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Science,English,and math Ⅱ

レス500 HIT数 5796 あ+ あ-

燻し銀三( 50代 ♂ Oe38xe )
19/02/13 18:26(更新日時)

【Science,English,and math Ⅱ】

I'm going to start a new thread from now on. It seems that the description on the book of which title,『トリセツ.カラダ』will be over befere long,so after that I'll express on the math,I hope so.

I've wanted to be a splendid English speaker someday,so I've expressed some of English thread,but I'm afraid I find it hard that my hope comes true,but I won't give up,for it seems that I love English.

There are lots of things which I don't know in the world,so I've wanted to express the unknown world for me as much as I can.

No.2441339 17/03/05 20:17(スレ作成日時)

新しいレスの受付は終了しました

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No.51 17/06/25 20:14
燻し銀三 ( 50代 ♂ Oe38xe )

【What is a function?】

《A vending machine for a soft drink or a ticket for a train are also the function》

Needless to say the x is the time and the y is the number of the car which is produced. There seems to be lots of functions around us.

There are some members in a group. If each age corresponds to each member,it's also the function. At first each name of us is the function. It's the function which each people corresponds to each name. Then we can't have two kinds of name at the same time. It's limited to either the name in a family register of a pen name.

There is a vending machine in a station. If inserting some money into the vending machine and pushing buttons for our destination,we can get its ticket. In short the correspondence from the button to the ticket for the destination is the function. Other vending machines for the soft drink or liquor are the same.

There are some research workers in a university. Each number is assigned each of them,for when using a name,there…

No.52 17/06/25 20:51
燻し銀三 ( 50代 ♂ Oe38xe )

【What is a function?】

《Vending machines for a soft drink or for a ticket in a station are also the function》

Each number is assigned to each research worker,for when using a name,if being the same name it doesn't go well,but as to the numbers for the research worker,different numbers are assigned to them,so any mistake won't happen. The function which different one corresponds to other different one is the function of 1 to 1.However two buttons sometimes one kind of soft drink,so in general all the functions aren't 1 to 1.

Our everyday life seems to be filled with the functions like this. In other words,even if it seems to be complicated,if showing it with the function,it's put in order and we may find it easy to recognize.

【If using a graph,we can see right away】

《A straight line for the primary function,a parabola 放物線 for the quadratic one,and hyperbola 双曲線 for an inverse proportion 反比例》

In general,it seems to be easy to understand the function with a graph,for the graph…

No.53 17/06/25 21:27
燻し銀三 ( 50代 ♂ Oe38xe )

【if using a graph,we'll see it at once】

《A straight line for the primary function,parabola for the quadratic function,and hyperbola for an inverse proportion》

General speaking,it seems to be easy to understand the function with a graph ,for the graph appeals us more visually than a formula. I'm going to adopt some basic functions and think over each of graph.

There are plenty of functions,and the one which belongs to the easiest one to understand is the primary function. Its formula is y =ax + b. If showing it with the graph it becomes a straight line. When the b is 0,it's y =ax. The formula shows us a relation that the a is in proportion to the b. It's a straight line which goes through the origin on the graph.

The proportion has been adopted for calculations in various reactions on chemistry,besides the proportion is necessary in varied scene everyday life.

For example,a spring balance バネばかり has nature that weight is in proportion to stretch of the spring. When its density …

No.54 17/06/29 23:32
燻し銀三 ( 50代 ♂ Oe38xe )

【If using a graph,we'll see it immediately】

《A straight line for the primary function,a parabola for the quadratic function,and a hyperbola for the inverse proportion》

When the density is fixed,the cubic volume and the weight is in proportion,and if the electric resistance is the same,the electric current is proportion to the voltage.

There is the quadratic function close to us. If dropping a little stone from the rooftop of a building,its falling distance is the quadratic function of the time. If taking no account of the air resistance,it's shown with the next formula.

y=4.9t squared. It means the whole of 4.9 isn't squared but the t alone is squared.

If wanting to know the depth of a well or the height of a bridge,we have only to drop the little stone from the bridge or the well. Then we need to see how much it takes for the stone to reach the surface of the water and make use of the formula.

If showing the quadratic function with a graph,it becomes the parabola. In general…

No.55 17/07/01 23:34
燻し銀三 ( 50代 ♂ Oe38xe )

【If using a graph,we'll see it at once】

《A straight line for the primary function,parabola for the quadratic function,and hyperbola for the inverse proportion.》

In general,a locus 軌跡 of throwing something is the graph of the quadratic function,so we call the graph the locus.

The volume of a sphere is the function of its radius,so its formula is V = four-thirds multiple πr. Then the r alone is cubed to the third power.

By the way whenever I show a fractional number or an involution 累乗,I find it complicated,for I've wanted to express everything in English,but I don't think it's easy to show. I wish I could show it more easily.

As for the inverse proportion,we can find lots of its examples. When moving from one point to other in car,its speed is inverse proportion to its time. When electric resistance is fixed,the electric current is also inverse proportion to the voltage.

The inverse proportion is showed with the formula,y=a xth,and we call the curve on the graph the hyperbola.

No.56 17/07/02 00:25
燻し銀三 ( 50代 ♂ Oe38xe )

【The innermost secret of the solution for an equation】

《The solution of a question starts from setting up an equation as it is.》

A function only looks like an equation,though they are related deeply. The equation is a formula of two functions which is tied with =. Solving the equation means finding the value which meets the equation.

Speaking of math in Ancient Greece or in Ancient Egypt,it almost means geometry. On the other hand it is said that Diophantine is an early settler 草分け with regard to algebra in Alexandria from the 3rd to the 4th century.

It is said there are some sentences on his tombstone like the next.

