3 14 full. Calculation with the desired accuracy of the number Pi. Pi in society

In mathematics there are infinite many different numbers. Most of them completely do not attract attention. However, some, at first glance, absolutely uninteresting numbers are known so much that they even have their own names. One of these constants includes the irrational PI, studied in school and used to calculate the area or perimeter of the circle at a given radius.

From the history of Constanta

Interesting facts about the number Pi - history of study. The existence of a constant is counting on about 4 thousand years. In other words, she is a little younger than mathematical science.

The first testimony that the number of Pi was known in Ancient Egypt, is the papyrus Akhmes, one of the oldest found tasks. The document is dated approximately 1650 BC. e. In the papyrus, the constant was taken equal to 3,1605. This is a fairly accurate value, if we consider that other peoples used 3 to calculate the length of the circle by its diameter.

A little more precisely, the number of Pi calculated Archimedes, ancient Greek mathematician. He managed to approximately present the value in the form of ordinary fractions 22/7 and 223/71. There is a context that he was so busy with the calculations of the constant, which did not pay attention to how the Romans seized his city. At that moment, when the warrior approached the scientist, Archimeded shouted to him so that he did not touch his drawings. These words mathematics became the last.

The founder of algebra al-Khorezmi, who lived in VIII-IX centuries, worked on the calculations of the constant. With a short error, he received the number Pi, equal to 3,1416.

After 8th centuries, 36 characters after a comma were correctly defined by Mathematics Lyudolf van Zeilen. For this achievement, the number of pi is sometimes called the Lyudolphic constant (other well-known names - the Archimedean constant or the circular constant), and the numbers received by scientists were knocked out on its gravestone.

At about the same time, the permanent began to apply not only for the circumference, but also to calculate complex curves - arches and hypocycloid.

Only at the beginning of the XVIII century, the constant began to call the number Pi. The designation in the form of the letter π is chosen non-random - it is from her that 2 Greek words begin, meaning a circle and perimeter. The name was proposed by scientist Jones in 1706, and already after 30 years, the image of this Greek letter is firmly used among other mathematical designations.

In the XIX century, William Shenks worked on the calculation of the first 707 constant characters. He failed to fully achieve the task - an error was crushed into calculations, and 527 digit turned out to be incorrect. However, even the result was a good achievement for the science of that time.

At the end of the XIX century, the incorrect value of the number equal to 3.2 was almost accepted at the state level in Indiana. Fortunately, mathematics managed to oppose the bill and prevent a mistake.

In the XX-XXI centuries. With the use of computing technology, the accuracy and rate of calculation of the constant rose thousands of times. By 2002, in Japan, with the help of a computer, more than 1 trillion digits permanent were determined. After 9 years, the accuracy of the calculation was already 10 trillion after a semicolon.

In art and marketing

Despite the fact that Pi is a mathematical constant, for many years people have tried to use irrational and mysterious meaning and in other areas of life, including in works of art.

The very first signs of constant were found in the architectural monument in Giza. In determining the size of the Great Pyramid, it turned out that the ratio of the perimeter of its base to height is equal to π. It is not known only if I wanted to use my knowledge about this number, or such a relation happened by chance.

Currently, the number Pi is also not deprived by attention in creativity. For example, if you designate every note of the minor gamma with a number from 0 to 9, and then play the resulting sequence in the form of a PI number on a musical instrument, you can enjoy an unusual melody with an interesting sound.

The constant also did not bypass the cinema. The dramatic film called "Pi: Faith in Chaos" received a reward for the best director at the Sandrence Film Festival. According to the plot, the main character is in search of simple and understandable answers to questions about Constant, which as a result almost brought it to madness. Mentioned numbers are also found in other movies and serials.

Your number application has found even in such an unexpected area as marketing. Thus, the company Hivenchi was released cologne called "PI".

Constant and Society

Some features of the number:

1. The constant is an irrational value. This means that it is impossible to present in the form of two numbers. In addition, there is no regularity in his record.
2. Repeating in a row signs in constant are not uncommon. So, for every 20-30 characters, at least 2 digits running in a row. Sequences of 3 characters are already rare, they come across with a frequency of about 1 repetitions by 150-300 characters. And at 763, a chain of 6 walks in a row begins. This place in the record even has its own name - Feynman's point.
3. If we consider the first million symbols, then according to statistics with the rare figures in it will be 6 and 1, and the most frequent - 5 and 4.
4. The figure 0 appears in the sequence later than the rest, only on the 31st sign.
5. In trigonometry angle of 360 degrees and constant are closely connected. Oddly, the number 360 is located at 358, 359 and 360 after the comma position.

