eat the insects also learns to fly in a low altitude.

When listening to the rational explanation, I find it convincing, at the same time I frequently find it pleasant. The author said and continued.

I understand that it doesn’t always mean that everyone has the same sensation, but a way of logical thought which was formulated systematically through the Elements of Euclid in Ancient Greece has been accepted and developed ceaselessly until the modern era, for it means that not...

【Something beautiful in math exists in internal pleasure】

《Rationality》

...for it means that not a few people have found that the rationality is something pleasant.

Except for the conviction, there is other reason why I’m fond of the rationality. It’s that even if its course is different, it reaches the same conclusion.

For example, there are two right angle triangles. They are the same figure altogether. The right angle is between two sides. The one’s length is 4 and the other is 3, and the oblique side was 5. There is a vertical line from the oblique side and it reaches the triangle...

【Something beautiful in math exists in internal pleasure】

《Rationality》

A question is asking the value on the length of the oblique side.

I’ve said there are two of the same right angle triangles. The one is the side of which length is 4 is on the ground if there is the ground there.

On the other hand, as to the other right angle, its oblique side of which length is 5 is on the ground if there is the ground there.

Needless to say, two of the value on the area of the right triangles are the same, but two ways of asking the value are expressed.

The one is 4×3÷2, and the other is 5 × the...

【Something beautiful in math exists in internal pleasure】

《Rationality》

3×4÷2＝5 which is the length of the oblique side ×...

Oh! I’ve forgotten to express one more thing. There is the vertical line from the oblique line to the right angle.

When the base is the oblique line, the height of the right angle triangle is equal the straight line from the oblique line to the right triangle, so I call its length Z.

As a result, as to the area on the value of the two right angle triangles is the next one.

3×4÷2＝5×Z÷2, so Z ＝two and two-fifths. We paid attention for its area this...

【Something beautiful in math exists in internal pleasure】

《Rationality》

We paid attention for its area this time, but in the next we do the similar two triangles.

When the length of the base is 4 on the ground if there is the ground, the triangle is expressed with A,B,C, and D.

The base is AB, the oblique side is AC, and the last one is CB. There is a vertical line from the oblique side to the right angle. As to its starting point from the oblique side, it’s D.

Then the two of the triangles, △ ABC and △ADB are similar, so AC：CB＝AB：BD, 5：3＝4：Z, so 5Z ＝12, Z＝two and two-fifths.

【Something beautiful in math exists in internal pleasure】

《Rationality》

By the way, as you know, I hate math and I’ve never studied math so much, as a result, to my sorrow, I have little knowledge on math, so I was forced to be worried about. Two of the triangles, △ ABC and △ABD are similar, why?

But two of them are right angle triangles, and ∠ DCB and ∠ACB are the same angle. It means each of the two angles are the same, so the last angles of each of them are also the same, as a result, two of the triangles are similar.

The author continued.

In other words, being rational means that...

【Something beautiful in math exists in internal pleasure】

《Rationality》

In other words, being rational means everyone can reach the same conclusion as long as each of them adopts a logical thought, and the way of choosing is free as long as it’s logical, so I’m happy for it. The author said like that.

For example, let’s suppose that you learned to go to a cooking school and its instructor was irrational one, and the instructor forced you to adopt everything as the way the instructor was pleased.

As to a way of washing vegetable, the way of cutting it, measuring way of amount of the food...

【Something beautiful in math exists in internal pleasure】

《Rationality》

...and an order of putting in a seasoning. The instructor showed you each way minutely. It didn’t allow to adopt other way at all. Even if your way was different a little, the instructor was enraged at it.

In addition, even if the same cooking, in the next time, the contents of its direction was different. All of the students attending the cooking school would be intolerable for it. They had always to be sensitive to the mood of the instructor, so the cooking school was rigid and stressful. No one can enjoy there.

【Something beautiful in math exists in internal pleasure】

《Rationality》

If the instructor in the cooking school is rational, it will allow you to adopt various ways. In fact, a course of cooking delicious dish shouldn’t be the only one alone. If by any chance, you the students may come up with a good idea which makes the dish better than a recipe which the teacher prepared for.

If the instructor is rational, it’ll be willing to accept the device, and praise you. If you can go to the cooking school like that, you’ll have a fun. Every time you go to the cooking school, you will look ...

【Something beautiful in math exists in internal pleasure】

《Rationality》

Every time you go to the cooking school, you will look forward to the class.

Being rational is connected with freedom on thinking, so the internal pleasure is caused.

《Elements of surprising》

When learning math, we often discover unexpected fact.

For example, if going on adding odd number, 1 ＋ 3 ＋ 5 ＋ 7...wherever stopping it, its value is a square measure. It seems that few people find it natural at once.

It’s expressed with an illustration and simple equations, and I want to express in English, but I find it hard.

【Something beautiful in math exists in internal pleasure】

《Elements of surprising》

How is the square measure made up? The book said like the next.

It’s made up with L, but the direction of the L is turned. When adding the odd number, the size of the L is bigger a little.

The first odd number is 1, and 3 is added. The L is turned and is made up with three elements, and the square measure is made up with the four elements.

After that, 5 is added, and other L is added like the same way, but the size of L is bigger a little this time. It’s made up with five elements, and the other square...

【Something beautiful in math exists in internal pleasure】

《Elements of surprise》

...then the other square measure made up with nine elements.

As a result, the figure is always the square measure, and its elements are square measure as well.

At first, when we can’t find a thing by intuition, but we’re convinced with a logical explanation, then we’re surprised and impressed with it. It’s one of internal pleasure.

On the other hand, if a thing which we find it natural is harped from the first to the end, we find it boring, at least we can’t find it anything pleasant there.

【Something beautiful in math exists in internal pleasure】

《Surprise of elements》

When a mathematical thought made us find a fact unexpectedly and it caused the internal pleasure for us, so we may find something beautiful. It may be natural.

《Simplicity》

The biggest reason why we find the math is beautiful may be something simple.

It seems that there is a phrase that less is more. The phrase has been used from a long ago in the world of design. Its origin is an English poet used it in his work.

It means that when designing simple is better than decorating too much. It resembles a...

【Something beautiful in math exists in internal pleasure】

《Simplicity》

It resembles a phrase, simple is best.

Leonardo da Vinci also said being simple is a refinement extremely.

When we need to follow a transient fashion, we sometimes should add various elements and decorate, but if seeking common beauty beyond the times, simplicity is indispensable.

For example, Golden Gate Bridge in San Francisco was built about eighty years ago, but we call it the most beautiful bridge in the world even at present. It's so simple that if something is removed from the bridge, it doesn’t exist as bridge.

【Something beautiful in math exists in internal pleasure】

《Simplicity》

We’re forced to feel like that, for kinds of decoration are removed from the bridge.

A Japanese famous designer said it’s aesthetics made up with subtraction.

By the way, I’ve never seen the Golden Gate Bridge, so I’m not sure whether or not it’s beautiful.

At first even if someone famous said something splendid, it doesn’t always mean that I can sympathy with it, but if saying like that, this thread can’t go ahead, but I’m forced to say it.

Not only mathematicians but scientists have wanted to give a clear...

【Something beautiful in math exists in internal pleasure】

《Simplicity》

Not only mathematicians but scientists have wanted to give clear explanation to the common truth on the cosmos, and it’s the most original motivation, and they have thought that the common truth is simple and beautiful.

In fact lots of the theorems and formulas which the mathematicians found in the past are simple. The author said he was going to show one of them.

When counting the number on vertex, edge, and face of a cube, taking from its initial and each of numbers is expressed with V, E, F, then the next ..

【Something beautiful in math exists in internal pleasure】

《Simplicity》

...then the next simple equation stands up. V— E＋F＝2. We call it the theorem on polyhedron by Euler.

There are illustration of a tetrahedron, a rectangular, and a pentagonal prism on the book, and everyone can see each number of vertex, edge, and face, so the equation holds water.

Thus it’s seemingly complicated, but simple in its essence. It frequently happens in math. The simplicity and the common truth are influenced each other, we find it beautiful when seeing it.

The author continued.

A feeling of wishing to...

【Something beautiful in math exists in internal pleasure】

《Simplicity》

A feeling of wishing to be mathematic resembles the one of wishing to be beautiful. If wishing to be beautiful, we need a heart in which we feel something beautiful, so wishing to be mathematic, we need to polish a sensitivity in which we can feel that being mathematic is fantastic and beautiful.

It doesn’t always mean that I want to be mathematic, but I find math is interesting at least.

【Pythagoras and a secret art on numbers】

《Numbers have each character?》

Everyone has number which each of them feel a sense of ...

【Pythagoras and a secret art on numbers】

《Numbers have each character》

Everyone has numbers which each of them has a feeling of closeness. For example, it’s the number of birthday, the other which we’re fond of from the long ago, or another of the uniform number on a baseball player who you like.

The author said he likes 8. When the number is expressed with a Chinese character, it fans our downward, so it’s auspicious, but he was absorbed in playing baseball in childhood, his favorite professional baseball player was Tatsunori Hara, and his uniform number was 8, so he learned to like 8.

【Pythagoras and a secret art on numbers】

《Numbers have each characters?》

7 which positions in front of 8, and is frequently said lucky and is popular among us, but the author said he has an image on 7 that it keeps itself above all the vulgarity around it, and gives off aura which doesn’t let anyone get close to it, so he doesn’t have a feeling of closeness to 7.

9 which is after 8, and doesn’t always make friends with everyone, but is very reliable among its pals, so if being in difficulty, the author feel like being helped from the number.

The author said he has an imagination on...

【Pythagoras and a secret art on numbers】

《Numbers have each of characters》

The author said he has an image on various numbers like that. He doesn’t established it by force but it was done automatically.

He said he was afraid that everyone may think he was eccentric, but he said the ones who are good at numbers have a natural tendency in common like that, though it doesn’t always mean that he listened to each of them and made sure of it.

Even if each of them has an original image for each number, each of them has different image to 7,8,9.

The ones who love music can distinguish between...

【Pythagoras and a secret art on numbers】

《Numbers have each characters》

The ones who love music can distinguish difference between good and bad musical performance, and the ones who good at cooking recognize difference the right amount of salt between good and bad one, or difference the right heat between good one and right one.

The ones who love math are sensitive to difference between characters on numbers like that.

《A discovery by Pythagoras》

The prosperity on Pythagoras and his disciples in the Ancient Greece started from the time when walking,Pythagoras was aware something strange.

【Pythagoras and a secret art on numbers】

《A discovery by Pythagoras》

When a smith stroke the iron, there was sound of iron, but the ones reverberated good and the others didn’t like that. Pythagoras and his disciples visited the smith and examined why there was difference between the sound.

It turned out that difference between the weight on the hammer used by each of smith caused the phenomenon, and after minute research, unexpected fact was discovered.

When the sound was reverberated good, the proportion on the weight of the hammer was simple proportion of the integral number, 2：1 or 4：3

【Pythagoras and a secret art on numbers】

《A discovery by Pythagoras》

We easily imagine that Pythagoras and his disciples were surprised and impressed with the fact very much. A musical interval which means difference two sounds between high and low. We the human being thought it was beautiful automatically, and it was expressed with simple integral numbers.

They had a feeling as if they had discovered a trick made by the God, didn’t they? If they thought the numbers which the God used for the trick, it means the integral number, was the words from the God, it may be a matter of course.

【Pythagoras and a secret art on numbers】

《Adding the numbers on the date of its birth and telling the fortune》

Actually Pythagoras and his disciples learned to think the source from everything was the number soon and to have a faith in the integral numbers themselves as if they had been the God. It developed into the secret art on numbers by Pythagoras. What was it?

The secret art gave meaning numbers from 1 to 10. The secret art on numbers is one of fortune telling like the fortune telling in Western and the art of divination. Each of meanings on the secret art of numbers in modern era...

【Pythagoras and secret art on numbers】

《Adding number on the date of its birth and telling the fortune》

Each meaning on the number defined by the modern secret art in the modern era is different, according to its school. The meaning done by Pythagoras and his disciples was like the next.

1 reason 2 woman 3 man 4 justice, truth 5 marriage 6 love and soul 7 happiness 8 essence and love 9 ideal and ambition 10 holy number

The most general way of telling fortune using the secret art on numbers by Pythagoras is adding all the numbers on the date of one’s birth, and we got a number...

【Pythagoras and secret art on numbers】

《Adding all the numbers on the date of one’s birth and telling fortune》

...and we got a number in two-digit, and still more adding the number of each digit and thinking of the meaning on the last number.

For example, there is the one who was born on the 18th day June in 1974, 1＋9＋7＋4＋7＋1＋8 =37⇒3＋7=10. It means the holy number and the book says it means the perfect, or cosmology.

We can use the number and calculate. For example, 2 ＋ 3 = 5 , a man plus a woman is equal a marriage. 2 × 3 ＝ 6, multiple a man by a woman is equal a love. 4 ＋ 5 = 9...

【Pythagoras and the secret art on numbers】

《Adding all the numbers on the date of one’s birth and telling the fortune》

...and 4 ＋ 5 = 9 means justice adds to marriage is equal an ideal.

The author said it was a good device, wasn’t it? And said he wants us to try to have enjoy other combination.

Needless to say we don’t always have to take the meaning each number on the secret art by Pythagoras as very serious, but if we success in being familiar with the numbers like that, it will be meaningful.

《Why was his disciple murdered?》

While the numbers had been deified, Hippasus who was one...

【Pythagoras and secret arts on numbers】

《Why was his disciple murdered?》

While the numbers had been deified, Hippasus who was one of disciple of Pythagoras was aware something important. He discovered he couldn’t express an oblique side of a right angle isosceles 2等辺 triangle with any numbers which they discovered until then.

The numbers means rational numbers which are expresses with fraction of rational number on both a denominator and a numerator.

Besides, ironically it turned out that the existence of the number was clear by the Pythagoras theorem.

When hearing the report from...

【Pythagoras and secret arts on numbers】

《Why was a disciple of Pythagoras murdered?》

When hearing the report from Hippasus, Pythagoras and all his entire disciples tried to verify and reached a conclusion that the report from Hippasus was right.

There is another version of the story which holds that when hearing it, Pythagoras was very frightened at it, and he ordered all his disciples not to leak out the existence of the numbers anyone, and to kill Hippasus....

We call the numbers which don’t belong to the rational numbers irrational numbers at present. As to the irrational number...

【Pythagoras and secret arts on numbers】

《Why was his disciple murdered?》

If it was the fact, why did Pythagoras have to murder his disciple then?

As to an irrational number, for example when trying to express the square root of seven with numerical numbers, it’s 1.141421356...as irregular numbers continue in decimals forever, so expressing its number exactly is hard, but the irrational number exists definitely in our world as the length of the oblique side on the right isosceles angle triangle, of which triangle is between the same length of two sides.

On the other hand, in the era of...

【Pythagoras and his secret art on numbers】

《Why was his disciple murdered?》

On the other hand an era of the Ancient Greece was the time when various facts had been verified exactly by math for the first time. The author said he’s sure that lots of the people thought there was nothing more definite than math.

Pythagoras was in the center of the time, so he couldn’t forgive the existence of the irrational number which he couldn’t understand its precise value.

To be exact, the number which continued irregular ones decimals forever must not have existed for the great mathematician, for...

【Pythagoras and secret arts on numbers】

《Why was his disciple was murdered?》

...for it was against his sense of beauty because math should be simple for him.

By the way, it proved that irrational numbers are by far more than the rational numbers at present.

《An unexpected relation between mathematicians and a tune》

Pythagoras and his disciples invented musical scales like what is called do re mi fa singing method from the discovery on a relation between musical interval which was resonant wonderfully and mysterious integral numbers.

Putting musical scales from the first one, do to the...

【Pythagoras and secret arts on numbers】

《An unexpected relation between mathematicians and tunes》

Putting various sounds between the first do to the last do from the musical scales according to the degree on the sounds. We call its rule a tune.

There was a musical interval of do and so when the proportion on the weight of the hammers was 3 to 2, and as being based on the musical interval, Pythagoras and his disciples invented tunes.

To tell the truth, a way of specifying tunes is various. There have never been perfect tunes until at present in other words.

The perfect tunes mean that...

【Pythagoras and secret arts on numbers】

《Unexpected relation between mathematicians and tunes》

The perfect tunes mean that when several sounds resonate at the same time, a pleasure of the resonating and the one when listening to as melody are consistent.

If trying to invent the tunes, we need knowledge on geometric progression 等比数列, or radical root, so there is a deep relation between mathematicians and tunes.

In fact, Kepler and Euler left original tunes, and Japanese mathematician named Genkei Nakane 中根元圭 who was born in Edo era invented tunes divided into 12 octaves. It’s temparment.

【Pythagoras and secret arts on numbers】

《An unexpected relation between mathematicians and tunes》

Temperament means 平均律, though I can express it in English, but to my sorrow I have none of knowledge on it. First of all, as to the music, all we have to do is enjoying ourselves, so we don’t need theory which is hard to understand, I’m sure.

【Math used to be music, or astronomy?】

《An origin of a word on math》

The author said he was going to express on the origin of the word of math.

数学 was used as translation from mathematics in a foreign book written in China in the 19th century for the...

【Math used to be music or astronomy?】

《An origin on a word of math》

The word of 数学 was used as a translation from a foreign book written in China in the 19th century for the first time.

The word was used in 英和対訳袖珍辞書 which was issued the first orthodox English Japanese dictionary in 1862.

How should we translate the word of math? It seemed that there had been various opinions then.

As to its meaning, arithmetic resembles math, and it’s frequently translated into 算数, but it’s not accurate.

Arithmetic is one of fields on math in which study on the calculation using four basic operation.

【Math used to be music and astronomy?】

《An origin on a word of math》

The four basic operation is addition, subtraction, multiplication, and division.

《The translation which doesn’t fit nicely somehow》

The author said that he could understand how the people took pains when translating then, but the translation on 数学 from math struck him as incongruous, for the field which the math handles hasn’t been limited numbers alone.

The number is a conception which shows us an order or an amount on something. It started from natural numbers like 1.2.3..., but it’s expanded its area like decimal...

【Math used to be music or astronomy?】

《The translation which doesn’t fit nicely somehow》

...but it’s expanded its area like decimal, a fraction, and irrational number and it means real number and the whole imaginary numbers at present. As the number is a conception which is abstracted, so it doesn’t have any unit.

On the other hand, quantity is an objection for measure like length, area, volume, angle, weight, time and speed. The quantity has an unit basically. In geometry when asking the value on length of a side or on area of a figure, it means that we measure the quantity exactly.

【Math used to be music or astronomy?】

《The translation which doesn’t fit somehow》

It’s sure that not only the number but the quantity have been in the center on math from the time of Ancient Greece. Then should we translate it into a learning on number and quantity? The author said it’s not enough yet.

When a function which some amounts cause other amounts appears in 17th century, lots of mathematicians began to take in interest for studying change between them closely.

When saying y is the function for x, it means that y is the number which is fixed with x. The author says when imaging...

【Math used to be music or astronomy?】

《The translation which doesn’t fit nicely somehow》

The author said if imaging that putting into a value of x on an instrument, we can get other value of y. It’s easy to understand.

As the x is changed into the y with the instrument of the function, the conception on function was born, as a result, lots of the people, including mathematicians and other ones who aren’t always interested in math started pay attention to the change.

After that, a big field of mathematical analysis of which center is differential and integral calculus has been established.

【Math used to be music or astronomy?】

《The translation which doesn’t fit nicely somehow》

Then the area which math handles has expanded so much that it covers natural science, so it is no exaggeration to say that dealing with the change has established the present position on the math.

Euclid made rules on nature which the space has as axiom for the first time in his book, the Elements of Euclid, in the time of Ancient Greece.

For example, the shortest distance between two points is a straight line, or two straight lines parallel each other don’t cross forever.

It’s so natural that it...

【Math used to be music or astronomy】

《The translation doesn’t fit nicely somehow》

It’s so natural that it seems that we don’t have to say, but mathematicians can define other special space which doesn’t have the nature like that, so thinking of the space which doesn’t belong to the Elements of Euclid led to a new math and new science.

I’m wondering what kind of space it is.

《Area where math handles is by far larger》

Considering those things, the translation of 数学 from math isn’t enough, the author said.

When being a student in high school, he used to have some doubts why the word of...

【Math used to be music or astronomy】

《An area where math deals with is by far larger》

When being a student in high school, the author used to have some doubts why 数学 was adopted.

Instead of numbers, some letters have been used in equations. It’s algebra, 代数学 He knew it, so 代数学 was shorten and it changed into 数学 he thought like that, but it means that function, geometry and probability aren’t included. He seemed to be worried about it.

Actually the math deals with the number, quantity, space, change and construction, and math has been put to practical use at the wide variety.

There is a...

【Math used to be music and astronomy】

《The area where math deals with is by far larger》

When there is a question that what math is, if trying to answer, it is hard not only mathematically but philosophically at present.

There is a field of learning that philosophical study on an object or a way for math has been done. We call it mathematical philosophy.

《Arithmetic, music, geometry and astronomy》

In that meaning, the origin of the word on math is something which they should learn in Greek, I’m interested in it, the author said like that.

Something which they should learn in the..

【Math used to be music or astronomy】

《Arithmetic, music, geometry and astronomy》

Something which they should learn in the Ancient Greece is arithmetic which is quiet number, music which is active number, geometry which is quiet quantity, and astronomy which is active quantity.

Platoon used to open an academia for himself, and when fixing its curriculum, he thought the ones whose age from 16 to 17 should be trained so as to learn philosophical questions and answers, so he prepared for those four subjects.

As he was influenced over by Pythagoras and his sects so greatly that he made much...

【Math used to be music or astronomy】

《Arithmetic, music, geometry and astronomy》

Pythagoras and his sects had such a tremendous impact on Platoon that Platoon made much of those four subjects. Why were they divided into four?

At first Pythagoras divided into two, number and quantity, and after that separated into something quiet and something active for each of two.

Quiet means itself and active is the one which changes according to its relation with other.

An ability on algebra which they learn number itself, which means something quiet, is the base on everything, so it’s one of...

【Math used to be music or astronomy】

《Arithmetic, music, geometry and astronomy》

...so it’s one of things which they should learn, none of us raise an object against it.

Pythagoras and his disciples accomplished remarkable outcome in the field of geometry where they learn on figures which means something quantitative, at the same time established a logical way on thinking.

Then learning geometry means learning a logical way of thinking of something then, if trying to learn philosophy, geometry was indispensable. It was natural for them then.

《Mystery on numbers and music in the...》

【Math used to be music or astronomy】

《Something mysterious on numbers and music in celestial sphere》

Pythagoras and his disciples were aware that something mysterious was hiding in beautiful consonance, and expounded everything was numbers. They enlightened that the cosmos was made of harmony which was caused a relation between numbers.

Their enlightenment made the people in the Ancient Greece think the foundamental principle on the cosmos was ムジカ and its harmony was ハルモニア, ムジカ means music, and ハルモニア harmony in English.

To tell the truth, music had been thought as rather something...

【Math used to be music or astronomy】

《Something mysterious on numbers and music in celestial sphere》

In fact music had been thought as something symbolic for for order and harmony rather than an entertainment until the Middle Ages. As a result, they learned music so as to grasp the principle in cosmos. In other words, music was indispensable for them to understand the language from the God.

Besides, Pythagoras created an idea of a music on celestial sphere.

Then they thought all the stars were fixed to a gigantic spherical surface of which center was the earth, and the rotation of the...

【Math used to be music or astronomy】

《Something mysterious on numbers and music in celestial sphere》

...and its rotation on the spherical surface made all the stars move, they thought like that then and it’s Ptolemaic theory 天動説.

In the viewpoint of the Ptolemaic theory, a movement on planets was so mysterious, and when trying to express it with a relation between figures which means active quantity, they need to contrive a complicated geometry on spherical surface.

Even if it looked very complicated, a heavenly body revolved on a harmonious orbit, so the cosmos should be filled with...