"I grew up as a boy for one sixth of my lifetime and grew up as a youth for one twelfth of my lifetime. I was unmarried for one seventh of my lifetime. After 5 years my marriage,my son was born,and my son was dead 5 years ago. Then my son's age was as half as mine. When being dead how old was I?"

If wanting to solve the question like that,we fix the value which…

No.57 17/07/02 04:08
燻し銀三 ( 50代 ♂ Oe38xe )

【The innermost secret of the solution for an equation】

《The solution of a question starts from setting up the equation as it is.》

If wanting to solve a question like that,without thinking anything,it's just we have only to adapt the sentence in the question to the equation as it is. Without thinking of various answers,nor having wicked thought,we have only to make the equation as the sentence in the question showed. It's a key to solving the problem.

If being able to make the equation,it's just that we have only to solve it. We frequently have to change the equation into other one so as to solve it. Then we add,subtract,multiple,or divide the same number on the both side of the equation,without using 0.

The way of thinking is very simple and easy. As to the transformation into plus or minus we call it transposition of a term,but without thinking of transposition,we do the same thing both on the side of the equation. It is easy for us to understand. There is no room for …

No.58 17/07/08 10:37
燻し銀三 ( 50代 ♂ Oe38xe )

【The innermost secret of solution for an equation】

《The solution of a question is starting from setting up its equation》

There is no room for misunderstanding then.

An equality is thought to be a situation which two things are evenly balanced on a pair of scales. When putting same weight both on sides of the scale or removing the same weight from the pair of scale,it is evenly balanced. The equality is also the same theory.

Actually a mathematician who is famous for the education for math has devised a teaching material,called algebra balance. If using it,we can measure a weight of minus,according to the book,数学のしくみ,written by 川久保勝夫.

As you know,I have little knowledge on math,so I can't imagine how the weight of minus and the algebra balance at all.

The author set up an equation,being based on the sentence on the tombstone of the mathematician.

He lived to be the age of X. One-sixth of his life he led as boy. One -twelfth of his life he led as youth. One-seventh of his life he…

No.59 17/07/09 01:52
燻し銀三 ( 50代 ♂ Oe38xe )

【The innermost secret of solution for an equation】

《The solutio of the problem starts from setting up the equation》

Diophantus lived for one-twelfths of his life as a youth. He lived a single life for one-seventh of his life. His son was born after 5 years he got married. His son was dead and he was also dead after 4 years. When his son was dead the son's age was
equal to half of his father's lifetime. How old is Diophantus?

Its equation is the next.

One-sixths・X + one-twelfths ・X + one-twelfths・X+5+half・X+4 = X

Seventyfive-eightyfourths・X +9=X

9=nine-eightyfourths・X

X = 84

I hate to say,but I can't understand why the answer is 84 clearly. When reducing the fractions to a common denominators I can understand. After that I can't grasp why it is. In short I hate math,but I'm going to go on expressing on math…maybe.

【If adopting the method of calculation based on fuguring the number of cranes and tortoises from the totals of their legs,simultaneous equation is more useful】

No.60 17/07/09 08:03
燻し銀三 ( 50代 ♂ Oe38xe )

【If adopting the method of calculation based on figuring the number of cranes and tortoises from the totals of their legs,simultaneous equations are easier連立方程式】

《Even if it's very complicated,we can solve it at once with the the simultaneous equations》

I've expressed on the equation last time,then unknown quantity was one,but when the question is complicated,the unknown quantity isn't always single. If the unknown quantity is two kinds the method of calculation based on figuring the number of cranes and tortoises from the totals of their legs,what is called 鶴亀算.

By the way when trying to express it in English,it's long a little,so I make it a rule to call it the method of calculations.

When there are two kinds of unknown quantity,we call it an equation with two unknowns 二元方程式. Then without being two equations we can't solve the question. I'm going to use the equation with two unknowns with the method of calculations.

There are cranes and tortoises. Their total number is 8.

No.61 17/07/15 18:44
燻し銀三 ( 50代 ♂ Oe38xe )

【If adopting a method of calculation on figuring the number of cranes and tortoises from the total of their legs,a simultaneous equation is easier】

《Even if it's complicated,we can solve it with the simultaneous equation》

When adding their legs its total numbers are 22. Then how many are the cranes? And how many are the tortoises?

Without using any algebra,we used to solve the question like the next long ago.

The crane has two legs and the tortoise four legs. If the all the tortoises pull their two legs,the total numbers of the legs of the crane and the tortoises are 16. 2×8=16,but the original numbers of the legs used to be 22. The difference is 6. All the tortoises pulled their two legs. 6÷2=3,so the tortoises are 3,as a result,the cranes are 5.

The way of solving the question is making the tortoise pull its two legs,but we don't have to do it with the algebra. The crane is X,and the tortoise is Y,so the number of the leg of the crane is 2X,and the tortoise is 4Y.

After that…

No.62 17/07/15 19:22
燻し銀三 ( 50代 ♂ Oe38xe )

【If adopting a method of calculation based on figuring the number of cranes and tortoises from the totals of their legs,a simultaneous equation is easier.】

《Even if it's complicated,we can solve it right away with the simultaneous equation》

After that without thinking anything,it's just that all we have to show an equation with the question and solve it.

X + Y = 8. The total number of the crane and the tortoise are 8.

2X + 4Y = 22. The total number of their legs are 22.

I'm going to solve the equations. At first multiple 2 by both sides of the former equation. 2X + 2Y = 16. Then subtract the the second equation from the third equation on both sides,so 2Y = 6,Y = 3.

When substituting the answer to the first equation,X + 3 = 8,so X = 5. The tortoises are 3 and the cranes are 5.

If using a matrix 行列,we can solve the simultaneous equation at once. I'm going to express it its chapter later,but the problem is whether or not we can reach there…

【The solution of the formula is helpful】

No.63 17/07/15 20:00
燻し銀三 ( 50代 ♂ Oe38xe )

【The solution of the formula is very helpful】

《We can distinguish the property of the solution in a discriminant of a quadratic equation二次方程式》

An equation with two unknown and a quadratic one resemble when saying in Japanese.