In order to exchange information about discoveries, a Pi Club was established. Those who wish to join it have to withstand a difficult exam: the future member of the mathematical community should correctly name the memory as much as possible constant signs.

Of course, learning a long numerical sequence that does not have patterns and repetitions is quite difficult. To facilitate the task, various texts and poems are invented, in which the number of letters in the Word corresponds to a specific constant number. This memory method is popular with members of the Pi Club. One of the longest stories contained the 3834 first signs of the number.

Monument at the Art Museum in Seattle

However, recognized postcard records are, of course, the residents of China and Japan. So, the Japanese Akira Haraguti was able to learn over 83 thousand digits after the comma. And Chinese Liu Chao became famous as a person who was able to name 67890 characters of the Pi number for a record time - 24 hours. At the same time, the average speed amounted to 47 characters per 1 minute. Initially, its goal was to name 93 thousand digits, but they were mistaken, after which he did not continue.

To emphasize the value of the constant, in Seattle in front of the Museum of Arts, a monument was erected in the form of a huge Greek letter π.

In addition, since 1988, every 14 March marks the day of the number of PI. The date coincides with the first signs of constant - 3.14. Celebrate him after 1:59. On this day, interested people are treated with cakes and cookies with a symbol of Pi, after which there are various mathematical contests and quizzes. By the way, A. Einstein, Astronon Skiaparelli and Cosmonaut Sernan, were born on this day.

The number of Pi is an amazing constant that has found its use in a wide variety of areas, ranging from technology and construction and ending with the areas of art. Like any other value that is often applied and which it is impossible to calculate completely, it will always attract the attention of mathematicians, physicists and other scientists.

Today is the birthday of the number of Pi, who, on the initiative of American mathematicians, is celebrated on March 14 at 1 o'clock and 59 minutes of afternoon. This is due to the more accurate value of the PI number: we all accustomed to consider this constant as 3.14, but the number can be continued as follows: 3, 14159 ... Transferring it to the calendar date, we get 03.14, 1:59.

Photo: AIF / Nadezhda Uvarova

Professor of the Department of Mathematical and Functional Analysis of the South Ural State University, Vladimir Zalyapin says that "Day of Pi" should still be considered July 22, because in the European dates format this day is recorded as 22/7, and the value of this fraction is approximately equal to the value of Pi .

"The history of the number that gives the ratio of the circumference of the circumference to the diameter of the circle, goes into distant antiquity," says the shoal. - Already, Sumerians and Babylonians knew that this this attitude does not depend on the diameter of the circle and is constant. One of the first references to the number Pi can be found in the texts egyptian scribe Akhmesa (about 1650 BC. e.). Ancient Greeks, many borrowed from the Egyptians, contributed to the development of this mysterious value. According to the legend, Archimedesit was so passionate about the calculations that he did not notice how Roman soldiers took his native city of Syracuse. When the Roman soldier approached him, Archimedes shouted in Greek: "Do not touch my circles!". In response, the soldier is broken with his sword.

Platoreceived the rather accurate value of the PI number for its time - 3,146. Ludolf baths Zeileni spent most of my life above the calculations of the first 36 digits after the semicolons of the pi number, and they were engraved on his tombstone after death. "

Irrational and abnormal

According to the professor, at all times the pursuit of the calculation of new decimal signs was determined by the desire to obtain the exact meaning of this number. It was assumed that the number Pi is rational and, therefore, can be expressed by a simple fraction. And this is rooted incorrectly!

The number Pi is also more because it is a mystical. Since ancient times, a religion of the confent holders existed. In addition to the traditional value of Pi - the mathematical constant (3,1415 ...), expressing the ratio of the circumference of it to its diameter, there is a mass of other values \u200b\u200bof the number. Curious facts are curious. In the process of measurements of the size of the Great Pyramid in Giza, it turned out that it has the same ratio of height to the perimeter of its base, as a radius of the circle to its length, that is, ½ pi.

