The history of the origin of pi. Amazing number pi

Tatiana Durimanova

I created on Facebook page b called it “Language as a philosophy of life.” Actually, I wanted to call it “Notes from a Madhouse,” because what other than a madhouse is our modern life? No, I’m not going to talk about the fact that everyone is running somewhere, doesn’t have time to do something, is always missing something: time, money, etc. That we are overwhelmed by a wave of misunderstanding of what is happening around us, where the world is heading...
We spin like squirrels in a wheel. We feel like we are running in a vicious circle. We lose our circle of friends, we end up in vicious circle... Sound familiar? And morning-day-evening-night, and again in a circle. Spring-summer-autumn-winter, and again in a circle.
By the way, who can say exactly at what time morning will give way to night, winter, spring? Is it even possible to draw a clear line between the chicken and the egg, and are they separable? It might be better to recognize that an egg is a potential chicken, a chicken is a potential egg, and they are not separable. Where do I end and my problems begin, the problems of my children, friends, etc., who become mine, simply because we live in the same apartment, house, city, world? Did the Lord God tell us that zero hours should be determined by Greenwich, that I should be called Tatyana, and a chair a chair? Where does the real (substantial) world end and the world invented by us begin?
The Earth rotates around its axis and in orbit (circle, ellipse - what's the difference?). Galaxies rotate. Scientists have discovered torsion fields, proven that ... “according to Albert Einstein’s theory of relativity, the world is not structured exactly like [as we were taught and taught at school]), there is a curvature of space in it, so that two straight lines, which are parallel in a given area of ​​space, at some segment of their length, they can intersect. Recently, Einstein’s assumption about the curvature of space was confirmed experimentally” (Alexander Babitsky).
And we all move from point A to point B, believing that they are on a straight line.
And why did this bring me, a linguist, into physics, you ask? Yes, because everything around us, and within ourselves, is physics. Language is physics. Doesn't sound belong to the realm of physics? Now tell me, what is a vowel sound? I offer you a “cute” definition of sounds for the 21st century: “We pronounce and hear sounds, and we write and see letters. When pronouncing a vowel sound, the air does not encounter any obstacles: [a], [o], [u], [i], [s], [e]. When pronouncing a consonant sound, the air encounters an obstacle: lips, teeth, tongue. A consonant sound is pronounced with voice and noise or only with noise.”
In principle, everything is correct. You can simply hum with a “vowel sound” without opening your lips. Cheers to your health. But if you open your lips, then you get the sounds familiar to us all, “a”, “e”, which differ only in the degree of rounding, stretching or pulling the lips into a tube. Do you agree? It's like a watermelon, which can be cut into slices, cubes, figures, but it still remains a watermelon!!! And at what point does the sound “a” turn into “o”? Is there a clear boundary? Of course, the quality of the vowel sound can be affected by the position of the tongue (back sounds), lowering the jaw, again with the corresponding position of the tongue, but it’s still the same watermelon, cut into shapes.
The consonant sound is a barrier to the vowel sound. How can such a barrier be created? Read above: lips, teeth, tongue. In other words, the tools of speech are quite limited, but what an abundance of languages!!! (How do you like 7 notes and such an abundance of music?)
Now let's think about it: a cat has this toolkit, a dog, a dolphin, and fish in general, etc....
“Well, I stopped by,” you say. Yes, I'm here! Wasn't there a time when the Earth was considered a pancake? Doesn't electricity exist simply because we don't see or hear it? If it is proven that there is no vacuum, then everything is there, but all this can be distinguished, again, depending on the tools that we use to examine and study the object. As it improves, we learn more and more new things that we could not even imagine before.
Language is the formalization of thought. Where is the thought formalized? What do we know about our world, about ourselves? We are looking for other worlds without knowing our own! This is precisely the problem!
What do we know about language, except that it is formalized in sounds. Please formalize – kummmmarama. What is this? Nothing, because a vowel sound can “carry” only a certain number of consonant sounds, just as I, with my weight of 50 kg, cannot lift a load of 150 kg. Physics, you know!
Now let's turn to the curvature of space and the circle with which we started. Let's say we doubt that language develops not in a spiral (in terms of context), but in a straight line, and I tell you that “in our big city there is a main street that crosses the entire city, on which there are many trees and many people walking...". Stupidity, tell me, where are the punctuation marks? Where are the commas and periods?
But what are punctuation marks? They are the signs of separation between the subject-predicate complement (with related definitions) of one sentence and the beginning of another. The participle is nothing more than multiplication: which passes = passing, while the expansion of “passing” into “which passes” is already division. And this is mathematics! Nothing surprising. The world is indivisible. This is integrity. Language is also integrity. It's just time for us to look at everything in a new way. Wake up and look around. Teach children non-rules, like “There is a separate group of words - predicates (or state category). These are words denoting a non-dynamic state and acting as the main member (predicate, predicate) of a one-component impersonal offer. Scientists have still not decided on the status of state category words. So the word NEED, along with other words (sorry, hunting, lack of time, time, etc.) is included in this group of words.”
Do you understand what this is about? Me not! Who is this written for? Probably for the students. Poor students! If even scientists still haven’t understood something there, how should children understand it? I wonder if the teachers at least learned this definition by heart?
This is why I created my YouTube channel, to simply (in human language) talk about the main thing - about language.
If, after reading, all this (written, by the way, hastily) seems like nonsense to you, do not rush to tell me that I am abnormal. I called it notes from a madhouse. If this seems abnormal to you, then you live in the opposite house. I'm not going to define it. We live in a country of victorious democracy and... values. Everyone has the right to their opinion.

