A Brief History of Pi. What is the number "Pi", or how mathematicians swear

One of the most mysterious numbers, known to mankind, of course, is the number Π (read - pi). In algebra, this number reflects the ratio of the circumference of a circle to its diameter. Previously, this quantity was called the Ludolf number. How and where the number Pi came from is not known for certain, but mathematicians divide the entire history of the number Π into 3 stages, into the ancient, classical and era of digital computers.

The number P is irrational, that is, it cannot be represented as a simple fraction, where the numerator and denominator are integers. Therefore, such a number has no end and is periodic. For the first time, the irrationality of P was proved by I. Lambert in 1761.

In addition to this property, the number P cannot also be the root of any polynomial, and therefore is a number property, when it was proved in 1882, it put an end to the almost sacred dispute of mathematicians “about the squaring of the circle”, which lasted for 2,500 years.

It is known that the first to introduce the designation of this number was the Briton Jones in 1706. After Euler's work appeared, the use of such a designation became generally accepted.

To understand in detail what Pi is, it should be said that its use is so widespread that it is difficult to even name a field of science in which it would be dispensed with. One of the simplest and most familiar school curriculum values ​​is the designation of the geometric period. The ratio of the length of a circle to the length of its diameter is constant and equal to 3.14. This value was known even to the most ancient mathematicians in India, Greece, Babylon, Egypt. The earliest version of calculating the ratio dates back to 1900 BC. e. More close to contemporary meaning P was calculated by the Chinese scientist Liu Hui, in addition, he invented and fast way such a calculation. Its value remained generally accepted for almost 900 years.

The classical period in the development of mathematics was marked by the fact that in order to establish exactly what the number Pi is, scientists began to use methods mathematical analysis. In the 1400s, the Indian mathematician Madhava used the theory of series to calculate and determined the period of the number P with an accuracy of 11 digits after the decimal point. The first European, after Archimedes, who investigated the number P and made a significant contribution to its justification, was the Dutchman Ludolf van Zeulen, who already determined 15 digits after the decimal point, and wrote very entertaining words in his will: "... whoever is interested - let him go further." It was in honor of this scientist that the number P received its first and only nominal name in history.

The era of computer computing brought new details to the understanding of the essence of the number P. So, in order to find out what the number Pi is, in 1949 the ENIAC computer was used for the first time, one of the developers of which was the future "father" of the theory of modern computers J. The first measurement was carried out on for 70 hours and gave 2037 digits after the decimal point in the period of the number P. The mark of a million characters was reached in 1973. In addition, during this period, other formulas were established that reflect the number P. So, the Chudnovsky brothers were able to find one that made it possible to calculate 1,011,196,691 digits of the period.

In general, it should be noted that in order to answer the question: "What is the number Pi?", Many studies began to resemble competitions. Today, supercomputers are already dealing with the question of what it really is, the number Pi. Interesting Facts associated with these studies permeate almost the entire history of mathematics.

Today, for example, world championships are held in memorizing the number P and world records are set, the latter belongs to the Chinese Liu Chao, who named 67,890 characters in a little over a day. In the world there is even a holiday of the number P, which is celebrated as "Pi Day".

As of 2011, 10 trillion digits of the number period have already been established.

Ever since people had the ability to count and began to explore the properties of abstract objects called numbers, generations of inquisitive minds have made fascinating discoveries. As our knowledge of numbers increased, some of them attracted Special attention, and some were even given mystical meanings. Was, which stands for nothing, and which, when multiplied by any number, gives itself. There was, the beginning of everything, also possessing rare properties, prime numbers. Then they discovered that there are numbers that are not integers, and sometimes are obtained by dividing two integers - rational numbers. Irrational numbers, which cannot be obtained as a ratio of integers, and so on. But if there is a number that has fascinated and caused the writing of a mass of works, then this is (pi). A number that despite long history, was not called as we call it today, until the eighteenth century.

Start

The number pi is obtained by dividing the circumference of a circle by its diameter. In this case, the size of the circle is not important. Large or small, the ratio of length to diameter is the same. Although it is likely that this property was known earlier, the earliest evidence of this knowledge is the Moscow Mathematical Papyrus of 1850 BC. and the papyrus of Ahmes, 1650 B.C. (although it is a copy of an older document). It has a large number of mathematical problems, in some of which it approximates as , which is slightly more than 0.6% different from the exact value. Around the same time, the Babylonians considered equal. IN Old Testament, written more than ten centuries later, Yahweh does not complicate life and establishes by divine decree that it is exactly equal to .