If you calculate the length of the Earth's equator using the Pi number with an accuracy of the ninth sign, the error in calculations will be only about 6 mm. Thirty nine marks after a semicolons Among the pi is enough to calculate the circumference of the circumference, encircling the well-known space objects in the universe, with an error not greater than the radius of the hydrogen atom!

The study of PI is involved in the mathematical analysis. Photo: AIF / Nadezhda Uvarova

Chaos in numbers

According to Professor of Mathematics, in 1767 Lamberti installed the irrationality of the number Pi, that is, the inability to present it with the attitude of two integers. This means that the sequence of decimal signs of the pi is chaos, emitted in numbers. In other words, in the "tail" of decimal signs contain any number, any sequence of numbers, any texts that were, there are and will be, but only to extract this information is not possible!

"The exact value of the number of PI is impossible to find out - Vladimir Ilyich continues. - But these attempts are not left. In 1991. Musicovsky A new constant signs have achieved, and in 1994 - 4044000000. After that, the number of faithful signs of the number pi increased avalanche-like ".

World record for memorizing the number of pi among the Chinese Liu Chao.Who managed to remember 67890 decimal places without error and reproduce them within 24 hours and 4 minutes.

About "golden section"

By the way, the relationship between PI and another amazing value - the golden cross section is actually never proven. People have long noticed that the "golden" proportion is the same number f - and the number of Pi, divided into two, differ in less than 3% (1.61803398 ... and 1,57079632 ...). However, for mathematics, these three percent - the difference is too significant to consider these values \u200b\u200bidentical. In the same way, it can be said that the PI number and the number of fi are relatives of another known constant - the number of Euler, as the root of it is close to half of the number Pi. One second pi - 1, 5708, fi - 1,6180, root of e - 1, 6487.

This is only part of the pi value. Photo: Screenshot

Birthday Py.

In the South Ural State University, the birthday constants celebrate all teachers and mathematics students. So it was always - it is impossible to say that interest appeared only in recent years. The number 3.14 greets even a special festive concert!

If you compare the circles of different sizes other than each other, then you can see the following: the dimensions of different circles are proportional. This means that with an increase in the diameter of the circle, the length of this circle increases at the same number of times. Mathematically write it as possible:

C. 1		C. 2
	=
d. 1		d. 2	(1)

where C1 and C2 are the length of two different circles, and D1 and D2 are their diameters.
This ratio works in the presence of a proportionality coefficient - the constant already familiar to us. From the relationship (1) we can conclude: the length of the circle C is equal to the product of the diameter of this circle to the proportionality non-circumference:

C \u003d π d.

Also, this formula can be written in other form, expressing the diameter D through the radius R of this circle:

C \u003d 2π R.

Just this formula is a conductor into the world of circles for seventh graders.

Since more antiquity, people tried to establish the value of this constant. For example, the residents of Mesopotamia calculated the area of \u200b\u200bthe circle by the formula:

Where π \u003d 3.

In ancient Egypt, the value for π was more accurate. In 2000-1700 BC, the scribe, called Akhmes, was a papyrus, in which we find recipes for solving various practical problems. So, for example, to find the area of \u200b\u200bthe circle, it uses the formula:

			8			2
S.	=	(		d.	)
			9

What considerations did he get this formula? - Unknown. Probably, based on its observations, however, as other ancient philosophers did.

In the footsteps of Archimedes

Which of the two number is more 22/7 or 3.14?
- They are equal.
- Why?
- Each of them is equal to π.
A. A. Vlasov. From the exam ticket.

Some believe that the fraction 22/7 and the number π is identically equal. But this is a delusion. In addition to the above incorrect response to the exam (see Epigraph), you can also add one very entertaining puzzle. The task says: "Put one match so that equality becomes faithful."

The solution will be such: you need to form a "roof" for two vertical matches on the left, using one of the vertical matches in the denominator on the right. It turns out a visual image of the letter π.

Many know that the approximation π \u003d 22/7 determined the ancient Greek mathematician Archimedes. In honor of this, such an approximation is called "Archimedean" number. Archhimeda managed not only to establish an approximate value for π, but also to find the accuracy of this approximation, namely, find a narrow numerical gap to which the value of π belongs. In one of his works, Archimedes proves the chain of inequalities, which would look like this in modern way:

	10		6336					14688				1
3		<			<	π	<			<	3
	71			1					1			7
			2017					4673
				4					2

it can be written easier: 3,140 909< π < 3,1 428 265...

