Science,English,and math Ⅱ

>> 148 To a person who sent its message in 147. 147にメッセージを送ってくれた方に。 Tha… そうだと思います！！
当時も先生とお呼びしていました。

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To 151

151さんへ

I've never seen you for a long time. How are you?

久しぶり！元気？😊

I remember you called me a teacher,but I'm not so great as to be called the teacher…

先生と呼んでくれたのは覚えてるけど、先生と呼ばれるほど偉くないよ…😓

When you visited my thread for the first time,I was very hard on you. I apologize for it.

初めて来てくれた時に、邪険な態度取っちゃったね。ゴメン💦

Even so,I miss you.

それにしても懐かしいな☺️

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>> 152 はい、懐かしいです。
先生を見付けた時は嬉しかったです(*^▽^*)ではお邪魔しました

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To 153

153さんへ

Thanks.😊

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【The probability of each ball.】

《As for the number of each case,we can count it easily as Long as we can make a list.》

When thinking over the probability,we often mention to a ball,but it doesn't always mean that we often come across a question of the ball everyday life. There are lots of similar questions,and if replacing them with the ball and thinking it over,we can understand it easily.

Making an appropriate question for a test so easily that the question on the ball often appears in an entrance exam in an university. I'm going to give an instance and to think it over.

There are 4 white balls and three red balls in a box. We pick up one of them at random and put it back in the box and pick up one of them again at random. Then there are various probabilities.

1: Both of them are red balls.

2: Both of then are white balls.

3: The first time is a white ball and a red one at the second time.

4: At least one of them is the white ball.

As to the question,if making a list …

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【The probability of each ball.】

《As to the number of each case,we can count it easily as long as we make a list.》

As to the question,if making a list,we can see its whole image and then we can understand what happens clearly.

As there are 4 white balls,we number each of them,W1,W2,W3,W4,and number each of red balls,R1,R2.R3. When putting back the ball,there are 49 combinations,W1,W1・W1,W2・W1,W3・W1,W4・W1,R1・W1,R2・W1,R3…then the first one is W2 and its combination is repeated.

I've said that as long as we make a list,we can understand easily,but I can't do it here,so I'm forced to omit it. I'm sorry for it.

As a result the combination of both of them are white balls are 16,so the probability is sixteen-49ths. It's the answer for the 1.

The combination of both of them are red balls are 9,so the probability is nine-49ths. It's the answer for the 2.

The combination that first one is a white ball and second one is a red ball are 12,so the probability is twelve-49ths. It's the answer…

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【An average in relation to earnings and loss.】

《If thinking over the value which we hope when gambling,it means that we’ll always be lose.》

As to the last description,it’s halfway,but I don’t feel like expressing myself in English any more,for I did it about three weeks ago,so it's destroyed my interest,so I’m going to start a new chapter.

When buying a lottery ticket,or gambling in casino,we tend to know whether how much it is profitable for us,or how much it’s advantageous for us.

Not only when gambling,but when being forced to choose something serious in our job,if we can get information on how much we can get profit from it,we can choose it easily. The concept responds what we hope something probitable responds,we call it a expected value.

When planning various strategies,the expected value shows us how much we’ll gain profit quantitatively. For example,the principle of insurance is based on the concept of the expected value. I’m going to think over an example concretely.

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【An average in relation to profit and loss.】

《If thinking over the expected value when gambling,it means that we'll always lose.》

There are two white balls,three red balls,and five black balls in a box. When taking out one of them,if it’s the white ball,we can get ¥300,the red ball ¥100,but if the black ball we have to pay ¥200. It’s a kind of a gambling. Is the gambling profitable or not?

If calculating like the next,we can get its answer. It’s the one which seeks the average on both of profits and losses. In short,it’s a calculation on its probability which it happens and an addition of all the money which we get when it happens,needless to say,when paying money we calculate it as minus. Its calculation is the next one.

Two-tenths multipled by three hundred plus three-tenths multipled by two hundred plus five-tenths multipled by minus two hundred is equal to minus 10,in short,it’s loss of ¥10 on average.

As to the gamble or the public lottery,we always lose with a point of…

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【An average in relation to profit and loss.】

《If thinking over the expected value when gambling,it means that we’ll always lose.》

As to the gamble or the public lottery,we’ll always lose with a point of view on the expected value,but we buy our dream,if thinking so it may be all right.

There is something important which we have to be careful on the expected value and at the same time it’s the attention on the concept of the whole probability,we must not interpret the expected value that we can always get the profit with only one trial. It’s the average on the profit and loss when repeating the same trial. It’s just that we can hope or expect it.

“When calculating the expected value,I can recognize that it proves out to be profitable and I’ve betted all my fortune but unexpectedly I’ve lost. What should I do?” If someone said like that I have nothing to do with that. Please don’t blame me for it,the author said like that.

