Science,English,and math 4th

【Statistics has changed nations】

《The most advanced statistics》

As a result, we can put into the Bays theory into practice a thing which we can’t handle before.

As to 2, the calculation is complicated, almost all of everyone can use an computer at present, so there is no problem.

Statistics started in the ancient country, and developed into the Bays statistics supported by math and computer.

The expression on Bays statistic continued in the book, but I have none of knowledge on statistics, so it doesn’t appeal to me at all, but the author continued like the next.

Number and judgement...

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【Statistics has changed nations】

《The most advanced statistics》

Numbers judgement is a ground for an estimation through the statistics in the present day life. Numbers express the society and change the society.

What’s the use of learning math? Statistics is one of its answers.

What’s the use of learning English? As for me, then I have fun, so if I can think like that, I’ll learn math, maybe, though I can’t promise it.

【Let’s overcome a big number with N ary】

《When taking a glance, how many are there bars?》

N ary means N 進法, but the phrase isn’t included in my electric dictionary, so...

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【Let’s overcome a big number with N ary】

《When taking a glance, how many are there bars?》

...so I looked up it in the internet, so there is N ary, N進法.

I’m afraid some of people are confused with my English, so I’m sorry for it. Though I’ve done my best, I read again before contributed, and I find it perfect, then I contributed my expression to this bulletin board, but sometimes it didn’t go well as I expected.

To return to my main topic, I’m going to start again.

There are two illustrations on bars in the book.

The first illustration is there are 13 bars. They stand vertically and...

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【Let’s overcome a big number with the N ary】

《When taking a glance, how many are there bars?》

...and they stand vertically and parallel each other.

The second illustration is also on bars, and the number of bars are the same, but the way of expressing is different.

It was expressed with three groups.

There are four bars and they stand vertically and parallel each other and a single bar is crossed over four bars, and there are more five bars like that. There are three bars and they stand vertically and parallel each other but any bar isn’t crossed them.

There are the same numbers of...

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【Let’s overcome a big number with N ary】

《When taking a glance, how many are there numbers?》

In short, there are two illustration of the same numbers on bars.

When taking a glance at the first illustration, can you see how many numbers are there bars at once? It is said the number which we can recognize at once is a few, at best four, without counting on their fingers, lots of people find it hard to count the number of bars.

As to Roman number on 1,2,3, it’s expressed with Ⅰ,Ⅱ,Ⅲ, but 4 isn’t expressed with ⅠⅡⅠ, it was Ⅳ, for if using ⅠⅡⅠ, there are some people who can understand it’s 4...

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【Let’s overcome a big number with N ary】

《When taking a glance, how many are there bars?》

...for there are some people who can’t understand it was 4 at once, though 4 used to be expressed with ⅠⅡⅠ in the Roman number before.

If using the way on the second illustration, without counting on fingers, we can count easily.

When countering numbers, getting together by five and using the mark on the second illustration was done widely before, and we call the mark a five-bar gate, according to the book.

We the Japanese use 正, and count by every five, but if the number...

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【Let’s overcome a big number with N ary】

《When taking a glance, how many bars are there?》

...but when the number increases, it doesn’t always mean it’s useful. For example, if trying to express 96 with 正, I’m sure almost all of us will be tired of it, so we’ve contrived the position of decimal point. 位取り

Five is expressed four bars which vertically and a bar is crossed over them. If the mark is five, two bars are crossed, then 341 is three of two bars are crossed and four of a bar is crossed and a fraction 端数 is one.

《A reason why the decimal system has spread over among us》

N ary N進法...

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【Let’s overcome a big number with N ary】

《A reason why the decimal system has spread over among us》

The N ary is in the position of decimal point when a number gather to some extent, we call it N temporarily, we regard it a lump of number, and move to the second digit. We call the situation the number was carried up 繰り上がる.

I wanted to show it with an example in relation to the word of carry up, but there is no example for it, so I can’t show you the example. I’m sorry for it.

As to the way of showing 341, when 5 is gathered, it’s one of the lump of number and the number is carried up...

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【Let’s overcome a big number with N ary】

《A reason why the decimal system has spread over among us》

...and the number is carried up to the next digit, so it’s the quinary 五進法.