As you can see from inequalities, Archimedes found a rather accurate value with an accuracy of 0.002. The most surprising thing is that he found the first two signs after the comma: 3.14 ... It is exactly that we most often use in simple calculations.

Practical use

Two ride in the train:
- Here look, the rails are straight, the wheels are round.
Where is the knock?
- How come from? Wheels are round, and the area
Circle Pi Er Square, here is a square and knocks!

As a rule, get acquainted with this awesome number in 6-7 grades, but more thoroughly they are engaged in the end of the 8th grade. In this part of the article, we present the main and most important formulas that will use you in solving geometric tasks, only to begin with, we agree to take π for 3.14 for the convenience of counting.

Perhaps the most famous formula among schoolchildren in which π is used, this is the formula of the length and area of \u200b\u200bthe circle. The first - the formula of the area of \u200b\u200bthe circle - is written as follows:

	π D. 2
S \u003d π R 2 \u003d
	4

where S is the area of \u200b\u200bthe circumference, R is its radius, D is the diameter of the circle.

The length of the circle, or, as it is sometimes called, the perimeter of the circle is calculated by the formula:

C \u003d 2. π R \u003d π D,

where C is the circumference length, R radius, D is the circle diameter.

It is clear that the diameter D is equal to two radius R.

From the circumference length formula, you can easily find a circle radius:

where D is the diameter, C - the length of the circle, R is the circle radius.

These are the basic formulas that each student must know. It also sometimes has to calculate the area of \u200b\u200bnot the entire circumference, but only its part sectors. Therefore, we present it to you - a formula for calculating the area of \u200b\u200bthe circle sector. She looks like this:

			α
S.	=	π R 2.
			360 ˚

where S is the sector area, R is the radius of the circle, α is the central angle in degrees.

Such a mysterious 3,14

True, it mysteriously. Because in honor of these magical numbers, holidays are arranged, films take off, spend public actions, they write poems and much more.

For example, in 1998, the film of the American director Darren Aronofski called "Pi". The film received many awards.

Every year on March 14 at 1:59:26 Persons interested in mathematics, celebrate the "day of Pi". By the holiday, people prepare a round cake, sit down for a round table and discuss the number Pi, solve problems and puzzles associated with PI.

Attention, this amazing number did not go and poets, an unknown wrote:
It is only necessary to try and remember everything as it is - three, fourteen, fifteen, ninety-two and six.

Let's entertain!

Your attention is offered interesting rebuses with the number of Pi. Sold down the words that are encrypted below.

1. π R

2. π L.

3. π K.

Answers: 1. PIR; 2. inscribed; 3. Pick.

1. Pi was known for 4 thousand years ago.

In the Babylonian kingdom, the number was 25/8 (or in decimal form 3,125), in ancient Egypt - 256/81 (about 3,1605), in ancient India - 339/108 (approximately 3,1389).

Ancient Greek scientist Archimeda first suggested using the method of calculating the number π, which determines the dependence of the circumference length from its diameter. So the world received the first approximate value of the number π, equal to ~ 22/7 (about 3,14286).

2. The number of pi is carved on the tombstone of the Dutch mathematics

At the beginning of the XVII century, Netherlands Scientific Ludolf Van Ceilen spent ten years to calculate π up to 35 characters with a row. It is not surprising that the scientist bequeathed that the number π with this accuracy was carved on his tombstone. In honor of Van Ceilein, the Pi number was also sometimes called the "Lyudolphic Number."

3. 22,400,000,000,000 first numbers of the number of Pi are known.

With the development of civilization grew the need for accuracy of calculations, which required and improving the accuracy of the number π. True, the forty signs after the comma are quite enough, for example, in order to calculate the circumference with the size of the galaxy, and with an accuracy of one ten million dollars of the millimeter!

At the same time, the accuracy of calculating π, which is implemented by modern computers, exceeds all imaginable needs. But the race for accuracy, apparently, do not stop. In 1973, the number π was calculated with an accuracy of a million signs, and in 2011, the Japanese engineer Sigar Condo calculated π with an accuracy of 10,000,000,000 (trillion) signs. His record broke in 2016, Czech physicist Peter Trub - 22,400,000,000,000 characters. Both used a program for calculating

Among the pi is a lot of riddles. Or rather, it is not even a riddle, but some kind of truth, which no one has ever solved in the entire history of mankind ...