Ever since humans were able to count and began exploring 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 stands for nothing, and which when multiplied by any number gives itself. Was, the beginning of everything, also possessing rare properties, prime numbers. Then they discovered that there are numbers that are not integers, but are sometimes obtained by dividing two integers - rational numbers. Irrational numbers that cannot be obtained as a ratio of whole numbers, etc. But if there is a number that has fascinated and caused a lot of writing to be written, it is (pi). A number that, despite a long history, was not called what 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. Big 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 Ahmes papyrus 1650 BC. (although this is a copy of an older document). It contains a large number of mathematical problems, in some of which it approaches as , which is slightly more than 0.6\% different from the exact value. Around this time, the Babylonians considered equals. In the Old Testament, written more than ten centuries later, Yahweh keeps things simple and establishes by divine decree what exactly equals .

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 by experimental measurements. Archimedes was the first to understand how to theoretically evaluate its significance. The use of circumscribed and inscribed polygons (the larger one is circumscribed around the circle in which the smaller one is inscribed) made it possible to determine what is greater and less. Using Archimedes' method, other mathematicians obtained better approximations, and already in 480 Zu Chongzhi determined that the values ​​were between and . However, the polygon method requires a lot of calculations (remember that everything was done manually and not in modern system reckoning), so he had no future.

Representation

It was necessary to wait until the 17th century, when a revolution in calculation took place with the discovery of the infinite series, although the first result was not close, 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 by various methods. It turns out that some of these series converge to or some quantity related to . In order for a series to converge, it is necessary (but not sufficient) that the summed quantities tend to zero as they grow. Thus, the more numbers we add, the more accurate we get the value. Now we have two options for getting a more accurate value. Either add more numbers, or find another series that converges faster, so that you can 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 until 1945, when it was discovered that he had made a mistake and all the numbers, starting with , were incorrect. However, his approach was most accurate before the advent of computers. This was the penultimate revolution in computing. Mathematical operations that would take several minutes to perform manually are now completed 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.

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 with the same computing power. The current record is just over 10 trillion correct digits. Why calculate so accurately? Considering that, knowing 39 digits of this number, you can calculate the volume known universe down to the atomic precision, nothing... yet.

Some interesting facts

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

Perhaps the biggest problem associated with , is the famous squaring of the circle problem, the problem of constructing, using a compass and ruler, a square whose area is equal to the area of ​​a given circle. The squaring of the circle tormented generations of mathematicians for twenty-four centuries until von Lindemann proved that it is a transcendental number (it is not a solution to any polynomial equation with rational coefficients) and, therefore, impossible to grasp the immensity. Until 1761 it was not proven that the number is irrational, that is, that there are no two natural numbers and those that . Transcendence was not proven until 1882, but it is not yet known whether the numbers or ( is another irrational transcendental number) are irrational. Many relationships appear that are not related to circles. This is part of the normalization factor of the normal function, apparently the most widely used in statistics. As mentioned earlier, a 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 magnetic field. In a number system with any base (decimal, binary...), the numbers pass all tests of randomness, there is no 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.

Math enthusiasts around the world eat a piece of pie every year on the fourteenth of March - after all, it is the day of Pi, the most famous irrational number. This date is directly related to the number whose first digits are 3.14. Pi is the ratio of the circumference of a circle to its diameter. Since it is irrational, it is impossible to write it as a fraction. This is an infinitely long number. It was discovered thousands of years ago and has been constantly studied since then, but does Pi still have any secrets? From ancient origin until the uncertain future here are some of the most interesting facts about the number Pi.

Memorizing Pi

The record for memorizing decimal numbers belongs to Rajvir Meena from India, who managed to remember 70,000 digits - he set the record on March 21, 2015. Previously, the record holder was Chao Lu from China, who managed to remember 67,890 digits - this record was set in 2005. The unofficial record holder is Akira Haraguchi, who recorded himself on video repeating 100,000 digits in 2005 and recently published a video where he manages to remember 117,000 digits. The record would become official only if this video was recorded in the presence of a representative of the Guinness Book of Records, and without confirmation it remains only an impressive fact, but is not considered an achievement. Math enthusiasts love to memorize the number Pi. Many people use various mnemonic techniques, for example poetry, where the number of letters in each word matches the digits of Pi. Each language has its own versions of similar phrases that help you remember both the first few numbers and the whole hundred.

There is a Pi language

Mathematicians, passionate about literature, invented a dialect in which the number of letters in all words corresponds to the digits of Pi in exact order. Writer Mike Keith even wrote a book, Not a Wake, which is entirely written in Pi. Enthusiasts of such creativity write their works in full accordance with the number of letters and the meaning of numbers. This has no practical application, but is a fairly common and well-known phenomenon in the circles of enthusiastic scientists.