【Math used to be music or astronomy】

《Something mysterious on numbers and music on celestial sphere》

...so the cosmos should be filled with beautiful music on the celestial sphere which we the humankind couldn’t listen to, Pythagoras and his disciples thought like that.

Platoon set arithmetic, music, geometry, and astronomy as required subject in the Academia where he established for himself, but the one who had a strong spirit for being a leader in the next era should learn with its own free will, so it shouldn’t be forced to do from anyone, Platoon thought like that.

The four required...

【Math used to be music or astronomy】

《Something mysterious on numbers and music on celestial sphere》

Platoon set the required subjects like arithmetic, music, geometry and astronomy, and it was called マテマータ in the Ancient Greek and it was an origin of word on mathematics, and the マテマータ learned to be called liberal arts in English later. It means several technical skills which should be acquired with free will.

IT and AI have tried to change the industrial structure in a large scale at present, so some people seem to call it the fourth industrial revolution, so math is indispensable which...

【Math used to be music and astronomy】

《To liberal arts》

...so math is indispensable which we should learn. It’s returned to its original meaning.

It seems that math used to tend to be learned by the ones who adopted science course or others who were good at it alone when learning math or making use of it as if they had hidden from others people. What we call オタク in Japanese slang.

But from now on the ones who adopted the humanities and others who were not good at math won’t be able to avoid math. The author continued.

But I don’t think I want math to be the one which is forced to do ...

【Math used to be music or astronomy】

《To liberal arts》

I don’t think I want to math to be the one which is forced to learn from others. When learning everyone can find something fun from math, and I want math to be the liberal arts which we’re ready to tackle. I’ve believed in that math has something tolerable.

I’m sure the author has loved math.

【Welcome to a museum on curves】

《Nature is a scene which creates curves》

Hideki Yukawa who accepted a Nobel prize for physics as the Japanese for the first time used to say that nature creates curves, and we the humankind does straight lines.

【Welcome to a museum on curves】

《Nature is a scene which creates curves》

If looking around us, it is certain that almost all of the things of the straight line is something artificial like a pen, an end of a desk, or an outline on electric appliance, but needless to say, there is a thing which looks straight in the natural world like a cedar 杉, for its origin of the word is a straight tree, according to the author, though I’ve never heard of it until now.

But strictly speaking, the cedar isn’t the straight line. Both a stone, a flower, a mountain, and a cloud are made of complicated...

【Welcome to a museum on curves】

《Nature is a scene which creates curves》

Both a stone, a flower, a mountain and a cloud are made with complicated combination by curves, but in spite of it, except for a circle, we the humankind have never paid attention to other curves for a long time.

After Pythagoras, it has been dots, street lines and circles which geometry treated, except for a single exception, Aplllonios, who played an active role in the era of the Ancient Greece.

He’d studied a cut end of a cone when it was cut not through its vertex minutely, and named three kinds of curves which...

【Welcome to a museum on curves】

《Nature is a scene which creates curves》

...and named three kinds of curves which appeared as the cut end of the cone then. The one is ellipsis, and the other parabole and the last was hyperbole. Each of them is the origin of word, ellipse 楕円, parabola, and hyperbola in English.

Why did Apollonios call like that?

There is an illustration on the cone when it was cut, then the cone was illustrated as an isosceles triangle 二等辺三角形, and the illustration was the one when we look at the each of cut end from right at our side.

When cutting the isosceles triangle...

【Welcome to a museum on curves】

《Nature is a scene which creates curves》

Roughly speaking,when the isosceles triangle was cut, three other isosceles triangles appear, according to the cut end. As to the three isosceles triangles, when comparing the angle of the cut end with the one of the base angle, there are three situations.

First is both of the angles are the same, and the second is the angle of the cut end is smaller than the one on the base angle, and the third one is the cut end is larger. Then we can distinguish difference on each curve.

When the angle on the cut end was small...

【Welcome to a museum on curves】

《Nature is a scene which creates a curve》

When the cut end was smaller than the base angle, its curve was an ellipse, the cut end was the same, its curve was a parabola, and it was larger, a hyperbola 双曲線.

When the cut end was parallel to the base, a circle appears in the cut end, but the circle has been thought as a kind of ellipse in math, the author said like that.

《Equations on curves》

Collecting, a circle, an ellipse, a parabola, and hyperbola together, and we call them quadratic curves, for equations for each of curves are expressed with quadratic...

【Welcome to a museum on curves】

《Equations on curves》

...for the equations for each of curves are expressed with quadratic equations of x squared or y squared.

The author expressed equations on curves, but we can do because Descartes introduced coordinates 座標, a coordinate axis, and variables into math in the 17th century, so we can express the curves with the equations.

The coordinates means a single point on a plain or in the space are expressed with a pair of numbers or each of combination. The single point on the plain is expressed with a pair of numbers（2.1）and the one in the space...

【Welcome to a museum on curves】

《Equations on curves》

...and the single point in the space like （2.4.3）is expressed with three combinations on numbers.

The single point and the coordinates are dealt with one to one, and there is a straight line on its base, and we call it the coordinate axis.

We’ve learned the way of expressing the point on the plain or in the space using the coordinates in junior high school or in high school, so we find it natural at present.

However every kind of point in the plain or in the space is able to be expressed with a pair of numbers or there combinations...

【Welcome to a museum on curves】

《Equations on curves》

However every kind of point on the plain or in the space is able to be expressed with a pair of numbers or three combinations on numbers was a novel and epoch-making then.

Descartes gave some letters each role which contained some values, and called the values variables 変数.

For example, x and y are variables, let’s suppose there was an equation, x ＋ y = 2, then there are some coordinates on equation of x ＋ y = 2 like the next.

（x,y）＝（1,1）,（x,y）＝ （2、0）,（x, y）＝ （0, 2）

When connecting each of three points, there is a straight line on...

【Welcome to a museum on curves】

《Equations on curves》

When connecting each point on the coordinates, there was a straight line, so Descartes thought the equation of x ＋ y ＝ 2 is the straight line expressing the numerical formula.

At last we the humankind succeeded in connecting a figure with an equation. It was so epoch-making that it was said it was an evolution on the history of math.

Descartes has not only made us express curves which have already been existed in the world but made us create characteristic curves on mathematically or physically from numerical formulas.

【Welcome to a museum on curves】

《Gaudiy and Sagrada Familia》

Apollonios discovered curves on cone which appears as cut end of cone and about 1700 years passed after it, so it learned to be expressed with a numerical formula, and one of them is a parabola, and it turns out that it corresponds to an orbit of throwing something.

While as to the Pythagorean theorem, if using a circle, it seems to be easy to understand it, though I have little knowledge on it, if trying to demonstrate a theorem on Fermat, curves on ellipse is deeply related to it.

What is the theorem on Fermat?

When some ..

【Welcome to a museum on curves】

《Gaudiy and Sagrada Familia》

When some number is bigger than 3, natural numbers of x, y, and z which meet an equation of x cubed ＋ y cubed = z cubed don’t exist.

Though it took no less than 350 years so as to demonstrate the theorem on Fermat, the great feat was accomplished because numerical formulas and curves were connected.

I’ve wanted to express it some day, but to my sorrow I find it impossible with my own ability on math at present.

The author said he was going to show us two curves and each of equation, which we’ve never learned in junior high...

【Welcome to a museum on curves】

《Gaudiy and Sagrada Familia》

...which never appears in junior high school nor in high school.

『Catenary』

When holding both ends of a string or chain of which density is fixed, there is a curve, and we call it a catenary. The density means a unit on weight for a single cube. Catenary is 懸垂曲線 in Japanese.

When holding both ends of the chain, there is a curve made by the chain, and we’d thought it was a parabola for a long time, but Christian Huygens in Holland demonstrated it wasn’t the parabola but the catenary.

We can see lots of catenaries in both...

《Gaudiy and Sagrada Familia》

『Catenary』

We can see lots of catenaries in artificial things created by us the humankind or natural thing which were created by nature like a power transmission line, a suspension bridge, and a spiderweb.

When the catenary is turned upside down it prove to be stable physically, so when holding both ends of the chain, there is the curve, and the curve is turned upside down, it’s stable.

As a result a design of shape is arch that the catenary is turned upside down has been adopted for lots of buildings, especially Antonio Gaudiy who is a typical architect...

《Gaudiy and Sagrada Familia》

『Catenary』

...especially Antonio Gaudiy who is a typical architect from Spain adopted catenary for lots of his works and it has been well known.

Gaudiy thought a beautiful shape has a stable structure, so we have to learn the structure from nature, so his design wasn’t decided by a desk plan but actual experiments.

When designing curves on buildings, he used numberless weights made of sandbag and string, and created curves of catenary, and he thought the curves of catenary were the most natural and strongest structure against vertical weighting.

He had ...

《Gaudiy and Sagrada Familia》

『Catenary』

Gaudy had a absolute confidence on design of the structural strength, so when workmen feared for accumulating tremendous stone like arch, he got rid of scaffoldings for himself, and demonstrated the design was safe.

An experimental instrument which Gaudy used for the design was called Fujikura, and we can look at it in archives beside the Sagrada Familia.

The equation is expressed in the book, but it’s too complicated to express here. It’s regrettable.

《A roller coaster and spirals on Euler》

『Clothoid』

It was in 1895 when Flip Flap appeared in...

《A roller coaster and curves on Euler》

『Clothoid』

It was in 1895 when Flip Flap appeared at suburbs of New York in America for the first time. Flip Flap was a roller coaster which rotates vertically.

A crowd who loved something new gathered and after the roller coaster started to drive with passengers, some of them suffered from whiplash injury むちうち症, and others were hurt on the neck, so lots of them were damaged.

Why was it caused?

Because the shape on the rail which rotated a single time vertically was almost a circle. If connecting the part of the circle with the part of the...

《A roller coaster and curves on Euler》

『Clothoid』

If connecting the part of the circle with the part of the straight line, passengers suffered from violent burden in the place where the rail changed from the curve to the straight line.

Instead of the circle, a curve called Clothoid was adopted for the roller coaster which rotated vertically in order to prevent the dangerous situation form happening.

Leo hard Euler in Germany studied the curves minutely, so it’s also called spirals by Euler. Clothoid started from a straight line, and the more it go ahead, the sharper the way of curving.

《A roller coaster and curves on Euler》

『Clothoid』

If comparing the Clothoid with driving a car, when driving at a fixed speed and turning the steering wheel with the fixed ratio, curves are drawn by the car. We call the curve the Clothoid.

If we have to drive form a straight line to curve composed of a circle and to the straight line again, the driver has to cut the steering wheel in a hurry both at first and the last of the curve.

Unless the driver decreases its speed so less, not only driving the car is hard to do but physical burden to the passengers is serious.

On the other hand...

《A roller coaster and curves on Euler》

『Clothoid』

On the other hand, there is a corse from the straight line to the Clothoid to the circle and to Clothoid to the straight line. When going into the Clothoid from the straight line, the driver has only to cut the steering wheel with the fixed pace gradually.

When driving on the circle, as long as the driver keeps the steering wheel as it was, it’s all right, so it’s easy to drive and the passenger find it comfortable, so we find the Clothoid pleasant.

It’s Autobahn in Germany where the Clothoid has been adopted for roads on cars for the...

《A roller coaster and curves on Euler》

『Clothoid』

It is Autobahn in Germany where the Clothoid has been used for roads on cars for the first time. The Autobahn is a highway express.

Almost all of the highway expresses have adopted the Clothoid all over the world at present.

《Two giants and breaking away from a circle》

After Pythagoras in the Ancient Greece, orbits on stars in the cosmos which was the symbol for a complete harmony used to be believed firmly that it must have been a circle.

Not only in the Ptolemaic theory 天動説 that the earth was the center on the cosmos, but in the...

【Welcome to a museum on curves】

《Two giants and breaking away from a circle》

...but the Copernican theory 地動説 advocated by Copernicus in Poland used to think an orbit on starts was a circle, but being based in the circular orbit and trying to express the movements on stars, they needed very complicated theory in both of Ptolemaic and of Copernican.

In addition the place on stars which was done by a large amount of calculation were often different from actual ones.

On the other hand, Johannesburg Kepler who supported the Copernican theory studied the result from observation carefully and...

【Welcome to a museum on curves】

《Two giants and getting away from a circle》

...and formulated a hypothesis that an orbit on planet was an ellipse. If having thought, he discovered that he could explain movements on planets by far more simply and more accurately rather than regarding the orbit on planets as a circle.

Kepler framed that when all the planets which rotated around the sun including the earth, all of the orbits were ellipse, and foresaw the movements on the planets ahead of several years and created a list of astronomy. We call it the Rudolphine Tables.

Its precision was...

【Welcome to a museum on curves】

《Two giants and getting away from a circle》

As its precision is thirtieth times as accurate as in the past, the predominant position on Copernican theory was strengthened.

Moreover being based on the theory by Kepler, Newton arrived at common physical laws including the law of universal gravitation, and succeeded in expressing from a pebble on the earth to the movements on planets with an unified way, as a result, the Ptolemaic theory was forgotten completely.

Two giants of Descartes and Kepler who showed up in the former part of the 17th century have...

【Welcome to a museum on curves】

《Two giants and getting away from a circle》

Two giants of Descartes and Kepler who turned up at the former part in the 17th century have made us the humankind get away from the circle. It means that looking at nature as it is and expressing mathematically. It’s indispensable for modern science.

【Math in which tiles spread all over】

《Geometric design at Alhambra palace 》

There is Alhambra palace at Granada of ancient city in Spain. It used to be a symbol on power and fortune of Muslims who ruled over there, and was a tragic stage where they were defeated...

【Math in which tiles spread all over】

《Geometric design in Alhambra palace》

...and a tragic stage where Muslims were defeated by the Christians with the battle of Reconquista. It was a the battle on restoration of their territory by the Christians.

Its premises is no less than one hundred and fifty thousand square. It’s equal to three of Tokyo Dome which is a baseball stadium. Not only a king but some two hundred of the nobles used to live there then.

As to the origin of a word on Alhambra, it’s Islam language which means a red fortress. Several magnificent building are lined up in the...

【Math in which tiles spread all over】

《Geometric design in Alhambra palace》

Several magnificent buildings stand in a line in a tremendous site, and each of them are masterpiece which gathers the best that Islamic architecture embodied. It’s so splendid that it was said the king completed the palace by magic then.

As worshipping an idol is prohibiting by dogma in Islam, decoration wasn’t done with a motif of a persons or animals. Instead geometric designs have developed. We can see the tradition everywhere in Alhambra palace, especially a wall or a ceiling were spread all over by basic...

【Math in which tiles spread all over】

《Geometric design in Alhambra palace》

...especially a wall or a ceiling are spread all over by basic figures all round, on the contrary, tiles spread all over by different design one by one. They are so excellent that everyone is forced to watch it with its eyes wide open in astonishment.

Dutch painter, and a woodblock artist, Maurice Escher is one of them. He copied the decoration in the palace carefully in full three days, and his inspiration was stimulated greatly by the beauty continues limitlessly.

Speaking of Escher, he is famous for the art...

【Math in which tiles spread all over】

《Geometric design in Alhambra palace》

Speaking of Escher, he is famous for the recognized authority on the art of illusion like one of his works, a waterfall as if it looks the water flew from lower to upper part, and as to his style in other work, its canvas is filled with different pattern which changed one after another, and it shows us striking originality.

It is no exaggeration to say that his visit Alhambra palace caused to create a group of works on the latter one like the series on Metamorphose.

In fact the first work on Metamorphose was...

【Math in which tiles spread all over】

《Geometric design in Alhambra palace》

In fact his first work on Metamorphose was done in the next year after he visited the Alhambra palace. Escher adopted a motif on animal which wasn’t done in Islam and created a work by designs of reputation, so whoever watches the work is attracted by the original world.

《What is a polygon which is filled with a plain?》

To tell the truth, when wanting to fill a plain which spread all over limitless all round, it doesn’t always mean that every shape is all right. For example, if trying to fill the plain with a...

【Math in which tiles spread out all over】

《What is a regular polygon where a plain is filled with?》

To tell the truth, if trying to fill a plain with a regular pentagon, it’s impossible, for one of angles on the regular pentagon is 108 degree, so gathering three of the regular pentagons in a single apex, 108 × 3 = 324, it’s smaller than 360 degree, so there is a space.

If gathering four, it’s over 360 degree, so one of the regular pentagon is on top of the other.

If thinking over with the same way, the regular polygon which filled with the plain all around is limited the next one.

【Math in which tiles spread all over】

《What is a regular polygon in which the plain is filled with?》

A scale of a single angle × an integral number = 360 degrees. The regular polygon means the one of which all angles are equal each other, and that kind of regular polygons are regular triangle, regular square, and regular hexagon. It’s just the three kinds.

In general, filling the plain with several kinds of plain figures all around without being the one on top of the other is called tessellation or tiling.

Except for regular triangle, how about other triangle?

In fact every kind of...

【Math in which tiles spread all over】

《What kind of regular polygon in which a plain is filled with?》

In fact the plain is filled with every kind of triangle as long as it’s the triangle, for preparing for the same shape of two triangles, and turning the one upside down and put together, it changes into a parallelogram.

If the parallelogram is spread out up and down, right and left, the plain is filled with the parallelogram all around.

In fact, as to a square, even if it’s an every kind of a square, the plain is able to be filled with. When turning the one square upside down, and...

【Math in which tiles spared all over】

《What is the regular polygon in which a plain is filled with?》

In fact when turning a square upside down and put together it with the same shape of the square, it changes into a parallel hexagon of which each side faces each other is parallel, and the plain is filled with the parallel hexagon.

《Inspection on a pentagon》

It turns out that triangle and square which we can choose as we like are the figures, the ones which a plain is be able to filled with until now, and it was already clear in the time of Ancient Greece.

If the triangle, and square...

【Math in which tiles spread all over】

《Inspection on pentagons》

If the triangle, and square are all right, everyone tends to try to pentagons, but then the situation suddenly becomes complicated.

As the author described, the plain isn’t filled with the regular pentagon, the plain isn’t filled with every kind of the pentagon, but if it’s a warped pentagon which has a character, the plain is filled with the pentagon. For example it has pair of sides which are parallel.

15 kinds of concave pentagons in which a plain is filled with have been discovered until November in 2019. Please look up...

【Math in which tile spread all over】

《Inspection on pentagons》

Please look up it into pentagonal tiling in Wikipedia if you are interested in it.

A student in German announced a graduation thesis, five kinds of concave pentagons which are able to fill with a plain in university in 1918, and after that the tessellation has been argued mathematically.

He demonstrated that the plain is filled with three kinds of concave hexagon alone in relation to polygon. Then the polygon means the one which has more angle than pentagon. It doesn’t exist in any concave heptagons.

On the other hand...

【Math in which tiles spread all over】

《An inspection on pentagon》

On the other hand, as to the concave pentagons, he said it was unclear whether or not there were any other ones, except for five kinds which he discovered.

Then a question on tesselation 平面充填 by a single kind of a concave polygon is limited to the pentagon alone.

《A difficult problem which a housewife solved》

After fifty years from the graduation thesis by the German student in 1968, three kinds of concave pentagons were discovered, so the concave pentagon which fills with the plain becomes eight kinds.

The question in....

【Math in which tiles spread all over】

《A difficult problem which a housewife solved》

A question of filling the plain with the concave pentagon was such a difficult problem that it took no less than a half century to discover three more kinds, but five kinds have been discovered one after another from 1975 to 1977.

Besides, it wasn’t done by any mathematicians, and the one was done by a computer scientist, and the other four were done by a housewife. Her favorite thing was a patchwork. She took interested in a question that the field is filled with tiles by polygons introduced in a magazine.

＞＞170

To the person who sent me a message in 170.

170にメッセージをくれた方に。

Hi, you’re welcome.

こんにちは、ようこそ。

I don’t know why you feel a little inferior to a person who can speak English, but everyone tends to feel like that for something, I’m sure. It doesn’t always mean that it limits to English alone.

あなたが何故英語を話せる人に負い目を感じるのかは知らないけど、英語に限らず、誰でもそう感じる事はあると思います。

Even if being able to speak English a little, it’s not so useful as I expected. It’s just that I have a fun when expressing myself in English.

多少英語を話せたとしても、思ったほどには役に立つわけでは無いです。ただ、英語が好きなんで、その時は楽しいと思うだけですよ。

＞＞169

【What I’ve thought】

Without being able to show its illustration, I’m afraid I can’t send others what I thought, so I’m going to give up the chapter, math in which tiles spread out, so I’ll start the next chapter.

【A correspondence by one to one and a string of Hideyoshi】

《Making a pebble and a thing correspond》

An English word of calculation is made up with calc and ation, and the calc means a pebble, and the ation means doing something.

Calculus means 微分積分 in Japanese, and has also meaning of a stone in the kidney.

Thus calculation and calculus are related to the stone, for...

【A correspondence by one to one and a string of Hideyoshi】

《Making a pebble and a thing correspond》

...for when we the human race started to use numbers, the stone used to be a tool for counting numbers.

As our ancestors in the ancient time, they couldn’t count more than three, they thought of three, thirty, and a hundred as many, though there are various theories on it.

But if leading every day life, we frequently come across a scene when we have to count more than three.

For example, there was a farmhouse which kept some cattle, and a routine for the head of the farmhouse was...

【A correspondence one by one and a string of Hideyoshi】

《Making a pebble and a thing correspond》

...and the routine of the head of the farmhouse was corresponding each of cattle to a stone before putting the cattle out to the pasture. When the cattle returned, the head make sure whether or not all the cattle returned, using the stones.

When we the human race couldn’t use any of large number, we corresponded the one which we wanted to count to the stone by one to one, so it is said using the stone learned to mean the calculation.

What is the correspondence by one to one?

When there are...

【A correspondence by one to one and a string of Hideyoshi】

《Making a stone and a thing correspond》

When there are two kinds of set, A and B, every element on A corresponds with only element on B, at the same time, every element on B corresponds with only element on A. It’s the correspondence by one to one.

For example, a system on my number has started in Japan from 2015. All individuals who have its certificate of residence in Japan are given different number in twelve digit number.

As long as the one has the certificate of residence in Japan, everyone corresponds with one of my number...

【A correspondence by one to one and a string of Hideyoshi】

《Making a pebble and a thing correspond》

...and if choosing one of my number, everyone can pinpoint the individual, so the set of the individuals who have certificate of residence in Japan and the set of my number correspond by one to one.

On the other hand a set of students who belong to A class in high school, the total number on students are 40, and the set of birthday, one year is made up with 365 days, so there are 365 elements. Those of two set don’t correspond by one to one.

A birthday for a student is only one, but while...

【Correspondence by one to one and a string of Hideyoshi】

《Making a pebble and a thing correspond》

As to a single student’s birthday, it’s only one, but while the number of the students are 40, the number of the birthday is 365, as a result, as to the rest of 325 of the birthday, none of the students correspond with them.

In addition there is a possibility that two or more students correspond with one birthday. It means that there is a possibility that two or more students have the same birthday. The author may add that the probability of having the same birthday in a class made up forty...

【Correspondence by one to one and a string of Hideyoshi】

《Making a pebble and a thing correspond》

The probability that having the same birthday in a class made up forty students is 89%, the author said. Its high unexpectedly.

《Hideyoshi has a marvelous mathematical sense》

It is well known that Hideyoshi Toyotomi had a splendid mathematical sense, and the author said he is going to show us his episode when he used to be a subject of Nobunaga Oda. Hideyoshi used the correspondence by one to one skillfully and he was put more confidence by Nobunaga then.

Nobunaga ordered his common foot...

【Correspondence by one to one and a string of Hideyoshi】

《Hideyoshi has a marvelous mathematical sense》

Nobunaga ordered his common foot soldiers 足軽 to count the number of the trees on the mountain behind his territory for an investigation one day.

His subjects obeyed the order from their master and started to count but they fell into a confusion at once, for they separated into groups and counted the number of the trees, but they were not sure that who counted which trees.

When looking at the confusion, Hideyoshi said the subjects like the next.

There are a thousand of strings here.

【Correspondence by one to one and a string of Hideyoshi】

《Hideyoshi has a marvelous mathematical sense》

You don’t have to count the trees, but tie a string a tree together one by one, all the trees.