However, the great explorers of this number were the ancient Greeks such as Anaxagoras, Hippocrates of Chios and Antiphon of Athens. Previously, the value was determined, almost certainly, using experimental measurements. Archimedes was the first to understand how to theoretically evaluate its significance. The use of the circumscribed and inscribed polygons (the larger one is circumscribed near the circle in which the smaller one is inscribed) made it possible to determine what is greater and less than . With the help of Archimedes' method, other mathematicians obtained better approximations, and already in 480, Zu Chongzhi determined that the values ​​​​are between and . Nevertheless, the polygon method requires a lot of calculations (recall that everything was done manually and not in modern system reckoning), so he had no future.

Representation

It was necessary to wait for the 17th century, when with the discovery of the infinite series, a revolution in calculation took place, although the first result was not nearby, it was a product. Infinite series are the sums of an infinite number of terms that form a certain sequence (for example, all numbers of the form where takes values ​​from to infinity). In many cases the sum is finite and can be found various methods. It turns out that some of these series converge to or some quantity related to . In order for the series to converge, it is necessary (but not sufficient) for the summable quantities to tend to zero with growth. So than more numbers we add, the more precisely we get the value of . Now we have two possibilities for getting a more accurate value. Either add more numbers, or find another series that converges faster so that you add fewer numbers.

Thanks to this new approach, the accuracy of the calculation increased dramatically, and in 1873 William Shanks published the result of many years of work, giving a value with 707 decimal places. Fortunately, he did not live to see 1945, when it was discovered that he had made a mistake and all the numbers, starting with , were wrong. However, his approach was the most accurate before the advent of computers. This was the penultimate revolution in computing. Mathematical operations, which would take a few minutes to execute manually, are now completed in a fraction of a second, with virtually no errors. John Wrench and L. R. Smith managed to calculate 2000 digits in 70 hours on the first electronic computer. The million-digit barrier was reached in 1973.

Last (on this moment) advance in computing - the discovery of iterative algorithms that converge to faster than infinite series, so that much higher accuracy can be achieved for the same computational power. The current record is just over 10 trillion correct digits. Why calculate so precisely? Considering that, knowing 39 digits of this number, it is possible to calculate the volume of the known Universe with an accuracy of an atom, there is no reason ... yet.

Some interesting facts

However, calculating a value is only a small part of its history. This number has the properties that make this constant so curious.

Perhaps the most big problem, associated with , is the well-known problem of squaring a circle, the problem of constructing, using a compass and a ruler, a square whose area is equal to the area of ​​the given circle. The squaring of a circle tormented generations of mathematicians for twenty-four centuries, until von Lindemann proved that is a transcendental number (it is not a solution to any polynomial equation with rational coefficients) and, therefore, it is impossible to grasp the immensity. Until 1761, it was not proved that the number is irrational, that is, that there are no two natural numbers and such that . Transcendence was not proven until 1882, but it is not yet known whether numbers or ( is another irrational transcendental number) are irrational. Many relationships appear that are not related to circles. This is part of the normalization coefficient of the normal function, apparently the most widely used in statistics. As mentioned earlier, the number appears as the sum of many series and is equal to infinite products, it is also important in the study of complex numbers. In physics, it can be found (depending on the system of units used) in the cosmological constant (Albert Einstein's biggest mistake) or the constant constant magnetic field. In a number system with any base (decimal, binary...), the digits pass all tests for randomness, there is no apparent order or sequence. The Riemann zeta function closely relates number to prime numbers. This number has a long history and probably still holds many surprises.

The history of the number "pi"

The history of the number p, which expresses the ratio of the circumference of a circle to its diameter, began in ancient Egypt. Area of ​​circle diameter d Egyptian mathematicians defined as (d-d/9) 2(this entry is given here in modern symbols). From the above expression, we can conclude that at that time the number p was considered equal to a fraction (16/9) 2 , or 256/81 , i.e. p= 3,160...
In the holy book of Jainism (one of ancient religions that existed in India and arose in the VI century. BC) there is an indication from which it follows that the number p at that time was taken equal, which gives a fraction 3,162...
Ancient Greeks Eudoxus, Hippocrates and other measurements of the circle were reduced to the construction of a segment, and the measurement of the circle - to the construction of an equal square. It should be noted that for many centuries, mathematicians from different countries and peoples have tried to express the ratio of the circumference of a circle to its diameter by a rational number.