The average life expectancy is the base on charge on…

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『Continuation of the last response』

The calculation of average of remaining days in life is base of the charge of insurance fee. It’s an expected value of years of the one who live after that. A fire insurance is a little more complicated,but its calculation is the same basically.

【There is a deep sense in nonsense.】

《Random numbers have been used in various places in society.》

When lining up some numbers from 1 to 9 at random,we call them random numbers. When exchanging signs between a battery in the professional baseball,a list of random numbers had been used at one time.

For example,the random numbers are made by a dice of regular icosahedron 正二十面体, noise in an electron tube,or a computer. With a rapid progress of the computer,even a gigantic and complicated problem has been solved with a way of the list of random numbers called Monte Carlo simulation.

As you know,the Monte Carlo is the name of a city of Monaco which has an official gambling house. In the middle of the …

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【There is a deep sense in nonsense.】

《Random numbers have been used in various places in society.》

In the middle of the World War Ⅱ,some physicists had studied on atomic bombs. They couldn’t solve a problem on the movement of neutron and were in trouble. Then some mathematicians whose center was John von Neuman replaced the problem with a roulette and succeeded in getting an approximate value.

John von Neuman named the way Monte Carlo simulation,and it’s learned to be used in various areas in varied field since then.

Not only in natural science but in economic and social problem,some analyses by varied experiments have been done. The most typical way is the Monte Carlo simulation. In a word,the Monte Carlo simulation is trying to deduce conclusion by lots of experiments. As a result the experimenters can recognize what kind of phenomena happen.

Then if the experiment is done in so special situation,its result may be different from original one so much,so they need to verify an…

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【There is a deep sense in nonsense.】

《Raodom numbers have been used in various areas in society.》


…so they need to verify a case as ordinary as possible,so the random numbers are used. The random numbers have been used in everywhere else. For instance,in a cryptogram 暗号文,alphabets are changed into numbers and random numbers are added and is the cryptogram is conveyed.

There is a researching a part of people and supposing the whole image as a research of rating audience on TV. We call the way a sample survey. If researching such the special people,there may be fear of not reflecting the whole image,so it’s important how we choose the people who we research.

As a result we use the list of random numbers and pick out the sample at random,for example,choosing a test question for a driver’s license of a car,license examination for a calculation on the abacus 珠算,random sampling for a mass production,objects on research for an opinion poll,a marketing research for a product,and so on.

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【The way of understanding statistics】

《Average and standard deviation 標準偏差. The way of dealing with the notorious the standard deviation value.》

When gathering varied statistics,if looking at the data aimlessly,we can’t grasp its whole image in the least,so we should analyze the data and catch them quantitatively. Its way is statistic theory. At first the one which we adopt is a mean value 平均値.

The mean value has frequently been used for an average like a height,weight,examination,temperature,and it’s one of index which shows us a character of the whole data.

When grasping the data in numbers,we often adopt a standard deviation with the mean value. The standard deviation shows us the unevenness of the whole data. For example there are two graphs which shows the statistics of result of a test.

The vertical line 縦軸 of the graph is the number who take the test and the horizontal line is the grade of the test. Both of the two graphs are shown with curves,but its shape is different.

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【The way of understanding statistics.】

《Average and standard deviation 標準偏差. The way of dealing with the notorious standard deviation value》

To put in an extreme way,the curve of the one of graph is high,but its width isn’t large very much. On the other hand one more curve of the other graph is low but its width is larger than the other. Both of the shapes are symmetric of which center is the score of 50. Both of the average score is 50.

As to the distribution of the score,the former gathers around the mean value 平均値 and the latter is uneven. If calculating the standard deviation,it turn out to be that the latter is larger than the former.

By the way,the distribution of the shape of a temple bell 釣鐘 on the graph is called a normal distribution. When gathering plenty of data,the distribution appears the most naturally.

A student preparing for taking an entrance exam tend to be worried about the standard deviation value,in additon,the standard deviation value is a golden rule for…

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【The way of understanding statistics.】

《Average and standard deviation. The way of dealing with the notorious standard deviation value well.》

The standard deviation value is a golden rule for the parents of the student preparing for taking an entrance exam. On the other hand,the standard deviation value which ranks a person in number is called the source of every evil which get rid of our human nature.

It is certain that the standard deviation value seems to be apt to be attached too much. Even if trying to do its best from now on,if being said“Your rank is around this,it’s not so high objectively.”I find it natural not to feel like doing its best.

However if dealing with the standard deviation value well,it is possible that we can make friend with it. At first we have to understand the way of thinking the standard deviation value.