Hideyoshi said like that, and the common foot soldiers obeyed and went to the mountain behind the territory again.

After an hour, all the common foot soldiers returned, and Hideyoshi made them collect the rest of the strings and count them. If the rest of the strings were 220, it turned out that the number of the trees were 780.

The more trees are, the harder to count, but Hideyoshi made...

【Correspondence by one to one and a string of Hideyoshi】

《Hideyoshi has a marvelous mathematical sense》

...but Hideyoshi made the trees which are hard to count to strings which are easy to count correspond, and succeeded in counting the number of the trees in the mountain behind the territory.

It is said that after that not only Nobunaga but all the other subjects raised their hats to him more and more.

When there a thousand of invited guests in a wedding ceremony, we can see whether or not all of the invited guests were present there at a glance, for seats for the invited guests alone ...

【Correspondence by one to one and a string of Hideyoshi】

《Hideyoshi has a marvelous mathematical sense》

...for the number of seats for invited guests alone are prepared for in an usual wedding ceremony, so if there is an absentee, we have only to see a list of the seating order. We can see who is the absentee right away.

When wanting to know the number of the one who go to a movie theater, if going into there and counting them one by one actually is hard, and some of them may go to a restroom, so we don’t have to do it. There are stubs 半券 from admission tickets at an entrance, so we have...

【Correspondence by one to one and a string of Hideyoshi】

《Hideyoshi has a marvelous mathematical sense》

We have only to count the number of the stubs.

Thus the correspondence by one to one has been widely used at present so as to count numbers efficiently.

『Supplementation. A way of seeking probability of people who have the same birthday in other forty』

At first we seek for the probability that none of the forty have the same birthday. We choose a one at random, and as to its birthday, even if which birthday it is, it’s all right.

The probability that the next people is different...

《Hideyoshi has a marvelous mathematic sense》

『Supplementation. A way of seeking the probability that having the same birthday in forty people』

The probability that the next person is different from the first one on the birthday is three hundred sixty four three hundred sixty fifths.

The probability that the third person is different from the other two on the birthday is three hundred sixty three three hundred sixty fifths.

The probability that the fortieth person is different from the other 39 is three hundred and twenty six three hundred sixty fifths.

In the next, multiplying each of...

《Hideyoshi has a marvelous mathematical sense》

『Supplementation. A way of seeking the probability of having the same birthday in forty people』

In the next, multiplying each of the probability one by one each other, and its value is 0.1087, and it’s the probability that none of them have the same birthday in forty people.

As a result, the probability that having the same birthday in forty people is 1−0.1087 = 0.8912, so it’s about 89%.

To my sorrow, I can understand why it is. Could someone kind please tell me why?

I think Arabian number is by far better than English this time.

【Correspondence by one to one and a string of Hideyoshi】

《The device by Descartes》

A situation in which the correspondence by one to one isn’t the one alone when counting numbers.

As introducing coordinates, we’ve learned to express a figure and a graph with a numerical formula, for a pair of points on the graph like （1.2）corresponds with one to one.

When looking at the process which Descartes introduced the coordinates, we can understand it’s more significant than counting numbers.

When thinking over geometry, we find it difficult, but Descartes changed the difficulty into a ...

【Correspondence by one to one and a string of Hideyoshi】

《Device by Descartes》

...but Descartes changed into the difficulty into a mechanical work of changing a numerical formula and handling it.

When being a students it seems that there are lots of people who could solve an equation easily but are poor at geometry.

The author said when looking at his students in his cramming school, almost all of them can solve a linear equation, a quadratic equation, and a simultaneous equation as long as the author taught them how to solve.

Actually as a way of solving those basic equations has been...

【A correspondence by one to one and a string of Hideyoshi】

《Device by Descartes》

Actually as a way of solving those basic equations has been established, we can get the answer relatively easily as long as we know its process, but a question on figures don’t go ahead so easily.

Even if we understand how to solve a question, we frequently tend to be worried about other question from the scratch. As a result, if trying to solve the question on figures, we need a sense or an inspiration.

Descartes was worried about it as well, so he thought of coordinates which means figures and numbers.

【A correspondence by one to one and a string of Hideyoshi】

《Device on Descartes》

Descartes thought of correspondence, it means making use of the figure and the number and make them correspond by one to one in order to change of thinking of a question on geometry into solving an equation.

In short, he tried to change something difficult into something simple.

When there is a question which is difficult to solve, taking advantage of the correspondence by one to one, and changing the difficult one into simple one. It has been done in algorithm which is a process of calculating in computer.

【Correspondence by one to one and a string of Hideyoshi】

《Device on Descartes》

Algorithm makes a computer decrease its burden and lead to a quick outcome accurately. Furthermore, the correspondence by one to one is important in the world of function.

The author said he is going to go over the function here.

If saying y is a function to x, it means that the value on the y is fixed by the value on the x functionally.

Function is 関数 in Japanese, and the word came from China originally. Then 函数 used to be adopted. As to the essence on the function, 函数 is easier to understand than 関数.

【Correspondence by one to one and a string of Hideyoshi】

《A device on Descartes》

The word of 函 has a meaning of a box. If saying y is a function to x, it means that x was input to a box 函 and a value came out of the box. It’s y, but the box isn’t an inaccurate one. When pushing the specified input button, it corresponds with the specified output goods, it’s a reliable box as if it were a vending machine in a town.

But as to some ordinary vending machine, some buttons often correspond with a single goods alone which is popular among us, so we can’t specify the goods which button we pushed.

【Correspondence by one to one and a string of Hideyoshi】

《Device on Descartes》

The situation resembles the next one.

Each of cause corresponds with a outcome, but we can’t specify the cause from the single outcome.

《Imaging from a secret tool by Doraemon》

When our sweetheart was in a bad mood, but we didn’t know why it was, then we were forced to be worried about it.

Or when a score on golf was getting worse, if we know why the score became bad, we can improve the situation.

If it’s fixed that something causes anything definite, it’ll be helpful for us. For example, our sweetheart...

【Correspondence by one to one and a string of Hideyoshi】

《Imaging from a secret tool on Doraemon》

For example, our sweetheart is in a bad mood in the morning, or when shooting with a driver, if straining himself, he is apt to put slice on a ball.

In addition if we can specify the cause from the outcome, we’ll be encouraged with it.

When we can specify the outcome from the cause, and we can do the cause from the outcome, then the correspondence by one to one is formed between the cause and outcome.

At the same time, when y is a function to x, and at the same time x is the function to y...

【Correspondence by one to one and a string of Hideyoshi】

《Imaging from a secret tool on Doraemon》

...then x and y corresponds with one to one and it is said the inverse function exists in math.

As to a function, if the inverse function exists, after x is changed into y through the box 函, and returning through the box, y changed into x again.

If imaging the Gulliver tunnel which is one of secret tools by Doraemon, it may be easy to understand. It has two entrance. The one is big and the other small. If going into from big one, we are smaller, but if going into the small one, we return...

【Correspondence by one to one and a string of Hideyoshi】

《Imaging one of secret tools on Doraemon》

...if returning from small one to big one, we return to the same size.

By the way, I have little knowledge on Doraemon, and I’ve never heard the Gulliver tunnel until now. Others call me strange, but I’m not so eccentric.

The correspondence by one to one on the function has been used in computer for compression when exchange some data between computers.

A way of compress the data is two, the one is a Lossless compression and a Lossy compression.

After compressing a file, we can restore...

【Corresponding by one to one and a string of Hideyoshi】

《Imaging from a secret tool on Doraemon》

After compressing a file, we can restore on original file from the compressed one. It’s called the lossless compression.

On the other hand, once the file is compressed, we can’t restore the original file. It’s the lossy compression.

By the way, I have little knowledge on computer, so when expressing, I find it boring, so I’m going to give it up.

Math has developed with the correspondence by one to one before numbers were born, so being able to understand the correspondence by one to one...

【Correspondence by one to one and a string of Hideyoshi】

《Imaging from a secret tool on Doraemon》

...so being able to understand the correspondence by one to one means having the most basic ability on math like being able to understand a relation between orders, to observe, or to abstract.

【An estimation on Fermi and something approximate】

《Estimating a scale of market on jeans》

The author said he remembered a conversation with his friend when he was a freshman in university. The one who started to talk was the author.

The author said, how many jeans do you have?

His friend, oh!...

【An estimation on Fermi and something approximate】

《Estimating a scale of market on jeans》

His friend, oh! Suddenly, what’s up?

The author, I bought a pair of jeans yesterday and I’m wondering how many jeans others have.

His friend, I’ve had three at present. When having worn out, I’ll throw it away.

The author, I’m the same one, by the way, how many jeans do you buy in a year?

His friend, Mmmm....a pair of jeans in a year.

The author, Are the others like that?

His friend, There are the ones who buy more, but there are others who rarely buy, but as to our generation, we are the average

【Estimation on Fermi and something approximate】

《Estimating a scale of market on jeans》

His friend, There is a possibility that the older we are the less we buy, so considering the whole nation, a person buy a pair of jeans in two years.

The author, Oh, yes, the whole population in Japan is about one hundred twenty million, so multiply one hundred twenty million by zero point five is equal sixty million, so the whole Japanese buy sixty million pairs of jeans a year. How much is the average price on a pair of jeans?

His friend, Some of them were priced at ten thousand yen like Levis or...

【Estimation on Ferm and something approximate】

《Estimating a scale of market on jeans》

His friend, Some of their price is ten thousand yen like Levis and Edwin, but others are more reasonable like for children, and the ones which aren’t big name brands, if an average price on a pair of jeans is five thousand yen, the scale of market on jeans in Japan is multiply sixty million by five thousand is equal three hundred billion, so it’s about three hundred billion.

The author, GDP, the gross domestic product a year in Japan is five trillion, so the scale of market on jeans is equal to about...

【Estimating on Fermi and something approximate】

《Estimating a scale of market on jeans》

The author, ...so the scale of market on jeans is equal to about 0.6% on the GDP, and is equal to a thousand of Ochiai who is a professional Japanese baseball player. By the way how about the whole world?

An annual salary of Ochiai was three hundred million and both the author and his friend love baseball.

His friend, Mmmm...the whole world? There is a cultural sphere where no one puts on jeans, so the number of jeans they buy a year become less. One to ten people buy a pair or two pairs jeans a year?

【Estimation on Fermi and something approximate】

《Estimating a scale of market on jeans》

The author, If one of ten people buy two pair of jeans a year, as the whole population in the world is about sixty billion, so multiply sixty billion by zero.point five by five thousand is equal to six trillion. So the scale of market on jeans in the whole world is six trillion.

The conversation was done in 2000. The author said he found interesting that the estimated value on the scale of market both in Japan and the whole world on jeans was led from a conversation nonchalantly, so he remembers it.

【Estimation on Fermi and something approximate】

《Estimating a scale of market on jeans》

By the way the scale of market on jeans in Japan was about a hundred billion, and six trillion in the world in 2020

As to the scale of the domestic market is less half than the one which the author and his friend calculated, but the scale of market in the world is six trillion.

The author said he read an article on a magazine and the article said that a tendency of being away from putting on jeans among the youth has gone ahead. Some of jeans shop of which sales have dropped to half twenty years ago.

【Estimation on Fermi and something approximate】

《Estimating a scale of market on jeans》

The author said the three hundred billion isn’t beside the point. First of all this kind of presumption is all right unless it had a wrong number of digits.

The scale of market on the whole world is exactly right, but the whole population in the world is about 7.5 billion at present, so it must have been more then. In either case, it’s not beside the point.

《A father of nuclear power and estimation on Fermi》

Estimating approximate value like the conversation is called the estimation on Fermi.

【Estimation on Fermi and something approximate】

《A father of nuclear power and estimation on Fermi》

A question like how many manholes are there in Tokyo prefecture? has been prepared in an entrance exam for various companies recently, as a result, the estimation on Fermi seems to be indispensable skill when trying to get a job for new graduates.

There was no phrase of estimation on Fermi twenty years ago when the author was a student, but students who chose a course of science and math carried out it from long ago, estimating approximate value.

When making an experiment, they...

【An estimation on Fermi and something approximate】

《A father of nuclear power and estimation on Fermi》

When making an experiment, they formulate a hypothesis in the beginning. The hypothesis is an opinion which is set up temporarily so as to express a natural phenomenon. After the hypothesis is recognized officially by the experiment, it will be a new law or new theory then.

How much can we get a value from the experiment? We need to estimate it beforehand. Otherwise, we don’t know an experimental instrument to prepare for. How much precise does the experimental tool need? We need to know.

【An estimation on Fermi and something approximate】

《A father of nuclear power and an estimation on Fermi》

Furthermore, if estimating in advance, when getting value which was unusual, which means that the value was more different than we expected, we can guess it has possibility of making a mistake when making an experiment.

The origin of the assumption on Fermi is Enrico Fermi who is well known for a father of nuclear power. He received a Nobel prize for physics. He got a remarkable results as theoretical physicist and experimental physicist, and was a master of estimating approximate value

【An estimation on Fermi and something approximate】

《A father of nuclear power and estimation on Fermi》

Fermi had an episode like the next.

When a bomb exploded, he dropped a sheet of tissue paper, then the tissue paper whirled in the air by the blast. Fermi watched the orbit of the tissue paper and estimate of the amount of the gunpowder on the bomb roughly.

He went to Chicago University for a lecture, and he prepared for a question to freshmen like the next. Its famous.

How many piano tuners are there in Chicago?

Why did Fermi prepare for the question to the freshmen who adopted...

【An estimation on Fermi and something approximate】

《A father of nuclear power and an estimation on Fermi》

If living through in the world of physics, an ability on estimating something unknown is very important. The question was the message from Fermi to the freshmen.

The target here isn’t always mean that reaching the right value, which means the right number of the piano tuners, for if wanting to grasp the exact number on the piano tuner in Chicago, they have only to phone to a committee on the piano tuners, though it isn’t clear whether there is the committee like that, but therefore ..

【An estimation on Fermi and something approximate】

《A father of nuclear power and the estimation on Fermi》

...and the important thing for us is even if it’s unknown value, whether or not we can ask an approximate value with our data logically which we have already.

The author expressed a procedure on the estimation of Fermi, adopting the number of piano tuner in Chicago.

1『Formulating a hypothesis』

Framing a hypothesis that a demand and supply on the piano tuner is well balanced in Chicago and thinking over the number on the piano tuners who are indispensable in Chicago so as to tune...

《A father of nuclear power and the estimation on Fermi》

1『Formulating a hypothesis』

...and thinking over the number of the piano tuners who are indispensable so as to tune all the pianos in Chicago.

2 『Dividing a matter into several elements』

What is a necessary data and an amount of presumption we need so as to think over the question?

The population in Chicago.

The number of people in a single household.

A ration of the household which has the piano in the whole Chicago.

A number of times on tuning the piano for a year.

A number of times on a single piano tuner who does his own....

《A father of nuclear weapon and the estimation on Fermi》

2『Dividing a matter into several elements』

A number of times on the single piano tuner who does his own job for a year.

3『Making use of data which we’ve already known』

The indispensable data which we estimate the number of piano tuners in Chicago is the population in Chicago. Its population is about three million, though the number isn’t familiar with us, it seems to be common knowledge for the students who go to the Chicago University.

4『Calculating the amount of presumption on each element』

Amount of presumption on the number...

《A father of nuclear weapon and the estimation on Fermi》

『Taking advantage of data which we’ve already known』

Amount of presumption, 1, the number of the people in a single household.

How many households in a big city of which population is about three million? Needless to say, a household is one and the other is four and another is ten, but we set up the average number on the single household is three.

Amount of presumption 2, the ration on the household which possesses the piano.

How many households do they have the piano in the whole Chicago?

Though the situation may have been...

《A father of nuclear weapon and the estimation on Fermi》

『Making use of the data which we’ve already known』

Amount of presumption 2, the ration on the households which possesses the piano.

The situation may have been different between Japan and America, but please think over how many children they learned the piano in the class when you were in elementary school.

If the class is coeducational and made up with forty children, there are lots of cases that four or five children who learned to play the piano in the class, the author said, though the elementary school which the author went...

《A father of nuclear weapon and the estimation on Fermi》

3『Calculating the presumption number on the each element』

Presumption number 2, the ration on the household which possesses the piano.

...though the school which the author went to wasn’t coeducational, so the one who learned to piano was a few in the class.

So we set up that the households which possesses the piano make up for 10% in the class.

When being in junior high school or being in high school, lots of people seemed to stop learning to play the piano, so we should exclude the piano which no one played, for it doesn’t have...

《A father of nuclear weapon and the estimation on Fermi》

『Calculating the amount of presumption on each element 』

The amount of presumption 2, the ration on the households which possesses the piano.

... for the piano doesn’t have any chance of being tuned, so the number of the piano seems to be a little more, but except for the households, there are pianos in school, a public hall, so it’s all right.

The amount of presumption 3, the number of times on a single piano which is tuned a year.

A piano is usually necessary to be tuned once a year.

The amount of presumption 4, the number of ...

《A father of nuclear weapon and the estimation on Femir》

4『Calculating the amount of presumption on each element』

Amount of presumption, 4, the number of times on a single piano tuner who does its own job a year.

Thinking over the number of times on the piano for the single piano tuner tunes a year. How many of the piano are tuned?

Tuning the piano is a hard labor and takes a lots of time. Even if the piano tuner works harder, it can tune three pianos a day at most.

If the piano tuner is on a five-day week, and it works 250 days a year, 3×250＝750, so the number of the piano which the...

《A father of nuclear weapon and the estimation on Fermi》

4『Calculating the amount of presumption on each element』

...so the number of piano which the single piano tuner can tune a year 750.

5『A conclusion』

As being based on those things, we can estimate the number of the piano tuners in Chicago.

The population on Chicago is 3 million, and the number of the people who made up with the single household is three, 300÷3＝100, so the number of households in Chicago is a million.

The number of the pianos in Chicago is 10% of the households, so 100÷10＝10, so the number of pianos are a hundred...

《A father of nuclear weapon and the estimation on Fermi》

5『Conclusion』

...so the whole number of the piano in Chicago is a hundred thousand.

The number of times on the piano which is tuned a year is once a year, 100.000×1＝10, so the tuning on the piano is done a hundred thousand times a year in Chicago.

The number of the times on the piano tuner who can do tune three times a day and it works is 250 days a year on the condition that it adopts the five-day week, 3×250＝750. 100.000÷750＝133...so we can estimate the number of the piano tuners in Chicago is 133.

But the author said it’s his...

《A father of nuclear power and the estimation on Fermi》

5『Conclusion』

But the author said it’s his own estimation, so it doesn’t alway mean that the number of 133 alone is the right answer.

If the data which we’ve already known is combined with the amount of presumption appropriately, even if it’s not 133, its estimation seems to be right.

《The estimation on Fermi isn’t beside the point so much》

If trying in various ways, we can improve the way on Fermi estimation, so the author suggested we estimate numbers around us, for example, the number of cars which are sold a year, the amount...

【An estimation on Fermi and something approximate】

《The estimation on Fermi isn’t beside the point so much》

...for example, the number of cars which are bought a year, the amount of domestic consumption on wine, or distance in which a soccer player ran in a match

To tell the truth, if trying to estimate the value on Fermi in related to those things, its value isn’t beside the point from the true value.

It seems to be strange, but it means mixing each of element mutually. It hardly happens that all of the amount of presumption on each element is too much or to less.

A knack on the...

【An estimation on Fermi and something approximate】

《The estimation on Fermi isn’t beside the point so much》

A knack on the estimation of Fermi is dividing the question as minutely as possible.

The bigger the amount of presumption is, the more there is a risk that the amount of estimation is beside the point from the true value.

The author said he recommended that if the one who reads this sentence is weak at numbers, and has wanted to be good at numbers, it should challenge at the estimation on Fermi.

It seems to be harder a little, but the calculation which we adopt is simple, and we...

【An estimation on Fermi and something approximate】

《The estimation on Fermi isn’t beside the point so much》

...and we can try it with the way of easygoing and it’s all right as long as we don’t have the wrong number of digits.

Besides, if getting used to it a little more, we can estimate a thing which we have little knowledge pretty close, so we have fun.

The author says he will be happy if the one who read his book learns to like math because of the challenge.

【The number which comes to the head most frequently】

《What is the law on Benford》

There have been varied numbers around us.

【A number which comes to the head most frequently】

《What is the law on Benford?》

When reading a newspaper, a book or contents on the internet, the number always appears in front of us. Needless to say, a business performance, a telephone number, an address, a population, and a price on stock are expressed with some numbers.

Of course all the values are made up with a combinations of numbers. As to the number on the head, it means a number of the highest digit, it’s limited from 1 to 9.

By the way what is the number at the head of value in every conceivable kind of numbers?

Some people...

【A number which comes to the head most frequently】

《What is the law on Benford?》

Some people thought as a variety of number appears each of them has the same probability.

Others thought that it’s irregular, and it depends on, so It’s different according to its time and place, so we can’t foresee it.

But the way of appearing the number in the head is regular conspicuously. The author said he was going to show us here it from now on.

In fact it turns out that a ration on the way of appearing the number isn’t equal. The one which appears in the head the most frequently is 1, and the....

【A number which comes to a head most frequently】

《What’s the law on Benford?》

...and a ration of the value which starts from 1 accounts for about 30%.

If each of numbers from 1 to 9 appears equally, its probability is it’s one-ninth, and one-ninth≒11%, so 30% is high ration. As it’s the head number 0 is excluded.

The larger the number of the head is, the less its ration of which the number appears there was.

The ration of the value which starts from 9 makes up for no more than 20% in the whole number. We call it the law on Benford.

There is a graph and a list in the book. Numerical...

【The number which comes to a head most frequently】

《What is the law on Benford?》

A ration on the numerical value of which head is one from three is over 60% in the whole number, according to the graph and the list.

It was Frank Benford who showed the law for the first time in 1938. He was an American physicist. It is said he collected samples which was over twenty thousand like an amount of molecule, a population or a number which appears at articles in newspaper, and reached the conclusion.

He seemed to gather various kinds of sample. For example, loss on HP, or a black body.

【The number which comes to a head most frequently】

《What is the law on Benford?》

As to the loss on HP, HP is heat pump and it’s an instrument which collects heat and the loss on HP means that the energy on the instrument is lost.

The black body is a material which never reflects the light.

To my sorrow, I don’t know anything on physics, and as to the two of them, it’s just that I’ve expressed as the book says.

Average value on the samples which were over twenty thousand were close to the value on the theory of Benford.

When paying attention one by one, some distribution on numbers ...

【The number which comes to a head most frequently】

《What is the law on Benford?》

If paying attention one by one, the distribution on numbers like the area of a river’s basin, numbers which appear at articles in newspaper, a pressure, a design, and address are close to the values on the theory.

On the other hand, physical constant 物理定数, the amount of molecule, and the amount of atom, they have a bigger margin of error comparatively 誤差.

By the way when having read until now, I’m afraid you aren’t satisfied with my expression, but the ground on Benford is from now on. Please just a moment.

【The number which comes to a head most frequently】

《Germs increases in w way of exceptional function 指数関数》

Some of numbers correspond to the values on theory, but others don’t. Why?

Let’s think over a reason intuitively why the law on Benford stands up.

For example, some number becomes double within the set time in natural world like germs increase. It’s not unusual.

If it becomes a double in a year, when there is a hundred, it becomes two hundred in the next year. After two years, four hundred. After three years, eight hundred, and after four years, one thousand six hundred.

This way...

【The number which comes to a head most frequently】

《Germs increase in a way of exceptional function》

This way of increasing is called an increase of exceptional function.

In this case, it takes a year to increase from 100 to 200, and then its head of the number remains 1 until the next year.

On the other hand, if the head number is 5, for example when the number increases from 500 to 600, it’s no more than three months, but it doesn’t always mean that I’ve calculate if for myself, but the book said so, it takes three months.

In the same way, if increasing from 1000 to 2000, it takes...