While the equation with two unknown means that there are two unknown numbers such as x and y in the equation,there is one unknown number in the quadratic one,but the quadratic one includes the square of x.

General form of the quadratic equation is ax(only part of the x is squared)+ bx + c =0. As for the quadratic equation,we can solve it with factorization 因数分解 as long as it's a special case,however in general we can't always solve it with the factorization,so someone contrived the solution of the formula.

The solution of the formula is familiar with the math in high school. If using it,our mathematical ability improves so much,maybe…even if the a,b,and c are what kinds of number we don't have to mind them.

As to the solution of the formula,√ appears,and…

No.64 17/07/22 01:53
燻し銀三 ( 50代 ♂ Oe38xe )

【The solution of formula is very helpful】

《We can distinguish the property of the solution in a discriminant of a quadratic equation.》

I've tried to express on the solution of the formula,but I hate to say it's too hard to understand. In short,when it's value is minus,it means that the solution of the equation isn't a real number.

The number with which we usually deal is the real number,so then there is nothing with regard to the solution,but even if its is minus of,it seems to be all right in a wide sense in mathematic.

As you know,I'm not good at math at all,but when there is a thing which we can't understand,we need one more thing in order to understand the situation. I'm wondering if it's a complex number. Its square root is minus.

If it's irrational and strained,the phrase of the complex number must have disappeared,I'm sure. The book of 数学のしくみ uses a graph and expresses on the complex number though…

No.65 17/07/22 04:32
燻し銀三 ( 50代 ♂ Oe38xe )

【The secret of solution with regard to a cubic equation】

《As for the solution,it's the most interesting episode in the history of math》

It seems that the solution of liner equation and quadratic one has been familiar with us from long ago,and the intellectual curiosity of the human being has moved forward to the solution of the cubic and biquadratic solution naturally.

There is an episode of the solution and announcement of the solutio which is very interesting. It happened in Italy in the 16 century.

The one who dealt with the solution was スキピオ デル デル フェロ in Bologna universiy. He discovered a special solution on the cubic equation,and he told it for his disciple,but he didn't make public his discovery.

On the other hand,ニコロ フォンタナ discovred an ordinary solution. He was born into a very poor peasant,but he studied math with educationally,and getting over lots of difficulty and he won the solution of the cubic equation.

Mathematical game had been popular in those day. フォンタナ …

No.66 17/07/22 05:22
燻し銀三 ( 50代 ♂ Oe38xe )

【The secret of solution with regard to a cubic equation】

《As to the solution,it's the most interesting episode in the Mathematical history》

フォンタナ challenged フェロ to the mathematical game. The mathematical game was done on view to general public. They prepared 30 questions each other and they had tried to solve the questions each other.

While フォンタナ solved all フェロ's questions in 2 hours,フェロ couldn't solve フォンタナ's questions at all. フォンタナ tried not to open the solution to the public because of the manners and customs then,but カルダン in Milan was interested in the solution very much.

カルダン talked フォンタナ into showing him the solution. He made a promise not to leak the solution to others and succeeded in getting the solution out of フォンタナ. It caused a heated dispute over two of them.

カルダン broke his promise and showed the solution in his book. Needless to say フォンタナ stamped his foot in frustration and protested to カルダン about breaking the promise,but カルダン pretended ignorance.

フォンタナ was in…

No.67 17/08/27 14:39
燻し銀三 ( 50代 ♂ Oe38xe )

【The secret solution with regard to a cubic equation】


《As to the solution,it's the most interesting episode in mathematical history》

フォンタナ who was in the extreme distress challenged カルダン to do an opened mathematical match,but instead of カルダン,フェラリwho was カルダン's pupil appeared the place where the match was done. フェラリwas young and spirited mathematician. Unfortunately カルダン was defeated then.

In this way,the one who discovered the solution of the cubic equation becomes カルダノ. His name isn't カルダン but カルダノ. I'm sorry for it.

As for the solution of quartic 四次方程式,フェラリhas done.

This is the famous episode in relation to the cubic equation or quartic in mathematical history,but this kind of thing isn't always in the past. Similar incident seems to happen sometimes at present.


【is there a solution of quintile 五次方程式?】

Both of the solution of the cubic equation and quartic was discovered in ITaly in 16th century,so everybody thought the solution of quintile was also done easily,but …

No.68 17/08/27 15:32
燻し銀三 ( 50代 ♂ Oe38xe )

【Is there a solution of quintile? 】

However the most shocking incident in the history of mathematic has happened.

A young Norwegian mathematician アーベル demonstrated that there is no formula of solution in relation to the quintic. Then he was 21 years old.

Being unable to solve absolutely. It doesn't occur to everyone at ease. It seems to be one of the hardest things to do…oh! I've tried to describe the episode but I've lost interest in it. To begin with I hate math. I'm not interested in the quintic at all.

Even if being able to solve or not,it doesn't matter. It's none of my business.

【If turning into a triangle…】

《As to a space of a figure which is surrounded by straight lines, we can find it easily as long as it divides into triangles》

As It is said that Egypt is the gift from the Nile,the Nile river supported the Egyptian culture and flooding of the Nile river which happened almost every year stimulated development technique of surveying. The word of geometry is made up …

No.69 17/09/02 07:41
燻し銀三 ( 50代 ♂ Oe38xe )

【If turning it into triangles …】

《As to a space of figure which is surrounded by straight lines,we can find it easy as long as it's divided into triangles》

The word of the geometry is made up with geo and metry. It means a land and its measurement. It shows the two relations between them plainly.