Archimedes in the 3rd century BC. substantiated in his short work "Measurement of the circle" three positions:

Every circle is equal right triangle, whose legs are respectively equal to the circumference and its radius;

The areas of a circle are related to a square built on a diameter, as 11 to 14;

The ratio of any circle to its diameter is less than 3 1/7 and more 3 10/71 .

The last sentence Archimedes substantiated by successive calculation of the perimeters of regular inscribed and circumscribed polygons by doubling the number of their sides. First, he doubled the number of sides of regular inscribed and inscribed hexagons, then dodecagons, and so on, bringing the calculations to the perimeters of regular inscribed and circumscribed polygons with 96 sides. According to precise calculations Archimedes the ratio of circumference to diameter is between the numbers 3*10/71 And 3*1/7 , which means that p = 3,1419... The true meaning of this relationship 3,1415922653...
In the 5th century BC. Chinese mathematician Zu Chongzhi a more accurate value of this number was found: 3,1415927...
In the first half of the XV century. observatories Ulugbek, near Samarkand, astronomer and mathematician al-Kashi computed p with 16 decimal places. He made 27 doublings of the number of sides of the polygons and came up with a polygon with 3*2 28 angles. Al-Kashi made unique calculations that were needed to compile a table of sines with a step of 1" . These tables have played an important role in astronomy.
Half a century later in Europe F.Viet found a number p with only 9 correct decimal places by doing 16 doublings of the number of sides of the polygons. But at the same time F.Viet was the first to notice that p can be found using the limits of some series. This discovery had great importance, since it allowed us to calculate p with any accuracy. Only 250 years later al-Kashi his result was surpassed.
The first to introduce the notation for the ratio of the circumference of a circle to its diameter with the modern symbol p was an English mathematician W. Johnson in 1706. As a symbol, he took the first letter Greek word "periphery", which means in translation "circle". Introduced W. Johnson the designation became common after the publication of works L. Euler, who used the entered character for the first time in 1736 G.
At the end of the XVIII century. A.M. Lazhandre based on works I.G. Lambert proved that the number p is irrational. Then the German mathematician F. Lindeman based on research Sh. Ermita, found a rigorous proof that this number is not only irrational, but also transcendental, i.e. cannot be a root algebraic equation. It follows from the latter that using only a compass and a ruler to construct a segment equal in circumference, impossible, and hence there is no solution to the problem of squaring the circle.
The search for the exact expression for p continued even after the work F. Vieta. At the beginning of the XVII century. Dutch mathematician from Cologne Ludolf van Zeulen(1540-1610) (some historians call him L. van Keulen) found 32 correct signs. Since then (publication year 1615), the value of the number p with 32 decimal places has been called the number Ludolf.
TO late XIX c., after 20 years of hard work, an Englishman William Shanks found 707 digits of the number p. However, in 1945 it was discovered with the help of a computer that Shanks in his calculations he made a mistake in the 520th sign and his further calculations turned out to be incorrect.
After the development of methods of differential and integral calculus, many formulas were found that contain the number "pi". Some of these formulas allow you to calculate "pi" in ways other than the method Archimedes and more rational. For example, the number "pi" can be reached by looking for the limits of some series. So, G. Leibniz(1646-1716) received in 1674 a number

1-1/3+1/5-1/7+1/9-1/11+... =p /4,

which made it possible to calculate p in a shorter way than Archimedes. Nevertheless, this series converges very slowly and therefore requires rather lengthy calculations. To calculate "pi" it is more convenient to use the series obtained from the expansion arctg x with the value x=1/ , for which the expansion of the function arctan 1/=p /6 in a series gives equality

p /6 = 1/,
those.
p= 2

Partially, the sums of this series can be calculated by the formula

S n+1 = S n + (2)/(2n+1) * (-1/3) n,

while "pi" will be limited by a double inequality:

An even more convenient formula for calculating p got J. Machin. Using this formula, he calculated p(in 1706) with an accuracy of 100 correct characters. A good approximation for "pi" is given by