In a word,the standard deviation value is the one which changes the score of a test into a standard one. The way of changing it setting the average…

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【The way of understanding statistics】

《Average and standard deviation. The way of dealing with the notorious standard deviation value well.》

The way of changing it is setting average score is 50 and the standard deviation value which show us it unevenness is 10. It seems to be a natural way of mending,the author said like that,but to tell the truth,I’m not sure of it.

【Calculation of astronomical numbers.】

《An exceptional function 指数関数 which spreads out from the microscopic world to the microcosm マクロの世界》

We can watch an interesting scientific short film named 〈Powers of ten〉in an aerospace museum in Washington of America. If translating the title word for word,it’s 10 の累乗.

The movie started from a person who relaxed on the lawn in a park. The camera was faded away by powers of second from the ground every second more and more.

A figure of the person became smaller and smaller in an instant,the scene of the whole town appeared on the screen,and it became a vast extent like…

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【A calculation of astronomical numbers.】

《An exceptional function 指数関数 which spreads out from microscopic world to microcosm.》

…and the scene became a vast extent like an aerial photograph,and before long the whole scene of the earth,planets and galaxy appeared on the screen. It’s as if we rode in a rocket of which acceleration was terrible and inspected the scenes. The speed of the acceleration is the powers of ten.

There was the next part in the movie. It also started from the person on the lawn in the park,but this time the camera approached the person closer and closer. The smaller part was magnified by powers of ten every second.

At first the skin of the person appeared on the screen,it changed into a cell,an atom,and atomic nucleus. There used to be a movie named ミクロの決死圏. I’m wondering if the member of the party in the movie could have experienced that kind of scene.

In this movie,the scale of the screen became ten times,a hundredth times,a thousandth times in the first…

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【A calculation of astronomical number.】

《An exceptional function which spreads out from microscopic world to microcosm.》

In this movie,the scale of the scene became ten times,a hundredth times,and a thousandth timesin the first half,on the other hand it became one-tenths,one-hundredths.and one-thousandth in the latter half of the movie.

The 0 increased one by one every second,after 100 seconds,no less than a hundred of 0 was to be written down. I find it tiresome,so the concept of the power has been contrived like the title of the movie.

The number which a hundred of zero are added to 1 is the one which multiply 10 by 100. Instead of writing no less than a hundred of zero,if using the power,it’s useful. The powers of ten is shown with number of 10 and small size of 100 is upper right of the 10.

I’ve wanted express everything in English,but sometimes I’m sure I don’t have to like this case,but I can’t express it the number alone,so I can’t help it.

In general when some number…

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【A calculation of astronomical number.】

《An exceptional function which spreads out from microscopic world to macrocosm.》

In general when some number multiplied by other number alone repaeatedly like a × a × a × …is described like a adding little n upper right on the a,and we call it an involution 累乗 of the a.

For example,a × a is described as a squared and a adding 2 upper right on the a with tiny size,and a × a × a is a cubed and a adding 3 upper right on the a with small size. In general we call them the involution of the a,and the small numbers on the right upper is called exponents 指数.

The exponents aren’t use only in natural numbers but in integral numbers,rational numbers,and even real numbers. In this way, y ＝ x to the nth power,and it’s a function of the x,and we call it an exceptional function.

Bit seems to be used in computer frequently like 8 bit,and it shows the ability of handling information of the computer. Exactly 8 bit means 2 to the eighth power,so the computer…

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【The way of Gauss who was extremely talented in math.】

《We can find the sum of an arithmetic progression 等差数列 easily with this way.》

8 bits is 8 to the eighth power,so when using in the word in the computer,it means that the computer can handle no less than 256 pieces of information at once.

Geniuses often appear in the history of math,and Gauss is also one of them,but no one can compare with his precocity 早熟. This is his episode when he was a school boy in an elementary school.

A teacher put a question to his students. It’s adding numbers from 1 to 100,for the teacher tried to make his students study by themselves for a while,but to his surprise,Gauss solved the question instantly. What kind of way did he adopt?

There were numbers from 1 to 100,which were lined from little one to large one in turn,and there were other numbers below the first one,and the second one were lined from 1 to 100,but from large one to little in number.

Instead of adding numbers from 1 to 100,adding…

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【The way of Gauss who was extremely talented in math.】

《We can find the sum of arithmetic procession 等差数列 easily with this way.》

Instead of adding from 1 to 100 in number,adding two numbers,like 1 ＋ 100, 2 ＋99, 3＋98…so each sum is 101,and the number of the sums are 100,so 100 × 101＝ 10,100,but the value which we want is double,so the value need to be divided by 2,so it’s 5050.

When expressing the sum of the progression of numbers,I’ve done like 1,2,3,4,…100,its way is easy to understand but when being used to do,we find it tiresome,so the sign of Σ is often used. The Σ also means the sum. It’s a Greek letter in Latin.