What is the number of pi? The number of PI is a mathematical "constant" expressing the ratio of the circumference of it to its diameter. First, by ignorance it (this attitude) was considered to be equal to three, which was rudely approximately, but they were enough. But when the prehistoric times were changed by times ancient (i.e., there were no limit to the surprise of the inquisitive minds: it turned out that the number three very inaccurately expresses this ratio. Over time and the development of sciences, this number began to belong to the twenty two seventh.

English Mathematician August de Morgan called somehow the number of Pi "... mysterious number 3,14159 ..., which climbs into the door, in the window and through the roof." Tireless scientists continued and continued to calculate the decimal signs of the number of PI, which is actually a wildly nontrivial task, because it is simply so in the column it does not compute it: the number is not only irrational, but also transcendent (this is just such numbers that are not calculated by simple equations).

In the process of calculating these most signs, many different scientific methods and entire sciences were opened. But the most important thing - in the decimal part of the number Pi there is no repetitions, as in the usual periodic fraction, and the number of signs after the comma is infinite. Today it is checked that 500 billion signs of the number of repetitions are really not. There is reason to believe that they are not at all.

Since there are no repetitions in the sequence of signs of the number of PI numbers - this means that the sequence of signs of the number Pi is obeyed by the theory of chaos, more precisely, the number Pi is also a chaos recorded by numbers. Moreover, if desired, you can present this chaos graphically, and there is an assumption that this chaos is intelligent.

In 1965, American mathematician M. Ulum, sitting on a boring meeting, from nothing to do began to write figures on the cellular paper included in the number Pi. Putting in the center 3 and moving along the spiral counterclockwise, he wrote 1, 4, 1, 5, 9, 2, 6, 5 and other decimal numbers. Along the way, he drank all the simple numbers with circles. What was his surprise and horror when the mugs began to line up along the straight lines!

In a decimal tail, you can find any intended sequence of numbers. Any sequence of numbers in decimal signs of the number Pi is sooner or later. Anyone!

So what? - you ask. And that. Count: If there is your phone there (it is), then there is also a phone of that girl who did not want to give you her room. Moreover, there are both credit card numbers, and even all the values \u200b\u200bof the winning numbers of the tomorrow's tracing lottery. What is there, in general, all lotteries on many millennia ahead. The question is how to find them there ...

If you encrypt all the letters in numbers, then in the decimal decomposition of the Pi number, you can find all the world's literature and science, and the recipe for the manufacture of sauce Bezamel, and all the sacred books of all religions. This is a strict scientific fact. After all, the sequence of infinite and combinations among the pi do not repeat, therefore it contains all combinations of numbers, and it has already been proven. And since everything is all. Including those that comply with the book you have chosen.

And this again means that there is not only the whole world literature, which is already written (in particular, both the books that burned, etc.), but also all the books that will still be written. Including your articles on sites. It turns out that this is the number (the only reasonable number in the Universe!) And manages our world. It is only necessary to consider more signs, find the right site and decipher it. This is something akin to the paradox with a chimpanzee flock by keyboard. With a long time (you can even estimate this time) the experiment, they will print all the plays of Shakespeare.

Immediately, an analogy is suggested with periodically appearing messages that in the Old Testament, allegedly, sent to the descendants that can be read by reading through ingenious programs. Make such an exotic feature of the Bible is not entirely wise, the boredom boredoms are engaged in searching for such prophecies, but I would like to give a message of one researcher, which with the help of a computer found in the Old Testament the word that there are no prophecies in the Old Testament. Most likely, in very large text, as well as in the infinite figures of the number of pi, you can not only encode any information, but also to "find" phrases, originally not embedded there.