Exponential growth

Pi is an infinite number, so by definition people will never be able to establish the exact digits of this number. However, the number of decimal places has increased greatly since Pi was first used. The Babylonians also used it, but a fraction of three whole and one eighth was enough for them. Chinese and creators Old Testament and were completely limited to three. By 1665, Sir Isaac Newton had calculated the 16 digits of Pi. By 1719, the French mathematician Tom Fante de Lagny had calculated 127 digits. The advent of computers has radically improved human knowledge of Pi. From 1949 to 1967 the number known to man digits skyrocketed from 2037 to 500,000. Not long ago, Peter Trueb, a scientist from Switzerland, was able to calculate 2.24 trillion digits of Pi! It took 105 days. Of course, this is not the limit. It is likely that with the development of technology it will be possible to establish an even more accurate figure - since Pi is infinite, there is simply no limit to accuracy, and it can only be limited technical features computer technology.

Calculating Pi by hand

If you want to find the number yourself, you can use the old-fashioned technique - you will need a ruler, a jar and some string, or you can use a protractor and a pencil. The downside to using a can is that it needs to be round and accuracy will be determined by how well a person can wrap the rope around it. You can draw a circle with a protractor, but this also requires skill and precision, as an uneven circle can seriously distort your measurements. A more accurate method involves using geometry. Divide a circle into many segments, like a pizza into slices, and then calculate the length of a straight line that would turn each segment into isosceles triangle. The sum of the sides will give the approximate number Pi. The more segments you use, the more accurate the number will be. Of course, in your calculations you will not be able to come close to the results of a computer, nevertheless these simple experiments allow you to understand in more detail what the number Pi actually is and how it is used in mathematics.

Discovery of Pi

The ancient Babylonians knew about the existence of the number Pi already four thousand years ago. Babylonian tablets calculate Pi as 3.125, and an Egyptian mathematical papyrus shows the number 3.1605. In the Bible, Pi is given in the obsolete length of cubits, and the Greek mathematician Archimedes used the Pythagorean theorem, a geometric relationship between the length of the sides of a triangle and the area of ​​the figures inside and outside the circles, to describe Pi. Thus, we can say with confidence that Pi is one of the most ancient mathematical concepts, although the exact name of this number appeared relatively recently.

New look at Pi

Even before the number Pi began to be correlated with circles, mathematicians already had many ways to even name this number. For example, in ancient mathematics textbooks one can find a phrase in Latin that can be roughly translated as “the quantity that shows the length when the diameter is multiplied by it.” The irrational number became famous when the Swiss scientist Leonhard Euler used it in his work on trigonometry in 1737. However, the Greek symbol for Pi was still not used - this only happened in a book by a lesser-known mathematician, William Jones. He used it already in 1706, but it went unnoticed for a long time. Over time, scientists adopted this name, and now it is the most famous version of the name, although it was previously also called the Ludolf number.

Is Pi a normal number?

Pi is definitely a strange number, but how much does it follow normal mathematical laws? Scientists have already resolved many questions related to this irrational number, but some mysteries remain. For example, it is not known how often all the numbers are used - the numbers 0 to 9 should be used in equal proportion. However, statistics can be traced from the first trillions of digits, but due to the fact that the number is infinite, it is impossible to prove anything for sure. There are other problems that still elude scientists. It is quite possible that further development science will help shed light on them, but at the moment it remains beyond the scope of human intellect.

Pi sounds divine

Scientists cannot answer some questions about the number Pi, however, every year they understand its essence better and better. Already in the eighteenth century, the irrationality of this number was proven. In addition, the number has been proven to be transcendental. This means no a certain formula, which would allow us to calculate Pi using rational numbers.

Dissatisfaction with the number Pi

Many mathematicians are simply in love with Pi, but there are also those who believe that these numbers are not particularly significant. In addition, they claim that Tau, which is twice the size of Pi, is more convenient to use as an irrational number. Tau shows the relationship between circumference and radius, which some believe represents a more logical method of calculation. However, it is impossible to unambiguously determine anything in this matter, and one and the other number will always have supporters, both methods have the right to life, so this is just an interesting fact, and not a reason to think that you should not use the number Pi.

Essay

Amazing number pi

Introduction

March, Pi Day is celebrated all over the world. This holiday was invented in 1987 by San Francisco physicist Larry Shaw, who noted that in the American date system (month/day), the date March 14 (3.14) and the time 1:59 coincide with the first digits of the date π = 3.14159). Typically, Pi Day is celebrated at 1:59 pm local time (12-hour clock). For the holiday they bake (or buy) pies (cakes), because in English π Pronounced “pie”, which sounds the same as the word pie (“pie”). Special celebrations take place in scientific societies and educational institutions. Interestingly, the holiday of Pi, celebrated on March 14, coincides with the birthday of one of the most prominent physicists of our time, Albert Einstein.