The sign is as same as the words. While it’s useful,but it’s hard to understandsometime,so we don’t have to be worried about it so please look at the example which is used in the book,said the author in the book,but if trying to use it here,I find it impossible.

There are lots of episode on Gauss. This is one of them which shows us his precocity 早熟. When his father…

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【A geometric progression 等比数列 far exceeds our imagination.】

《We should be careful of a fishy story in which there is a chance of making big money.》

The newspapers are sometimes full of stories about a pyramid investment scheme,what is called ねずみ講. It’s a fraudulent business practices 悪徳商法 which cheats its customers ingeniously,saying,“you can make a profit more and more.”,but without recognizing its trick,everyone will be stabbed in the back.

Roughly speaking,its trick is like this,though it seems to be more shrewd actually. For example,let’s suppose that the membership fee for a member in an association was ¥100,000. If each of the member solicits new members by 10,people,a child member for each member increases by 10 people,and as for a grandson member,it increases by 100 people.

There is a rule on the association,the member can get no less than 80 percent of the membership fee from the grandson member. 20 percent of the membership fee is paid to the sponsor of the association.

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【A geometric progression far exceeds our imagination.】

《We should be careful of a fishy story in which there is a chance of making a big money.》

If there are a hundred of grandson members,the member can get ¥ eight hundred million. The rule book for soliciting says like the next.“If giving up soliciting new members halfway,other members will suffer from damage. As long as the member accomplish its duty,the one who takes part in the association can make money of ¥7.9 million.”

“No one suffers from damage at all seemingly,so it isn’t the fraudulent business practices in the least,is it?” Someone may think like that,but if thinking so,the one is easy to be deceived. There is no doubt that no one suffers from damage as long as the system is maintained,but the problem is the number of the member.

The first member has 10 of child members and 100 of grandson members. The number of the member increases by 10 times. If there is the 10th member,its number reaches no less than 10 billion.

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【Geometric progression far exceeds our imagination.】

《We should be careful of a fishy story in which there is a chance of making big money.》

Both of the number of the people and the money in the society are limited,so the system always come to a deadlock,and is forced to face a miserable situation. If you are the one who faces to the deadlock,you will suffer from the loss.

The pyramid investment scheme gives us an illusion as if it went on forever,but it’s in the wrong. Even if being aware that it’s limited,the one who is cheated never think it faces to the deadlock. The number of the eighth member reaches close to the Japanese population,so we can recognize how optimistic to go on taking part in the pyramid investment scheme.

In the pyramid investment scheme,the number of the member increases by 10 times,and in general we call a progression which increases by r times a geometric progression of r times on the common ratio 公比. If the r is larger than 1,it becomes terrible …

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『Continuation of the last response』

If the r is larger than 1,it becomes the terrible geometric progression,so we should be careful of it.

【The geometric progression which is close to us.】

《There is the geometric progression in bank savings,in interest rates,and even in the world of sound.》

I’m going to pick up some of the geometric progressions which are close to us.

The first one which we should indicate is calculation in the interest rates in a bank savings. As it's been familiar with us,the interest rates are calculated with a compound interest 複利. There are several ways of counting its term like the compound interest of a year or the one of a half year,so we should be careful of it.

For instance,if comparing half year compound rates of 3 % with a year compound rates of 6%,the former one is more profitable. Needless to say,if borrowing some money,the latter is more profitable.

If it’s the compound rates by the number of days like loan shark,the debt snowballs,so we should…

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【The geometric progression which is close to us.】

《There is the geometric progression in bank savings,in interest rates,even in the world of sound.》

If it’s the compound interest by the number of days like shark loan,it snowballs,so we should bear its horror in our mind. By the way,when calculating the compound interest,if making use of a table of logarithms 対数表,it’ll be useful. The author said like that.

For example,let’s suppose that we deposited ¥10 million for a decade with the half year compound interest of 3%. Its equation is 1000 × （1＋0.03）to the 60th power. The part of （）and other phrase means （）の60乗,but its calculation is tiresome,so the table of logarithms is taken advantage of. I’m going to describe it in the near future.

The geometric progression isn’t only used for in the economic world,but in the sensuous world of sounds. For example,the music scales like do,re,mi,fa,so,ra,si,and do have been fixed with the geometric progressions.

There are white keys and black…

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【The geometric progression which is close to us.】

《There is the geometric progression in bank savings,in interest rates,even in the world of sound.》

There are white keys and black keys in a piano. 1 octave is divided into 12 with both of the white and black keys. In short there are 12 semitones 半音. The vibration of the sound becomes double a octave,so the 2 is divided into 12. It’s not done by an arithmetic progression but by geometric progres.