However, it should be remembered that this equality should be considered as approximate, since the right side of it is an algebraic number, and the left side is a transcendental one, therefore, these numbers cannot be equal.
As pointed out in their articles E.Ya.Bakhmutskaya(60s of the XX century), back in the XV-XVI centuries. South Indian scientists, including Nilakanta, using the methods of approximate calculations of the number p , found a way to expand arctg x into a power series similar to the series found Leibniz. Indian mathematicians gave a verbal formulation of the rules for expanding into series sinus And cosine. By this they anticipated the discovery of the European mathematicians of the 17th century. Nevertheless, their isolated and limited by practical needs computational work has no effect on further development science was not provided.
In our time, the work of calculators has been replaced by computers. With their help, the number "pi" was calculated with an accuracy of more than a million decimal places, and these calculations lasted only a few hours.
In modern mathematics, the number p is not only the ratio of the circumference to the diameter, it is included in a large number of different formulas, including the formulas of non-Euclidean geometry, and the formula L. Euler, which establishes a connection between the number p and the number e in the following way:

e 2 p i = 1 , where i = .

This and other interdependencies allowed mathematicians to further understand the nature of the number p.

On March 14, a very unusual holiday is celebrated all over the world - Pi Day. Everyone has known it since school days. Students are immediately explained that the number Pi is a mathematical constant, the ratio of the circumference of a circle to its diameter, which has an infinite value. It turns out that a lot of interesting facts are connected with this number.

1. The history of number has more than one millennium, almost as long as the science of mathematics exists. Certainly, exact value numbers were not calculated immediately. At first, the ratio of the circumference to the diameter was considered equal to 3. But over time, when architecture began to develop, it took more precise measurement. By the way, the number existed, but it received a letter designation only at the beginning of the 18th century (1706) and comes from the initial letters of two Greek words meaning “circumference” and “perimeter”. The mathematician Jones endowed the number with the letter "π", and she firmly entered mathematics already in 1737.

2. In different eras and at different peoples pi has different meaning. For example, in ancient Egypt it was 3.1604, among the Hindus it acquired the value of 3.162, the Chinese used the number equal to 3.1459. Over time, π was calculated more and more accurately, and when it appeared Computer Engineering, that is, a computer, it began to have more than 4 billion characters.

3. There is a legend, more precisely, experts believe that the number Pi was used in the construction of the Tower of Babel. However, it was not the wrath of God that caused its collapse, but incorrect calculations during construction. Like, the ancient masters were mistaken. A similar version exists regarding Solomon's temple.

4. It is noteworthy that they tried to introduce the value of Pi even at the state level, that is, through the law. In 1897, a bill was drafted in the state of Indiana. According to the document, Pi was 3.2. However, scientists intervened in time and thus prevented an error. In particular, Professor Purdue, who was present at the legislative assembly, spoke out against the bill.

5. It is interesting that several numbers in the infinite sequence Pi have their own name. So, six nines of Pi are named after an American physicist. Once Richard Feynman was giving a lecture and stunned the audience with a remark. He said he wanted to learn the digits of pi up to six nines by heart, only to say "nine" six times at the end of the story, hinting that its meaning was rational. When in fact it is irrational.

6. Mathematicians around the world do not stop doing research related to the number Pi. It is literally shrouded in mystery. Some theorists even believe that it contains a universal truth. In order to share knowledge and new information about Pi, they organized the Pi Club. Entering it is not easy, you need to have an outstanding memory. So, those wishing to become a member of the club are examined: a person must tell as many signs of the number Pi from memory as possible.

7. They even came up with various techniques for remembering the number Pi after the decimal point. For example, they come up with whole texts. In them, words have the same number of letters as the corresponding digit after the decimal point. To further simplify the memorization of such a long number, they compose verses according to the same principle. Members of the Pi Club often have fun in this way, and at the same time train their memory and ingenuity. For example, Mike Keith had such a hobby, who eighteen years ago came up with a story in which each word was equal to almost four thousand (3834) first digits of pi.

8. There are even people who have set records for memorizing Pi signs. So, in Japan, Akira Haraguchi memorized more than eighty-three thousand characters. But the domestic record is not so outstanding. A resident of Chelyabinsk was able to memorize only two and a half thousand numbers after the decimal point of Pi.

"Pi" in perspective

9. Pi Day has been celebrated for more than a quarter of a century, since 1988. One day, a physicist from the Popular Science Museum in San Francisco, Larry Shaw, noticed that March 14 was spelled the same as pi. In a date, the month and day form 3.14.