The one which we usually is the decimal system. If writing 324, it means 3 × 100 ＋ 2 × 10 ＋ 4. If expressing the number with the Chinese numerals, it’s 三百二十四, and it’s easy to understand.

The third digit is a lump of number which is 10 is gathered 10, so it’s a position of 10×10.

As to the N ary, let’s suppose there was a number of abc. Then c is a fraction, b is the second digit, and a the third digit, so the...

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【Let’s overcome a big number with N ary】

《A reason why decimal system has spread over among us》

...so 324 is 3×10 squared ＋2×1＋4, so the number of three digit of abc is a×N squared＋b×N＋c.

When expressing numbers, a reason why the decimal system has been used the most general is our fingers are 10 in total, adding to both of hands.

If the number of the fingers on the both of hands were 8 in total like the Mickey Mouse, we could have used the octal 8進法.

Considering the number of a single hand is five and a limit of the number which we the human being can grasp at once is four, it isn’t...

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【Let’s overcome a big number with N ary】

《A reason why the decimal system has been used》

...it isn’t strange that there is a society where the quinary 五進法 has been used, for the quinary has been used in Ilongot in Philippine and in Indonesia and South America partly at present.

In addition the Sumerian in ancient time adopted sexagesimal 60進法, and the Babylonians took it over, so they contrived a measure of time, 60 seconds is one minute, and 60 minutes is an hour. Why did they choose 60?

It is said that the 60 has such the lots of divisors that they could count easily.

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【Let’s overcome a big number with N ary】

《A reason why the decimal system has been spread over among us》

So they chose 60 as a lump of numbers. In the same reason, mathematicians in Arab adopted 60 when calculating in astronomy.

《The binary system and Bacon who is a philosopher》

Except for the decimal system, there is a trace that other way of counting number was used in our everyday life, and we can see it unexpectedly.

For example, 1 dozen is 12, 1 gross is 12 dozens, and a year is twelve months. They are remains of duodecimal 12進法. Moreover, 80 is called quartre-vingits in France.

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【Let’s overcome a big number with N ary】

《The binary system and Bacon who is a philosopher》

It means quartre 4× vingt 20, so there is 20 ary 20進法 there.

By the way, I can’t speak French at all. When expressing other way of counting except for the decimal system, the author happened to adopt the French way, and I expressed it as the author says.

I’m afraid there is a mistake somewhere, but please don’t blame me for it, for I have none of knowledge on the French.

To return to my main object, I’ll going to start again.

They aren’t a mainstream but just remains. Except for the decimal ...

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【Let’s overcome a big number with N ary】

《The binary system and Bacon who is a philosopher》

Except for the decimal system, there is a world where other way of counting numbers has driven back the decimal system and played an active role is the world of computer. Mainly the binary system and hexadecimal 16進法 have been used there.

Capacity on memory of UBS is expressed with gigabyte, and number of 16, 32, 64, 128, and 256 are lined up. There isn’t any good number to count like 20 or 100 in the decimal system.

When buying a golf ball in a unit of box, we seem to be forced to buy the next...

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【Let’s overcome a big number with the N ary】

《The binary system and Bacon who is a philosopher》

When buying a golf ball by the unit of box, we seem to buy it by every number of 12, 24, 36, 48...for a dozen is the base. A multiple of 12 is the number which is a good place to count numbers in duodecimal 12進法.

Hexadecimal 16進法 has been used in the world of computer and its good place to count numbers is 16 × integral numbers, but why has the hexadecimal 16進法 been used in the world of the computer? It gets along well with the binary system.

By the way, I’ve expressed the binary system, but ...

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《The way of counting numbers in the binary system》

I’ve expressed the binary system until now, but I’m not sure of the way of counting numbers in the binary system, so I’m going to show it from now on.

The binary system is a way of expressing all the numbers, using 0,1,2 alone.

As we use usually the decimal system everyday life, so we can’t imagine of using the binary system. When a number which is expressed with a one digit increases, and its digit also increases, the number is expressed with two-digit then.

In the decimal system, the next number of 9 is 10, for the 9 is carried up 繰り上がる.