Its equation is X to the 12th power is equal 2. The X is a radical root of the 2. It’s an irrational number,about 1.06. If a sound is higher than other one by a semitone,it means that the number of vibration of the sound is as same number of vibration of…oh…I find it complicated. I don’t feel like expressing on the number of the vibration any more.

I’m sure the music isn’t calculating its number of the vibration,but listening to it or playing it. When enjoying music we have fun,though fortunately,no one is forced me to calculate it.

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【An invitation to the world of logarithm 対数】

《Logarithm is an index 指数 which is expressed from an opposite direction》

Logarithm and index are alwaya handled as a pair.

On hearing the word of logarithm,there seem to be lots of people who are disgusted by it,for the sign of the logarithm,log,makes its image become worse.

Why do we handle the index and logarithm as a pair? There is a reason for it. For example,from the definition of the index,10 cubed is 10×10×10＝1000. Then when thinking of it from an opposite direction,if asking,what times 10 make 1000?

The question is log 10 1000 and its answer is cubed,so it’s expressed log 10 1000 ＝3.The number of the 10 is usually written smaller than this one,and I’ve also wanted to describe like that,but I can’t here,so I can’t help it. I’m afraid it’s hard to understand,I’m sorry for it.

In short the logarithm is the one which is expressed with index,and it’s just that we see it from the opposite direction.

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【Complicated calculations become easy ones.】

《Even if the calculation of compound interest is tiresome,we can find its answer easily with the logarithm.》

As I’ve expressed in the former response,the logarithm is the index which we look at it from the opposite direction,and it’s the change of conception,but it’s caused revolution in varied calculation.

When calculating the orbit of a heaven body 天体,they used to need calculate of multiplication and involution 累乗 with large number,what is called astronomical figures,and it was hard to calculate.

But,when being invented the logarithm,those calculations have been easier so much. It’s symbolized with the word,“The logarithm has expanded the life span of astronomers. “The logarithm has expanded the life span of astronomers.” I’m going to translate the law of index into the world of the logarithm.


10 to the nth power times 10 to the mth power make 10 to the addition of nth and mth of the power. It’s log 10 NM＝log10 N ＋log 10 M.

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【A complicated calculation becomes a easy one.】

《Even if the calculation of compound interest 複利 is tiresome,we can find its answer easily with the logarithm.》

10 to the nth power divided by 10 to the mth power is equal to 10 the subtraction that we subtract N from M the power. There is other example which is expressed with parenthesis ,（）but I find it complicated,so I omit it. Please look at the next equitation.

Log 10 100000000000＝12

In this way,the logarithm has power in which a large number changes into small one. The merit for using the logarithm is the way of changing calculation. A multiplication changes into an addition,division turns into subtraction,and an involution is multiplication.

The addition and subtraction are easier to do than multiplication and division,and a multiplication is easier than an involution.
When using the logarithm,the calculation becomes easier by 1 rank,and it’s the largest merit we use the logarithm.

It could have been easy to express,but…

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【Human’s perceptively is a sense for logarithm.】

《Grade on stars,dynamics on passage of music,or magnitude on an earthquake…》

Glittering stars in the night sky have each grade like a star of the first magnitude,second…it has started in the time of Ancient Greece,and the brightest star is the star of the first magnitude,and other star which we can barely manage to watch with our naked eyes is the star of the sixth magnitude.There are other magnitudes between the first and sixth.

When being invented a telescope,we’ve learned to need to add other magnitude for darker star than the sixth,so the magnitude has expanded to 8th. The definition for the magnitude depends on the human sense,so it wasn’t firm very much.

Hershel found that the magnitude of the star of the first is about 100th times of the one of the 6th,and whenever the magnitude rises by one rank,its luminous becomes 2.5 times. In short the difference between the magnitude is a geometric procession 等比数列 on the luminous.

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【Human’s perceptively is a sense for logarithm.】

《Magnitude on stars,stress on passage of music,and magnitude on an earthquake and so on.》

Pogson proposed that they define the magnitude of stars like the next,being based on the discovery.

Difference between the two stars and their brightness is 0,4 × difference between the magnitude ＝ log 10,and the log 10 is the proportion of the brightness. Then the number of the 10 is usually written smaller on the lower right of the log.

When the differece between the magnitude is 1,its value I s 2.5 times,for the proportion of the brightness is 0.4 ＝log 10,so 10 ・0.4 ＝2.51,then the number of 0.4 is usually written smaller on the upper right of the 10.

In this way,it means that we actually feel that the logarithm of the brightness of the stars as difference between the brightness. In short,the human’s perceptively inherits function which recognizes the logarithm unconsciously.

There is the law of Fechner,and it shows the relation difference…

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【Human’s perceptivity is a sense of logarithm.】

《Magnitude on stars,stress on passage of music,magnitude on earthquake and so on.》

There is a law of Fechner and it shows the relation of difference between the magnitude of stimuli and the percetivity which catches the stimuli. The law indicates the sense for logarithm itself by the human race.