We were interested in this number. Who was the first to guess about the relationship between the circumference of a circle and its diameter? Who was the first to calculate its value? What is the history of this number? Why was this number called " π»?

Purpose of work: get to know the number π, study the history of its discovery, methods of finding

study the history of the discovery of the number π;

Learn methods for finding numbers π;

Draw conclusions.

1. Number designationπ

We know who built the first airplane, who invented the radio, but no one knows who was the first to guess about the connection between the length of a circle and its diameter. But it is known when the first designation of a given number by a letter appeared. It is believed that this designation was first introduced by the English teacher William Johnson (1675-1749) in his work “Review of the Achievements of Mathematics,” published in 1706. Even earlier, in 1647, the English mathematician Oughtred used the letter π to indicate the circumference of a circle. It is assumed that he was prompted to this designation by the first letter of the Greek alphabet of the word περιφερια - circle. But the international standard designation π for the number 3, 141592 ... became after it was used by the famous Russian academician, mathematician Leonhard Euler in his works in 1737. He wrote: “There are many other ways of finding the lengths or areas of a corresponding curve or plane figure, which can greatly facilitate practice.

. History of the numberπ

It is believed that the number π was first discovered by Babylonian magicians. It was used in the construction of the famous Tower of Babel, the story of which was included in the Bible. However, insufficiently accurate calculations led to the collapse of the entire project. It is also believed that the number Pi was at the basis of the construction famous Temple King Solomon. History of numbers π went in parallel with the development of all mathematics. Some authors divide the entire process into 3 periods: ancient period, during which π studied from the perspective of geometry, the classical era, which followed the development of mathematical analysis in Europe in the 17th century, and the era digital computers.

Ancient period

Any schoolchild now calculates the circumference of a circle by diameter much more accurately than the wisest priest ancient country pyramids or the most skillful architect of great Rome. In ancient times, it was believed that the circumference was exactly 3 times longer than the diameter. This information is contained in cuneiform tablets from the Ancient Interfluve. The same meaning can be seen in the text of the Bible: “And he made a sea cast of copper, ten cubits from edge to edge, completely round... and a string of thirty cubits hugged it all around.” However, already in the 2nd millennium BC. Mathematicians of Ancient Egypt found a more accurate relationship. In the Rhind Papyrus, which dates to around 1650 B.C. for number π the value given is (16/9) 2, which is approximately 3.16. The ancient Romans believed that a circle is 3.12 longer than its diameter, while the correct ratio is 3.14159... Egyptian and Roman mathematicians established the ratio of the circumference to the diameter not by strict geometric calculation, like later mathematicians, but found it simply from experience. But why did they make such mistakes? Couldn't they just wrap a thread around something round and then straighten the thread and just measure it?

Take, for example, a vase with a round bottom with a diameter of 100 mm. The circumference should be 314 mm. However, in practice, measuring with a thread, we are unlikely to get this length: it’s easy to make a mistake by one millimeter, and then π will be equal to 3.13 or 3.15. And if we take into account that the diameter of the vase cannot be measured quite accurately, that even here an error of 1 mm is very likely, then for π This results in fairly wide ranges between 3.09 and 3.18.

We decided to conduct several experiments. To do this, we drew several circles. Using a thread and a ruler, we measured the length of each circle and its diameter. Then divide the circumference of the circle by its diameter. We got the following results.

No. Circumference Diameter π 114.5 cm5 cm2.9231 cm10 cm3.1310 cm3 cm3, (3)419.5 cm6.5 cm3516.5 cm5 cm3.5618 cm6 cm3735 cm11 cm3, (18)820.5 cm6.5 cm3.15922 cm6.9 cm3.191021 cm3 cm31113 cm4 cm3.25126 cm1.7 cm3.51312 cm4 cm31412.5 cm4 cm3, 1251526 cm8 cm3.251638 cm12 cm3.2 mathematical pi number digit

Average - 3.168

Defining π using the indicated method, you can get a result that does not coincide with 3.14: one time we get 3.1, another time 3.12, the third 3.17, etc. By chance, 3.14 may be among them, but in the eyes of the calculator this number will not have more weight than the others.

This kind of experimental path cannot give any acceptable value for π. In this regard, it becomes more clear why ancient world did not know correct attitude circumference to diameter.

From 4th century BC mathematical science developed rapidly in Ancient Greece. Ancient Greek geometers strictly proved that the circumference of a circle is proportional to its diameter, and the area of ​​a circle is equal to half the product of the circumference and radius S = Ѕ С R = π R2 . This proof is attributed to Euclid of Cnidus and Archimedes.

Archimedes, in his essay “On the Measurement of a Circle,” calculated the perimeters of regular polygons inscribed in a circle and circumscribed around it - from the 6-gon to the 96-gon. Taking the diameter of a circle to be one, Archimedes considered the perimeter of the inscribed polygon as a lower bound for the circumference of the circle, and the perimeter of the circumscribed polygon as an upper bound. Considering the regular 96-gon, Archimedes arrived at the estimate

Thus, he established that the number π contained within

3,1408 < π < 3,1428. The value 22/7 is still considered quite good approximation numbers π for applied tasks.