I'm going to handle a space of a figure which is surrounded by complicated curve at integration 積分,and I'm going to deal with the one which is surrounded by straight lines this time.

In general,as to the figure which is surrounded by straight lines,its shape will never be fixed if each length of sides is given,for a quadrangle 四角形 of the length of all sides is 1 is just a lozenge ひし形,so both its shape and space is varied.

On the other hand,if being fixed each length of the three sides,its shape is also done in relation to a triangle. It's one of the congruent conditions 合同条件 on the triangle,the two triangles of which three sides are the same are congruent. As to the condition,the …

No.70 17/09/02 08:30
燻し銀三 ( 50代 ♂ Oe38xe )

【If turning it into a triangle …】

《As to a space of a figure which is surrounded by straight lines we can find it easy as long as it's divided into triangles.》

This is a point which is remarkably different from other figures with regard to the triangle.

If divided into triangles and fixed each length of sides,its shape is done and as a result it's space is done. In short,when showing a figure,if using a triangle as one of piece,we can show the figure precisely.

In this way a land is divided into triangles and is drawn on a plan of a registry book. The way of thinking is applied to measurement of lands,its triangular surveying. I'm going to express its detail at the trigonometric functions.

There is one more application,it's a diagonal brace すじかい. When building a house it's put between posts diagonally. Without putting the diagonal brace,the house is easily broken. Not only the house but all the constructions like abridge or a tower is constructed,being based on a triangle,for…

No.71 17/09/02 09:25
燻し銀三 ( 50代 ♂ Oe38xe )

【Geometry will revive】

《What's called 五心 in a triangle》

All the constructions are based on triangles,it's not for beauty but for it's adopted so as to keep balance on the construction.

It's been a long time since geometry disappeared from textbooks in high school but lots of people seem to want to it revive,for not only it's beautiful and fascinating as material but the demonstration is important for a proof. Plenty of people have learned to recognize it. Geometry seems to be taught in high school. I'm relieved that I 'm not a student in high school any more.

What is called 五心 is also included in the geometry. The five of them are in a triangle,it's a center,inner center,circumcenter,excenter 傍心,orthocenter 垂心.

Dividing each three angle of a triangle into half and drawing lines from each triangle,and there is a point where the three lines intersect. It's the center of the triangle.


Drawing a perpendicular垂線 line from each side to each angle and there is a point where the three …

No.72 17/09/02 20:27
燻し銀三 ( 50代 ♂ Oe38xe )

【The shape which the God has created】

《They calculate as long as there is π there. The drama of the π.》

I've tried to express what's called 5心,for the author insists that the geometry is beautiful and the 五心 is also included,but I don't think so,in addition without illustrating them,I find it hard to make others understand,so I give it up. I'm going to express other topic.

No figure is more beautiful than a circle. Ancient people seemed to think the circle was a perfect one which the god created.


The circle is also mysterious,for it's always the same shape however it is,so the length of the circumference and its diameter is the same ratio whether the size of the circle is big or small. It's the ratio of the circumference of a circle to its diameter, the π. It is Euler who used the mark of the π for the first time. The mathematician appears in various field of math.

The calculation of the π has long history. Ancient Babylonia people used to adopt an approximate value of the π…

No.73 17/09/14 22:39
燻し銀三 ( 50代 ♂ Oe38xe )

【The shape which the God has created】

《They calculate as long as there is π there. The drama of the π》

Ancient Babylonian people used to adopt 3 as the approximate value of the π. It's said that Ahmed is the oldest mathematic book of the ancient Egypt. The book said that the value of the π is 3.16049…so it's value is very close to the π.

It's Archimedes who tackled the calculation of the π theoretically. He was a Greek mathematician. He adopted a regular polygon 正多角形 which was inscribed 内接 or circumscribed 外接 a circle and calculated the π both from the bigger polygon and the small one.

As to the regular polygon which is inscribed to the circle is made up with 6 regular triangles of which side is 1,so it's circumstance is 1×6 =6,and its diameter is 2,so 6÷2=3.

On the other hand other regular polygon which is circumscribed the circle is made up with 6 right triangles. The two sides which put the right angle between them is 1 and route 3. Its oblique side 斜辺 is 2 and the oblique…

No.74 17/09/16 17:34
燻し銀三 ( 50代 ♂ Oe38xe )

【The shape which the God has created.】

《They calculate as long as there is the π there. The drama of the π.》

Archimedes divided both the triangles which is inscribed and circumscribed into 96. The circumference of inscribed triangles is 3.1408…and the circumscribed ones is 3.1428…,so the π is between them. It's showned as an inequality like the next.

3.1498 < π < 3.1428.

To my surprise,Archimedes calculated the value of the π as an approximate value of about 3.14 then.It's the one which we usually use the approximation at present.

After that mathematicians have learned to calculate the π as if it's were a famous saying of the mountaineer,'We climb because there is the mountain there.' so they continue calculating the π there is the π there.

New ways of calculating the π have been found after differential and integral calculus 微分積分 was born in the 17th century. In addition when being invented a computer,needless to say the calculation of the π has made rapid progress.

As for …

No.75 17/09/16 18:18
燻し銀三 ( 50代 ♂ Oe38xe )

【Spreading all over the same shape】

《When tiling,can we do it what kind of regular polygon?》

As for an efficiency of a computer what number of digit can the computer calculate? It's a kind of barometer for the computer ,so the calculation of the π has gone on at present.

When walking along the streets which are lined with old neat rows of stores and houses in Europe,some people tend to be fascinated with geometric pattern spreading all over the road made of stone,without realizing it,they can't take their eyes off it.


The stylish tile spreading all over the floor in a fashionable hotel or company takes part in creating a good atmosphere. The way of tiling seems to be varied.For example, the one which spread the same pattern all over and two kinds of pattern is done. I'm going to describe the way of tiling the same regular polygon here.