There are plenty of examples which indicate the sense for logarithm,and I’m going to show you some of them.

Whenever an earthquake happens,the scale of magnitude of the earthquake is always reported. The magnitude is the scale of energy from a hypocenter 震源,and it’s defined with the use of the logarithm.

Decibel is an unit which is used for electric communication and sound,and phon is also the unit for scale of the sound. Both of them are defined with the logarithm.

【There is the index and logarithm in the natural world.】

《Both of the index and logarithm appear in the natural world naturally.》

Various dinosaurs used to swagger on the…

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【There is the index and logarithm in the natural world.】

《Both the index and logarithm appear in the natural world naturally.》

Various dinosaurs used to swagger on the earth as if they had owned the place millions of years ago. Fossils like the teeth and bones of the dinosaurs are proof of the existence of the dinosaurs.

How do we search the fossils like the dinosaurs on the era when they became the fossils? In short,how do we measure the fossils of the plants and animals on the era when they become the fossils?

We take advantage of the collapse of radioactive materials. For example,the radioactive carbon is so unstable chemical element that it gives off beta rays and collapses. The amount of the radioactive carbon in the body both of the plants and animals keep balance between the outer world as long as the creatures live.

However when the creatures die,the supply for the radioactive carbon in the body stops and the radioactive material continues to collapse.

In general…

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【Threr is the index and logarithm in the natural world.】

《Both of the index and logarithm appear in the natural world naturally.》

In general,the radioactive material has property that it collapses with the speed which is proportion to the amount,so as long as we can measure how much it collapsed,we can recognize how many years have passed since the creature died.

The speed of the collapse for the radioactive carbon is so slow that its half life period 半減期 is 5370 years. In short,the amount of the atom of the radioactive carbon become half after 5370 years. Either of the index or logarithm is made use of in the measurement for the radioactive carbon.

There are some equations on the collapse for the radioactive carbon on the book,but I can’t express them here and even if I can,to my sorrow I can’t understand them at all,so I omit them.

The word of biotechnology has been familiar with us lately,and multiplication of a microbe 微生物 is expressed with an exponential function. Both the…

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『Continuation of the last response and so on.』

Thus,both of the index and logarithm appear in the natural world naturally,but to my sorrow I can’t recognize them very much. I’ve expressed on math until now,and I used to study them in high school. Then I used to think and complain like next. “What’s the use of studying them? As long as being able to read and write,and to reckon like addition,subtraction,multiplication,and division,it’s enough.”

Those hard things to understand seem to be useless,but they have made our daily life comfortable one,I’m wondering,though I can’t say it concretely.

I’m sure learning is accumulation. For example,the index and logarithm have developed by themselves. Without other elements,not only the math but our daily life would never have developed like this.

I wish I would study harder when I was young,for studying something is interesting and wonderful. Learning something makes me feel happy,though even if lots of people have already had the knowledge.

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I’m going to express on an useful story of the trigonometric function 三角関数 like sine cosine.

【Everyone who has antipathy to the trigonometric function.】

《They look as if they were close friends who were a trio of sine,cosine,and tangent.》

The trigonometric function which is known as sine,and cosine. On hearing the phrases,there seem to be lots of people who are displeased with them.

As long as they understand the motivation why they think of the trigonometric function,the trigonometric function proves to be very simple and useful over a wide area,the author said like that.

For example,putting the theory of the trigonometric function to practical use for the most advanced technology like a cd which reappears a splendid tone,or a ct scanner which plays an active part in medical treatment can’t be stopped by anyone.

There is difference between a triangle and other figures like a square or a pentagon. What is it? I’m going to start from there,though it isn’t so hard,it’s very…

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【Everyone who has antipathy to the trigonometric function.】

《They look as if they were close friends who were a trio of sine,cosine,and tangent.》

The difference between a triangle and other figures like a square and pentagon isn’t hard to understand,and it’s very simple and natural. As for the triangle, if once being fixed each of the three angles,the triangles become similar ones. As a result,the proportion of the three sides of the triangle is also fixed.

Let’s limit a situation on a right triangle. The total of the interior angles on a triangle is 180 degrees. As long as fixing an angle which isn’t the right angle,the other angle is also done,and the proportion of the three sides are also done.Then the fixed angle is called θ,sheeter.

When there are two right triangles and if two of their θ is the same angle,the proportion of each side of the two right triangles is fixed and it’s expressed with sin θ,cos θ,tangent θ,though I’m afraid my expression isn’t enough. I’m sorry for it.