＞＞231

To the person who responded 231

Thank you for your message and congratulation on being a student in Hosei University. To be exact, I’m not ぬし,for instead my sister built thread for me.

It’s just that I’m interested in English, so I’ve started to study English at the age of 45. I’ve never been to any university, for my school record wasn’t so good.

It seems that you will be in difficulty for the time being from now on, but lots of people around expect you. Please good luck.

I’m not sure whether or not my English so wonderful as to encourage you, but I’m going to do my best.

＞＞231

231にレスした方へ。

メッセージありがとう、そして法政大学在学おめでとう御座います。正確に言えば、主は自分ではなく、妹が自分のためにスレを立ててくれました。

英語に興味があったので、45歳から英語の勉強を始めました。学校の成績は良くなかったので、大学には行けなかった。お恥ずかしい。

これから暫くは大変ですね、でもあなたの周りの大勢の人があなたに期待しているんですね、頑張って下さい。

あなたを励ませるかどうかは分からないけど、自分も英語の勉強を頑張ります。

- << 238 有り難うございます。 頭が私すごーく悪いため、 仕事、家事手伝いが不可能となりました。実はIQ76しかないんです。 (詳しくは私のスレで！ せんか慎重派さんのハンドルネームです。) だから、学校に行く以外、ないのです。 こういうとりえがなんもない。 スタイルも163センチ、75キロ以上の学習障害、発達障害の、中学校にてあまりに低いIQだから、大人になり自力で生きる力がありません。 進学校進学 専門学校 短大、大学ましてや院。 高校生アルバイトや、大学からの正社員の仕事もみんな不可。担任にボーダーのIQを心配され、親が呼び出された三者面談がありました。親は教室や公共交通機関の中でも、声にあげておいおい泣き伏し、家庭に帰宅ののち、すっごい剣幕でおこられ、生むんじゃなかった、元気ないからよ。シャキッとしなさい。見た目も。同時にお父さん、おばあちゃん、お姉ちゃんにも。15才の時のはなしです。私は地頭がとてもよくないから、一生懸命学歴を手にして、いじめられたり、囃し立てられたり、下に見られ、もうなめられたりしたくはないから、 長々、ヒトのスレ内にすみませんでした。

>> 236
＞＞231
231にレスした方へ。
メッセージありがとう、そして法政大学在学おめでとう御座います。正確に言えば、主は自分ではなく…
有り難うございます。

頭が私すごーく悪いため、

仕事、家事手伝いが不可能となりました。実はIQ76しかないんです。

(詳しくは私のスレで！

せんか慎重派さんのハンドルネームです。)

だから、学校に行く以外、ないのです。

こういうとりえがなんもない。

スタイルも163センチ、75キロ以上の学習障害、発達障害の、中学校にてあまりに低いIQだから、大人になり自力で生きる力がありません。

進学校進学

専門学校

短大、大学ましてや院。

高校生アルバイトや、大学からの正社員の仕事もみんな不可。担任にボーダーのIQを心配され、親が呼び出された三者面談がありました。親は教室や公共交通機関の中でも、声にあげておいおい泣き伏し、家庭に帰宅ののち、すっごい剣幕でおこられ、生むんじゃなかった、元気ないからよ。シャキッとしなさい。見た目も。同時にお父さん、おばあちゃん、お姉ちゃんにも。15才の時のはなしです。私は地頭がとてもよくないから、一生懸命学歴を手にして、いじめられたり、囃し立てられたり、下に見られ、もうなめられたりしたくはないから、

長々、ヒトのスレ内にすみませんでした。

【The number which comes to a head most frequently】

《Germs increase in a way of exceptional function》

In the same way, when increasing from 1000 to 2000, it takes a year, but when doing from 5000 to 6000, it takes no more than three months.

In other case, as to the change on increase of exceptional function, the period when the head number is 1 above all longer than any other period in other numbers come.

《Even if its unit changes, is it the same nature?》

Even if it doesn’t change in a way of exceptional function, the law on Benford is applied to some cases. They are the ones in which...

＞＞236,238

To the person who responded 236,238

Thank you for your message. I’m wondering if it is mortifying for you until now.

236, 238 にレスしてくれた方へ。

メッセージ有難う御座います。今まで悔しい思いをしてこられたのだろうと思います。

I used to express it in English but do in Japanese for the first time here, at ミクル. I’m sure I’m suffering from what is called Asperger Syndrome.

英語では言った事があるけど、日本語ではここ、ミクルでは初めてです。自分はアスペルガーだと思います。

I’m afraid you have been distressed by your situation and will be in difficulty from now on, but I’m also on the same boat and we shouldn’t be daunted.

あなたは、今まで辛い思いをしてこられ、これからも大変だと思います。でも、自分も同じです。挫けちゃいけない。

＞＞241

Good luck and please do your best.

頑張れ〜💪

＞＞239

【The number which comes to a head most frequently】

《Even if an unit on measure is changed, is it the same nature?》

This is the case in which the numbers are lined up in order from the 1 like membership number. Then the number which starts from 0 is excluded.

For example, let’s suppose there was a fan club made up with five thousand of members. Then each of membership number of which the head number is either 5,6,7,8,or 9 is extremely less than the each one of which number is 1,2, or 3.

【The number which comes to a head most frequently】

《Even if an unit on measure is changed, is it the same nature?》

There is a list on the book.

When the number of membership is from a thousand to ten thousand, researching the head number of the numbers of membership at a place where is good to count numbers like 1000, 2000, or 3000, to 10000l.

The number of which the head is 1 is more than any other number of membership of each place like 1000, 2000. It has been showed on the list, but to tell the truth, I’m not sure of the way of looking at the list, so I’m not sure of it.

【The number of which head comes most frequently】

《Even if an unit on measure is changed, is it the same nature?》

Except for the number of memberships of which number increases in order, in a case in which the numbers are scattered in some extent with almost the same ways like a population or a length of a river, the same phenomenon occurs, so the law on Benford is applied well.

As to the number of membership, I can understand, but I’m not sure in relation to the population and the length of the river.

A line of numbers which is determined by other rule like a telephone number, and a...

【The number of which head comes most frequently】

《Even if an unit on measure is changed, is it the same nature?》

...and data controlled by normal distribution like the score in the National Center Test for University Admission センター試験 isn’t applied to the law on Benford. The normal distribution is the most important part in statistics. Its shape becomes a temple bell.

In addition, as to a set for numbers of which value is limitless and are showed at random, it isn’t applied to the law on Benford.

However if picking up from some data which isn’t applied to the law on Benford like numbers...

【The number of which head comes most frequently】

《Even if an unit on measure is changed, is it the same nature?》

But if picking up some data at random from some distributions which aren’t applied to the law on Benford, it proved out that they are applied to the law on Benford.

Even if I can understand the law on Benford means, I can’t grasp why it is until now, except for the numbers of membership.

《A knack on judging injustice with regard to numbers》

How is the law on Benford useful in our society?

Hal Varian who designed a model for advertising which was a source of income at the...

【The number of which head comes most frequently】

《A knack on judging injustice with regard to numbers》

Hal Varian who designed a model for advertising on source of income in Google at the dawn is said to be a economist who made Google a number one company in the world.

He said if putting the law practical use, we can see a window-dressing settlement 粉飾決算.

Without knowing the law on Benford, the one who tries to disguise an amount of money at an account book in company, it tends to express the numerical value with a distribution of being too much equal head number, on the other hand it’s...

【The number of which head comes most frequently】

《A knack on judging injustice with regard to numbers》

...on the other hand, it’s apt to express the distribution which leans some places alone too much.

Then the ration in which the value starts from 1 is against the law on Benford, so we can discover it’s a false data.

In fact, the next thing happened actually in the beginning of 1990s.

An instructor in an accounting school gave his students a lesson. He said his students to make sure whether or not each number of biggest digit on earnings and expenses of a company corresponded to the law.

【The number of which head comes most frequently】

《A knack on judging injustice in regard to numbers》

There was a student whose relative managed a hardware store, and the number on the account book was different from the law on Benford so much, so the student discovered it and it led to the discovery of injustice on the account book.

Except for auditing 会計監査, the law on Benford has been adopted for verification on unfair voting in an election as well at present.

【How to find a way of useful information】

《Digging up unfair usage from a mountain of data》

The author was sure that lots of...

【How to find a way of useful information】

《Digging up unfair usage from a mountain of data》

The author said he was sure that lots of people have heard of a word, data mining. It has suddenly learned to be used a few years in the past with other phrase, big data.

If translating the data mining word for word, it means mining potential needs from data. It originally started to be used in a studying field called Knowledge Discovery in Database from the latter half in 1990s.

After that the internet has spread over the world due to the IT revolution and knowledge on computer has expanded by...

【How to find a way of useful information】

《Digging up unfair usage from a mountain of data》

...and ability on computer has extended by leaps and bounds, so what is called a big data has been piled up in the world of business. As a result, analyzing enormous data in ordinary society, drawing from useful information which was unclear on its value, and a phrase of data mining which has such a nuance was born.

It was in 2010 when the term of big data was introduced in an English business magazine, Economist for the first time. From then a profession called data scientist was on the rise.

【How to find a way of useful information】

《Digging up unfair usage from a mountain of data》

The data scientist is the one who specialized in analysis on swollen data and rendered great service to a company or to the society.

It is said an article in a magazine of Wall Street Journal in which the thing on data mining was Introduced for the first time. It was issued on the 23rd day December in 1992. Its content was like the next.

It turned out that when a major supermarket analyzed data on a cash register in America, it turned out to be the next thing.

A customer who bought a disposable...

【How to find a way of useful information】

《Digging up unfair usage from a mountain of data》

A customer who bought a disposable diaper from five to seven in the evening tended to buy a beer together then.

We can consider from the fact that in a house which has a child, a wife asked her husband to go shopping for the disposable diaper in the evening, and her husband happened to buy the beer then.

Besides the ones who are engaged in the supermarket can expect if displaying the beer and the disposable diaper side by side, both of the sales increase.

The author said his credit card used to...

【How to find a way of useful information】

《Digging up unfair usage from a mountain of data》

The author said his credit card used to be make use of illegally, but fortunately it wasn’t serious damage, for the company which dealt with his credit card made inquiry to him by telephone like the next.

You took advantage of your credit card in the store of I Tunes store, and spent three thousand yen on some articles then. Are you sure of it?

The inquiry added to the day when the author went shopping.

The author seemed to go shopping there, but he didn’t buy the articles, and he said so.

【How to find a way of useful information】

《Digging up unfair usage from a mountain of data》

The staff who made inquiry said.

I see. Then as a illegal usage on your credit card was brought to light, we’re going to make your credit card invalid, but there is no request for illegal usage on your credit card, so please don’t be worried about it.

The author said he was relieved to choose the company who handled her credit card, but how did the company which handled her credit card find the illegal use? The amount of money on which she spent was no more than three thousand yen, and the site...

【How to find a way of useful information】

《Digging up a illegal usage from a mountain of data》

...and the site in which she took advantage of was the one she used in the past, but finding something illegal was exactly done by data mining.

The author said he usually went shopping in both actual shop and in the shop of the internet. Some of them are the ones where the author went shopping for the first time, and the amount of money on which he spent is various.

But if analyzing his record on shopping in the past, there is a fixed rule though the author isn’t aware of it, and the company....

【How to find a way of useful information】

《Digging up illegal usage from a mountain of data》

...and the company can pick up articles from which is different in his record in the past.

All the data on the credit card of which customers made use in the past has been accumulated in the company which handled the credit card. They are very important information not only for discovery on illegal usage but for a marketing of the company which handled the credit card.

Linking a profile of an address, age, sex gender and occupation on a customer with the record on shopping in the past, so the...

【How to find a way of useful information】

《Digging up illegal usage from a mountain of data》

...for example,so the company can deduce a man who lived in Yokohama city and whose age was forty, and who was engaged in a free lance and whose tendency on shopping. What kind of things do the customer buy?

Needless to say, it leads to efficient advertisement and development on goods which fills a niche.

《A mutual relationship and a casual relationship》

In general when the one increases, the other does, we call that rough relation a mutual relationship. It’s a positive mutual relationship.

＞＞259

To the person who sent the message in no.259.

259にレスしてくれた方へ

Thank you for your message and it seems to be a lot of trouble to go to university in various way from now on. I hope you can carry out your original purpose.

メッセージ有難う、大学に行くのも色々大変みたいだね。初志貫徹される事を願います。

By the way, I have a favor to ask of you. If you have something to talk with me, instead of responding here, please sending your message to other thread, 初心者英語講座 or Let’s enjoy English eleventh.

ところで、お願いがあるんだけど、もし話したい事があるならここじゃなくて、初心者英語講座か、Let’s enjoy English eleventh の方にメッセージを送って欲しい。お願いします。

- << 293 了解しました。

【How to find a way of useful information】

《A mutual relationship and casual relationship》

When there is a tendency that the one increases but the other decreases, we call it a negative mutual relationship.

If we can find the mutual relation in an unexpected combination like the disposable diaper and beer in the article of Wall Street Journal, the people who work in the supermarket may be able to expect its sales improves.

A discovery on the mutual relationship is a mainstay of data mining, but when researching the mutual relation, we have to be careful of two things.

The one is the...

【How to find a way of useful information】

《A mutual relation and a casual relation》

The one is the relation which we got is the result on the investigated object. The author continued.

There is the positive mutual relation in students of his cramming school like the next.

The better the English record is, the better the record on math is.

However it’s just that there is the tendency like that, so there is also an exception, and it’s not clear whether or not the tendency corresponds to the students in high school all over Japan.

When happening to find the mutual relation from...

【How to find a way of useful information】

《A mutual relation and a casual relation》

When happening to find the mutual relation in an unexpected combination or to get a result as we expected, we tend to speak loudly, we’ve found a marvelous or favorable law, but without all the populations 母集団, we have to be especially careful of our judgement.

One more thing.

Even if we can find the mutual relation between two of things, we can’t conclude they have a relation on a cause and a result.

If there is the relation on the cause and the result between X and Y, there is the mutual relation...

【How to find a way of useful information】

《A mutual relation and a casual relation》

If there is the relation on the cause and the result between X and Y, we always recognize the mutual relation, but we can’t say its opposition is always right.

《There is a person who read a newspaper. Is its annual income high?》

If there is the positive mutual income between X and Y, there are five possibilities like the next.

1 There is a relation between the cause which is X and the result which is Y.

2 There is a relation between the cause which is Y and the result which is X.

3 Both of X and Y...

【How to find a way of useful information】

《There is a person who reads a newspaper. Is its annual income high?》

3 Both of X and Y are results from Z which is a mutual cause of both X and Y.

4 There is a more complicated relation.

5 It’s an accidental thing.

As to 1 and 2, let’s suppose that there was a positive mutual relation between buying newspaper and annual income. Then we may have expected that if reading the newspaper, our annual income may will.

But there is a possibility that it’s not reading newspaper leads to be high annual income, but being high annual income leads to...

【How to find a way of useful information】

《There is a person who reads a newspaper. Is its annual income high?》

...but there is a possibility that being high annual income leads to raising its annual income.

As being high annual income and its social position improving the person need to read the newspaper for a topical gimmick 話題作りas sociability.

As to 3, if a person thinks if the profit on a zoo increases, the profit on a beauty parlor 美容院 also increases, so the profit on the beauty parlor was caused by the profit on the zoo increase. The person was in the wrong clearly.

Both of the...

【How to find a way of useful information】

《There is a person who reads a newspaper. Is its annual income high?》

Both of the zoo and the beauty parlor are crowded on holidays rather than weekdays.

Since it’s the third cause of being on holidays, so both of the zoo and the beauty parlor increased, so it’s appropriate to think over that there is no relation between both of the profits in relation to the cause and result.

Being on holidays made the profits on the zoo and the beauty parlor increase. The single cause of being on holidays leads both of them profits, though they have nothing to...

【How to find a way of useful information】

《There is a person who reads a newspaper. Is its annual income high?》

...though they have nothing to do with, so two results form the single cause were linked together. We call the mutual relation in appearance a suspected mutual relation.

As to 4 or 5, for example, a number of students in elementary school who take exam on junior high school tends to increase in a metropolitan area after 2015.

In addition the ones who make use of Instagram on SNS have increased in these five years in the past, but the cause that the ones who takes exams in...

【How to find a way of useful information】

《There is a person who reads a newspaper. Is its annual income high?》

...but the cause that ones who take exams on junior high school in the metropolitan area leads to a result that the ones who take advantage of instagram increase, or its opposition, the cause that the ones who make use of instagram increase leads to the result that the ones who take exam on junior high school increase in metropolitan area.

No one think like both of them, and it is hard to think that there is the third cause which is in common to both of them.

The ones who take...

【How to find a way of useful information】

《There is a person who reads a newspaper. Is its annual income high?》

The ones who take exams on junior high school in a metropolitan area have increased. There seem to be some factors in its background.

For example, educational cost per a single child has increased owing to dwindling birth rate, anxiety on unclear reform of taking exam on university, or expectation on private junior high school which displays its originality and takes care of students have increased.

Besides, the ones who make use of instagram have increased. In its background...

【How to find a way of useful information】

《There is a person who reads a newspaper. Is its annual income high?》

In the background that the ones who make use of instagram, a smartphone has spread over, lots of people have learned to use hashtag, or there is a vogue word of what is called インスタ映え.

Increase on the number of the ones who take exam on junior high school in the metropolitan area and the ones who take advantage of instagram have increase in the same period. Both of them may have been connected closely in a complicated way or it’s just an accidental thing.

Anyway, it’s very hard...

【How to find a way of useful information】

《There is a person who reads a newspaper. Is its annual income high?》

Anyway it's very hard to make sure whether or not the relation between a cause and result is formed, in particular as to a mutual relation in which we got by researching a part of population, so we have to very careful of it.

《A correct statistics and a wrong statistics》

There is phrase like the next.

There are three kinds of lie in society. An ordinary lie, a downright one and a statistics.

A result by statistics is sometimes showed us through a graph or numerical value...

【How to find a way of useful information】

《A correct statistics and a wrong statistics》

...then it has a persuasiveness overwhelmingly.

In fact, if being said it was the statistics, lots of people who felt an atmosphere in which none of them can refute, didn’t they?

But the statistics isn’t always right, for its data lean in one place, the date wasn’t handled appropriately, or in a worse case, the data itself was sometimes falsified, but without being died out, the wrong statistics sometimes got out of control due to its persuasiveness.

When Trump was elected as President in the U.S.A....

【How to find a way of useful information】

《A correct statistics and a wrong statistics》

When Trump was elected as President in the U.S.A.for the first time, almost none of the mass media foresaw he won in advance.

Lots of the mass media, from the New York Times, referred a public opinion poll as statistics, and reported as if Hillary Clinton had won certainly all together, but she couldn’t win.

But as we can’t ignore a numerical value on account of what is called AI, a correct statistics and a wrong one will be jumbled up more and more in the near future, so we should master an ability...

【How to find a way of useful information】

《A right statistics and a wrong statistics》

...so we have to master a true ability of literacy on statistics. It’s the ability of reading and understanding information correctly and determine a rational will. Then we need to be the one who can withdraw indispensable and correct information from data which is a mountain of treasures.

【Statistics have changed nations】

《There is statistics where there is a nation》

Roughly speaking, math is divided into two things. Pure math and applied math. Pure math is the one of studying abstract conception...

【Statistics has changed nations】

《There is statistics where there is a nation》

Pure math is the one of studying an abstract conception with a strict logical thought.

Applied math is the one of studying a way or way of putting practical use. It’s put in practical use into natural science, social science and industry with the theory cultivated by the pure math.

In plain language, a goal for the applied math is studying how making math be useful in the society actually.

As to the pure math, there are main three fields.

The one is a solution on equations and algebra which is a solution...

【Statistics has changed nations】

《There is statistics where there is a nation》

The one is algebra which is a solution on a equation and liner algebra, number theory and group theory have been studied in algebra.

The other is analysis in which center is differential and integral calculus 微積分 and studying the whole function.

The last one is geometry which studies a figure and nature on space.

On the other hand, applied math has expanded several learning area so much that the subject on study are various, and it made an impression of expanding day by day, especially statistics which has...

【Statistics has changed nations】

《There is statistics where there is a nation》

...especially statistics which has been paid attention recently and its usefulness has received high opinion.

How important statistics is for us who have lived in the present age, it has been said repeatedly in public opinion, so the author said he was going to express an outline that how statistics was born and how it has developed.

Understanding the history on statistics is useful so as to understand what statistic is, the author said like that.

The word of statistics is originated from Latin language...

【Statistics has changed nations】

《There is statistics where there is a nation》

The word of statistics is originated from Latin language of status. The word was born so as to investigate an actual situation on the nation by a ruler. For example a population on the nation.

There is a record in Bible that parents of Jesus Christ stayed in Bethlehem before Christ was born. In spite of being pregnant, Maria who was the mother of Christ had to go to Bethlehem, for the Ancient Roman Empire ordered the people to return a town where their ancestors were from in order to...

【Statistics has changed nations】

《There is statistics where there is a nation》

...for the Ancient Roman Empire ordered the people to return to a town from which their ancestor was from in order to investigate the population.

Maurice Bloch who was a French statistician in the 19th century said there is statistics where there is a nation.

Actually there is a record in the Ancient Egypt of investigation on the population and on the land was done so as to build the pyramid.

In Japan the investigation which was related to area of rice field was done in Asuka era.

Hideyoshi Toyotomi also...

【Statistics has changed nations】

《There is statistics where there is a nation》

Hideyoshi Toyotomi made an investigation on family registration in the whole Japan so as to grasp a military power to send troops to Korea.

Judging from those things, the author said he was sure that statistics has developed as the one which is indispensable for ruling nations.

If the one who ruled over the nation tries to collect tax or to draft people for a war, the ruler had to know how many people there were, or what kind of things they produced. It's natural.

《A modern nation has made much of statistics》

【Statistics has changed nations】

《Modern nations have made much of statistics》

From the 18th century to 19th century when modern nations were formed, rulers recognized statistics was indispensable as base to rule countries more and more in each country, so arrangement on structure or investigations for statistics were done actively.

Then a modern national census for all the people who lived in the country was done. Population and contents of each household were investigated.

Napoleon used to say statistics is a budget for things. Without founding the budget, there is no public welfare.

【Statistics has changed nations】

《Modern nations have made much of statistics》

A statistics bureau was set up in France in 1801. It was done first in the world.

An investigation which is done for all the groups as object without exceptions like the national census in modern nation is called a completely survey or census.

It was John Graunt in England who opened up a world on new statistics which draws a line between the investigation on population in the ancient nation a few thousands years ago.

Then churches preserved a number of dead people as data. As being based on the data, John...

【Statistics has changed nations】

《Modern nations have made much of statistics》

As being based on the data, John gathered together a death rate which was preserved with each of age into a booklet. He analyzed the data and it turned out that the death rate was higher in childhood or it was higher in a city than a province.

Then the population on London was thought to be about two million, but he estimated it three hundred eighty four thousand through the data. He showed us we can estimate the whole number from the limited data.

Not only gathering together data simply, but by observing...

【Statistics has changed nations】

《Modern nations have made much of statistics》

Not only gathering together the data simply, but by observing the data, we can find a law among complicated things which was apparently disorder, Grant showed us it, so his analysis was epoch-making.

The technical skill of Grant was succeeded to Edmond Halley in England who is well-known the one who discovered the comet of Halley.

Halley made Newton write Principia which is a famous book in the century, and he published it at his own expenses. He had lots of ...

【Statistics has changed nations】

《Modern nations have made much of statistics》

Halley has lots of scientific achievements, and was the one who made a list for life for the first time among us the humankind, being based on data of birth and death in a town.

He showed us clearly there is a law on death of people, and a premium 保険料 in life insurance should be calculated, being based on death rate from each of age in a book which he published for himself.

There were some of companies of life insurance in England then, but the premium was established at random.

With the achievement...

【Statistics has changed nations】

《Modern nations have made much of statistics》

With his achievements, the companies of life insurance have learned to be able to calculate rational premium at last.