10. Pi Day is celebrated not only in an original way, but in a fun way. Of course, scientists involved in the exact sciences do not miss it. For them, this is a way not to break away from what they love, but at the same time to relax. On this day, people gather and cook different goodies with the image of Pi. Especially there is a place for confectioners to roam. They can make pi cakes and cookies similar shape. After tasting the treats, mathematicians arrange various quizzes.

11. There is an interesting coincidence. On March 14, the great scientist Albert Einstein was born, who, as you know, created the theory of relativity. Be that as it may, physicists can also join in the celebration of Pi Day.

Pi- a mathematical constant equal to the ratio of the circumference of a circle to its diameter. The number pi is, the digital representation of which is an infinite non-periodic decimal fraction - 3.141592653589793238462643 ... and so on ad infinitum.

100 decimal places: 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 347211.

The history of refining the value of pi

In every book on entertaining mathematics, you will certainly find a history of refining the value of pi. At first, in ancient China, Egypt, Babylon and Greece, fractions were used for calculations, for example, 22/7 or 49/16. In the Middle Ages and the Renaissance, European, Indian and Arabic mathematicians refined the value of pi to 40 digits after the decimal point, and by the beginning of the computer age, the number of digits was increased to 500 by the efforts of many enthusiasts.

Such accuracy is of purely academic interest (more on that below), and for practical needs within the Earth, 10 decimal places are enough. With a radius of the Earth of 6400 km or 6.4 10 9 mm, it turns out that, having discarded the twelfth figure of pi after the decimal point, we will be mistaken by several millimeters when calculating the length of the meridian. And when calculating the length of the Earth's orbit around the Sun (its radius is 150 million km = 1.5 10 14 mm), for the same accuracy it is enough to use the number pi with fourteen decimal places. The average distance from the Sun to Pluto, the most distant planet solar system- 40 times the average distance from the Earth to the Sun. To calculate the length of Pluto's orbit with an error of a few millimeters, sixteen digits of pi are enough. Yes, there’s nothing to trifle about, the diameter of our Galaxy is about 100 thousand light years (1 light year is approximately equal to 10 13 km) or 10 19 mm, and yet back in the 17th century 35 pi signs were obtained, redundant even for such distances.

What is the difficulty in calculating the value of pi? The fact is that it is not only irrational, that is, it cannot be expressed as a fraction p / q, where p and q are integers. Such numbers cannot be written exactly, they can only be calculated by the method of successive approximations, increasing the number of steps to obtain greater accuracy. The easiest way is to consider regular polygons inscribed in a circle with an increasing number of sides and calculate the ratio of the polygon's perimeter to its diameter. As the number of sides increases, this ratio tends to pi. This is how, in 1593, Adrian van Romen calculated the perimeter of an inscribed regular polygon with 1073741824 (i.e. 2 30) sides and determined 15 signs of pi. In 1596, Ludolf van Zeulen obtained 20 signs by calculating an inscribed polygon with 60 x 2 33 sides. Subsequently, he brought the calculations to 35 characters.

Another way to calculate pi is to use formulas with an infinite number of terms. For example:

π = 2 2/1 (2/3 4/3) (4/5 6/5) (6/7 8/7) ...

π = 4 (1/1 - 1/3) + (1/5 - 1/7) + (1/9 - 1/11) + ...

Similar formulas can be obtained by expanding, for example, the arc tangent in a Maclaurin series, knowing that

arctg(1) = π/4(because tg(45°) = 1)

or expanding the arcsine in a series, knowing that

arcsin(1/2) = π/6(leg lying against an angle of 30 °).

In modern calculations, even more effective methods. With their help today.

pi day

The day of the number pi is celebrated by some mathematicians on March 14 at 1:59 (in the American date system - 3/14; the first digits of the number π = 3.14159). It is usually celebrated at 1:59 pm (in the 12-hour system), but those who adhere to the 24-hour system of light of time consider it to be 13:59 and prefer to celebrate at night. At this time, they read eulogies in honor of the number pi, its role in the life of mankind, draw dystopian pictures of the world without pi, eat pie ( pie), drink drinks and play games that begin with "pi".

• Pi (number) - Wikipedia

Before talking about history of pi , we note that the number Pi is one of the most mysterious quantities in mathematics. You will now see for yourself, my dear reader...

Let's start our story with a definition. So the number Pi is abstract number , denoting the ratio of the circumference of a circle to the length of its diameter. This definition is familiar to us from the school bench. But here's where the mysteries begin...