For practice, there are 11 characters after a point within the land. Then, knowing that the radius of the Earth is 6400 km or 6.4 * 1012 millimeters, it turns out that we, throwing the twelfth digit among the PI after the point when calculating the length of the meridian, will be mistaken for several millimeters. And when calculating the length of the earth orbit during rotation around the Sun (as it is known, R \u003d 150 * 106 km \u003d 1.5 * 1014 mm) for the same accuracy, it is enough to use the number of pi with fourteen signs after the point, and that it is too small - the diameter of our Galaxies about 100,000 light years (1 light year is approximately 1013 km) or 1018 km or 1030 mm., And in the XVII century, 34 characters of the number of pi, redundant for such distances, were obtained, and they are currently calculated up to 12411th trillion signs!!!

The absence of periodically repeated numbers, namely, based on their formulas, the circumference of the circle \u003d PI * D, the circle does not closes, as there is no finite number. This fact can also be closely associated with a spiral manifestation in our life ...

There are also a hypothesis that all (or some) universal constant (constant bar, the number of Euler, the universal gravitational constant, charge of the electron, etc.) over time they change their meanings, as the curvature of space changes due to the redistribution of matter or For other reasons not known to us.

Risching to join the anger of the enlightened community, we can assume that the number of PI is considered today, reflecting the properties of the universe, can change over time. In any case, no one can prohibit us to re-find the value of the number of PI, confirming (or confirming) the existing values.

10 interesting facts about the number Pi

1. The history of the number does not have one millennium, almost as much as there is a science mathematics. Of course, the exact value of the number was calculated not immediately. At first, the ratio of the circumference length to the diameter was considered to be 3. But over time, when the architecture began to develop, it took more accurate measurement. By the way, the number existed, but it received the letter designation only at the beginning of the XVIII century (1706) and comes from the initial letters of two Greek words meaning "circle" and "perimeter". The letter "π" number was put on Mathematics Jones, and she was firmly in mathematics already in 1737.

2. In different epochs and different peoples, the number Pi had a different meaning. For example, in ancient Egypt, it was equal to 3,1604, the Indians have acquired the value of 3,162, the Chinese used a number equal to 3,1459. Over time, π was calculated more accurately, and when computing equipment appeared, that is, the computer, it began to have more than 4 billion characters.

3. There is a legend, more precisely, the specialists consider that the number Pi was used in the construction of the Babylonian tower. However, the wrath of God caused its collapse, and the wrong calculations during construction. Like, the ancient masters were mistaken. Such version exists regarding the temple of Solomon.

4. It is noteworthy that the value of the PI number tried to introduce even at the level of the state, that is, through the law. In 1897, Bill prepared Bill in Indiana. According to Document PI, 3.2. However, scientists intervene in time and prevented the error in this way. In particular, Professor Perdy acted against Bill, who was present at the Legislative Assembly.

5. Interestingly, there are several numbers in the infinite sequence of Pi. So, six nine numbers pi carry the name of American physics. Somehow Richard Feynman read a lecture and stumbled upon the public with the remark. He said that he would like to take an examination of the numbers of Pi up to six nine only in order to under the end of the story six times "nine" hinting that his meaning is rational. Whereas in fact it is irrational.

6. Mathematics around the world do not cease to conduct research related to the number of PI. It literally shrouded some secret. Some theorists even believe that the universal truth is enclosed in it. To share knowledge and new information about Pi, organized a Pi Club. It is not easy to join it, you need to have an extraordinary memory. So, who want to become a member of the club examine: a person must be able to tell a person as much as possible the signs of the number of Pi.

7. Even various techniques for memorizing the number of pi after a comma. For example, invent whole texts. In them, words have the same number of letters as the corresponding digit after the comma. To still simplify the memorization of such a long number, poems compound on the same principle. The members of the Pi Club are often entertained in this way, and at the same time they train memory and intelligence. For example, such a hobby was Cate Mike, who invented the story, each word in which was almost four thousand (3834) of the first signs of the number of Pi.

8. There are even people who have set records to memorize the signs of Pi. So, in Japan, Akira Haragii learned more than eighty three thousand characters. But the domestic record is not such an outstanding. A resident of Chelyabinsk managed to speak only two and a half thousand numbers after the semicolon.

9. The day of the number Pi is celebrated more than a quarter of a century, since 1988. Once, a physicist from a popular science museum in San Francisco Larry Shaw noted that on March 14, writing coincides with the number of Pi. In the date of the month and the number form 3.14.

10. There is a curious coincidence. On March 14, the great scientist Albert Einstein was born, which created, as you know, the theory of relativity.