Making an investigation, gathering together data, and arranging them into a numerical value, a list and a graph and grasping the tendency and nature shown by the whole data like the booklet gathered together by Grant or list for life by Halley. We call it descriptive statistics.

But the way which Halley and Grant adopted was extremely so simple that it is hard to say it leads to the modern ...

【Statistics has changed nations】

《Modern nations have made much of statistics》

But the way which Grant and Halley adopted was extremely so simple that it is hard to say it leads to the modern descriptive statistics direct.

As being based on data which they got actually, even if they tried to solve an actual situation or mechanism on social phenomena, math itself had never developed enough then.

After that, theory on probability or normal distribution such as mathematical preparation was done by Pierre-Simon Laplace in France and Gauss in Germany, the one who tried to put into it ...

【Statistics has changed nations】

《Modern nations have made much of statistics》

...the one who tried to put the theory on probability or normal distribution into practical use for the society appeared. It was Quetelet in Belgium.

The concept of average used to be used in the world of natural science alone until then, but Quetelet adopted it in the human society for the first time.

He thought we the humankind was also a part of the cosmos which should be harmonious with everything, even if each of individuals took into action with each own free will which wasn’t restricted from anything...

【Statistics has changed nations】

《Modern nations have made much of statistics》

...if collecting data, we can find a scientific order in the whole society, Quetelet thought like that.

As he applied a statistical way which leads to the present age to the society for the first time, he is called a father on modern statistics.

If the history on statistics stopped at the descriptive statistics, statistics won’t be so important learning as at present.

As statistic inference 推計統計 has developed in the 20th century statistics becomes indispensable for in a life or a study at present age.

【Statistics has changed nations】

《Modern nations have made much of statistics》

While we know its tendency and nature from data on hand, it’s the descriptive statistics, we presume the nature on the whole thing from the data which we gathered. It’s inference statistics.

It resembles when ladling up a spoonful of miso soup from the miso soup which was stirring in a pot and tasting its seasoning.

When having wanted to know the result on an election, sending out questionnaires to all the eligible voters, or if having wanted to control a quality on industrial goods, inspecting all of them.

【Statistics has changed nations】

《Modern nations have made much of statistics》

Both of them aren’t realistic.

Then instead of giving it up, without inspecting all of them, we can’t know it, inspecting some of them and if being able to say something on its probability clearly, it is useful.

《An experiment on milk tea》

The statistics inference has started by Ronald Aylmer Fisher who was an English statistician.

He made a famous experiment at a tea party.

He held a tea party at a garden and enjoyed himself with some of his friends at the end of 1920s. Then a lady who love the tea said.

【Statistics has changed nations】

《An experiment on a tea》

She said when making a milk tea, if pouring milk into a cup first or pouring tea into the cup first, it’s important, for its taste is different.

When hearing it gentlemen who joined in the tea party turned their noses up at her, and said, it couldn’t have been it, for if being mixed up once, it’s the same, and none of them took her seriously.

Then Fisher said, let’s make an experiment and proposed like the next.

They prepare for eight cups of milk tea where the lady didn’t look at. Four of them are pouring milk the first, and ...

【Statistics has changed nations】

《An experiment on a tea》

...and other four are pouring tea first.

In the next offering the lady the eight cups of milk tea at random and make her guess each of milk tea, which the first one is, tea, or milk one by one, but the lady is informed to be offered two kinds of milk tea at random, and four of them which are pouring milk the first, and other four are tea the first beforehand.

When answering, she was perfect. She guessed whether the tea was the first or the milk was the first right, and other gentlemen around her said she happened to succeed in it...

【Statistics has changed nations】

《An experiment on a tea》

...so the gentlemen tried not to admit they lost, but Fisher recognized that when she said nonsense, the probability of guessing all of eight cups of milk tea right was no more than 1.4%, so he concluded that she didn’t happen to guess them right. She could tell difference between the taste on milk tea.

Statistical inference has two mainstay.

The one is an estimation. Investigating a sample and estimating the characteristic on a population as probability.

The other is 検定. We get a data from the sample and verify its difference...

【Statistics has changed nations】

《An experiment on a tea》

...and verifying the difference on the data from the sample is an accidental error or a difference which is something meaningful.

For example, an audience rating and a quick report of election results are an estimation and 検定 is proving a reliability on hypothesis that two cups of coffee prevents a cancer from generating.

The experiment which Fisher made was 検定 itself, and it’s the most famous experiment on statistical inference.

By the way, I’m sure I should express 検定 in English, but the one in math seems to be different ...

【Statistics has changed nations】

《An experiment on a tea》

...but the word of 検定 which was adopted in math and the one which we used every day life seem to be different, so I express it in Japanese.

《The most advanced statistics》

It’s what is called Bays statistics which is a big current in the field on statistics in the 21st century. It’s based on a theorem on Bays which Thomas Bays in English contrived.

The Bays statistics is the most advanced statistics in 21st century, but Bays himself was a clergyman and a mathematician in the beginning of the 18th century, but he’s paid....

【Statistics has changed nations】

《The most advanced statistics》

...but he has been in the limelight in the 21st century. The theory which is put into practice in the modern society has been buried more than two hundred years. Why?

It is said it’s because the Bays statistics has two of the next natures.

1 An arbitrary choice is allowed. It means the statistic doesn’t need logical necessity.

2 Its calculation is sometimes complicated.

The nature on 1 the theorem on Bays had been criticized by mathematicians who loved strictness for a long time, but what we don’t need logical necessity...

【Statistics has changed nations】

《The most advanced statistics》

...but what we don’t need logical necessity means, in other words, it has a merit that the theory is able to be put into practice for a situation which isn’t strict. It has turned out recently. We can’t always establish all the conditions strictly at present age.

But in the Bays statistics, we’re allowed to establish a parameter, being based on our experience and common knowledge.

The parameter is some number which changes.

As a result, the Bays statistics allowed to adopt an intuition, so we can put into the Bays theory...

【Statistics has changed nations】

《The most advanced statistics》

As a result, we can put into the Bays theory into practice a thing which we can’t handle before.

As to 2, the calculation is complicated, almost all of everyone can use an computer at present, so there is no problem.

Statistics started in the ancient country, and developed into the Bays statistics supported by math and computer.

The expression on Bays statistic continued in the book, but I have none of knowledge on statistics, so it doesn’t appeal to me at all, but the author continued like the next.

Number and judgement...

【Statistics has changed nations】

《The most advanced statistics》

Numbers judgement is a ground for an estimation through the statistics in the present day life. Numbers express the society and change the society.

What’s the use of learning math? Statistics is one of its answers.

What’s the use of learning English? As for me, then I have fun, so if I can think like that, I’ll learn math, maybe, though I can’t promise it.

【Let’s overcome a big number with N ary】

《When taking a glance, how many are there bars?》

N ary means N 進法, but the phrase isn’t included in my electric dictionary, so...

【Let’s overcome a big number with N ary】

《When taking a glance, how many are there bars?》

...so I looked up it in the internet, so there is N ary, N進法.

I’m afraid some of people are confused with my English, so I’m sorry for it. Though I’ve done my best, I read again before contributed, and I find it perfect, then I contributed my expression to this bulletin board, but sometimes it didn’t go well as I expected.

To return to my main topic, I’m going to start again.

There are two illustrations on bars in the book.

The first illustration is there are 13 bars. They stand vertically and...

【Let’s overcome a big number with the N ary】

《When taking a glance, how many are there bars?》

...and they stand vertically and parallel each other.

The second illustration is also on bars, and the number of bars are the same, but the way of expressing is different.

It was expressed with three groups.

There are four bars and they stand vertically and parallel each other and a single bar is crossed over four bars, and there are more five bars like that. There are three bars and they stand vertically and parallel each other but any bar isn’t crossed them.

There are the same numbers of...

【Let’s overcome a big number with N ary】

《When taking a glance, how many are there numbers?》

In short, there are two illustration of the same numbers on bars.

When taking a glance at the first illustration, can you see how many numbers are there bars at once? It is said the number which we can recognize at once is a few, at best four, without counting on their fingers, lots of people find it hard to count the number of bars.

As to Roman number on 1,2,3, it’s expressed with Ⅰ,Ⅱ,Ⅲ, but 4 isn’t expressed with ⅠⅡⅠ, it was Ⅳ, for if using ⅠⅡⅠ, there are some people who can understand it’s 4...

【Let’s overcome a big number with N ary】

《When taking a glance, how many are there bars?》

...for there are some people who can’t understand it was 4 at once, though 4 used to be expressed with ⅠⅡⅠ in the Roman number before.

If using the way on the second illustration, without counting on fingers, we can count easily.

When countering numbers, getting together by five and using the mark on the second illustration was done widely before, and we call the mark a five-bar gate, according to the book.

We the Japanese use 正, and count by every five, but if the number...

【Let’s overcome a big number with N ary】

《When taking a glance, how many bars are there?》

...but when the number increases, it doesn’t always mean it’s useful. For example, if trying to express 96 with 正, I’m sure almost all of us will be tired of it, so we’ve contrived the position of decimal point. 位取り

Five is expressed four bars which vertically and a bar is crossed over them. If the mark is five, two bars are crossed, then 341 is three of two bars are crossed and four of a bar is crossed and a fraction 端数 is one.

《A reason why the decimal system has spread over among us》

N ary N進法...

【Let’s overcome a big number with N ary】

《A reason why the decimal system has spread over among us》

The N ary is in the position of decimal point when a number gather to some extent, we call it N temporarily, we regard it a lump of number, and move to the second digit. We call the situation the number was carried up 繰り上がる.

I wanted to show it with an example in relation to the word of carry up, but there is no example for it, so I can’t show you the example. I’m sorry for it.

As to the way of showing 341, when 5 is gathered, it’s one of the lump of number and the number is carried up...

【Let’s overcome a big number with N ary】

《A reason why the decimal system has spread over among us》

...and the number is carried up to the next digit, so it’s the quinary 五進法.

The one which we usually is the decimal system. If writing 324, it means 3 × 100 ＋ 2 × 10 ＋ 4. If expressing the number with the Chinese numerals, it’s 三百二十四, and it’s easy to understand.

The third digit is a lump of number which is 10 is gathered 10, so it’s a position of 10×10.

As to the N ary, let’s suppose there was a number of abc. Then c is a fraction, b is the second digit, and a the third digit, so the...

【Let’s overcome a big number with N ary】

《A reason why decimal system has spread over among us》

...so 324 is 3×10 squared ＋2×1＋4, so the number of three digit of abc is a×N squared＋b×N＋c.

When expressing numbers, a reason why the decimal system has been used the most general is our fingers are 10 in total, adding to both of hands.

If the number of the fingers on the both of hands were 8 in total like the Mickey Mouse, we could have used the octal 8進法.

Considering the number of a single hand is five and a limit of the number which we the human being can grasp at once is four, it isn’t...

【Let’s overcome a big number with N ary】

《A reason why the decimal system has been used》

...it isn’t strange that there is a society where the quinary 五進法 has been used, for the quinary has been used in Ilongot in Philippine and in Indonesia and South America partly at present.

In addition the Sumerian in ancient time adopted sexagesimal 60進法, and the Babylonians took it over, so they contrived a measure of time, 60 seconds is one minute, and 60 minutes is an hour. Why did they choose 60?

It is said that the 60 has such the lots of divisors that they could count easily.

【Let’s overcome a big number with N ary】

《A reason why the decimal system has been spread over among us》

So they chose 60 as a lump of numbers. In the same reason, mathematicians in Arab adopted 60 when calculating in astronomy.

《The binary system and Bacon who is a philosopher》

Except for the decimal system, there is a trace that other way of counting number was used in our everyday life, and we can see it unexpectedly.

For example, 1 dozen is 12, 1 gross is 12 dozens, and a year is twelve months. They are remains of duodecimal 12進法. Moreover, 80 is called quartre-vingits in France.

【Let’s overcome a big number with N ary】

《The binary system and Bacon who is a philosopher》

It means quartre 4× vingt 20, so there is 20 ary 20進法 there.

By the way, I can’t speak French at all. When expressing other way of counting except for the decimal system, the author happened to adopt the French way, and I expressed it as the author says.

I’m afraid there is a mistake somewhere, but please don’t blame me for it, for I have none of knowledge on the French.

To return to my main object, I’ll going to start again.

They aren’t a mainstream but just remains. Except for the decimal ...

【Let’s overcome a big number with N ary】

《The binary system and Bacon who is a philosopher》

Except for the decimal system, there is a world where other way of counting numbers has driven back the decimal system and played an active role is the world of computer. Mainly the binary system and hexadecimal 16進法 have been used there.

Capacity on memory of UBS is expressed with gigabyte, and number of 16, 32, 64, 128, and 256 are lined up. There isn’t any good number to count like 20 or 100 in the decimal system.

When buying a golf ball in a unit of box, we seem to be forced to buy the next...

【Let’s overcome a big number with the N ary】

《The binary system and Bacon who is a philosopher》

When buying a golf ball by the unit of box, we seem to buy it by every number of 12, 24, 36, 48...for a dozen is the base. A multiple of 12 is the number which is a good place to count numbers in duodecimal 12進法.

Hexadecimal 16進法 has been used in the world of computer and its good place to count numbers is 16 × integral numbers, but why has the hexadecimal 16進法 been used in the world of the computer? It gets along well with the binary system.

By the way, I’ve expressed the binary system, but ...

《The way of counting numbers in the binary system》

I’ve expressed the binary system until now, but I’m not sure of the way of counting numbers in the binary system, so I’m going to show it from now on.

The binary system is a way of expressing all the numbers, using 0,1,2 alone.

As we use usually the decimal system everyday life, so we can’t imagine of using the binary system. When a number which is expressed with a one digit increases, and its digit also increases, the number is expressed with two-digit then.

In the decimal system, the next number of 9 is 10, for the 9 is carried up 繰り上がる.

《The way of counting numbers in the binary system》

In the same way, after the digit increases it becomes two-digit in the binary system, but the next number of 1 is 10.

When the digit changed, the number is 10 in the both of decimal system and binary system, but its value is different, whether the number is the one in the binary system or the one in the decimal system, so we have to be careful of it.

For example, 10 in the decimal system is 10 in the decimal system, but 10 in the binary system is 2 when changed into the decimal system.

There is the number of 24 in the decimal system.

《The way of counting in the binary system》

I’m going to change 24 from the decimal system to the binary system. It’s in the next ways.

Dividing the number in the decimal system with 2 and express its remainder. If it’s divided with two, we express 0 then. Adding a quotient 商 and the remainder. We are going to divide 24 by 2 continuously.

24 ÷ 2＝12、12÷2＝6, 6÷2＝3、3÷2＝1, its remainder 1.

When 24 in the decimal system is expressed with the binary system is 11000. First three of equation is divided and there is no remainder, and the last one is its quotient and remainder 1, so it’s like that

【Let’s overcome a large number with N ary】

《The binary system and Bacon who is a philosopher》

The development on the binary system has been influenced by Bacon who is an English philosopher. He made a wise remark, knowledge is power, and said the knowledge from which we got our experience and observation will reach the truth.

In the middle of trying to contrive a new cipher which was called the cipher by Bacon later, he concluded an idea.

When preparing for a cypher which has two kinds of situations like a large letter and a small letter, ◯ or ×, so 2 raised to the power of 5, 2の5乗...

【We the humankind has searched for pi】

《A famous question on the entrance exam from Tokyo University》

When expressing the decimal and binary system, I don’t think I have fun, so I’m going to change the topic.

Have you ever thought why pi is 3.14?

Tokyo University used to make a question for entrance exam like the next.

Demonstrate the pi is bigger than 3.05.

It’s probably the most famous question in math for the entrance exam from Tokyo University, so some people may have heard of it.

In the first place, what is the pi?

A circumference is multiplying the diameter by the pi. It’s ...

【We the humankind has searched for pi】

《A famous question on the entrance exam from Tokyo University》

It’s an equation which we learn in elementary school. In other words, the pi is the ratio of the circumference of a circle to its diameter. It means that the length of a circumference is three times as big as the diameter. Exactly it’s a little over three times.

Needless to say, as all the circles are similar it stands up all the circles.

The length of the circumference of a circle is shorter than the diameter which is three times or it’s four times as long as the diameter isn’t possible.

【We the humankind has searched for pi】

《A famous question on the entrance exam from Tokyo University》

In other words, if we can search for the ratio on the length to the circumstance to the diameter of a circle, it’s the pi.

《Archimedes thought like that》

However searching for the length on a circumference isn’t easy. A primitive way is measuring it actually.

For example, painting the surface on a tire and rolled the it, trying not to slip, and after the tire turned around wholly once, we measure the length of the trace of the tire.

Or we drive in a stake on the ground and tie up one...

【We the humankind has searched for pi】

《Archimedes thought like that》

...and tie up one end of a string to the stake and other end to a stick of which top is sharp, in short we make a thing like a compass, and draw a circle. The length of the rope is twice becomes the length of the diameter, so we measure the ratio to the length of the diameter to the circumference.

Actually the ratio on the length of the conference to the diameter was sought with the latter way in Babylonia around BC 2000, and it had been thought to be about 3.125. Babylonia is southern area in Iraq at present.

【We the humankind has searched for pi】

《Archimedes thought like that》

But when measuring, it comes together an accidental error. As long as we depend on the measure, we can hardly reach an exact value, so Archimedes in the Ancient Greece used a regular polygon and thought of an estimation on the circumference from a calculation.

There is an illustration on the book, and there is a regular square of which one side is 2 in the illustration. A circle is inscribed to the regular square. A regular hexagon is also inscribed to the circle and is divided into six equilateral square 正三角形.

When...

【We the humankind has searched for pi】

《Archimedes thought like that》

When looking at the illustration, the circumference of the regular hexagon ＜ circumference on the circle ＜ the circumference of the regular square is clear, for the regular hexagon is composed of six equilateral triangles of which side is 1, so its conference is 6.

The side of the regular square and the diameter of the circle is 2, so the circumference on the circle is 2 ×pi = 6.24. The circumference on the regular square is 8, so we can demonstrate pi is bigger than 3 and smaller than 4.

But the circumference on the...

【We the humankind has searched for pi】

《Archimedes thought like that》

But the circumference both of the regular square and regular hexagon is different from the circumference on the circle so much that the estimation wasn’t done so exactly.

If improving the precision on the estimation, we should increase the vertex on the regular polygon, then the space between the circle and the regular polygon is smaller, the length of the regular polygon approaches the length of the circle more closely.

By the way, the question on the entrance exam from Tokyo University in the beginning is thinking....

【We the humankind has searched for pi】

《Archimedes thought like that》

As to demonstrating pi is bigger than 3.05, if thinking of a regular dodecagon 12角形 which inscribed to a circle, it’s solved. Its answer example is on the book, but we can’t understand nor express here with my technical skill and knowledge on the key board.

Archimedes thought of a regular 96角形 which inscribed and circumscribed to a circle, and reached a conclusion that pi is bigger than 3.1408 and smaller than 3.1429. He succeeded in searching for the exact value to the second decimal place.

【We the humankind has searched pi】

《An irrational number continues forever》

Pi is so to speak an irrational number. The irrational number isn’t able to be expressed with a fraction of which numerator or denominator with integral number. It means that an irregular number continues forever in decimals.

On the other hand, it’s able to be expressed with the fraction of which numerator or denominator with integral number, the number of decimals is limited, or if it continues forever, it has something regular.

As pi is the irrational number, the line of number doesn’t come to an end. Its last...

【We the humankind has searched pi】

《An irrational number continues forever》

Its last number doesn’t exist, so no one can answer it, for it has nothing regular. Aristotle expected pi is the irrational number in BC 4th century, but it was latter in the 18th century when it was demonstrated actually.

Though Archimedes estimated pi by regular polygons, it means that it closely resembles that something continues limitless is expressed with something limited, so it’s limited naturally.

So it has been contrived that it’s expressed with a multiplication forever. What kind of the fraction is it?

【We the humankind has searched pi】

《An irrational number continues forever》

Its denominator is pi and numerator is 2, and it’s expressed with other fraction of which denominator is 2 and numerator is √2, and the 2 in the √ of the numerator is substituted with 2 ＋ √2 one after another forever.

After that, the calculation on pi is expressed with an numerical formula which continues forever.

《It has been sought by 31 trillion 400 billion》

Irregular number continues forever in decimal means that every kind of line of number is included as long as it’s limited, so not only four digit ...

【We the humankind has searched pi】

《It has been sought by 31 trillion 400 billion》

...not only four digit of the date of our birth but even if those of every kind of people in the world, there is a line of number of eight digit in pi.

I’m wondering if taking the Christian Era into account, it’ll be eight digit.

In addition, when trying making computer understand the information on letters, and the letters are changed into numbers, if the whole passages on Hamlet written by Shakespeare is changed into numbers, we will be able to find the same lines of number on Hamlet in the pi.

【We the humankind has sought for pi】

《It has been sought by 31trillion 400 billion》

It makes us the humankind that being infinite is limitless, but if the story stands up all right, the line of numbers on pi need to be at random completely. We call the line of numbers random numbers.

If investigating pi and frequency on each number from 0 to 9 which appears in the line of numbers of pi, it’s almost the same, so the line of numbers on pi seems to be the random numbers, but it has never been demonstrated mathematically yet.

Google announced Haruka Ema Iwao who was from Japan succeeded in...

【We the humankind has searched for pi】

《It has been sought for by 31 trillion 400 billion》

Google announced Haruka Ema Iwao who is from Japan succeeded in calculating pi 31 trillion 400 billion decimal on pi on 14th March in 2019. The day was for pi. Its previous record was made in 2016, but from then no less than 9 trillion has been updated. It’s a marvelous record.

Pi is the number which we won’t understand, but it appears in numerical equations on all kinds of math and natural science including other fields which seem have nothing to do with pi. It’s mysterious and strange fixed number.

【We the humankind has searched for pi】

《It has been sought by 31 trillion 400 billion》

Pi is the number which we can’t know its exact numerical value absolutely.

In spite of being important so much, we can’t recognize its exact value. It’s not useful, the author said he wasn’t sure whether it was thought like that, but the numerical value on pi was about to be set up by law at Indiana State in America at the end of 19th century

An amateur mathematician who was a doctor wrote a thesis on pi that the circumference on a circle of which diameter 10 is 32, and presented to the National Assembly

【We the humankind has searched for pi】

《It has been sought by 31 trillion 400 billion》

Then a bill which supported the thesis was made. If the bill passed through in the National Assembly, pi would learn to be 3.2, moreover it was approved unanimously in the Lower House.

A mathematician who happened to visit the governor at the state knew it, and he said the precise numerical value on pi is never fixed to the members in the Upper House. He explained it to them minutely all night, so the bill the one which was put off indefinitely.

The author said it was in crisis. Was it so? I’m not sure.

【We the humankind has searched for pi】

《It has been sought by 31 trillion 400 billion》

Lots of people may have thought we don’t have to stick to its precise numerical value so much. 3.2 may have lots of accidental error, but if pi is 3.14, no one is in trouble actually, isn’t it?

《The numerical value on pi which supported the return ハヤブサ》

But if pi is 3.14, there was a national project in which failed.

There are lots of people who knew はやぶさ which is a Japanese space probe on planets, though I’ve never heard of it until now.

In the middle of the plan, the space probe on planets ...

【We the humankind has searched for pi】

《The numerical value on pi supported the return on はやぶさ》

In the middle of the plan, はやぶさ couldn’t correspond with the earth, but the people who concerned the plan on persistent support made the space probe succeed in returning to the earth. Its miraculous story was adopted in newscasting so much that it was made into a movie.

When calculating its numerical value on the orbit of はやぶさ,3.141592653589793 seemed to be adopted. It’s sixteen digit.

If adopting 3.14 for pi and was calculated with the numerical value, its orbit would swerve about a hundred...

【We the humankind has sought for pi】

《The numerical value on pi supported for return はやぶさ》

...the space probe would swerve from the orbit by a hundred fifty thousand kilometers if the worst came to worst, and even if the correspondence with the earth revived, はやぶさ couldn’t have returned to the earth.