The tile of which shape is a regular square is the most popular. We can see it in a bathroom. The other ones which occurred to us at once are…

No.76 17/09/18 19:51
燻し銀三 ( 50代 ♂ Oe38xe )

【Spreading all over the same shape.】

《When tiling,can we do it with what kinds of regular polygon?》

The other tiles which occur to us at once is a regular hexagon of which shape is a honeycomb,and a regular triangle. It seems that any kinds of shape is suitable for the tiling,but it doesn't,for if adopting a regular pentagon,there are some space between them.


When tiling,without being any space and spreading the same shape all over,it's limited to the regular triangle,the regular square,and the regular hexagon. If trying to do with a regular heptagon 7角形,it seems to be impossible.

If tiling with the regular polygon of the same shape without being any space,the one which becomes 360 degree when displaying is suitable.

By the way,the sum of interior angles of a polygon is found with an equation of (n−2)×180,then the n is the number of angles of the polygon,even if its shape changes its value is fixed,according to the book,though I don't know why. The number of the angle seems…

No.77 17/09/18 22:30
燻し銀三 ( 50代 ♂ Oe38xe )

【Spreading all over the same shape】

《When tiling,can we do it with what kind of regular polygons?》

The number of the angles decides the sum of the internal angles.

An equation of one of angles of a regular polygon is (n−2 )divided by n,and multiply its value by 180.

When collecting regular polygons of the same shape around one point without being any space,what's kind of the regular polygon? It's equation is multiply m by (n−2) divided by n,and multiply its value by is equal 360. The 360 means 360 degree. The m is the number of the regular polygon.

Though its calculation is showed in the book,but to my sorrow,I can't understand it very much. It's sub title is 入門ビジュアルサイエンス,so there are a lot of illustrations in the book. Without showing any illustrations,I find it hard to express what I want to say.

By the way,the m and n is more than 2,for there is no 2角形. If solving the equation,it's value is 3,4.or6,so the tilting of the regular polygon is regular triangle,regular square,and…

No.78 17/09/18 22:30
燻し銀三 ( 50代 ♂ Oe38xe )

【Spreading all over the same shape】

《When tiling,can we do it with what kind of regular polygons?》

The number of the angles decides the sum of the internal angles.

An equation of one of angles of a regular polygon is (n−2 )divided by n,and multiply its value by 180.

When collecting regular polygons of the same shape around one point without being any space,what's kind of the regular polygon? It's equation is multiply m by (n−2) divided by n,and multiply its value by is equal 360. The 360 means 360 degree. The m is the number of the regular polygon.

Though its calculation is showed in the book,but to my sorrow,I can't understand it very much. It's sub title is 入門ビジュアルサイエンス,so there are a lot of illustrations in the book. Without showing any illustrations,I find it hard to express what I want to say.

By the way,the m and n is more than 2,for there is no 2角形. If solving the equation,it's value is 3,4.or6,so the tilting of the regular polygon is regular triangle,regular square,and…

No.79 17/10/09 11:44
燻し銀三 ( 50代 ♂ Oe38xe )

【Spreading all over the same shape.】

《When tiling,can we do it with what kind of regular polygon?》

If solving the equation,it’s value is 3,4,or 6,so tiling of regular polygon is a regular triangle,a regular square,and a regular hexagon. If allowing for transformed hexagons,three are more various tilings.

【The beauty is a sense of balance.】

《We have been fascinated by golden ratio.》

If a circle is a flawless figure which the god has created,a rectangle of golden ratio which we the human has created is also beautiful and fascinating. It’s a size of paperback pocket edition.

If feeling that the rectangle is beautiful,the ratio of both height and length is important,and the golden ratio which we have been fascinated as literary gem for a long time. We call a rectangle of which ratio of height and length is golden one golden rectangle.

Dividing the length with the ratio is golden section,and painting,carving,and architecture have adopted it widely. It is said that Eudoxos who …

No.80 17/10/09 14:42
燻し銀三 ( 50代 ♂ Oe38xe )

【The beauty is a sense of balance.】

《We have been fascinated by golden ratio》
It is said that Eudoxos who thought up the golden ratio. He was in the era of a philosopher,Plato,and Italian genius,Leonardo da Vinci named it golden ratio later.

I’m going to express on the secret of charm of the golden ratio from now on. Unexpectedly it’s related to the solution of a quadratic equation.

There is a rectangle,ABCD here. Then we change it into a regular square and other rectangle. If the first rectangle and second one are similar in shape,two of them are golden rectangles and its ratio of height and length is the golden ratio.

I’m going to find the value of the ratio with calculation actually. Let’s suppose that the height was 1 and the length was X,and I’m going to set up its equation.

The equation shows that the first rectangle and the second one which is separated from the first one look exactly alike,though then we need to make the first one turn 90 degree. Its equation is X:1=1:…

No.81 17/10/09 16:05
燻し銀三 ( 50代 ♂ Oe38xe )

【The beauty is a sense of balance.】

《We have been fascinated by golden ratio》

The equation is X:1=1:(X−1). X(X-1)=1. X squared-X-1=0,so X=half of 1+ route 5,so X ≒ 1.62,according to the solution of the equation,though to my sorrow it doesn’t always mean that l calculate it for myself. It’s just that l made an exact copy of the book. To tell the truth I can’t understand the solution of the equation at all.

Leonardo da Vinci made use of the golden ratio for various areas in his artistic activity like art,industrial art,and architecture. Ancient art and architecture also adopted the golden ratio,but is it so beautiful? I’m wondering if it’s just the rectangle.

【Solve it with a ruler and compasses.】

《Greek three hardest problems,doubling the cube,dividing the angle into three equally,and squaring the circle.》

There are Greek three difficult problems in history. They are problems of drawing figures which we solve with a ruler and compasses.