It is impossible to calculate this value to the end, it is equal to 3,1415926535 , then after the decimal point - to infinity. Scientists believe that the sequence of numbers does not repeat, and this sequence is absolutely random...

Pi riddle it doesn't end there. Astronomers are confident that thirty-nine decimal places in this number is enough to calculate the circumference that encircles known space objects in the Universe, with an error in the radius of a hydrogen atom ...

irrationally , i.e. it cannot be expressed as a fraction. This value transcendent – i.e. it cannot be obtained by performing any operations on integers….

The number Pi is closely related to the concept of the golden ratio. Archaeologists have found that the height of the Great Pyramid of Giza is related to the length of its base, just like the radius of a circle is related to its length...

The history of the number P also remains a mystery. It is known that even builders used this value for design. Preserved, several thousand years old, which contained problems, the solution of which involved the use of the number Pi. However, the opinion about the exact value of this quantity among scientists different countries was ambiguous. So in the city of Susa, located two hundred kilometers from Babylon, a tablet was found where the number Pi was indicated as 3¹/8 . In ancient Babylon, it was discovered that the radius of a circle as a chord enters it six times, it was there that it was first proposed to divide a circle into 360 degrees. Let us note, by the way, that a similar geometric action was done with the orbit of the Sun, which led the ancient scientists to the idea that there should be approximately 360 days in a year. However, in Egypt, the number pi was equal to 3,16 , and in ancient india – 3, 088 , in ancient Italy - 3,125 . believed that this value is equal to the fraction 22/7 .

Pi was most accurately calculated by a Chinese astronomer. Zu Chun Zhi in the 5th century AD. For this he wrote twice odd numbers 11 33 55, then he divided them in half, put the first part in the denominator of the fraction, and the second part in the numerator, thus getting a fraction 355/113 . Surprisingly, the meaning coincides with modern calculations up to the seventh digit ...

Who gave the first official name this value?

It is believed that in 1647 mathematician Outtrade named Greek letterπ circumference, taking for this the first letter of the Greek word περιφέρεια - "periphery" . But in 1706 work came out English teacher William Jones "Review of the achievements of mathematics", in which he denoted by the letter Pi already the ratio of the circumference of a circle to its diameter. Finally, this symbol was fixed in the 20th century mathematician Leonhard Euler .

Ever since people had the ability to count and began to explore the properties of abstract objects called numbers, generations of inquisitive minds have made fascinating discoveries. As our knowledge of numbers has increased, some of them have attracted special attention, and some have even been given mystical meanings. Was, which means nothing, and which, when multiplied by any number, gives itself. There was, the beginning of everything, also possessing rare properties, prime numbers. Then they discovered that there are numbers that are not integers, and sometimes are obtained by dividing two integers - rational numbers. Irrational numbers that cannot be obtained as a ratio of integers, etc. But if there is a number that has fascinated and caused the writing of a mass of works, then this is (pi). A number that, despite its long history, was not called as we call it today until the eighteenth century.

Start

The number pi is obtained by dividing the circumference of a circle by its diameter. In this case, the size of the circle is not important. Large or small, the ratio of length to diameter is the same. Although it is likely that this property was known earlier, the earliest evidence of this knowledge is the Moscow Mathematical Papyrus of 1850 BC. and the papyrus of Ahmes, 1650 B.C. (although it is a copy of an older document). It has a large number of math problems, some of which approximate as, which is just over 0.6% off the exact value. Around the same time, the Babylonians considered equal. In the Old Testament, written more than ten centuries later, Yahweh does not complicate life and establishes by divine decree what is exactly equal.

However, the great explorers of this number were the ancient Greeks such as Anaxagoras, Hippocrates of Chios and Antiphon of Athens. Previously, the value was determined, almost certainly, using experimental measurements. Archimedes was the first to understand how to theoretically evaluate its significance. The use of the circumscribed and inscribed polygons (the larger one is circumscribed near the circle in which the smaller one is inscribed) made it possible to determine what is larger and smaller. With the help of Archimedes' method, other mathematicians obtained better approximations, and already in 480, Zu Chongzhi determined that the values ​​​​are between and. However, the polygon method requires a lot of calculations (recall that everything was done by hand and not in the modern number system), so it had no future.