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【Measuring the height with a stick.】

《The Ancient Egypt invited Thales,who adopted the way of measuring.》

While thus the definition on the trigonometric function is very easy,varied practical application has spread out from the easy definition. I’m going to express the practical applications from now on.

There is one of episodes which shows an origin of the trigonometric function.

It is said it's Ancient Greek mathematician,Thales who thought of the idea for the first time. The idea that if the θ is fixed,all the triangles become similar. He was invited in Egypt and measured the height of a pyramid. This episode of his is well-known.

At first he put up a stick on the ground vertically,and measured the length of the shadow. In the next,he measured the length of the shadow on the pyramid.

A triangle which was made up with the shadow of the pyramid and other triangle which was done with the stick is similar,so as long as being able to know the length of the stick,they can also…

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【Measuring the height with a stick.】

《The Ancient Egypt invited Thales,who adopted the way of measuring.》

…so,as long as being able to know the length of the stick,they can also know the height of the pyramid.

There is sine,cosine,and tangent in the trigonometric function,and each of threee isn’t dependent,but they have a deep relation each other. The connection is the theorem of Pythagoraas. If calculating with the theorem of Pythagoras,for example,cosine and tangent is expressed with sine,though I’ve not remembered the theorem.

Thales predicted a solar eclipse 日食,and it made him famous. He was so absorbed in an astronomical observation that he fell into a ditch. A girl who escorted him teased,“Though you can understand difficult thing like a heavy body,you can’t be aware of a thing at your foot.”

【Cosine which gets over obstacles.】

《When obstacles like mountains or a building prevent us from measuring,there is a way of measuring the distance.》

The trigonometric function is…

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【Cosine which gets over obstacles.】

《When obstacles prevent us from measuring,there is a way of measuring the distance.》

The trigonometric function is defined with a right triangle,so we tend to think that without being the right triangle,we can’t use the trigonometric theorem,but in fact we can make use of it every time.

For example,when trying to measure the distance between A and B,but there is obstacles like a mountain,building,or pond between them. The obstacles prevent us from measuring the distance. What should we do?Then the trigonometric function plays an active part as if it waited for its turn.

At first we choose the third place as C which we can see from both of A and B. As to the C,we can choose it as we like. In the next,we measure the distance between A and C,and between C and B.

As to the two of distance,we can measure them direct or we can do it at other place where we can do it easily. Then we can measure the angle of the C. When there is those information…

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【Cosine which gets over obstacles.】

《When obstacles prevent us from measuring,there is a way of measuring distance.》

As long as ther is the information,the distance between A and B is able to be sought by using the trigonometric function.

The way of measuring the distance is the next one. When we are aware of two length of the two sides on a triangle and the angle which is put between the two sides,we seek the length of the last side. It’s called the theorem of cosine. I’m going to find the answer by using the theorem.

By the way,as usual I’m forced to do without Illustration,so I’m afraid it’s hard to understand,so I recommend you should write the figure for yourself.

At first we draw a perpendicular line 垂線 from the apex 頂点 A to the side BC,and we regard its point of intersection as H. The perpendicular line is also called an additioanal line,and as long as we can draw it we can almost completer the theorem,for the line is the key point for the theorem.

The perpendicular line…

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【Cosine which gets over obstacles.】

《When obstacles prevent us from measuring,there is a way of measuring the distance.》

The perpendicular line caused the two right triangles,and taking advantage of the trigonometric function and the theorem of Pythagoras for the two right triangles and calculating a little,there is the theorem of cosine at once,according to the book.

To tell the truth,I’ve used the phrase of the theorem of Pythagoras many times,but it doesn’t always mean that I can understand the theorem perfectly,so I’m afraid that I can’t arrive the theorem of cosine at all.

I find it useful and interesting to learn math,but how should I study it? I don’t think I want to go to university in my age. When opening a textbook on math in high school and on looking at an equation,I’m sure I feel dizzy.

I’ve heard the one who is good at math is able to think logically,but I tend to become emotional,especially when coming across a problem on math,though everyone else find it easy.

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【The theorem of sine plays an active part in surveying land.】

《When having a wide view,we can recognize its distance and position exactly in triangular surveying.》

When making a map or surveying land,the trigonometric function is essential. The concept is setting up a triangle one after another as if it were a lattice 網の目,and fixing each point in each place, and we call it triangular surveying.

The principle of the triangular surveying is the next. Making use of one of sides and both ends of angles of the side in a triangle and finding the length of other two sides in calculation. We call it the theorem of sine.

I’m going to try to find the theorem of sine for myself.

There is a triangle of which apex is A,B,and C. I’m going to draw a perpendicular line 垂線 from the apex A to the side BC,and the we regard the point of intersection as H. The perpendicular line caused the two right triangles,ACH and ABH.