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《The way of counting numbers in the binary system》

In the same way, after the digit increases it becomes two-digit in the binary system, but the next number of 1 is 10.

When the digit changed, the number is 10 in the both of decimal system and binary system, but its value is different, whether the number is the one in the binary system or the one in the decimal system, so we have to be careful of it.

For example, 10 in the decimal system is 10 in the decimal system, but 10 in the binary system is 2 when changed into the decimal system.

There is the number of 24 in the decimal system.

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《The way of counting in the binary system》

I’m going to change 24 from the decimal system to the binary system. It’s in the next ways.

Dividing the number in the decimal system with 2 and express its remainder. If it’s divided with two, we express 0 then. Adding a quotient 商 and the remainder. We are going to divide 24 by 2 continuously.

24 ÷ 2＝12、12÷2＝6, 6÷2＝3、3÷2＝1, its remainder 1.

When 24 in the decimal system is expressed with the binary system is 11000. First three of equation is divided and there is no remainder, and the last one is its quotient and remainder 1, so it’s like that

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【Let’s overcome a large number with N ary】

《The binary system and Bacon who is a philosopher》

The development on the binary system has been influenced by Bacon who is an English philosopher. He made a wise remark, knowledge is power, and said the knowledge from which we got our experience and observation will reach the truth.

In the middle of trying to contrive a new cipher which was called the cipher by Bacon later, he concluded an idea.

When preparing for a cypher which has two kinds of situations like a large letter and a small letter, ◯ or ×, so 2 raised to the power of 5, 2の5乗...

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【We the humankind has searched for pi】

《A famous question on the entrance exam from Tokyo University》

When expressing the decimal and binary system, I don’t think I have fun, so I’m going to change the topic.

Have you ever thought why pi is 3.14?

Tokyo University used to make a question for entrance exam like the next.

Demonstrate the pi is bigger than 3.05.

It’s probably the most famous question in math for the entrance exam from Tokyo University, so some people may have heard of it.

In the first place, what is the pi?

A circumference is multiplying the diameter by the pi. It’s ...

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【We the humankind has searched for pi】

《A famous question on the entrance exam from Tokyo University》

It’s an equation which we learn in elementary school. In other words, the pi is the ratio of the circumference of a circle to its diameter. It means that the length of a circumference is three times as big as the diameter. Exactly it’s a little over three times.

Needless to say, as all the circles are similar it stands up all the circles.

The length of the circumference of a circle is shorter than the diameter which is three times or it’s four times as long as the diameter isn’t possible.

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【We the humankind has searched for pi】

《A famous question on the entrance exam from Tokyo University》

In other words, if we can search for the ratio on the length to the circumstance to the diameter of a circle, it’s the pi.

《Archimedes thought like that》

However searching for the length on a circumference isn’t easy. A primitive way is measuring it actually.

For example, painting the surface on a tire and rolled the it, trying not to slip, and after the tire turned around wholly once, we measure the length of the trace of the tire.

Or we drive in a stake on the ground and tie up one...

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【We the humankind has searched for pi】

《Archimedes thought like that》

...and tie up one end of a string to the stake and other end to a stick of which top is sharp, in short we make a thing like a compass, and draw a circle. The length of the rope is twice becomes the length of the diameter, so we measure the ratio to the length of the diameter to the circumference.

Actually the ratio on the length of the conference to the diameter was sought with the latter way in Babylonia around BC 2000, and it had been thought to be about 3.125. Babylonia is southern area in Iraq at present.

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【We the humankind has searched for pi】

《Archimedes thought like that》

But when measuring, it comes together an accidental error. As long as we depend on the measure, we can hardly reach an exact value, so Archimedes in the Ancient Greece used a regular polygon and thought of an estimation on the circumference from a calculation.

There is an illustration on the book, and there is a regular square of which one side is 2 in the illustration. A circle is inscribed to the regular square. A regular hexagon is also inscribed to the circle and is divided into six equilateral square 正三角形.

When...

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【We the humankind has searched for pi】

《Archimedes thought like that》

When looking at the illustration, the circumference of the regular hexagon ＜ circumference on the circle ＜ the circumference of the regular square is clear, for the regular hexagon is composed of six equilateral triangles of which side is 1, so its conference is 6.