Representation

It was necessary to wait for the 17th century, when with the discovery of the infinite series a revolution in calculation took place, although the first result was not nearby, it was a product. Infinite series are the sums of an infinite number of terms that form a certain sequence (for example, all numbers of the form where it takes values ​​from to infinity). In many cases the sum is finite and can be found by various methods. It turns out that some of these series converge to or to some quantity related to. For the series to converge, it is necessary (but not sufficient) that the summable quantities tend to zero with growth. Thus, the more numbers we add, the more accurate we get the value. We now have two possibilities for obtaining a more accurate value. Either add more numbers, or find another series that converges faster so that you add fewer numbers.

Thanks to this new approach, the accuracy of the calculation increased dramatically, and in 1873 William Shanks published the result of many years of work, giving a value with 707 decimal places. Fortunately, he did not live to see 1945, when it was discovered that he had made a mistake and all the numbers, starting with, were wrong. However, his approach was the most accurate before the advent of computers. It was the penultimate revolution in computing. Mathematical operations that would take several minutes to perform manually are now performed in fractions of a second, with virtually no errors. John Wrench and L. R. Smith managed to calculate 2000 digits in 70 hours on the first electronic computer. The million-digit barrier was reached in 1973.

The latest (so far) advance in computing is the discovery of iterative algorithms that converge to faster than infinite series, so that much higher accuracy can be achieved for the same computational power. The current record is just over 10 trillion correct digits. Why calculate so accurately? Considering that, knowing 39 digits of this number, it is possible to calculate the volume of the known Universe with an accuracy of an atom, there is no reason ... yet.

Some interesting facts

However, calculating a value is only a small part of its history. This number has the properties that make this constant so curious.

Perhaps the biggest problem associated with is the well-known problem of squaring the circle, the problem of constructing with a compass and straightedge a square whose area is equal to the area of ​​a given circle. The squaring of a circle tormented generations of mathematicians for twenty-four centuries, until von Lindemann proved that - is a transcendental number (it is not a solution to any polynomial equation with rational coefficients) and, therefore, it is impossible to grasp the immensity. Until 1761, it was not proved that the number is irrational, that is, that there are no two natural numbers and such that. Transcendence was not proven until 1882, however, it is not yet known whether the numbers are or (is another irrational transcendental number) irrational. Many relationships appear that are not related to circles. This is part of the normalization coefficient of the normal function, apparently the most widely used in statistics. As mentioned earlier, the number appears as the sum of many series and is equal to infinite products, it is also important in the study of complex numbers. In physics, it can be found (depending on the system of units used) in the cosmological constant (Albert Einstein's biggest error) or in the constant magnetic field constant. In a number system with any base (decimal, binary...), the digits pass all tests for randomness, there is no apparent order or sequence. The Riemann zeta function closely relates number to prime numbers. This number has a long history and probably still holds many surprises.

If we compare circles of different sizes, we can see the following: the sizes of different circles are proportional. And this means that when the diameter of a circle increases by a certain number of times, the length of this circle also increases by the same number of times. Mathematically, this can be written like this:

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

where C1 and C2 are the lengths of two different circles, and d1 and d2 are their diameters.
This ratio works in the presence of a proportionality coefficient - the constant π already familiar to us. From relation (1) we can conclude: the circumference C is equal to the product of the diameter of this circle and the proportionality factor independent of the circle π:

C = πd.

Also, this formula can be written in a different form, expressing the diameter d in terms of the radius R of the given circle:

C \u003d 2π R.

Just this formula is a guide to the world of circles for seventh graders.

Since ancient times, people have tried to establish the value of this constant. So, for example, the inhabitants of Mesopotamia calculated the area of ​​a circle using the formula:

Whence π = 3.

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

			8			2
S	=	(		d	)
			9

From what considerations did he get this formula? – Unknown. Probably based on their observations, however, as did other ancient philosophers.

In the footsteps of Archimedes

Which of the two numbers is greater than 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 are identically equal. But this is a delusion. In addition to the above incorrect answer in the exam (see epigraph), one very entertaining puzzle can also be added to this group. The task says: "move one match so that the equality becomes true."

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

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

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

can be written more simply: 3.140 909< π < 3,1 428 265...

As we can see from the inequalities, Archimedes found a fairly accurate value with an accuracy of 0.002. The most surprising thing is that he found the first two decimal places: 3.14 ... It is this value that we most often use in simple calculations.

Practical use

Two people are on the train:
- Look, the rails are straight, the wheels are round.
Where is the knock coming from?
- How from where? The wheels are round, and the area
circle pi er square, that's the square knocking!