After the Ancient Greece, lots of mathematicians in both West and East have challenged, and engineers on computer continue to do against pi, but its fight on pi never comes to an end.

【An imaginary number 虚数 and the quantum computer】

《When being squared, the one which becomes minus》

【An imaginary number and quantum computer】

《When being squared, the one which becomes minus》

There is a question like the next.

There is a rectangle of which length on vertical and horizontal is 10 in addition, and of which area is 24. Find the value the vertical and horizontal length on the rectangle.

How should we solve it?

The question seems to be prepared in a test, so we are apt to be lost in thought, but it’s not so difficult.

In short we have only to think of a pair of numbers when adding its total is 10, and when multiplying each other, it’s 24. We can do it with a mental...

【An imaginary number and quantum computer】

《When being squared, the number which becomes minus》

We can do it with a mental arithmetic calculation.

There is other question like the next. How should we solve it?

There is a rectangle of which length on vertical and horizontal is 10 in addition and its area is 20. Find its value of the rectangle on the length of vertical and horizontal.

This time solving it with the mental arithmetic calculation may be hard, but it’s a normal question for a student who is the third grade in junior high school.

【An imaginary number and quantum computer】

《When squaring, the number which becomes minus》

Regarding the vertical length as X and the horizontal length as Y and making a simultaneous equation 連立方程式 and if using a formula on the solution of a quadratic equation, we can find its answer.

But this time, the length on both vertical and horizontal is bigger than 5 or smaller than 5. The vertical length is smaller than 5 by the length of x, and the horizontal is bigger than 5 by the length of x.

Its equation is （5＋x）（5−x）＝20 and 25−x squared = 20, so x squared = 5, so x = √5

As a result, the...

【An imaginary number and quantum computer】

《When squared, the number which becomes minus》

As a result, the vertical length and horizontal length on the rectangle is 5 − √ 5 and 5＋√5

With adopting the same way, we try to find the numerical value on the next question.

When adding, its total is 10, and multiplying is...

By the way, I’m tired a little, so I don’t feel like going on this response any more for a while, so I have to take a rest, in addition the battery on my iPad is about to run out. Please just a moment for a while.

【An imaginary number and quantum computer】.

《When being squared, the number which becomes minus》

When adding it’s 10 in total, and when multiplying, it’s 40. We try to find its value with the same way. Then it’s x squared is −15. We are in trouble, for the number doesn’t exits. When being squared, none of numbers which become minus.

Girolamo Cardano of an Italian who left behind his reputation as a great mathematician for a formula on solution of a cubic equation faced to the same problem, but without giving up the solution and there was no solution...

【An imaginary number and quantum computer】

《When being squared, the number which becomes minus》

...instead he did other thing.

When x squared is equal to 5, x is √5. Cardano did the same thing, he put −15 into the √ by force, so x squared is equal to −15, and x is √−15. As a result the value which he found was 5＋√−15, and 5−√−15, and he wrote down like the next.

If disregarding a spiritual agony, its addition is 10, and multiplication is 40, but it’s a chop logic 詭弁, and even if expressing numbers minutely in math, we don’t have any practical use for it.

《A genius who challenged the...》

【An imaginary number and quantum computer】

《A genius who challenged the imaginary number》

Though it was half by force, and Cardano himself didn’t consider its existence positively, he was the first person who referred to the number which becomes minus when being multiplied and he wrote it down in his book.

There is no number of which addition is 10 and multiplication is 25. Why? Its maximum on that the addition is 10 and the multiplication is 40 is 25 on the condition, though it doesn’t meet the condition.

We call the minus number when being squared an imaginary number at present.

【An imaginary number and quantum computer】

《A genius who challenged the imaginary number》

As the imaginary number doesn’t exist actually, we can’t express any of them on the number line where real number is gathered, so it has a negative nuance in French and is changed into English.

As Descartes is the one who linked a numerical formula to a diagram, he may not have been hard to accept the imaginary number.

But a genius who tried to search for the imaginary number which we can’t express on the diagram appeared in the 18th century. He is Leonhard Euler in Swiss. He set up √−1 as an unit..

【An imaginary number and quantum computer】

《A genius who challenged the imaginary number》

Euler set up √− Ⅰ as an unit for the imaginary number and took its initial i and made it a rule to express the imaginary number as i. Then he reached the numerical formula on Euler at the end of his study for a long time finally.

It is said it’s the most beautiful numerical formula in the world. The word of i is included in the numerical formula, though I don’t think at all it’s beautiful, for I can’t understand it and nor find something regular in the numerical formula, I’m wondering.

After Euler...

【An imaginary number and quantum computer】

《A genius who challenged the imaginary number》

Even after Euler probed deeply the study on the imaginary number, lots of people didn’t try to recognize the existence of the imaginary number. It took a long time to accept even minus numbers in Europe, so it is natural for them to be skeptical on the imaginary number which doesn’t exist in the world.

《A discovery by Gauss》

But the situation has changed suddenly, for Gauss who is the greatest mathematician in Germany and other people contrived a number line where imaginary number gather.

【An imaginary number and quantum computer】

《A discovery by Gauss》

The number line where the imaginary number gathers called an imaginary axis, and intersects with other number line where real number gathers.

Gauss and others insisted that the imaginary number exists on the imaginary axis respectively, so the imaginary number becomes something visible for the first time and its existence has been recognized widely in the society after that.

Gauss called the one which was combined the real number with the imaginary number a complex number 複素数. Different plural elements of real number and...

【An imaginary number and quantum computer】

《A discovery by Gauss》

Different plural elements that real number and imaginary number is different altogether each other are combined and a new number is born then.

Gauss combined the real axis from the number line where real number gathers and imaginary axis from the imaginary number, and made each point on real number and imaginary number one to one on the plane of coordinates, and called it the plane of complex number.

The theory on the plane of complex number is expressed on the book, but it’s too hard to understand for me, so I’ll skip it.

【An imaginary number and quantum computer】

But even if the imaginary number on the plane of coordinates becomes visible for us, as the number doesn’t exist actually, what is the use of inventing the number and discussing about it? Lots of people may have thought like that, and I myself think so as well.

Quantum mechanics 量子力学 deals with a microscopic world, and a thing of which we can’t think from our common knowledge takes place there.

The material has both nature of a particle and a wave at the same time, a single material exists in other places at the same time. The material is...

【The imaginary number and quantum computer】

The material is born from a vacuum where there is nothing, or vanished there. It sometimes passes through a wall, so the complex number is indispensable so as to describe physics in the world of the quantum physics.

As you know I have none of knowledge on the quantum physics, so I’ve expressed the mysterious things, but it’s just that I did as the book showed. I want to express it why, but to my sorrow I can’t do at present.

The quantum physics is the base on modern scientific techniques. It is no exaggeration that without the quantum physics...

【The imaginary number and quantum computer】

It is no exaggeration that without the quantum physics neither smartphone nor personal computer were never born.

For example, a quantum computer about which we are much talked is put the theory on quantum physics to practical use and is made.

While the computer proceeded a calculation with 1 or 0 in the past, the quantum computer makes use of a situation that the number is 1 and 0, its speed on calculation becomes high speed.

Without the quantum physics, in other words, without the complex number, we the humankind couldn’t have established...

【The imaginary number and quantum computer】

Without the quantum physics, it means without the complex number, we the humankind couldn’t have established the modern civilization.

Not only we can describe the microscopic world with the complex number but without ending in failure the relative theory by Einstein we’ve succeeded in expressing the beginning on the universe by using the time of complex number, Dr. Hawking said like that.

To my sorrow, I can’t understand it in the least.

《Keen insight on Leibniz》

The author said that the complex number is indispensable in modern physics, but...

【Imaginary number and quantum computer】

《Keen insight on Leibniz》

I’m afraid I’ve confused the complex number and imaginary number. I’m sorry for it.

We need some numbers which don’t exist actually so as to express our real world, but there seem to be some people who think it doesn’t make sense to them.

Kronecker who is a mathematician said like the next.

While the God has created natural number like 1,2,3..., except for them, we the humankind has introduced new concepts and contrived all the other numbers like 0, minus numbers, a decimal, a fraction, or irrational numbers and we’ve ...

【Imaginary number and quantum computer】

《Keen insight on Leibniz》

...and we’ve expressed unknown world until now.

As we need a new kind of number of irrational number so as to express the length on the oblique side 斜辺 of a right angle isosceles triangle, in the same way, the complex number is indispensable and useful when trying to express the microscopic world in a compact way clearly.

It is said Leibniz said on the imaginary number like the next.

The God appears its figure in sublime way as miraculous product which drifts existence and non existence.

Leibniz was sixty years older...

【Imaginary number and quantum computer】

《Keen insight on Leibniz》

Leibniz is sixty years older than Euler, so he may have thought the significance on existence of the number which is squared is minus by far more earlier than Euler, before Euler studied on imaginary number in earnest and announced its income. If so he is awesome genius.

【A magic square is a good exercise for the brain】

《Simple and profound mathematical puzzle》

Pythagoras in the Ancient Greece said everything is number, Galileo said the universe is written by a language of math. It is certain that strictness and ...

【A magic square is a good exercise for the brain】

《A simple and profound mathematical puzzle》

It is certain that strictness and rationality in math is suitable to solve the truth on the universe.

Even if not being a scientist, if wanting to get to its own conclusion among various senses of values everyday life, mathematical ability of thinking way is necessary. Development on IT where technical skill on information has been made full use of and and the rise of artificial intelligence makes us feel something significant on math day by day.

If we’re the one who has lived in the modern era...

【A magical square is a good exercise for the brain】

《A simple and profound mathematical puzzle》

...we have to master the statistical literacy, but we’re sick and tired of hearing that, but it doesn’t always that math exists for the refined target alone, for math frequently appears in a thing like a game.

When gambling, if having some knowledge on probability, it is profitable for us. Besides when creating lots of puzzles, math has been taken advantage of then, especially a magical square is famous among the mathematical puzzle and has had a long history. What is the magical square?

【A magical square is a good exercise for the brain】

《A simple and profound mathematical puzzle》

There is a square, and there are some other small squares in the square, each of small square has each number which is different between each other. Even if adding up each of number of vertical, horizontal or diagonal line, its total number is the same, but we can’t use the same number twice.

There is an illustration on the magical square which is 3×3 in the book. When adding the number in every vertical line, in every horizontal line and in diagonal line,..

【A magical square is a good exercise for the brain】

《A simple and profound mathematical puzzle》

...it’s 15 in total. In general, the magical square on 3×3 is the magical square on the third order. The square is made up with other small nine squares, three lines each of vertical and horizontal.

If the magical square is made up with n×n, we call it nth order.

《A pattern on the shell of a holy tortoise》

How is the number of 15 kinds of number lined in the magical square? When turning it over or rotating, if regarding them as the same one, the magical square in which the number from 1 to 9...

【A magical square is a good exercise for the brain】

《A pattern on the shell of a holy tortoise》

...the magical square on which each number from 1 to 9 is used in the third order is one kind alone. The number of line from upper right,294753618. There is a comical variations of idioms 語呂合わせ on the numbers. 憎し 294 七五三、六一坊主に蜂が刺す.

As to other magical square on the fourth order 4×4 in which each number from 1 to 16 is used turned out to be 880 kinds.

Another one on the fifth order 5×5in which each number from 1 to 25 is proved to be about two hundred million seven thousand kind.

One more on ...

【A magical square is a good exercise for the brain】

《A pattern on the shell of a holy tortoise》

It turned out that the magical square on the sixth order of 6×6 is more than seventeen quintillion 1700 京 in which the number from 1 to 36 is used.

The bigger the ordinal number is, the more its kind increases by leaps and bounds.

The magical square is originated in China. A tortoise was picked up by a Chinese Emperor in the Yellow River, and it is said that there was the magical square of third order with the same number of points from 1 to 9 is described on the shell.

The Chinese Emperor ...

【A magical square is a good exercise for the brain】

《A pattern on the shell of a holy tortoise?》

The Chinese Emperor got together principles on politics and economy and divided it into nine, for it is said the pattern on the shell of a holy tortoise was divided into nine, and a fortune-telling of 九星術 was born from the pattern of the shell.

We’ve enjoy ourselves on the magical square as a mathematical puzzle at present, but it used to be thought to be something mysterious in the past.

The magical square was born and was spread over the Western area, but it was unclear how it was done then.

【A magical square is a good exercise for the brain】

《A pattern on the shell from a holy tortoise》

There are some variations on the magical squares like the one which is made up with square measures 平方積, and the other with prime numbers 素数 alone.

《A written challenge to readers on the magical square》

The author said he wanted us all of the readers to complete the magical square on the book, and he said he would be happy if we tackle it as a good exercise for the brain.

But even if trying to put each number in the magical square at random, we can’t complete it easily, so the author...

【The magical square is a good exercise for the brain】

《A written challenge to readers on the magical square》

...the author showed us some bases when tackling the magical square.

When creating the magical square of 4×4, each of the total number on the vertical, horizontal and diagonal line is always 34, and the author said he was going to say its reason later.

He has already put some of numbers in the magical square beforehand, and we have to be careful of numbers which we can use. It’s from 1 to 16, and we need to look for a line of which number is either extremely large or small...

【The magical square is a good exercise for the brain】

《A written challenge to readers on the magical square》

...and we need to look for a line of which number is either extremely large or extremely small., and to limit the candidacy for the number from there.

Why is the total number on the magical square of each vertical, horizontal, and diagonal line 34?

As to the fourth order of magical square, each number from 1 to 16 is used in the squares of 4×4, and if adding all the numbers, it’s 136. It means that the total number on the four lines are 136, so the total number on the single line...

【A magical square is a good exercise for the brain】

《A written challenge to readers on the magical square》

...so the total number on the single line is found dividing 136 by four, and it’s 34.

Some numbers which were lined on the magical square of 4×4 is like the next.

I recommend you describe the number actually, for it’s easy to understand.

The top on the horizontal line, there is 4 on the left corner and next to the 4 is vacant. There is no number there. The next place to the vacant is 15, and the right corner is also vacant.

The second line from the top. There is 8 in the left end...

【A magical square is a good exercise for the brain】

《A written challenge to readers on the magical square》

...and there are blank on other three small squares. There is 8 alone at the right end of the second line from the top.

The third line from the top. Its left side is blank and its next is 7, and other two are blank as well.

The bottom line. Its left corner is blank and its next is 2 and next to 2 is 3. The right corner is blank.

An example on a way of thinking.

There are some numbers which weren’t put in the magical square yet like 1, 5, 6, 9, 10, 11, 12, 13, 14, and 16.

We name...

【A magical square is a good exercise for the brain】

《A written challenge to readers on the magical square》

We need to name each of the blank square one by one. At the top from the left, between 4 and 15, we call it A, and the right corner at the top, it’s B.

At the second line from the top, there are three blank squares. We call C, D, and E from the left.

At the third line from the top, there are also three blank squares. The left side next to 7, we call it F. Other two which are right to 7, they are G, and H, from the left.

At the bottom, the left corner is I and the right corner is J.

【A magical square is a good exercise for the brain】

《A written challenge to readers on the magical square》

Please write down actually what I’ve said until now.

At first we pay attention to the bottom. Its total number should be 34, and there are two kind of numbers there, 2 and 3, so it turned out that I＋J＝29.

When looking at the numbers which have never used in the magical square, I,J is （16,13）or（13,16）, so at first we think of the case that I, J is 16, 13.

In the next, we think of the vertical line on the second from the left, and there is two kind of numbers there, 7 and 2, so the...

【A magical square is a good exercise for the brain】

《A written challenge to readers on the magical square》

...so the rest number in total, A＋D＝25, so A, D is 14,11 or 11, 14. If A,D is 11, 14, then B is 4, so it’s not appropriate, for 4 has already been used. Each of number is limited to use only once, so A,D is 14,11.

The rest of each number which has never been used is 1,5,6,9,10, and 12. If reaching until here, it’s easy, for each number in total on the rest line is 34, so we can fill the number in the magical square, so it’s completed.

【Do you know a pair of scales which is almighty?】

【Do you know a pair of scales which is almighty?】

《Look for a false coin!》

There is a question like the next. There are eight coins, and one of them is false, and the false one is a little heavier than the genuine ones.

A pair of scales has already been prepared for us so as to tell the coins between false and genuine.

Whichever we take any case, we want to distinguish the fake one from the genuine ones. How many times at least should we use the pair of scales to look for the fake one?

If you want to think it over slowly and carefully, I’m sorry for it, for the author said its answer...

【Do you know an almighty pair of scales ?】

《Look for a false coin!》

...the answer is twice. You may have thought, only twice? The author expressed its concrete way like the next.

The first time, we choose six coins at random and put them on the both of the scales by every three coins. If they are balanced, the fake is either of the rest two. If the scale is leaned on the one side, the fake one is among the three on the scale which was leaned.

The second time. When being balanced, we put the rest of the two coins on the scale one by one, and the one is leaned, so it’s the fake.

【Do you know an almighty pair of scales w】

《Look for a false coin!》

If leaning on the one side, taking the three coins form the leaning scale and put them on each of the scale again one by one. If being balanced, the rest one is the fake, and if being leaning on one side, the one on the leaning scale is the fake.

Then how about 20 coins? Even if the coin increases to 20, we can point out which coin is false as long as we used the pair of scales three times.

At the first time, we choose eighteen coins at random and put every nine coin on each of the scale. If being balanced, the rest...

【Do you know an almighty pair of balances?】

《Look for a false fake!》

If leaning on one side, the fake is among the nine coins on the leaning scale.

At the second time, if balancing at the first time, putting the rest of two coins one by one on the each scale, and the leaning one is the fake.

If leaning at the first time, taking nine coins from the leaning scale and put the coins three by three on each of the scale. If being balanced, the fake is among the rest of the three. If being leaning, the fake is among the three on the leaning scale.

At the third time, when being leaning at the...

【Do you know an almighty pair of balances?】

《Look for a fake coin!》

At the third time, if leaning in the first time, taking the three coins which turned out to be including the fake coin among them and put one by one on each of the scale. If being balanced, the rest one is the fake. If being leaned, the coin on the leaning scale is the fake.

To tell the truth, if thinking it with the same way, we can point out the fake coin until 27 coins as long as we can use the pair of scales. In addition, the number of the coin is 3 to nth power, it means that the number of coins is a multiple of 3...

【Do you know a almighty pair of scales?】

《Look for a fake coin!》

...it means that if the number of coins is a multiple of 3, we can point out the fake coin as long as we can use the pair of scales n times.

Except for the examples which were shown until now, the author recommended we try to count various number of coins, and think of the way of finding the fake coin. He said it will be a good exercise for the brain.

《A question in which a sense of mathematic is necessary for us》

The author said there is one more question and he said he wanted us to think it over. It’s in relation to the...

【Do you know an almighty pair of scales?】

《An question in which a sense of mathematics is necessary for us》

The question is relation to the pair of scales.

When using the pair of scales, and wanting to measure from 1 gram to 40 gram one by one gram, how many weight do we need at least? And how much is each of the weight?

If the one who is get used to it, it will answer we need six kinds of weight. 1 gram, 2 gram, 4 gram, 8 gram, and 16 gram.

If making use of the six kinds of weight, we can measure all the weight from 1 gram to 40 gram. All the weight is expressed with 2 to nth power.

【Do you know an almighty pair of scales?】

《A question in which a sense of mathematic is necessary for us》

1 is thought to be 2 to the 0th power in math. There is a list on the book, and what kind of weight like 1 gram, 2 gram, 4 gram, 8 gram, 16gram and 32 gram,is used for the weight is used is expressed on the list.

While the number which we usually use for every digit in the decimal system is from 0 to 9, the number which we usually used for the binary system is either 0 or 1 alone.

When 0 is adopted, it means that a thing isn’t adopted, and 1 is done, it means that something is used.

【Do you know an almighty pair of scales?】

《A question in which a sense of mathematics is necessary for us》

For example, when expressing 13 with the binary system, it's 1101, and it means that we measure something of which with each of single weight of 13 gram with a weight of 8 gram, 4 gram and 1 gram.

Though I can’t understand when 13 is expressed with the number of the binary system, 1101, why do we adopt those three kind of weigh? The total weight of the three kind is 13, I know why it is, but I’m not sure that the relation of the number which is expressed with the binary system and...

【Do you know an almighty pair of scales?】

《A question in which a sense of mathematic is necessary for us》

...but I’m not sure the relation between the number which is expressed with the binary system, 1101 and three kinds of weight, 1 gram, 4 gram and 8 gram.

The author continued, all the numbers in decimal system are expressed with those in the binary system, we have only to prepare for every weight of binary system one by one. It’s enough. Why?

To return to our main subject, I’ll start again.

But for example, how about in a case when we have to prepare for a weight of 4 to nth power?

【Do you know an almighty pair of scales?】

《A question in which a sense of mathematical is necessary for us》

Then we think it of with quaternary in which number from 1 to 3 in every digit. When trying to express 13 with the quaternary, it’s 31. Why?

Its numerical equation is 3×4＋1×0＝13

When trying to express with English, 3 multiplied by four to the first power plus 1 multiplied by four to the 0th power, then three means three weight is used. Then 1 is 4 to the 0th power, it means a single weight is used. It’s 31, so 13 is equal to 31 when changing from the decimal system to quaternary.

【Do you know an almighty pair of scales?】

《A question in which a sense of mathematic is necessary for us》

In the same way when expressing 31 with the binary system, 8＋4＋1, it means that 1×2 to third power ＋1×2 to the second power＋1×2 to 0th power. Three weight of 2 gram, two weight of the two grams, and a single weight of 1 gram is used. It’s 1101.

As to the binary system, if its digit is used, then putting 1, and if it’s not used, then putting 0.

The author continued his theory on the almighty pair of scales, but I don’t feel like expressing, for I find it boring. I’m sorry for it.

【A way of changing our both hands into an electronic calculator】

《There is few countries in which learning the multiplication table by heart》

We the Japanese learn the multiplication table in the second grade of elementary school. Learning the multiplication table may be the first deadlocked in arithmetic.

Everyone has memory of trying to learn it by heart with rhythmically, but a country where learning the multiplication table from 1 to 9 by force is few in the world.

For example, when learning it in the area of English speaking, a list of times table is used. The multiplication from...

【A way of changing both of our hand into an electric calculation】

《Few countries where leaning the multiplication tables by heart》

A list of multiplication table is used in the area of English spoken when learning the multiplication. There are multiplications to 12×12.

If going on using the list of the time table, they’ll learn it by heart automatically, and it’s all right, in America or Australia they’ve thought like that.

Why is it to 12×12?

1 feet is 12inch, and 1 dozen is 12, though it has been abolished, 1 shillings used to be 12pence in England, for the duodecimal system 12進法 is...

【A way of changing both of our hands into an electronic calculation】

《Few countries where learning the multiplication table by heart》

...for the duodecimal system is so frequently used in their everyday life that they have to adapt themselves to the duodecimal system.

《Japan where the electronic calculation has been forbidden》

Learning the multiplication table by heart has never been done by force in lots of foreign countries, for it may be related with the fact that they can use the electronic calculation at will in junior high school.

An questionnaire that whether or not they allow...

【A way of changing both of our hands into an electronic calculator】

《Few countries where learning the multiplication table by heart》

An questionnaire was done that whether they have allowed the students to use the electronic calculator in the class of math or arithmetic over some of foreign countries.

At the time of the fourth grade in elementary school, almost all of the countries, including Japan, have never allowed it, but the second grade in junior high school, the countries in which they allow to use the electronic calculation at will have increased so much.

After 10 years old...

【A way of changing both of our hands into an electronic calculator】

《Few countries where an electronic calculator has been prohibited》

After the age of 10 when cultivating an ability of logical thought, they want the children to spend on more time to think over something more logical rather than a simple work like the calculation.

On the contrary, teachers who allow the students to use the electronic calculator at will at the second grade in junior high school accounts for no more than 6% in Japan.

While the electronic calculator made in Japan has been used at school all over the world...