The side of the regular square and the diameter of the circle is 2, so the circumference on the circle is 2 ×pi = 6.24. The circumference on the regular square is 8, so we can demonstrate pi is bigger than 3 and smaller than 4.

But the circumference on the...

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【We the humankind has searched for pi】

《Archimedes thought like that》

But the circumference both of the regular square and regular hexagon is different from the circumference on the circle so much that the estimation wasn’t done so exactly.

If improving the precision on the estimation, we should increase the vertex on the regular polygon, then the space between the circle and the regular polygon is smaller, the length of the regular polygon approaches the length of the circle more closely.

By the way, the question on the entrance exam from Tokyo University in the beginning is thinking....

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【We the humankind has searched for pi】

《Archimedes thought like that》

As to demonstrating pi is bigger than 3.05, if thinking of a regular dodecagon 12角形 which inscribed to a circle, it’s solved. Its answer example is on the book, but we can’t understand nor express here with my technical skill and knowledge on the key board.

Archimedes thought of a regular 96角形 which inscribed and circumscribed to a circle, and reached a conclusion that pi is bigger than 3.1408 and smaller than 3.1429. He succeeded in searching for the exact value to the second decimal place.

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【We the humankind has searched pi】

《An irrational number continues forever》

Pi is so to speak an irrational number. The irrational number isn’t able to be expressed with a fraction of which numerator or denominator with integral number. It means that an irregular number continues forever in decimals.

On the other hand, it’s able to be expressed with the fraction of which numerator or denominator with integral number, the number of decimals is limited, or if it continues forever, it has something regular.

As pi is the irrational number, the line of number doesn’t come to an end. Its last...

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【We the humankind has searched pi】

《An irrational number continues forever》

Its last number doesn’t exist, so no one can answer it, for it has nothing regular. Aristotle expected pi is the irrational number in BC 4th century, but it was latter in the 18th century when it was demonstrated actually.

Though Archimedes estimated pi by regular polygons, it means that it closely resembles that something continues limitless is expressed with something limited, so it’s limited naturally.

So it has been contrived that it’s expressed with a multiplication forever. What kind of the fraction is it?

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【We the humankind has searched pi】

《An irrational number continues forever》

Its denominator is pi and numerator is 2, and it’s expressed with other fraction of which denominator is 2 and numerator is √2, and the 2 in the √ of the numerator is substituted with 2 ＋ √2 one after another forever.

After that, the calculation on pi is expressed with an numerical formula which continues forever.

《It has been sought by 31 trillion 400 billion》

Irregular number continues forever in decimal means that every kind of line of number is included as long as it’s limited, so not only four digit ...

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【We the humankind has searched pi】

《It has been sought by 31 trillion 400 billion》

...not only four digit of the date of our birth but even if those of every kind of people in the world, there is a line of number of eight digit in pi.

I’m wondering if taking the Christian Era into account, it’ll be eight digit.

In addition, when trying making computer understand the information on letters, and the letters are changed into numbers, if the whole passages on Hamlet written by Shakespeare is changed into numbers, we will be able to find the same lines of number on Hamlet in the pi.

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【We the humankind has sought for pi】

《It has been sought by 31trillion 400 billion》

It makes us the humankind that being infinite is limitless, but if the story stands up all right, the line of numbers on pi need to be at random completely. We call the line of numbers random numbers.

If investigating pi and frequency on each number from 0 to 9 which appears in the line of numbers of pi, it’s almost the same, so the line of numbers on pi seems to be the random numbers, but it has never been demonstrated mathematically yet.

Google announced Haruka Ema Iwao who was from Japan succeeded in...

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【We the humankind has searched for pi】

《It has been sought for by 31 trillion 400 billion》

Google announced Haruka Ema Iwao who is from Japan succeeded in calculating pi 31 trillion 400 billion decimal on pi on 14th March in 2019. The day was for pi. Its previous record was made in 2016, but from then no less than 9 trillion has been updated. It’s a marvelous record.

Pi is the number which we won’t understand, but it appears in numerical equations on all kinds of math and natural science including other fields which seem have nothing to do with pi. It’s mysterious and strange fixed number.