In the “Algebra” of the ancient Arab mathematician Mohammed ben Muz about calculating the circumference of a circle we read the following lines: “The best way is to multiply the diameter by 3 1/7. This is the fastest and easiest way. God knows best."

Zhang Heng clarified the meaning of the number in the 2nd century π, offering two equivalents: 1) 92/29 ≈ 3.1724..., 2) √10.

In India, Aryabhata and Bhaskara used the approximation 3.1416.

Brahmagupta in the 7th century proposed √10 as an approximation.

Around 265 AD mathematician Liu Hui from the Wei kingdom provided a simple and accurate algorithm for calculating π with any degree of accuracy. He independently carried out calculations for the 3072-gon and obtained an approximate value for π, π ≈3,14159.

Liu Hui later came up with a quick calculation method π and got an approximate value of 3.1416 with just a 96-gon, taking advantage of the fact that the difference in area of ​​successive polygons forms a geometric progression with a denominator of 4.

In the 480s, Chinese mathematician Zu Chongzhi demonstrated that π ≈355/113, and showed that 3.1415926< π < 3,1415927, using Liu Hui's algorithm applied to the 12288-gon. This value remained the closest approximation of the number π over the next 900 years.

Before the 2nd millennium, no more than 10 digits were known π.

Classical period

Further major achievements in studying π associated with the development of mathematical analysis, in particular with the discovery of series that make it possible to calculate π with any accuracy, summing up the appropriate number of terms of the series. In the 1400s, Madhava of Sangamagrama found the first of these series

This result is known as the Madhava-Leibniz series, or Gregory-Leibniz series (after it was rediscovered by James Gregory and Gottfried Leibniz in the 17th century). However, this series converges to π very slowly, leading to the difficulty of calculating many digits of a number in practice—about 4,000 terms of the series must be added to improve Archimedes' estimate. However, by transforming this series into

Madhava was able to calculate π as 3.14159265359, correctly identifying 11 digits in the number. This record was broken in 1424 by the Persian mathematician Jamshid al-Kashi, who in his work entitled “Treatise on the Circle” cited 17 digits of the number π, of which 16 are correct.

The first major European contribution since Archimedes was that of the Dutch mathematician Ludolf van Zeijlen, who spent ten years calculating the number π with 20 decimal digits (this result was published in 1596). Using Archimedes' method, he brought the doubling to an n-gon, where n = 60 229. Having outlined his results in the essay “On the Circle” (“Van den Circkel”), Ludolf ended it with the words: “Whoever has the desire, let him go further.” After his death, 15 more exact digits of the number were discovered in his manuscripts π. Ludolf bequeathed that the signs he found be carved on his tombstone. There is a number in his honor π sometimes called the "Ludolf number", or "Ludolf constant".

Around the same time, methods for analyzing and determining infinite series began to develop in Europe. The first such representation was Viète's formula, found by François Viète in 1593.

Another famous result was the Wallis formula: derived by John Wallis in 1655. Leibniz series, first found by Madhava of Sangamagram in 1400 In modern times for calculation π analytical methods based on identities are used. Euler, author of the notation π, got 153 faithful to the sign. The best result by the end of the 19th century was obtained by the Englishman William Shanks, who took 15 years to calculate 707 digits, although due to an error only the first 527 were correct. To avoid such errors, modern calculations of this kind are carried out twice. If the results match, then they are highly likely to be correct.

The era of digital computers

Shanks' bug was discovered by one of the first computers in 1948; he counted 808 characters in a few hours π.

With the advent of computers, the pace increased:

year - 2037 decimal places (John von Neumann, ENIAC),

year - 10000 decimal places (F. Genuis, IBM-704),

year - 100000 decimal places (D. Shanks, IBM-7090),

year - 10,000,000 decimal places (J. Guillou, M. Bouyer, CDC-7600),

year - 29360000 decimal places (D. Bailey, Cray-2),

year - 134217000 decimal places (T. Canada, NEC SX2),

year - 1011196691 decimal places (D. Chudnowski and G. Chudnowski, Cray-2+IBM-3040). They achieved 2260000000 characters in 1991, and 4044000000 characters in 1994. Further records belong to the Japanese Tamura Canada: in 1995 4294967286 characters, in 1997 - 51539600000. By 2011, scientists were able to calculate the value of the number π with an accuracy of 10 trillion decimal places!

3. The poetry of numbersπ

Let's take a close look at its first thousand characters, let us be imbued with the poetry of these numbers, because behind them stand the shadows of the greatest thinkers of the Ancient World and the Middle Ages, the New and the present.

8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989

Interesting data on the distribution of digits of a number π. Someone was not too lazy and calculated (for a million decimal places):

zeros - 99959,

units -99758,

twos -100026,

triples - 100229,

fours - 100230,

fives - 100359,

sixes - 99548,

sevens - 99800,

eight - 99985,

nines -100106.

Decimal digits π quite random. It contains any sequence of numbers, you just need to find it. This number contains in encoded form all written and unwritten books; any information that can be invented is already included in π. You just need to look at more signs, find the right area and decipher it. Here everyone can find their phone number, their date of birth or home address.