【A way of changing both of our hands into an electronic calculator】

《Japan where an electronic calculator has been prohibited》

...ironically the electronic calculation has hardly used in the class of Japan, but if the country where the electronic calculator is forbidden to use at will and learning the multiplication table by heart by force has higher mathematical ability, carrying out the education of Japanese style is significant, but unfortunately, there is none of the situation.

Hong Kong and Singapore where the electronic calculator has been used at will have higher mathematical ...

【A way of changing our both hands into an electronic calculator】

《Japan where the electronic calculation has been forbidden》

Hong Kong and Singapore where the electronic calculator has been used at will have higher mathematical ability in the questionnaire before. In addition there are lots of reports that if making the student use a tool for calculation like the electronic calculator or the list for multiplication table,it enhances their mathematical ability.

When trying to learn the multiplication table by heart, we Japan muttering rhythmically, but it seems to be strange for the....

【A way of changing both of our hands into an electronic calculator】

《Japan where the electronic calculator has been forbidden》

....but it seems to be odd for the Europeans and Americans who don’t have a habit of learning the multiplication table by heart. They thought we the Japanese looked as if we had muttered an incantation or something.

《Having a good command of the multiplication by counting on our fingers》

An expression on 九九 which is the multiplication table in Japan has been deep-rooted in Japanese life.

For example, 四六時中, 4×6=24, at the time of 24 hours a day, around the clock...

【A way of changing both of our hands into an electronic calculator】

《Having a good command of multiplication by counting on our fingers》

...18番 2×9=18, the one with expressed with the number of 2 and 9, it means that the one who we tend to hate, 29い奴, it has changed into an artistic skill on an actor who has popular among us.

28ソバ, ２×8＝16, the price of a bowl of soba used to be 16文 in Japan in the past.

A culture of making the people learn the multiplication table by heart has existed in Japan, China, India, and other countries in Asia from long long ago, but except for the countries...

【A way of changing both of our hands into an electric calculation】

《Having a good command of multiplication by counting on our fingers》

...except for the countries how did they multiple, especially at the time when there was no electronic calculator, for they couldn’t carry the list of the multiplication table with them everywhere around the clock.

Without learning the multiplication table by heart, nor carrying the list of the multiplication table with them, a way of being able to do an easy multiplication was contrived around 15 century. It’s the multiplication by counting on fingers.

【A way of changing both of our hands into an electronic calculator】

《Having a good command of multiplication by counting numbers on fingers》

Without clenching our fist, opening our hands, bending from the thumb to the little finger one by one when calculating , and an order on 8×6 is shown in the book like the next.

1 Counting 8 on fingers in one hand and 6 is with other hand.

２Adding the number of bending fingers on both hands. 2＋4=6

３Subtracting 6 from 10, and multiplying 10 by 4, 10−6=4, and 4×10=40

4 Multiplying each number of bending fingers, 4×2=8

5 Adding 40＋8=48.

At last we...

【A way of changing both our hands into an electronic calculator】

《Having command of multiplication by counting numbers on fingers》

At last we reach the answer. If trying to express with letters, we may have felt roundabout so much, but doing it with our fingers actually several times, we’ll learn to find its answer more quickly, the author said like that.

To tell the truth, I’ve never heard of the way of calculation until now, so I’ve never tried it. The author continued.

It's limited to multiplications of which number is bigger than 5.

The people long long ago who didn’t learn the...

【A way of changing both of our hands into a electric calculator】

《Having command of multiplication by counting numbers on our fingers》

The people long long ago when they didn’t learn to understand the multiplication table, they thought of the multiplication by small number each other like 2×4, they regarded as addition of 2 four times, 2＋2＋2＋2=8

But as to 8×6, when trying to think it of addition of 8 six times, they tend to get a little confused, so they contrived that way, the author said like that.

As to the multiplication table for 9, we can find its answer more easily.

【A way of changing both of our hands into an electronic calculator】

《Having command of multiplication by counting numbers on fingers》

I’m going to show the way on 9×3.

1 We spread out our both palms and direct them toward ourselves.

2 As it 9×3, we bend the middle finger alone in the left hand.

3 The left side of the two fingers from the bending middle finger is the second digit, so it’s two. The other fingers from the bending middle fingers are the first digit, so it’s 7, so it’s answer is 27.

Though it’s answer is correct, to my sorrow,I don’t understand why it is. I find it strange.

【A way of changing both of our hands into an electronic calculator】

《What should we do so as to decrease the people who are poor at math?》

Needless to say its remarks aren’t from me but from the author who teaches math, and he said like the next.

If making students calculate a multiplication of both three digits, or dividing some number on fourth digit by other number on two digit by writing, he doesn’t think it’s so significant, from the situation in which he teaches math for students in junior high school or high school or grown ups.

If the students solved questions in which that kind...

【A way of changing both of our hands into an electronic calculator】

《What should we do in order to decrease the people who think they are poor at math?》

When students solved the question in which that kind of calculations were included, he made it rule to tell them to use the electronic calculator, or saying the way of thinking is all right with it, so go ahead from that, for he wanted students to spend their time on solving other problem rather than the calculation.

But students who are good at an applied question in which they need an ability of thinking have an ability of calculation...

【A way of changing both of our hands into an electronic calculator】

《What should we do so as to decrease the people who think they are poor at math?》

But it is certain that students who are good at an applied question in which they need an ability of thinking have an ability of calculation more than ordinary people.

The author said he had never encountered any student who was poor at calculation but had a deep mathematical insight.

He hoped that the students should come to grip with a complicated calculation at least during a student of elementary school, and think it important for the...

【A way of changing both of our hands into an electronic calculator】

《What should we do in order to decrease the people who think they are poor at math?》

...and thinks it important that they have a feeling for something complicated on the calculation, for the feeling makes them look for a device on the calculation, or characteristic number makes them feel easy when calculating.

After that they will be fond of numbers gradually and be good at numbers in the future.

It’s all right whether they learn the multiplication table by heart or use the list of multiplication table. It’s all right...

【A way of changing both of our hands into an electronic calculator】

《What should we do in order to decrease the people who think they are poor at math?》

It’s all right either of them, the author said.

At any rate, he thinks students should handle lots of kind of numbers with their own hands at least in elementary school and should go through being familiar with numbers as more as possible then.

Without having a sensation of being familiar with numbers, numerical equation which is expressed with an alphabet which the students encounter in junior high school, they find it cold and distant.

【A way of changing both of our hands into an electronic calculator】

《What should we do in order to decrease the people who think they are poor at math?》

It doesn’t always mean that the higher ability on calculation they have, the easier they feel when encountering a difficult question in which abstract thought is necessary, so the education on math is hard.

But if being familiar with numbers through the calculation, we should be able to have an image easily on numerical formula which is abstract and is expressed with alphabetical way. Then they won’t be the one who think they are poor at..

【A way of changing both of our hands into an electronic calculator】

《What should we do so as to decrease the ones who think they are poor at math?》

Then they won’t be the ones who think they are poor at numerical formula at least, will they? The author said like that.

As for me, it’s too late?

Though this title has math, but to tell the truth, when expressing on math, I did it reluctantly, for math is too hard to understand, but once having its title, I hated to give it up, so I continued until now.

But I’m interested in math more than before due to this book. Too late? It’s all right.

【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as area》

Can you calculate 16×13 mentally at once? It seems that they make children learn the multiplication table until 19×19 during elementary school in India, but we the Japanese do until 9×9, so it doesn’t always mean that lots of us can calculate the multiplication 16×13 at once, but if thinking it over with an illustration, we can do it immediately, the author said like that, and there is an illustration like the next in the book.

There is a rectangle of which vertical side is 13 and horizontal...

【Calculating multiplication on two digit mentally at once】

《Thinking of the multiplication as the area of a rectangle》

The author said he wants us the readers to think of 16×13 as the area of a rectangle, and there is a figure on a rectangle of which horizontal length is 16 and of which vertical length is 13, and the rectangle is separated into a regular square of its side is 10 and three rectangles. How is situated the square and three other rectangles in the rectangle of 16×13？

The horizontal line on the rectangle of 16×13 is divided into 10 and 6, and there is a vertical line from the...

【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of a rectangle》

There is a rectangle of which horizontal length is 16 and a vertical line is drawn from the end of the regular square to the base of the rectangle of 16×13, as a result there is other rectangle of which horizontally length is 6 and vertical length is 13 next to the regular square.

The rectangle has the vertical line of which length is 19, and there is a horizontal line from the end of the regular square, and the horizontal line is parallel to the both sides of the rectangle.

【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of a rectangle》

I recommend you should write the figures actually.

As a result, the horizontal line starts from the end of the regular square in the rectangle of which vertical line is 13 and horizontal line is 19, there is another rectangle of which horizontal line is 19 and of which vertical line is three under the regular square.

Exactly there is two kinds of rectangle under the regular square. The one is just under the right square and the other is lower right from the regular square.

【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of a rectangle》

In other words, the vertical line from the regular square is expanded to the base of the rectangle which includes the regular square, so there is two kinds of rectangle. The one is just under the regular square, and the other is lower right from the regular square.

We cut out of the rectangle just under the regular square of which horizontal line is 10 and vertical line is 3, and turn its direction 90 degree, add it to right next to the figure which used to be the rectangle.

【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of a rectangle》

Though the rectangle just under the right square was cut out, so the rectangle of which horizontal length is 16 and vertical length is 13 doesn’t exist any more, we add the rectangle which is just under the right square to the right side of the figure which used to be the rectangle.

Its shape is like the next.

There is a new rectangle of which horizontal length is 19 and of which vertical length is 10, and there is other rectangle...

【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of a rectangle》

There is a new rectangle of which horizontal length is 19 and of which vertical length is 10 and there is other rectangle of which horizontal length is 6 and vertical length is 3.

Then the whole area on the figure is addition of the new rectangle of which horizontal length is 19 and of which vertical length is 10, 19×10=190and other that of the rectangle under the new one of which horizontal length is 6 and of which vertical length is 3, 6×3=18, 190＋18=208.

But as the ...

【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of a rectangle》

But as it’s just that the rectangle under the regular square was moved to the upper right, so the whole area is the same between before the rectangle is moved and after the rectangle is done, as a result, 16×13=190＋18＝208.

It means that （10＋a）×（10＋b）＝（10＋a＋b）×10＋ab.

If calculating, 12×14＝（12＋4）×10＋2×4＝168

In short, a multiplication on two digit until 19×19 is done like the next.

1 Adding the number in the first digit to the other number.

2 Ten times the value from the 1

【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of a rectangle》

3 Adding the product 積 by the number on the first digit.

We can calculate it easily with the order. We’ll be able to learn to calculate it mentally at once as long as we train a little, so let’s try, said the author like that.

If thinking of the multiplication as an area, it’s useful in various scenes.

For example dividing distance by speed is time, dividing total number by the number of thing is average, or dividing a portion by the whole is a proportion, there are ...

【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of a rectangle》

...there are lots of people who are poor at those formulas, but if changing into them in other form like, distance is multiplied speed by time, amount of numbers is multiplied a number of things by average, a portion is multiplied the whole by the proportion, and if using each of area and illustrating them, it is easy to understand.

So there are three kinds of illustration in the book. They are three kinds of rectangle, and each of the rectangles is shown as formula.

【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of a rectangle》

In the illustration, the distance is the area of the rectangle and its vertical line is speed and horizontal line is time. The part which we want to ask is shown with oblique line, and there is other illustration on the distance.

The oblique line is cut out and added to under the rectangle or right side. The oblique line is what we want to ask, so intuition told us how we should use the horizontal and vertical line and calculate.

Other amount of number and the portion is...

【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of rectangle》

The amount of number and the portion is shown with the same way.

Needless to say, not only learning a formula by heart, but we should understand why it is done enough, but if there is the one who is poor at changing the numerical formula into other form, the way of thinking is useful, the author said like that.

《A way of solving what is called 鶴亀算》

There is a question like the next.

There are cranes and tortoises and their total number is 10. The total number of two kinds...

【Calculating multiplication on two digit mentally at once】

《A way of solving what is called 鶴亀算》

The total number of two kinds of legs are 26. As to the number of the cranes and tortoises, how many are there?

As to the 鶴亀算, if thinking of multiplication as the area of a rectangle, we can show it with an illustration.

The number of crane on leg is two, and the leg of tortoise is four, so the total number of legs on two kinds of creature is 2 × number of crane ＋ 4 × number of tortoise= 26, according to the question.

We show 2 × number of crane and ...

【Calculating multiplication on two digit mentally at once】

《A way of solving what is called 鶴亀算》

We show 2 × number of crane, and 4 × number of tortoise as the area of each rectangle which is different shape each other.

To my sorrow, I’m not good at showing the illustration in English alone, so I’m afraid it is hard to understand. I’m sorry for it.

There is a rectangle on the crane. Its vertical line is two, for the number of leg on a single crane is two, and its horizontal line is number of the cranes, but its number isn’t written yet. It’s expanded horizontally.

【Calculating multiplication on two digit mentally at once】

《A way of solving what is called 鶴亀算》

As to the tortoise, there is also a rectangle. Its vertical line is four, for the number of the leg on a single tortoise is 4 and its horizontal line on the number isn’t written as well.

While the rectangle on the crane is expanded horizontally, the one on the tortoise is both short on vertical and horizontal line, though the vertical line on the tortoise is longer than the one on the crane.

The rectangle on tortoise is right side to the one on the crane, so the shape which two rectangle is...

【Calculating multiplication on two digit mentally at once】

《A way of solving what is called 鶴亀算》

The shape of which two different shape of rectangles is added is uneven, it resembles L which turns its direction 90 degrees to left, so the author suggested we add other shape to the uneven rectangle and changes it into other rectangle which isn’t uneven.

The new rectangle on the vertical line is 4 and on the horizontal line is 10. As to the horizontal line is total number of the crane and tortoise.

The area on the new rectangle is 40. It means that an additional rectangle is added to the...

【Calculating multiplication on two digit mentally at once】

《A way of solving what is called 鶴亀算》

It means that an additional rectangle is added to the uneven shape and the new rectangle which isn’t uneven is completed, so the area on new rectangle is 4 ×10= 40, so the area on the additional rectangle is 40−26=14

As to the additional rectangle, its vertical line is 4−2=2

4 is the number of tortoise on the leg, and 2 is on the crane.

The horizontal line on the additional rectangle is the number of the crane.

2 × the number of the crane is equal to the area on the additional rectangle...

【Calculating multiplication on two digit mentally at once】

《A way of solving what is called 鶴亀算》

...so dividing 14 by 2 is equal to 7, as a result it turned out that the number of the crane is 7, and the tortoise is 3.

Needless to say, without adopting the way, we can solve it as a question on simultaneous equations 連立方程式, but before establishing the way of solving an equation, in general we used to adopt the figure and solve the question.

By the way, the one who made the question on 鶴亀算 knew the each of total number of the tortoise and crane on their legs before the total number of...

The one who made the question on the 鶴亀算 knew each of the total number of tortoise and crane on the leg before each of the total number on two of them, I’m wondering.

Counting their legs is harder than counting their number, I’m sure.

【Calculating multiplication on two digit mentally at once】

《From a linear equation to a quadratic equation》

With the same way, we can solve a question on a quadratic equation 二次方程式. As an example, let’s think over x squared ＋10x −75＝0. It’s x squared ＋2x＝75, and we think of x squared as the area of a regular square, and 10x as the area of a rectangle.

【Calculating multiplication on two digit number mentally at once】

《From a linear to a quadratic equation》

As I’m not good at expressing on figure in English, so I recommend you actually write.

There is an illustration of the square of x squared and the rectangle of 10x which is added, and divided the rectangle of which area is 10x into two, and move the one of half of the rectangle to lower left under the square of which area is x squared.

Then adding a new regular square to one of half of the rectangle so as to complete a perfect regular square. The side of the new regular square is 5...

【Calculating multiplication on two digit mentally at once】

《From a linear to quadratic equation》

As the new square on the area is 25, for its side is 5, and the new perfect regular square on the area is 25＋75 =100, so we can find the value on x.

By the way, immediately after when reading my English, I can understand, but after several says, when reading, I find it hard to understand, even if it’s my own sentence, so if others try to understand it, I’m afraid it’s very hard to understand. I’m sorry for it.

I have to work harder, but without showing illustration, it’s very hard to do.

【When were the marks like ＋,−,×,÷ were born?】

《Origin on the mark on calculation which we don’t know unexpectedly》

The mark on calculation like ＋,−, ×, ÷, which we usually use every day life. Do you know the time when they began to use?

To tell the truth, the history on the marks isn’t so old. ＋ and − are at the end of 15th century, and × and ÷ are in the 17th century.

About 500 years ago from now on, Europe was in Age of Great Navigation, and commercial activity by ship was very active then.

As there was no radar, they had to do astronomical observation and to calculate a line for...

【When were the marks like ＋,−,x,÷, born?】

《Origin of the marks on calculation like ＋,−,x,÷, which we don’t know unexpectedly》

As there was no radar then, they needed to do astronomical observation and to calculate a line for ship so as to sail farther safely. They had to calculate astronomical number literally.

They’d wanted to calculate a number which was on a grand scale easily, so the marks on calculating was born.

《＋and −》

As to ＋ and −, some people say both of them were born when taking down in shorthand, and it seems to be reliable.

It is said ＋ is originated from et which was...

【When were the marks like ＋,−, ×,÷, born?】

《＋ and −》

It is said it's originated from et which was a Latin language and it means and , the and changed into ＋ gradually for a long long time.

On the other hand, − is m which is an initial on minus was abbreviated with a longhand 筆記体, so the mark on − has started.

There is other origin. It is said it was used by sailors for the first time. Sailors prepared for barrels in the ship and there were some water in barrels.

When using the water from the barrel, they drew the mark, −, on the barrel, and when adding water to the barrel, they drew a...

【When were the marks like ＋,−, ×, ÷, born?】

《＋ and −》

...when adding water to the barrel, they drew a vertical line on the horizontal line, ＋. The one which was used when decreasing, −, and the other which was used when adding, ＋, two of them become the marks when adding and subtracting, it’s the origin from the sailors.

《×》

As to the × and ÷, the one who used them for the first time has been clear. As to the x,it was an English mathematician, William Oughtred. He used the × in his book for the first time in 1631, and it’s the origin on ×.

But its origin on the shape, there are some....

【When were the marks, like ＋,−, ×, ÷, born?】

《×》

But its origin on the shape, there are some opinions.

The one is a cross on the Christianity. When the cross is inclined, it changes into x.

The other is it was taken from the national flag on Scotland.

By the way, William Oughtred is the one who used sin in trigonometric function 三角関数 for the first time.

Multiplication is expressed with other mark, • as well. In fact, x which is expressed with multiplication wasn’t popular among in European Continent so much.

A German mathematician, Leibniz used to say like the next.

He didn’t like....

【When were the marks like ＋,−, ×, ÷, born?】

《×》

He didn’t like x as mark for multiplication, for it is easy to mistake it for the one in alphabet. When expressing a multiplication, he used • between two numbers.

It seemed that the way of thinking was the main stream then.

After a typewriter and a personal computer have been spread out among the people, the x has not learned to use when calculating on multiplication, especially in alphanumeric half-size font character, it’s misleading.

When calculating on multiplication with a keyboard for a personal computer at present, there is no...

【When were the marks like ＋,−,×,÷, born?】

《×》

...there is no keyboard for x on multiplication. When calculating on multiplication with an excel, ＊ is used.

《÷》

A mathematician in Swiss used ÷ in his book in 1659 for the first time. A number which is expressed with a fractal number changed into ÷ during a long time.

When expressing a division, ：or ／ are used. ／ has an older history than ÷ and it has been used all over the world at present.

As to the ：, it is said Leibniz used it as a mark for division for the first time. ：has been used as the mark for the division in Germany and...

【When were the marks like ＋, −, ×, ÷ born?】

《÷》

While ：has been used as a mark for division in Germany or in France at present, it’s usually used as a mark for proportion in other countries.

In fact there aren’t so many countries where ÷ is generally used. It’s limited in a part of the world like England, America, Japan, Thailand or Korea. In general ／ has been used in other countries.

In ISO it is described when expressing the division, it should be expressed with ／ or a fractal number. No one should use ÷ as the mark for the division. It is said clearly.

÷ may be going to disappear...

【When were the marks like ＋, −, ×, ÷ born?】

《÷》

Possibly ÷ may be going to disappear from textbooks from all over the world in near future.

《A thing about which an almighty genius cared》

What is the first mark in arithmetic in childhood when starting to learn? If being asked like that, then lots of people may have answered it’s ＋ or =, but numbers like 1.2.3... are marks themselves.

An English mathematical philosophy said it needed a numberless years for us the humankind to notice that the number of 2 on February and two birds are the same two.

February is the second month. Two on...

【When were the marks like ＋, −,×, ÷, born?】

《A thing about which an almighty genius cared》

February is second month. It has number of two. Two in two birds, two in two meters, and two in twenty thousand yen, it means that under the same unit they are the same in relation to its essence, so we express they are two.

In short, getting rid of extra information from each of concrete examples, and we make each of essence abstract.

Whenever a new mark was introduced, making a thing become abstract is done in math. The history on math is the history of making the thing become abstract.

【When were the marks like ＋,−,x,÷ born?】

《A thing about which an almighty genius cared》

Though numerical formulas, including its numbers, have been expressed with marks alone which is abstract in high level, a part of people hate math so much.

Why has a new mark been sought with a new concept in math?

It has a side of character of wanting to an object on thought to be simple, but its biggest reason is preventing us from making a mistake.

When expressing, Leibniz appeared several times, and he cared about the marks so much.

He has been recognized as an authority who reached to essence ...

【When were the marks like ＋,−,x, and ÷ born?】

《A thing about which an almighty genius cared》

Leibniz has frequently been recognized as an authority who reached to essence on calculus 微積分, and contending with Newton for its position, but neither he is well known nor is appreciated like Newton at present. We tend to think like that.

But he had been admired as almighty, and a giant on intelligence then so much that his reputation had been well known all over Europe. Actually he was a versatile mathematician and got results which got fame for coming generations in several fields not only in...

【When were the marks like ＋,−,x, and ÷ born?】

《A thing about which an almighty genius cared》

His achievement reached not only in math but laws, history, logics and philosophy. Leibniz was a marvelous versatile.

He had put his heart and soul into a work from around age of 20 to just before being dead. His work is on marks. A general way of truth on all reason reduced to a single calculation and inventing the marks.

If it comes true, a reasoning in which we need to think it over with high level becomes a simple work like a calculation, and a wrong reasoning couldn’t have been occurred as...

【When were the marks like ＋,−,x, and ÷ born?】

《A thing about which an almighty genius cared》

...and a wrong reasoning couldn’t have been happened as principle, but unfortunately he was dead before accomplishing his achievement and his work was taken over by other mathematician.

《The work is done so as to understand the world precisely》

The marks on calculus which Leibniz invented is excellent. Its calculation has a background which isn’t easy to understand in mathematical field at all, but everybody can calculate as if it handled a fractal number even if it doesn’t understand its principle

【When were the marks like ＋,−,x, and ÷ born?】

《The work is done in order to understand the world precisely》

On the other hand, the marks which Newton invented are simple, those marks hardly lead the calculation anywhere, though it’s simple.

An mathematician in England who is well known as the founder on an electronically calculator said the marks which Newton contrived held up the proceeding the math in England by a century.

We use our language every day life and it has a variety of nuance in varied situation. Even if it has the same meaning, it has a different meaning in different scene.

【When were the marks like ＋,−,x, and ÷ born?】

《The work is done so as to understand the world precisely》

Some of the words are hard to define with regard to simple ones like right and left in the first place.

When using the words and going ahead with a discussion, a misunderstanding absolutely happens. It is possible that a reasoning which has ability of logical thought is led in a wrong way.

On the other hand, the marks have been contrived so as to express mathematical concept alone, we aren’t confused with ambiguous words nor nuance.

It means that we don’t make a mistake as long as...

【When were the marks like ＋,−,x, and ÷ born?】

《The work is done so as to make out the world exactly》

We don’t make a mistake as long as we understand the definition and rules on the marks, for the marks have been used in math there.