The first one is doubling the cube. It’s …

No.82 17/11/05 05:56
燻し銀三 ( 50代 ♂ Oe38xe )

【Solvie it with a ruler and compasses】

《Greek three hardest problems. Doubling the cube,diving an angel into three equally,and squaring the circle》

The first one is doubling the cube, if doubling the volume of a cube,what times should we make each side of the cube ? We call it Delian problem. If showing it in an equation,it’s X cubed =2A cubed.

In short it means that whether or not we can draw the solution of a cubic equation with a ruler and compasses.

The next one is dividing an angle into three equally. Then we also use the ruler and compasses alone. It also arrives at a conclusion of drawing a figure with the cubic equation.

The last one is squaring the circle. There is a circle. Drawing a square of which space is equal to the circle.

Why do we use the ruler and compasses alone? A straight line is done with the ruler. It’s shown with a linear equation. The circle is done with the compasses. It’s shown with a quadratic equation.

A point of intersection by drawing line is a …

No.83 17/11/05 06:33
燻し銀三 ( 50代 ♂ Oe38xe )

【Using a ruler and compasses and solve it】

《Greek three hardest problems. Doubling the cube. Dividing an angle into three equally,and squaring the circle.》

A point of intersection by drawing lines is a solution of simultaneous equation,so it’s a solution of the linear equation and the quadratic one. In a word,we use ー × ÷ + and find the value from each coefficient from the equations.

By the way,I can manage to understand the straight line means the linear equation,but why does the circle mean the quadratic equation? The point of intersection …

Lots of mathematicians have been worried about the problems for a long long long,no less than 2000 years,but on the other hand the problems have stimulated the development for mathematics which is related to the problems,but they reach the conclusion at last. To my surprise it was done in the 19th century. Unexpectedly it’s impossible.

Doubling the cube and dividing a triangle into three equally reach questions that we solve it by …

No.84 17/11/05 07:15
燻し銀三 ( 50代 ♂ Oe38xe )

【Use a ruler and compasses and solve it】

《Greek three hardest problems. Doubling a cube. Dividing an angle into three equally. Squaring a circle》

Doubling a cube and dividing an angle into three equally reach the conclusion of solving a cubic equation by drawing figures. In general we need the cubic root so as to solve a cubic equation,but when drawing the figures,it’s just that we can find a square root.

The problem of squaring a circle is different from the other two essentially. If trying to demonstrate it’s impossible,it’s by far harder than the others. Its key is the demonstration that pi is a transcendental number. What is the transcended number?

If thinking of an integral number as coefficient,it can’t be any solution of any algebraic equation. We can show a squared root with a figure but we can’t a cubed root.

By the way,when expressing,I am forced to recognize that we have little knowledge on math,but I don’t feel like studying math. It’s too late to do.

No.85 17/11/07 00:25
燻し銀三 ( 50代 ♂ Oe38xe )

【Euclid geometry】

《The best seller next to the Bible. It’s the base of mythology 方法論 on modern science》

It is said that geometry has started from surveying land in Ancient Egypt. It’s developed greatly in the era of Greece. Euclid formulated the perfect system of the geometry. It’s Euclid’s principles,but it’s not only the fruit of study by Euclid alone but by other Ancient Greek mathematician like Thales,Pythagoras,Hippocrates,and Eudoxos. It’s a classic which is said the best seller next to the BIble in Europe.

The best seller starts from definition which is an agreement with words like points,lines,surfaces,angles,or spheres. The definition changes into a self-evident truth 分かり切った真理,and gets to demonstration,for even if they tried to demonstrate,the chain of demonstration will never end,so they should start something which is accepted without any demonstration. Aristotle said like that,and Euclid recognized it very much.

The composition of definition,axiom,公理,theorem,and…

No.86 17/11/07 01:10
燻し銀三 ( 50代 ♂ Oe38xe )

【Euclid’s geometry】

《The best seller next to the Bible. It’s the base of methodology on modern science》

The composition of definition,axiom,theorem,and demonstration done by Euclid has been the base not only for math but methodology of modern science widely as good example of scientific way. For example,Principia which was done by Newton is a theory of dynamics and gravitation. It’s written with the style of Principia.

Great part of Euclid’s principles had been adopted in Europe as textbook as it is until the 19th century. In addition some schools had used it until only nowadays.

The whole Principles are made up with 13 volumes, and the famous Pythagoras theorem is in the last part of the 1st volume,and other parts are conditions of congruent 合同 triangle,drawing figures of a regular polygon. The book deals with five kinds of regular polyhedron 多面体 in the 13rd volume.

The descriptions are so various that they aren’t only geometry but comparison theory,the theory of integral …

No.87 17/11/07 02:00
燻し銀三 ( 50代 ♂ Oe38xe )

【Euclid’s geometry】

《The best seller next to the Bible. It’s the base of methodology on modern science》

The descriptions are so various that they aren’t only geometry but comparison theory,the theory of integral number,取りつくし方 which is base of integration 積分,but needless to say,to my sorrow,I know none of them.

It is said that the principles are immortal work on mathematic history,and it’s had influenced on not only math but thought and culture,however there is a point of issue about which lots of mathematicians have been worried in the Principles. It’s an axiom on parallel lines.

I’m going to express it in the next time at a slow speed,but before it I’m going to show five of Euclid’s axioms.

1 When there are 2 of any points,we can draw a line on the 2 points.

2 There is any segment 線分. We can extend both end of the segment.

3 If there is any point,and we regard it as a center,and we’ll draw a circle of which radius is any 任意

4 All the right angles are equal.

The last one is…

No.88 17/11/09 23:04
燻し銀三 ( 50代 ♂ Oe38xe )

【Euclid’ geometry】

《The bestseller next to the Bible. It’s the base of methodology on modern science》

There are two lines and other one which cross. Roughly speaking the two lines are horizontal ones and the other is vertical one. There are two interior angles at the place where the two horizfontal lines and vertical one cross. Let’s suppose that the sum of the two interior angles were less than 2 right angles.