Since there are no repetitions in the sequence of pi signs, this means that the sequence of pi signs obeys the theory of chaos, or more precisely, the number pi is chaos written in numbers.

Moreover, if desired, this chaos can be represented graphically, and there is an assumption that this Chaos is intelligent. In 1965, the American mathematician M. Ulam, sitting at one boring meeting, with nothing to do, began to write the numbers included in pi on checkered paper. Putting 3 in the center and moving counterclockwise in a spiral, he wrote out 1, 4, 1, 5, 9, 2, 6, 5 and other numbers after the decimal point. Along the way, he circled all the prime numbers. Imagine his surprise and horror when the circles began to line up along straight lines! Later, he generated a color picture based on this drawing using a special algorithm.

Long numbers that approximate meaning π, have neither practical nor theoretical value. If we wanted, for example, to calculate the length of the earth's equator with an accuracy of 1 cm, assuming that the length of its diameter is exact, then for this it would be enough for us to take only 9 digits after the decimal point in the number π. And taking twice as many numbers (18), we could calculate the length of a circle with a radius of the distance from the Earth to the Sun, with an error of no more than 0.0001 mm (100 times less than the thickness of a hair!).

For ordinary calculations with numbers π It is quite enough to fill in two decimal places (3.14), and for more precise ones - four decimal places (3.1416: we take the last digit 6 instead of 5 because what follows is a digit larger than 5).

Mnemonists love to remember numbers π. And they compete in the number of memorized digits of this infinite number. Record breakers different countries entered into the book of records. So the Japanese Hideaki Tomoyori can reproduce the number PI up to 40,000 characters. It took him about 10 years to memorize this number of numbers. Russian record in terms of memorizing the PI number, it is much more modest. Alexander Belyaev reproduced 2500 digits of the number PI. It took him an hour and a half to remember the numbers. Memorization time - one and a half months. The record for memorizing the number Pi belongs to the Ukrainian Andrey Slyusarchuk, who memorized 30 million decimal places. Since simply listing this would take a whole year, the judges tested Slyusarchuk in the following way - they asked him to name arbitrary sequences of the number Pi from any of the 30 million digits. The answer was checked against a 20-volume printout. Mnemonists remember the number π for one simple reason. If they simply reproduced a series of random numbers, then suspicions might arise that the person did not remember these numbers, but reproduced them according to some kind of system. But when a person reproduces an infinite number π, then any suspicion of dishonesty disappears, since there is no pattern in the sequence of numbers in the number π No. And the only way to reproduce these numbers is to remember them.

Small poems or colorful phrases remain in memory longer than numbers, so to remember any numerical value π they come up with special poems or individual phrases. In works of this type of “mathematical poetry,” words are selected so that the number of letters in each word sequentially coincides with the corresponding digit of the number π. There is a famous poem on English language- in 13 words, therefore giving 12 decimal places in the number π

See I have a rhyme assistingfeeble brain, its tasks off times resisting;

on German- in 24 words, and on French in 30 words. They are curious, but too big and heavy. There are such poems and sentences in Russian.

For example,

“I know this and remember it perfectly.”

“And many signs are unnecessary to me, in vain.”

“What do I know about circles?” - a question that hiddenly contains the answer: 3.1416.

“Teach and know in the number known behind the figure, note the figure as luck” (=3.14159265358).

Archimedean number

"Twenty-two owls were bored

On large dry branches.

Twenty-two owls dreamed

About the Seven Big Mice."

"You just have to try

And remember everything as it is:

Three, fourteen, fifteen,

Ninety two and six.

There is a monument to the number in the world π - it is installed in Seattle in front of the Museum of Art.

There are also Pi clubs, whose members, being fans of the mysterious mathematical phenomenon, collect new information about the number Pi and try to unravel its mystery. In 2005, singer Kate Bush released the album Aerial, which included a song about the number π. In the song, which the singer called “Pi,” 124 numbers from the famous number series were heard. But in her song the 25th number of the sequence was incorrectly named, and as many as 22 numbers disappeared somewhere.

Conclusion

While working on the abstract, we learned a lot of new and interesting things about the number π.

Number π has occupied the minds of scientists from ancient times to the present day. But it is not known who was the first to guess the connection between the length of a circle and its diameter. International standard designation π for the number 3, 141592 became after it was used by the famous Russian academician, mathematician Leonhard Euler in his works in 1737. History of numbers π can be divided into 3 periods: the ancient period, the classical era and the digital computer era. Various methods were used to calculate it. Number π Also called "Ludolfo number". Number π infinite non-periodic fraction. The numbers in its decimal representation are quite random. No other number is as mysterious as Pi, with its famous never-ending number series. In many areas of mathematics and physics, scientists use this number and its laws.

Some scientists even consider it one of the five the most important numbers in mathematics.

At the number π many fans not only among scientists. Exist

Pi - clubs for fans of this number, many sites on the Internet are dedicated to this amazing number.

“Wherever we turn our eyes, we see an agile and industrious number: it is contained in the simplest wheel and in the most complex automatic machine.” Kimpan F.