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【We the humankind has searched for pi】

《It has been sought by 31 trillion 400 billion》

Pi is the number which we can’t know its exact numerical value absolutely.

In spite of being important so much, we can’t recognize its exact value. It’s not useful, the author said he wasn’t sure whether it was thought like that, but the numerical value on pi was about to be set up by law at Indiana State in America at the end of 19th century

An amateur mathematician who was a doctor wrote a thesis on pi that the circumference on a circle of which diameter 10 is 32, and presented to the National Assembly

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【We the humankind has searched for pi】

《It has been sought by 31 trillion 400 billion》

Then a bill which supported the thesis was made. If the bill passed through in the National Assembly, pi would learn to be 3.2, moreover it was approved unanimously in the Lower House.

A mathematician who happened to visit the governor at the state knew it, and he said the precise numerical value on pi is never fixed to the members in the Upper House. He explained it to them minutely all night, so the bill the one which was put off indefinitely.

The author said it was in crisis. Was it so? I’m not sure.

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【We the humankind has searched for pi】

《It has been sought by 31 trillion 400 billion》

Lots of people may have thought we don’t have to stick to its precise numerical value so much. 3.2 may have lots of accidental error, but if pi is 3.14, no one is in trouble actually, isn’t it?

《The numerical value on pi which supported the return ハヤブサ》

But if pi is 3.14, there was a national project in which failed.

There are lots of people who knew はやぶさ which is a Japanese space probe on planets, though I’ve never heard of it until now.

In the middle of the plan, the space probe on planets ...

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【We the humankind has searched for pi】

《The numerical value on pi supported the return on はやぶさ》

In the middle of the plan, はやぶさ couldn’t correspond with the earth, but the people who concerned the plan on persistent support made the space probe succeed in returning to the earth. Its miraculous story was adopted in newscasting so much that it was made into a movie.

When calculating its numerical value on the orbit of はやぶさ,3.141592653589793 seemed to be adopted. It’s sixteen digit.

If adopting 3.14 for pi and was calculated with the numerical value, its orbit would swerve about a hundred...

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【We the humankind has sought for pi】

《The numerical value on pi supported for return はやぶさ》

...the space probe would swerve from the orbit by a hundred fifty thousand kilometers if the worst came to worst, and even if the correspondence with the earth revived, はやぶさ couldn’t have returned to the earth.

After the Ancient Greece, lots of mathematicians in both West and East have challenged, and engineers on computer continue to do against pi, but its fight on pi never comes to an end.

【An imaginary number 虚数 and the quantum computer】

《When being squared, the one which becomes minus》

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【An imaginary number and quantum computer】

《When being squared, the one which becomes minus》

There is a question like the next.

There is a rectangle of which length on vertical and horizontal is 10 in addition, and of which area is 24. Find the value the vertical and horizontal length on the rectangle.

How should we solve it?

The question seems to be prepared in a test, so we are apt to be lost in thought, but it’s not so difficult.

In short we have only to think of a pair of numbers when adding its total is 10, and when multiplying each other, it’s 24. We can do it with a mental...

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【An imaginary number and quantum computer】

《When being squared, the number which becomes minus》

We can do it with a mental arithmetic calculation.

There is other question like the next. How should we solve it?

There is a rectangle of which length on vertical and horizontal is 10 in addition and its area is 20. Find its value of the rectangle on the length of vertical and horizontal.

This time solving it with the mental arithmetic calculation may be hard, but it’s a normal question for a student who is the third grade in junior high school.

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【An imaginary number and quantum computer】

《When squaring, the number which becomes minus》

Regarding the vertical length as X and the horizontal length as Y and making a simultaneous equation 連立方程式 and if using a formula on the solution of a quadratic equation, we can find its answer.

But this time, the length on both vertical and horizontal is bigger than 5 or smaller than 5. The vertical length is smaller than 5 by the length of x, and the horizontal is bigger than 5 by the length of x.

Its equation is （5＋x）（5−x）＝20 and 25−x squared = 20, so x squared = 5, so x = √5

As a result, the...