Then if the two horizontal lines extend from the point where the interior angles are less than 2 right angles,the two horizontal lines cross. To tell the truth I find it natural,but it turned out to be one of the most important question.

《There is other geometry which doesn’t belong to the compendition of Euclid》

It’s started from parallel lines. When looking at the 5 axioms of Euclid in the last response they are easy and natural,except for the last axiom. On the other hand,the fifth one seems to be a little more complicated. The fifth one is so called the axiom of …

No.89 17/11/10 10:14
燻し銀三 ( 50代 ♂ Oe38xe )

【Euclid’s geometry】

《There is other geometry which doesn’t belong to the compendium 体系 of Euclid》

The fifth one is so called the axiom 公理 of parallel lines. If saying the axiom in a modern way,it’s the next one.

There is a straight line.There is other straight single line alone which passes on the one point outside the former line. The two straight lines are parallel. This is why the axiom is called the one of the parallel lines,but as you know,I’m not so intelligent and I’m not sure of it.

If there is a parallel line,there is two points at the both ends of the straight line,so if there is other straight line which is parallel to the former,there are two points at the both ends of the other one,so it means that the other one pass on two points,I’m sure,though I’m afraid I may make a blunder again.

To return to our main topic… The fifth axiom has been paid attention by plenty of mathematicians for a long time. For example they said,“I don’t think the fifth axiom is necessary …”

No.90 17/11/10 10:52
燻し銀三 ( 50代 ♂ Oe38xe )

【Euclid’s geometry】

《There is other geometry which doesn’t belong to the compendium of Euclid》

They said,“I don’t think the 5th axiom is necessary,for we can arrive at the 5th one from other four axioms.” There used to be lots of mathematicians who tried to demonstrate their theory,but none of them succeeded in it. It has been mysterious for no less than two millenniums.

However,it was solved in the 19th century unexpectedly. Even if they tried to deny the 5th axiom,they can construct other compendium of geometry like the one of Euclid which doesn’t contradict. Some genius on mathe discovered it independently. It’s what is called the birth of non Euclidean geometry.

There is illustrations on the difference between the Euclidean geometry and non Euclidean one on the book of which title,数学のしくみ,however to my sorrow,without any illustration,I can’t describe what they tried to say in my poor English,so if you’re interested in it,please look over some books or search and so on.

No.91 17/11/11 14:35
燻し銀三 ( 50代 ♂ Oe38xe )

【Euclid’s geometry】

《There is other geometry which doesn’t belong to the compendium of Euclid》

The non Euclidean geometry has been so different from others’ sense in everyday life that it wasn’t accepted by the society easily,in addition all the mathematicians who were engaged in studying the geometry led unfortunate lives.

Gauss was an influential figure in mathematician’s world,so he seemed to have great influence on other mathematicians,but even he avoided talking with others about the Non Euclidean Geometry in official place.

When some mathematicians were in difficulty because of the Non Euclidean Geometry,Gauss recognized it but he closed his eyes to their difficulty.

Instead of the 5the axiom,some mathematicians who were in difficulty like Lobachevskii and Bolyai adopted other axiom. It’s says like the next. “There is a straight.line. It’s A. There is a point outside the B. There are numberless straight lines which passes on the B and are parallel to the A.”

No.92 17/11/11 15:19
燻し銀三 ( 50代 ♂ Oe38xe )

【Euclidean geometry】

《What I’ve thought》

Some mathematicians say,“There is a straight line. It’s A. There is a point outside the A. It’s B. There is a single straight line alone which passes on the B and is parallel to the A” it seems to be true in Euclidean geometry. I’m wondering if it has no contradiction at all,but I’m not sure of it.On the other hand what other mathematicians said is different,but they say it has also no contradiction at all,the book said so.

I hate to say,but I’m so stupid that I can’t understand any problem at all,but the Euclidean geometry and non Euclidean geometry reminds me of an episode on 矛盾.

A merchant said his pike which is equivalent to 矛was so strong that no shield could defend the pike. He also said his shield was so strong that no pike could break the shield.A person told the merchant to use his pike to his shield,and what happened? The merchant couldn’t answer.

To tell the truth,I can’t understand what is what!

No.93 17/11/11 21:08
燻し銀三 ( 50代 ♂ Oe38xe )

【What is matrix 行列 and vector used for?】

《The matrix and vector are seemingly enumeration 羅列 of numbers,but it has deep meaning》

The problem is whether or not I can understand it…

It is often said that we the Japanese like a parade. When going to foreign countries,a group of the Japanese marched in,and they tried to act together.

There is the parade in math,but we don’t call it parade in math. It’s called matrix. As to the form itself,the people’s parade isn’t different from the matrix in math so much.

As to the matrix in math,the ones which are lined up are numbers,or letters which means some numbers,but if the numbers or letters are just lined up,there doesn’t seem to be mathematic significance very much. Then where is the significance of existence for the matrix?

Answering the question is the aim of this chapter,the author says like that. It looks like just an enumeration of numbers,but they make us feel things are magical. We learn the magical way and make use of it.

No.94 17/11/13 11:15
燻し銀三 ( 50代 ♂ Oe38xe )

【Non Euclidean geometry】

《A straight line which isn’t straight》

A straight line is the one which is connected between the two points by the shortest course. Let’s think of the shortest course which connects the big cities on the earth,Los Angels and Tokyo. It’s not the straight line,but a curve.…

I’ve tried say there are several things which we can make use of the Euclidean geometry but to my sorrow I find it impossible,I have little knowledge on math.