List of sources used

1.Zhukov A.V. "The omnipresent number π». - M: Editorial URSS, 2004, - 216s

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1. Relevance of the work.

IN infinite number numbers, just like among the stars of the Universe, individual numbers and their entire “constellations” of amazing beauty stand out, numbers with extraordinary properties and a unique harmony inherent only to them. You just need to be able to see these numbers and notice their properties. Take a closer look at the natural series of numbers - and you will find in it a lot of surprising and outlandish, funny and serious, unexpected and curious. The one who looks sees. After all, people won’t even notice on a starry summer night... the glow. The polar star, if they do not direct their gaze to the cloudless heights.

Moving from class to class, I became acquainted with natural, fractional, decimal, negative, rational. This year I studied irrational. Among the irrational numbers there is a special number, the exact calculations of which have been carried out by scientists for many centuries. I came across it back in 6th grade while studying the topic “Circumference and Area of ​​a Circle.” It was emphasized that we would meet with him quite often in classes in high school. Practical tasks on finding the numerical value of π were interesting. The number π is one of most interesting numbers encountered in the study of mathematics. It is found in various school disciplines. There are many interesting facts associated with the number π, so it arouses interest in study.

Having heard a lot of interesting things about this number, I myself decided by studying additional literature and searching on the Internet to find out how to more information about it and answer problematic questions:

How long have people known about the number pi?

Why is it necessary to study it?

What interesting facts are associated with it?

Is it true that the value of pi is approximately 3.14

Therefore, I set myself target: explore the history of the number π and the significance of the number π at the present stage of development of mathematics.

Tasks:

Study the literature to obtain information about the history of the number π;

Establish some facts from the “modern biography” of the number π;

Practical calculation of the approximate value of the ratio of circumference to diameter.

Object of study:

Object of study: PI number.

Subject of study: Interesting facts related to the PI number.

2. Main part. Amazing number pi.

No other number is as mysterious as Pi, with its famous never-ending number series. In many areas of mathematics and physics, scientists use this number and its laws.

Of all the numbers used in mathematics, science, engineering, and Everyday life, is given as much attention as is given to the number pi. One book says, “Pi is captivating the minds of scientific geniuses and amateur mathematicians around the world.” (“Fractals” for the Classroom").

It can be found in probability theory, in solving problems with complex numbers and other unexpected and far from geometry areas of mathematics. The English mathematician Augustus de Morgan once called pi “... the mysterious number 3.14159... that crawls through the door, through the window and through the roof.” This mysterious number, associated with one of the three classical problems of Antiquity - constructing a square whose area is equal to the area of ​​​​a given circle - entails a trail of dramatic historical and curious entertaining facts.

Some even consider it one of the five most important numbers in mathematics. But as the book Fractals for the Classroom notes, as important as pi is, “it is difficult to find areas in scientific calculations that require more than twenty decimal places of pi.”

3. The concept of pi

The number π is a mathematical constant expressing the ratio of the circumference of a circle to the length of its diameter. The number π (pronounced "pi") is a mathematical constant expressing the ratio of the circumference of a circle to the length of its diameter. Denoted by the letter "pi" of the Greek alphabet.

In numerical terms, π begins as 3.141592 and has an infinite mathematical duration.

4. History of the number "pi"

According to experts, this number was discovered by Babylonian magicians. It was used in the construction of the famous Tower of Babel. However, the insufficiently accurate calculation of the value of Pi led to the collapse of the entire project. It is possible that this mathematical constant underlay the construction of the legendary Temple of King Solomon.

The history of the number pi, which expresses the ratio of the circumference of a circle to its diameter, began in Ancient Egypt. Area of ​​a circle with diameter d Egyptian mathematicians defined it 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 the fraction (16/9) 2 , or 256/81 , i.e. π = 3,160...

In the holy book of Jainism (one of ancient religions, which existed in India and arose in the 6th century. BC) there is an indication from which it follows that the number p at that time was taken equal, which gives the fraction 3,162... Ancient Greeks Eudoxus, Hippocrates and others reduced the measurement of a circle to the construction of a segment, and the measurement of a circle to the construction of an equal square. It should be noted that for many centuries, mathematicians from different countries and peoples tried to express the ratio of the circumference to the diameter as a rational number.

Archimedes in the 3rd century BC. in his short work “Measuring a Circle” he substantiated three propositions:

Every circle is equal in size right triangle, the legs of which are respectively equal to the length of the circle and its radius;

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

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

According to exact calculations Archimedes the ratio of circumference to diameter is enclosed between the numbers 3*10/71 And 3*1/7 , which means that π = 3,1419... The true meaning of this relationship 3,1415922653... In the 5th century BC. Chinese mathematician Zu Chongzhi a more accurate value for this number was found: 3,1415927...

In the first half of the 15th century. observatory Ulugbek, near Samarkand, astronomer and mathematician al-Kashi calculated pi to 16 decimal places. Al-Kashi made unique calculations that were needed to compile a table of sines in steps of 1" . These tables played important role in astronomy.