After that,applying the theorem of sine to the two of triangles and …

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【The theorem of sine plays an active part in surveying land.】

《When having a wide view,we can recognize its distance and position exactly in triangular survey.》

After that applying the theorem of sine to the two right triangles and transforming them a little,so there is an equation of the theorem of sine at once,the author said like that.

To my sorrow,I can’t understand the theorem of sine at all,besides the sentence above one is a crude word for word translation,so I said transforming the theorem of sine,but I’m not sure how I should do it. First of all,I can’t understand the theorem of sine very well.

As a result I’m forced to express what I can understand in English. I’m sorry for it.

Drawing other perpendicular line from apex A to the side AC,we can get other equation on the theorem of sine. Both of the degrees of B and C is found by the theorem of sine,and as long as being aware of the degrees of each two angle,we can recognize the degree of the last one.

As a result the …

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【The theorem of sine plays an active part in surveying land.】

《When having a wide view,we can recognize its distance and position precisely in triangular survey.》

As a result,the length of AB and AC are found.

It is said that the theorem of sine is equivalent to one of congruent 合同 condition on triangles,the length of one side and the degrees of the each of two angles on the both end on the side is equal.

By the way,when doing the triangular survey actually,measuring the distance between 2 points which is the base minutely,precisely,very carefully. After that making triangles one after another,as long as measuring each of the angles,the lenght of each side is fixed by the theorem of sine.

In short,we don’t have to measure actually in the triangular surveying,as long as there is a basic line which is measured very accurately.

When measuring as long as being able to have a wide of view,we can measure in wide range,it fits for a measuring of a large scale. We can find the…

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『Continuation of the last response.』

The distance to the moon is found with the triangular surveying,the author said like that,but I want to say,so what? I’m sure I’m childish a little,but when looking at the theorem of the sine,or cosine over and over again,I can’t understand it at all.

By the way,I have little knowledge on a preposition,so the one who read may be hard to understand. I’m sorry for it.

June 3 is the day of measurement. Campaigns in which the importance of the measurement is shown,or events in which the necessity of making a map is shown are done every year.

【There are lots of sines in electricity.】

《I love the trigonometric function very much.》

The word of 《》isn’t my own coment,but the author’s.

We enjoy the blessing of electricity very much every day. The electricity which is essential to our everyday life is based on the theory of the trigonometric function.

The electricity is alternating current 交流 of 50 or 60 hertz. Roughly speaking,eastern Japan is 50…

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【There are lots of sines in electricity.】

《I love the trigonometric function very much.》

Roughly speaking,eastern Japan is 50 hertz and western Japan is 60 hertz. By the way,what is the alternating current of 50 hertz or 60 hertz?

It means the voltage of sine curve repeats 50 times or 60 times per second. What is the sine curve? There is a function,y＝sine x,and the function is expressed with a graph,but the sine used to be defined with triangles,the area which the x moves is larger than 0 and it’s smaller than 90.

Without limiting the area of the movement on the x,it’s defined like the illustration. It’s the function of sine,and its graph is the sine curve. The function of cosine is also defined like the same.

Without looking at the graph or illustration,it’s hard to understand,I’m afraid. Even if looking at them over and over again,I can’t grasp them very much.

Both of the sine and cosine are the shapes like a wave,and same shape is repeated every 360 degrees.

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【Practical application for the trigonometric function which spread infinitively.】

《A cd which reappear beautiful tone is done with combination of sine curve.》

When the electricity is used,voltage is changed according to its usage. Alternating current of sine curve is very useful then. When changing the voltage,a transformer is used. There is illustration of the transformer in the book.

Roughly speaking its shape is a square and it’s made of iron,and there is a big space of which shape is also the square in the transformer. Electric wires are wound on both sides of the square. The left one is the first coil and the right one is the second coil.

When the electricity of the sine curve is flowed in the first coil,and its power is in proportion to the bundle of magnetic power and the bundle of magnetic power changes.

There are some description on the sine curve,but too difficult to understand. The law of Faraday appeared in the description,so I’m going to express it at first.

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【Practical application for the trigonometic function which spreads infinitively.】

《A cd which reappears beautiful tone with a combination of sine curves》

I’ve said I was going to express the law of Faraday exaggeratedly,but it’s just that the next things.

When bringing a magnet close to a coil or moving the magnet from the coil,electric current flows in the coil.

I think the simpler the better when expressing something difficult as long as I have enough knowledge on it. Without having enough knowledge on it,if tried to express it,I’m forced to use technical terms which are unfamiliar with us. Then tried to express harder,the more complicated the situation is,though I recognize it…

As to my sentence in the last response,it’s hard to understand,though it’s just what I have done,so I’m going to express the strong point on the alternating current of the sine curve at first.

The alternating current of the sine curve has happened naturally,but it has varied merits.

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