As a rule, they get acquainted with this amazing number in the 6th-7th grade, but they study it more thoroughly towards the end of the 8th grade. In this part of the article, we will present the main and most important formulas that will be useful to you in solving geometric problems, but for starters, we will agree to take π as 3.14 for ease of calculation.

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

	π D 2
S=π R 2 =
	4

where S is the area of ​​the circle, R is its radius, D is the diameter of the circle.

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

C = 2 π R = πd,

where C is the circumference, R is the radius, d is the diameter of the circle.

It is clear that the diameter d is equal to two radii R.

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

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

These are the basic formulas that every student should know. Also, sometimes you have to calculate the area not of the entire circle, but only of its part - the sector. Therefore, we present it to you - a formula for calculating the area of ​​​​a sector of a circle. It looks like this:

			α
S	=	π R 2
			360 ˚

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

So mysterious 3.14

Indeed, it is mysterious. Because in honor of these magical numbers they organize holidays, make films, hold public events, write poetry and much more.

For example, in 1998, a film by American director Darren Aronofsky called "Pi" was released. The film received numerous awards.

Every year on March 14th at 1:59:26 am, people interested in mathematics celebrate "Pi Day". For the holiday, people prepare a round cake, sit down at round table and discuss pi, solve problems and puzzles related to pi.

The attention of this amazing number was not bypassed by poets either, an unknown person wrote:
You just have to try and remember everything as it is - three, fourteen, fifteen, ninety-two and six.

Let's have some fun!

We offer you interesting puzzles with the number Pi. Guess the words that are encrypted below.

1. π R

2. π L

3. π k

Answers: 1. Feast; 2. Filed; 3. Squeak.

The history of pi begins with ancient egypt and goes hand in hand with the development of all mathematics. We meet this value for the first time within the walls of the school.

The number Pi is perhaps the most mysterious of an infinite number of others. Poems are dedicated to him, artists portray him, and a film has even been made about him. In our article, we will look at the history of development and computing, as well as the areas of application of the Pi constant in our lives.

Pi is a mathematical constant equal to the ratio of the circumference of a circle to the length of its diameter. Initially, it was called the Ludolf number, and it was proposed to denote it by the letter Pi by the British mathematician Jones in 1706. After the work of Leonhard Euler in 1737, this designation became generally accepted.

The number Pi is irrational, that is, its value cannot be expressed exactly as a fraction m/n, where m and n are integers. This was first proved by Johann Lambert in 1761.

The history of the development of the number Pi has already been around 4000 years. Even the ancient Egyptian and Babylonian mathematicians knew that the ratio of the circumference to the diameter is the same for any circle and its value is a little more than three.

Archimedes proposed a mathematical method for calculating Pi, in which he inscribed in a circle and described regular polygons around it. According to his calculations, Pi was approximately equal to 22/7 ≈ 3.142857142857143.

In the 2nd century, Zhang Heng proposed two values ​​for pi: ≈ 3.1724 and ≈ 3.1622.

Indian mathematicians Aryabhata and Bhaskara found an approximate value of 3.1416.

The most accurate approximation of pi for 900 years was a calculation by the Chinese mathematician Zu Chongzhi in the 480s. He deduced that Pi ≈ 355/113 and showed that 3.1415926< Пи < 3,1415927.

Until the 2nd millennium, no more than 10 digits of Pi were calculated. Only with the development of mathematical analysis, and especially with the discovery of series, were subsequent major advances in the calculation of the constant made.

In the 1400s, Madhava was able to calculate Pi=3.14159265359. His record was broken by the Persian mathematician Al-Kashi in 1424. He in his work "Treatise on the Circumference" cited 17 digits of Pi, 16 of which turned out to be correct.

The Dutch mathematician Ludolf van Zeulen reached 20 numbers in his calculations, giving 10 years of his life for this. After his death, 15 more digits of pi were discovered in his notes. He bequeathed that these figures were carved on his tombstone.

With the advent of computers, the number Pi today has several trillion digits and this is not the limit. But, as noted in Fractals for the Classroom, for all the importance of pi, “it is difficult to find areas in scientific calculations that require more than twenty decimal places.”

In our life, the number Pi is used in many scientific fields. Physics, electronics, probability theory, chemistry, construction, navigation, pharmacology are just some of them that simply cannot be imagined without this mysterious number.

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According to the site Calculator888.ru - Pi number - meaning, history, who invented it.