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【An imaginary number and quantum computer】

《When squared, the number which becomes minus》

As a result, the vertical length and horizontal length on the rectangle is 5 − √ 5 and 5＋√5

With adopting the same way, we try to find the numerical value on the next question.

When adding, its total is 10, and multiplying is...

By the way, I’m tired a little, so I don’t feel like going on this response any more for a while, so I have to take a rest, in addition the battery on my iPad is about to run out. Please just a moment for a while.

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【An imaginary number and quantum computer】.

《When being squared, the number which becomes minus》

When adding it’s 10 in total, and when multiplying, it’s 40. We try to find its value with the same way. Then it’s x squared is −15. We are in trouble, for the number doesn’t exits. When being squared, none of numbers which become minus.

Girolamo Cardano of an Italian who left behind his reputation as a great mathematician for a formula on solution of a cubic equation faced to the same problem, but without giving up the solution and there was no solution...

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【An imaginary number and quantum computer】

《When being squared, the number which becomes minus》

...instead he did other thing.

When x squared is equal to 5, x is √5. Cardano did the same thing, he put −15 into the √ by force, so x squared is equal to −15, and x is √−15. As a result the value which he found was 5＋√−15, and 5−√−15, and he wrote down like the next.

If disregarding a spiritual agony, its addition is 10, and multiplication is 40, but it’s a chop logic 詭弁, and even if expressing numbers minutely in math, we don’t have any practical use for it.

《A genius who challenged the...》

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【An imaginary number and quantum computer】

《A genius who challenged the imaginary number》

Though it was half by force, and Cardano himself didn’t consider its existence positively, he was the first person who referred to the number which becomes minus when being multiplied and he wrote it down in his book.

There is no number of which addition is 10 and multiplication is 25. Why? Its maximum on that the addition is 10 and the multiplication is 40 is 25 on the condition, though it doesn’t meet the condition.

We call the minus number when being squared an imaginary number at present.

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【An imaginary number and quantum computer】

《A genius who challenged the imaginary number》

As the imaginary number doesn’t exist actually, we can’t express any of them on the number line where real number is gathered, so it has a negative nuance in French and is changed into English.

As Descartes is the one who linked a numerical formula to a diagram, he may not have been hard to accept the imaginary number.

But a genius who tried to search for the imaginary number which we can’t express on the diagram appeared in the 18th century. He is Leonhard Euler in Swiss. He set up √−1 as an unit..

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【An imaginary number and quantum computer】

《A genius who challenged the imaginary number》

Euler set up √− Ⅰ as an unit for the imaginary number and took its initial i and made it a rule to express the imaginary number as i. Then he reached the numerical formula on Euler at the end of his study for a long time finally.

It is said it’s the most beautiful numerical formula in the world. The word of i is included in the numerical formula, though I don’t think at all it’s beautiful, for I can’t understand it and nor find something regular in the numerical formula, I’m wondering.

After Euler...

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【An imaginary number and quantum computer】

《A genius who challenged the imaginary number》

Even after Euler probed deeply the study on the imaginary number, lots of people didn’t try to recognize the existence of the imaginary number. It took a long time to accept even minus numbers in Europe, so it is natural for them to be skeptical on the imaginary number which doesn’t exist in the world.

《A discovery by Gauss》

But the situation has changed suddenly, for Gauss who is the greatest mathematician in Germany and other people contrived a number line where imaginary number gather.

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【An imaginary number and quantum computer】

《A discovery by Gauss》

The number line where the imaginary number gathers called an imaginary axis, and intersects with other number line where real number gathers.

Gauss and others insisted that the imaginary number exists on the imaginary axis respectively, so the imaginary number becomes something visible for the first time and its existence has been recognized widely in the society after that.

Gauss called the one which was combined the real number with the imaginary number a complex number 複素数. Different plural elements of real number and...

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【An imaginary number and quantum computer】

《A discovery by Gauss》

Different plural elements that real number and imaginary number is different altogether each other are combined and a new number is born then.

Gauss combined the real axis from the number line where real number gathers and imaginary axis from the imaginary number, and made each point on real number and imaginary number one to one on the plane of coordinates, and called it the plane of complex number.

The theory on the plane of complex number is expressed on the book, but it’s too hard to understand for me, so I’ll skip it.

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