However even if the Euclidean geometry isn’t perfect,it’s a compendium 体系 which doesn’t have contradiction as geometry. It says the conditions of congruent 合同 triangles,the theorem of sine,five centroid of a triangle.

I can understand how great the Euclidean geometry it is somehow…maybe,I’m not sure of it…

It’s just that I was excited a little and I’m embarrassed. Even if whether or not a straight lien is parallel to other,I don’t mind it at all. It seems meaningless,but there are lots of people who have devoted themselves to it…

No.95 17/11/13 13:31
燻し銀三 ( 50代 ♂ Oe38xe )

【What is matrix 行列 and vector is used for?】

《The matrix and the vector are seemingly enumerations of numbers,but they have deep meanings》

The vector ranks with the matrix. While the vector is a special version of the matrix,it reacts with its own way. Both of the vector and matrix used to be a kind tool which we deal with natural scientific phenomena mathematically,but their practicable range has extended more and more,and they become the one which is connected with not only mathematic field but ordinary society.

Both of the matrix and vector has been put to practical use everywhere in the society as a means with which we arrange and analyze various information. They have played an active part in not only natural scientific field like physics,engineering,civil engineering,and architecture,but social science like business administration,statistics,and accounting.

In addition,they play an active role in the fiele which connects with business direct like marketing,production control…

No.96 17/11/13 14:22
燻し銀三 ( 50代 ♂ Oe38xe )

【Addition and subtraction of the vector】

《Rolfe of calculation draw out of the power from the matrix and vector》

If looking at the matrix and vector blankly,it’s just that they are enumeration of numbers. When bringing rules of calucuration there,mutual connection is closer,and their abilities are shown. For example,even if we have an electric appliance of multifunction,without reading its manual,we can’t use it at will.

However,it doesn’t always mean that we have to steel ourselves against it. It’s not so difficult,so we don’t have to be worried about it. It’s not as hard as so much,it’s just that I want you not to spare some kind of effort a little,the author said like that.

By the way,the vector is shown with an arrow. Its direction is the one of the vector. Its length is the size of the vector.

I’m going to start from addition of the vector. When there are two kinds of vector,A,B,its addition means that the parallelogram 平行四辺形 of each side is A and B.and its diagonal line is…

No.97 17/11/19 13:43
燻し銀三 ( 50代 ♂ Oe38xe )

【Addition and subtrsction on the vector】

《The rule of the calculation draws out of the power from the matrix and vector》

As for the last of my response on the subtitle,it’s not Rolfe but the rule. I made a mistake. Though it’s my own sentence,I was forced to be worried about it. I’m sorry for it.

When there are two kinds of vector,A,B,its addition means that a paralellgram of each side is A,and B,and its diagonal line is the vector,the sum of the addition of A and B on the vector. It’s a rule on the vector.

While there is the A of vector,there is other A which is minus. They are on the same line and its length is also same but its direction is opposite. It’s also the rule on the vector.

Being based on the rules,A-B defines as A+(ーB),it’s an addition. If showing it with a component,I’m wondering if it’s an arrow,it’s easier. As to the addition and subtraction of the vector,when being shown with the component,we can think of them as the addition and subtraction of the component.

No.98 17/11/19 14:27
燻し銀三 ( 50代 ♂ Oe38xe )

【Addition and subtraction on the vector】

《The rule of calculation draws out of the power from the vector and matrix》

The way of thinking of adding and subtracting of the component means a definition of addition and subtraction of the matrix as it is.

However there is one thing of which we should be careful. If each size of the vector and matrix isn’t the same,both of the addition and subtraction doesn’t stand up a bit.

【The way of multiplication on the matrix】

《The multiplication makes the vector and matrix move dyanamic》

If going on proceeding with the topic on the product 積 on the matrix,there is a few necessary words which I have to indicate.

Speaking of the 行列 in Japanese usually,it’s a single meaning which is the combination of the 行 and 列 ,but there is each meaning in two of the words in math. The 行 is a row which is a horizontal line. When there are a group of several lines,we call each row 1st row,second row…from the upper part.

As to a vertical the group of the …

No.99 17/11/19 16:00
燻し銀三 ( 50代 ♂ Oe38xe )

【The way of multiplication on the matrix】

《The multiplication makes the vector and matrix move dynamically》

As to the vertical line of the group of the several lines,we call each of them a column. We call each of them first column,second column …from the left. As you may be aware of it,its the origin of the word,行列.

When the number of the row is M,and the number of the column is N,we call the matrix the one of which row is M and of which column is N. If it’s more simplified,it’s the matrix of M×N.

When each number of row and column is same,and each component of both the row and column is the same,we call it a square matrix.

Oh! I’m afraid I’ve failed to say an essential and basic thing. Each of the row and column in the matrix is made up with some numbers which is lined at random,and we call each number the component or element. When saying,the component of the 3rd row,2nd column,or the element of(3rd,2nd),it means the element of the place where the 3rd intersects the 2nd.

No.100 17/11/25 20:02
燻し銀三 ( 50代 ♂ Oe38xe )

【The theory of vector makes us fly in the sky】

《A plane flies in the sky by the composition of lots of power from the vector》

A plane which is heavier than the air can fly in the sky. A pitcher throws a curve or screwball in a game of baseball. When serving a tennis player puts a spin on a ball and smashes the ball into its opponent’s court. All of these are the same theory.

The principle which is common to all of them is the theorem of Bernoulli,which have had a great influence on almost all of the sports in which the players handle the ball or which is basic principle in aeronautical engineering 航空力学.

However it doesn’t seem to be so difficult,maybe,I hope so.

The theorem is when there is flow of air and fluid,the more it’s speedy the less its pressure. It’s a very simple theorem,though the author said like that, the more it’s speedy the bigger the pressure which it receives,I’m wondering.

If using the theorem and the way of thinking on the composition of vector we can …

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