We may have felt the marks which are used in math is something inorganic and cold, but every mark is the fruit of effort and talent by people who had blood and at the same time, these marks are filled with ideal in which they tried to understand the world correctly.

The author said like that.

The math is over for the time being. I’ll start to the next topic.

【Doubting is start from science】

《Do you know the meaning the probability of rain is 30%?》

I’m going to a new talk, being based on a book, はじめてのサイエンス, written by Akira Ikegami.

When hearing of a word on science, we tend to think we have nothing to do with it, as for me, when being a student, my school record on science was poor, I don’t feel like reading a book on physics and chemistry filled with difficult marks and numerical formulas, besides, I’m not spiritless for doing it. It’s all right as long as we leave it to experts on science.

Some people have may thought so, including me.

【Doubting is start from science】

《Do you know the meaning that the probability of rain is 30%?》

But we’ve face to the way of watching scientifically every day life.

When going to the place for our job, we check the probability of rain today in weather forecast on the internet or TV, and we decide whether or not we should go with an umbrella.

Then let’s suppose that there was the probability of rain is 30%. As to the 30%, what does it mean?

A high pressure zone and a low pressure zone are written down in a weather chart. When forecasting the weather, the ones who are engaged in the...

【Doubting is a start from science】

《Do you know the meaning that the probability on rain is 30%?》

When forecasting the weather, the ones who are engaged in the weather forecast searched from a large amount of data.

When being in the same weather chart, how often did it rain in the past? When there were the same weather of hundred of the weather chart and it rained 30th time, it means that the probability of rain is 30% then.

Both of the principle on weather and the concept on the probability are typical scientific way of thinking. In short scientific idea has been rooted deeply in our...

【Doubting is a start from science】

《Do you know the meaning of the probability on raining is 30%?》

In short science has been rooted deeply in our every day life.

The author said he had never taken a science course in the past, and when looking at a technical book on science, he got cold feet, but he had to be in charge of a caster on the TV program in NHK, 週間こどもニュース, he couldn’t refuse it any more.

Lots of questions on weather were gathered to the 週間こどもニュース. It started from the meaning on the probability of raining to other question, what is the pressure at first?

The author had to...

【Doubting is a start from science】

《Do you know the meaning that the probability on raining is 30%?》

The author had to clear up the problems and to explain them to children one by one.

When a Japanese accepted Nobel prize on physics or chemistry, he had to comment on the most advanced technology in order to make the children understand.

Not only the weather but every kind of scientific field, he read technical books and listened to experts and tried to understand basic principles and mechanism for himself, and to explain those things to the children with his own words.

Thus drawing...

【Doubting is a start from science】

《Do you know the meaning that a probability on raining is 30%?》

Thus drawing science close to him, he was aware that scientific way or imagination has been related to not only the weather forecast in the scene of every day life but all kinds of situations from an international situation to the future on Japan closely.

A nuclear weapon has controlled the fate the international situation at present. Without progress on physics, it wouldn’t be developed.

It is said regenerative science 再生医療 decides the future on medical science, but without understanding...

【Doubting is a start from science】

《Do you know the meaning that the probability on raining is 30%?》

Without knowing biology, we can’t understand the essence on the regenerative medicine.

When thinking over an earthquake just direct under the metropolitan area which breaks out within 30 years with the probability of 70%, the knowledge on earth science is essential for us.

Scientific world has spread out in the background on serious news. It has never nothing to do with us.

We need to understand news which is reported everyday from TV and newspaper enough. We have to understand how...

【Doubting is a start to science】

《Do you know the meaning on the probability on raining is 30%?》

We have to understand from how our society is, how the international society is and to the future on the earth with our own brain, and to think it over sufficiently.

《What is science in the first place?》

At first what is science?

When hearing of the word on science, it may remind some people of a law or theory, for example, the law of universal gravitation when learning in school, or the principle of relativity. We are apt to think the law and theory is right completely and absolutely, but...

【Doubting is a start to science】

《What is the science at first?》

...but scientific law and its theory aren’t an absolute truth at all.

A TV program frequently said that a shocking truth is revealed. If hearing of the way, viewers thought there is an absolute truth which is correct with probability of 100%.

The author asked the staff who were in charged of the 週間こどもニュース not to use the way of saying, for a way of seeing a thing by us the human race isn’t perfect, so we can’t grasp the truth which is correct with probability of 100%. Otherwise we are an almighty God.

Science is the same.

【Doubting is a start to science】

《What is science at first?》

Science is the same. Possibly there may be the truth, if so we want to approach there as nearer as possible.

The author said he thought science is a work of approaching the truth with a limited recognized way little by little.

Then how do we approach the truth with science? The author said its first step is starting from doubting.

All of others think it’s true, but is it so? Or why does it happen?

Without swallowing opinions around himself or herself, the one who has a scientific attitude doubts whether or not it’s true and...

【Doubting is a start to science】

《What is science at first?》

...and issued a question, why? After issuing the question, then framing a hypothesis for the question so as to demonstrate the question.

Each of scientists frames the hypothesis in science and examines it whether or not it’s true, for the hypothesis is a theory which is built temporality, so the scientists have to make sure whether or not it’s right. They have to verify the hypothesis.

Verification has various ways. The one which is easy to understand is to make an experiment. When making the experiment, the hypothesis has an...

【Doubting is a start to science】

《What is science at first?》

When making an experiment and the experiment has an evidence to support the hypothesis, it turns out that the hypothesis is an explanation which is near the truth, but a result from the experiment is sometimes different from the hypothesis.

Then the scientists have to amend the hypothesis and verify whether or not the amended hypothesis is right. Thus hypothesis and verification are repeated and trying to approach the truth as nearer as possible. It’s the work on science.

《How do we make sure the hypothesis?》

When verifying...

【Doubting is a start to science】

《How do we make sure the hypothesis?》

When verifying the hypothesis, even if a single person succeeded in an experiment, it doesn’t always mean that the hypothesis is true.

In other words, if everyone takes the same procedure and reaches the same result in the experiment, the hypothesis is right.

Therefore even if insisting that there is STAP cell, other scientists all over the world can’t succeed in the experiment with the same procedure, it means that the hypothesis is in the wrong

Needless to say, a first experiment will be done by a single person alone

【Doubting is a start to science】

《How should we make sure a hypothesis?》

If succeeding in the experiment by the single person alone, clarifying conditions and procedures on the experiment, and making everyone make an experiment is possible.

When making the experiment over and over again, there was the same result, then the hypothesis is formed as truth temporarily. Its the scientific truth or its law.

There are lots of scientific laws and its theories in a textbook, but it doesn’t always mean that everyone recognized it as truth from the beginning. Lots of people verified it over and...

【Doubting is a start to science】

《How should we make sure a hypothesis?》

Lots of people repeated verifying over and over again, the hypothesis has learned to be recognized its true gradually.

As to the law of universal gravitation by Newton, it had been recognized as truth for a long time since it was discovered by Newton, but it has been replaced by a theory on gravitation proposed by Einstein in the beginning of the 20th century.

Einstein doubted the law of universal gravitation that every one thought it was right, and made science make progress...

【Doubting is a start to science】

《How should we make sure a hypothesis?》

In short Einstein doubted the truth which everyone thought and made progress science more forward. Truth isn’t absolute but temporary.

《What is abstract?》

Then how do scientists frame the hypothesis? When framing the hypothesis, it is important to make a thing become abstract.

Abstract means that selecting common elements from a concrete thing. We need to cut away extra minor points in order to select the common elements.

In short when scientists observe a thing, they cut away extra elements and make elements...

【Doubting is a start to science】

《What is abstract?》

When scientists watch a thing, they cut off side issues and make elements which become the hypothesis abstract.

For example, when an apple fell from the tree, an ordinary person thought the apple fell from the tree, it was natural, and didn’t stop so as to think over the phenomenon at all, but the scientists like Newton thought why a thing fell from the upper place to lower place, and framed a hypothesis on the reason why the thing fell.

Then the scientist paid attention to a movement that the apple falls, so cut off side issues on...

【Doubting is a start to science】

《What is abstract?》

...then scientists cut away side issues on the color or the perfume of the apple. With that scientists paid attention to movements that various things fell and thought over their reason. Why did they fell?

On the other hand, the moon doesn’t fall, so the scientists thought over its reason. They made concrete things abstract and reached the law of universal gravitation.

In this way it is said Newton discovered the law of universal gravitation when looking at the apple fall, but is his episode of this true? When verifying, it turned...

【What I’ve thought】

I’ve tried to continue my response on science, but I was hungry, I’ve wanted to eat something, and my concentration is about to run out, in addition the battery on my iPad is also in the same situation.

I have to work tomorrow. I can’t take the day off until Thursday in this week, though it’s my own option. No one forced me to work like that, it can’t be helped, but I want to rake a rest just now. I want to drink a little.

I’ll come here Thursday as long as we take day off.

When being hungry, I feel like drinking. Without studying English at all, I’ll be alcoholic.

【Doubting is a start to science】

《What is abstract?》

When verifying it turned to be very unclear.

Newton taught science at Cambridge University and there is a tree at the gate of the university at present, and the tree bears apples. It’s the descendant of the tree when Newton looked at the apple fall, and it’s a noted place for tourists, so the episode on Newton and the apple is an air of reality.

But when investigating, no one heard from Newton direct he discovered the law of universal gravitation because he looked at the apple fall from the tree. In fact the source of the informations...

【Doubting is a start to science】

《What is abstract?》

The source of the information were the ones who listened to the niece of Newton. It isn’t sure whether or not Newton said so actually. After Newton was dead some people may have said it humorously.

《If having a breakfast, its scholastic ability improves?》

Thus, scientific attitude is looking at a thing with a doubt and a question, and after observing the thing, making it abstract and framing a hypothesis. It’s a first step to science, but when framing the hypothesis, there is a mistake which we are apt to make. We confuse a relative...

【Doubting is a start to science】

《If having a breakfast, our scholastic ability improves?》

We frequently confuse a correlation with a casual relationship. The correlation is simply that the one thing has anything to do with the other, and the casual relation is two things connect with a cause and a result.

A researcher on educational policy in national organization announced on the result from a study that a child who had a breakfast enough everyday gets a good grade in school in 2003. Why did he know it?

The Ministry of Education has carried out a test on scholastic ability all over...

【Doubting is a start to science】

《If having a breakfast, our scholastic ability improves?》

The Ministry of Education has carried out a test on scholastic ability all over Japan every year, and at the same timer takes a questionnaire, and when checking the result against the questionnaire, it turned out that lots of children who got a good grade in school had had a breakfast enough. The result made some people insist that if having a breakfast enough, it would get a good record in school. They seemed to think like the next.

If having a breakfast enough, the nutrition spreads over the body...

【Doubting is a start to science】

《If having a breakfast enough, our scholastic ability improves?》

...and an energy is sent to the brain, so it makes children concentrate on its attention on studying enough from the morning, then its scholastic ability improves.

In short they thought that there was a casual relationship between the breakfast and the scholastic ability.

It is certain actually to have a breakfast enough is better, but can we call it the casual relationship actually?

Having a breakfast enough every day means that it leads everyday life regularly. If staying up until late...

【Doubting is a start to science】

《If having a breakfast enough, it improves our scholastic ability improves?》

If staying up until late at night and getting up in the morning barely in time, then children would go to school without having a breakfast.

Then what kind of children are they who lead every day life regularly? Probably the children have their parents who makes them lead the every day life regularly, don’t they?

Those parents not only make their children have a breakfast enough but may have had anything to do with their children on education positively, for example they order...

【Doubting is a start to science】

《If having a breakfast enough, our scholastic ability improves?》

...for example the parents order their children to study or make their children go to a cramming school. Then we need to think of the casual relation a home training by parents and the scholastic ability.

Or if teachers in school guide their children to have a breakfast enough, how about it? As those teachers devote themselves to education on their students so much that they teach their student the study enthusiastically, don’t they?

Then there is a possibility that there is the casual ...

【Doubting is a start to science】

《If having a breakfast enough, our scholastic ability improves?》

Then there is a possibility that there is the casual relationship between the enthusiasm on education by teachers and the scholastic ability on children.

In short we can say there is a correlation between having a breakfast and high scholastic ability, but we can’t always say there is the casual relationship because that having a breakfast enough makes the children improve their scholastic ability.

In the same example, OECD has carried out a test. Children whose age of 15 in several...

【Doubting is a start to science】

《If having a breakfast enough, our scholastic ability improves?》

Children whose age is 15 at several countries in the world take the test once three years. Children who read a newspaper have a high scholastic ability was led from analysis on the result from the test.

Newspaper companies are pleased with it very much, but if happening to say, reading the newspaper makes children improve their scholastic ability, it is far from being scientific attitude. In this case, they confuse the correlation with the casual relation.

It is certain that reading the...

【Doubting is a start to science】

《If having a breakfast, our scholastic ability improves?》

It may be certain that reading the newspaper makes the children improve their scholastic ability, but we can interpret it from an opposite direction.

In short, children who have high scholastic ability take interest in a news or in the newspaper.

When it turned out alone that children who have high scholastic ability are interested in news very much, it doesn’t always lead to the casual relationship that taking interest in the news makes them improve their scholastic ability.

《Even Aristotle...》

【Doubting is a start to science】

《Even Aristotle framed a wrong hypothesis》

Thus we tend to take the correlation with the casual relationship by mistake, so if judging something by its appearance, we are apt to frame a wrong hypothesis. It was the same with Aristotle who is a great philosopher in the Ancient Greece.

He was aware that sound is slower than light with regard to the speed in the fourth century B.C. Why was he able to be conscious of it?

For example, from the time that a lightning shines to it thundered, there is a time lag then. Though we see a lightning, it takes a time....

【Doubting is a start to science】

《Even Aristotle framed a wrong hypothesis》

Though we see a lightning, it takes time until it thunders.

Or when someone rows a boat with oar far away from us, the oar hits the water and we can see it instantly, but it takes time until the sound that the oar hit the water reach us.

When observing those things, Aristotle thought the speed on sound is by far slower than that on light.

There wasn’t any instrument which they could measure the speed on light then, so no one couldn’t know its concrete speed, but we know the hypothesis which is led from the ...

【Doubting is a start to science】

《Even Aristotle framed a wrong hypothesis》

...but we know the hypothesis from which Aristotle led from the observation is right, but observation alone sometimes leads a mistake.

He had a doubt why a thing fell. After thinking it over, he thought the thing had originally nature to move to lower direction.

For example, when dropping some soil, it fell to the ground, for it was natural for the soil to be on the lower part, so it was going to the natural place, he analyzed like that.

There was an opposite interpretation.

Spark flies toward upper direction...

【Doubting is a start to science】

《Even Aristotle framed a wrong hypothesis》

Spark goes up toward the sky, for it’s natural for the spark is in the sky, so it does like that, Aristotle established the theory like that.

It looks strange for us, for we who know gravitation and the law of universal gravitation .

Aristotle analyzed various natural phenomena in the 4th B.C.and framed each of hypothesis. While some of them are right like a relation between light and sound from the viewpoint on the modern science, as to the movement on a thing, he decided with his own way.

But neither of them...

【Doubting is a start to science】

《Even Aristotle framed a wrong hypothesis》

But neither of them reached the procedure of verification. We can frame varied hypothesis as long as we observe, but without demonstrating whether or not it’s right through a verification, it means that it’s just that a decision with its own way like the author expressed before.

《Science and theology》

But scientific way and its imagination were plentiful in the Ancient Greece when Aristotle played in active part. For example, natural philosophers like Socrates and Platoon observed nature and thought over and...

【Doubting is a start to science】

《Science and theology》

...they were wondering what kind of materials made up with the world around them. They framed in a variety of hypothesis, the world was made up with water, it was made from atoms and so on.

But the Ancient Roman Empire controlled the place where the Ancient Greece was, since then they learned to neglect the scientific attitude rapidly in the Middle age.

It is said that it has anything with that the Ancient Roman Empire adopted the Christianity as state religion in 392 A.D.

The world on the Ancient Greece is polytheism ...

【Doubting is a start to science】

《Science and theology》

The Ancient Greece was the world on polytheism 多神教, we know it from the Greek mythology in which various Gods appeared, but Christians thought the polytheism as a heathen 異教徒, and the way of thinking on the polytheism was in the way when the Christians thought of the God.

As a result, the Christians ordered to eliminate the Greek imagination and its learning and to think over the God on the Christianity alone.

In the Middle Age, learning on the Christianity itself became a way of rising in the world. Excellent students learned...

【Doubting is a start to science】

《Science and theology》

Excellent students learned a theology in Christianity then. Reaching the extremely truth in Christianity and being a scholar on Christianity, and the excellent students open the way of success in church.

Even if thinking of something scientific, no one can succeed, nor they become wealthy, as a result they learned to neglect science gradually. It seemed the people who believed in Christianity went through the process.

The author said he felt it actually when going to Iran.

There is a religious city called Qom in Iran. Khomeini who...

【Doubting is a start to science】

《Science and theology》

The revolution in Iran and Iraq was under the control of Khomeini, and he used to study at a theological school in Qom. When visiting there, excellent people who are at the top in Iran gathered.

The Shiites シーア派 account for the majority of Islam in Iran. Learning on the Shia is the most excellent thing in Iran, and the Iranian have learned the theology in Shia deeply and assemble a minute theory on the Shia.

When looking at them, the author thought if they were in America or in Europe, those excellent students would have gone ahead...

【Doubting is a way to science】

《Science and theology》

...they would have moved up to the way to science, the author said. If they major in the faculty of science or the faculty of engineering, they would contribute to development on science and technical skill from various points.

Instead that the excellent talent don’t go to science, it has been absorbed in theology in Iran.

Natural science developed in Islam world more than Europe in the Middle Age, but development on science and technical skill has been delayed after modern times. Its reason may have been there.

《But the earth rotates》

【Doubting is a way to science】

《But the earth rotates》

To return to the main topic and I’ll start again.

The Ancient Roman Empire adopted Christianity as state religion and after that the theology in Christianity was the way to a success, so they neglected to think scientifically, but Copernicus turned up in the 16th century, and changed the situation drastically.

The earth was the center in the universe, and the sun and other stars went around the earth.

The earth was the center in the universe in the world of Christianity, but Copernicus doubted it.

He thought the earth went around...

【Doubting is a way to science】

《But the earth rotates》

Copernicus thought the earth went around the sun actually. We call the change from the Ptolemaic theory to Copernican theory the change by Copernicus. It means that a way of thinking changes by 180 degrees.

The Christianity thought the earth was the center in the universe, on the other hand, Copernicus insisted that the earth went around the earth. The phrase of the change by Copernicus is originated from it.

The opinion on Copernicus was developed more still by Galileo who was born in the middle of the 16th century.

A telescope ...

【Doubting is a way to science】

《But the earth rotates》

A telescope was invented in the 17th century, and Galileo used the telescope at once and started astronomical observation.

When he looked at the moon with the astronomical telescope, the way of seeing the moon was different from the way of seeing until then altogether.

They’d thought the moon was a round ball, but when looking at it with the telescope, it’s uneven all over the surface.

Besides Galileo discovered that the Venus is full and is on the wane and a satellite of the Jupiter.

If thinking over the result on the observation...

【Doubting is a way to science】

《But the earth circles around》

But if thinking over the result of the observation with the Copernican theory, we can explain it to others smoothly. If framing the hypothesis that the earth goes around the sun, without any contradiction, we can understand the Venus is full and is on the wane, or the satellite on the Jupiter.

As a result, Galileo learned to be convinced that his thought was right.

Catholic in Christianity has a way of orthodox thinking, and the one who insisted against it was brought into a religious court, we call it an inquisition 異端審問.

【Doubting is a way to science】

《But the earth circles around?》

Eventually Galileo was brought into the religious court and was sentenced to imprisonment for life, though it’s just that he was confined leniently.

It is said that Galileo whispered that but the earth goes around the sun at the religious court, but was it true? Who heard of his whisper? If whispering such a thing in the middle of the religious court, he would be sentenced more heavily than the imprisonment for life.

As to the historical event like that, we should think, wait a minute, who heard of it? The scientific idea is...

【Doubting is a way to science】

《But the earth circles around?》

The scientific idea is significant, the author said like that.

There has never been any record that Galileo whispered like that, and some people say other people invented it after Galileo was dead, but there was none of record which supported the opinion.

There may have been the people like that truly or there may not have. There was none of the record, so we’re forced to say we can’t understand.

The episode on Galileo continued until recently. Catholic Church recognized that the religious court on Galileo was in the wrong...

【Doubt is a way to science】

《But the earth circles around?》

Catholic Church recognized the religious court on Galileo was in the wrong in 1992. The author said he impressed the Pope who recognized that the church was in the wrong favorably.

After Galileo was dead, 350 years have passed. Very long time has passed, and his innocence was fixated at last.

《At the time until he shouted he discovered》

After the time of Copernicus and Galileo, the one who turned up was Newton.

The author said his episode that when looking at an apple fall, he discovered the law of universal gravitation, but ...

【Doubt is a way to science】

《At the time until he shouted he discovered》

...but it’s unclear whether or not the episode is true, like the episode on Galileo when he was brought into the religious court.

Whether the episode is true or not, whenever Newton looked at something fall, he always continued to think why it fell, so he could frame the hypothesis of the law of universal gravitation.

Needless to say, lots of other people could have looked at the apple drop, but Newton alone thought, why didn’t the moon fall? from the phenomena, and reached the law of universal gravitation. What ...

【Doubting is a way to science】

《At the time until he shouted he discovered》

What difference is there between ordinary people and Newton?

He doubted all the things in the universe and always continued to think it over, why? It’s significant. Without thinking anything nor having any doubt originally, nothing will flash into our mind. The principle on Archimedes is the same.

He was ordered to research how much a crown included gold. He had to do it without breaking the crown. He continued to think it for a long time but was not sure what to do.

One day when he took a bath and see the...

【Doubting is a way to science】

《At the time until he shouted he discovered》

When taking a bath one day, he saw the hot water overflow from the bath tub, then something flashed into his brain and he shouted he discovered at last.

Both of Newton and Archimedes went on thinking a single thing for a long time, then something made them reach an idea which solve the mystery.

By the way, when Newton taught in Cambridge university, the plague had spread over England, so the university was closed.

As the university was closed and Newton was released from his duty of teaching students science.

【Doubting is a way to science】

《At the time until he shouted he discovered》

After that he went back to the house of his parents and could devote himself to his study enough. As a result it is said he could reach the law of universal gravitation.

《When a scientist say, I got it, excitedly, science developed》

Why did scientists continued to think over difficult problems? When scientists discovered various things, they have made our life wealthy, but it doesn’t always mean that a single discovery connects with a practical use at right away.

Actually there have been lots of researches...

【Doubting is a way to science】

《When a scientist says, he got it, science develops》

In fact there have been lots of researches which aren’t useful in our everyday life. In spite of it, scientists continue to study. Why?

Solving problems gives them a reward of a sense of satisfaction, doesn’t it?

Observing a natural phenomenon, and framing a hypothesis, then trying to challenge. When being able to express a thing in the world wonderfully with the hypothesis, each of the scientists would think, I did it! like Archimedes, and got a sense of satisfaction.

Various scientists studied for the...

【Doubting is a way to science】

《When a scientist says excitedly, I got it, science develops》

Various scientists have studied for the sense of satisfaction. Even if the scientist studied something, anyone doesn’t give it any money, but the sense of satisfactory that when the scientist expressed a thing in the world perfectly with its hypothesis is the driving force which makes science develop, isn’t it?

It’s not always mean that it is limited to natural science alone. In the world of economy, when an economist established a theory and he could express the movement on economy well, he...

【Doubt is a way to science】

《When a scientist says excitedly, I got it, science develops》

...the economist must have thought then, I got it.

The author suggested we the readers have a doubt at every scene in everyday life, why is it? and framing a hypothesis for ourselves and if being able to express lots of things well with the hypothesis, each of us must have thought, I got it. If enjoying the sense of satisfactory, it leads us to be used to and to be familiar with a scientific thought.

《Six subjects on modern science》

The author said he had commented on the scientific thought until now