A century and a half later in Europe F. Viet found pi with only 9 correct decimal places by doubling the number of sides of polygons 16 times. But at the same time F. Viet was the first to notice that pi can be found using the limits of certain series. This discovery was of great

value, since it allowed us to calculate pi with any accuracy. Only 250 years after al-Kashi his result was surpassed.

Birthday of the number “”.

The unofficial holiday “PI Day” is celebrated on March 14, which in American format (day/date) is written as 3/14, which corresponds to the approximate value of PI.

There is also Alternative option holiday - July 22. It's called Approximate Pi Day. The fact is that representing this date as a fraction (22/7) also gives the number Pi as a result. It is believed that the holiday was invented in 1987 by San Francisco physicist Larry Shaw, who noticed that the date and time coincided with the first digits of the number π.

Interesting facts related to the number “”

Scientists at the University of Tokyo, led by Professor Yasumasa Kanada, managed to set a world record in calculating the number Pi to 12,411 trillion digits. To do this, a group of programmers and mathematicians needed special program, supercomputer and 400 hours of computer time. (Guinness Book of Records).

The German king Frederick II was so fascinated by this number that he dedicated to it... the entire palace of Castel del Monte, in the proportions of which PI can be calculated. Now the magical palace is under the protection of UNESCO.

How to remember the first digits of the number “”.

The first three digits of the number  = 3.14... are not difficult to remember. And for remembering more signs there are funny sayings and poems. For example, these:

You just have to try

And remember everything as it is:

Ninety two and six.

S. Bobrov. "Magic bicorn"

Anyone who learns this quatrain will always be able to name 8 signs of the number :

In the following phrases, the number signs  can be determined by the number of letters in each word:

“What do I know about circles?” (3.1416);

“So I know the number called Pi. - Well done!"

(3,1415927);

“Learn and know the number behind the number, how to notice good luck.”

(3,14159265359)

5. Notation for pi

The first to introduce the modern symbol pi for the ratio of the circumference of a circle to its diameter was an English mathematician W.Johnson in 1706. He took the first letter as his symbol Greek word "periphery", which translated means "circle". Entered W.Johnson the designation became commonly used after the publication of the works L. Euler, who used the entered character for the first time in 1736 G.

At the end of the 18th century. A.M.Lagendre based on works I.G. Lambert proved that pi is irrational. Then the German mathematician F. Lindeman based on research S.Ermita, found strict proof that this number is not only irrational, but also transcendental, i.e. cannot be the root of an algebraic equation. The search for an exact expression for pi continued after the work F. Vieta. At the beginning of the 17th century. Dutch mathematician from Cologne Ludolf van Zeijlen(1540-1610) (some historians call him L. van Keulen) found 32 correct signs. Since then (year of publication 1615), the value of the number p with 32 decimal places has been called the number Ludolph.

6. How to remember the number "Pi" accurate to eleven digits

The number "Pi" is the ratio of the circumference of a circle to its diameter, it is expressed as infinite decimal. In everyday life, it is enough for us to know three signs (3.14). However, some calculations require greater accuracy.

Our ancestors did not have computers, calculators or reference books, but since the time of Peter I they have been engaged in geometric calculations in astronomy, mechanical engineering, and shipbuilding. Subsequently, electrical engineering was added here - there is the concept of “circular frequency of alternating current”. To remember the number “Pi,” a couplet was invented (unfortunately, we do not know the author or the place of its first publication; but back in the late 40s of the twentieth century, Moscow schoolchildren studied Kiselev’s geometry textbook, where it was given).

The couplet is written according to the rules of old Russian orthography, according to which after consonant must be placed at the end of the word "soft" or "solid" sign. Here it is, this wonderful historical couplet:

Who, jokingly, will soon wish

“Pi” knows the number - he already knows.

It makes sense for anyone who plans to engage in precise calculations in the future to remember this. So what is the number "Pi" accurate to eleven digits? Count the number of letters in each word and write these numbers in a row (separate the first number with a comma).

This accuracy is already quite sufficient for engineering calculations. In addition to the ancient one, there is also modern way memorization, which was pointed out by a reader who identified himself as Georgiy:

So that we don't make mistakes,

You need to read it correctly:

Three, fourteen, fifteen,

Ninety two and six.

You just have to try

And remember everything as it is:

Three, fourteen, fifteen,

Ninety two and six.

Three, fourteen, fifteen,

Nine, two, six, five, three, five.

To do science,

Everyone should know this.

You can just try

And repeat more often:

"Three, fourteen, fifteen,

Nine, twenty-six and five."

Well, mathematicians with the help of modern computers can calculate almost any number of digits of Pi.

7. Pi memory record

Humanity has been trying to remember the signs of pi for a long time. But how to put infinity into memory? A favorite question of professional mnemonists. Many unique theories and techniques for mastering a huge amount of information have been developed. Many of them have been tested on pi.

The world record set in the last century in Germany is 40,000 characters. The Russian record for pi values ​​was set on December 1, 2003 in Chelyabinsk by Alexander Belyaev. In an hour and a half with short breaks, Alexander wrote 2500 digits of pi on the blackboard.

Before this, listing 2,000 characters was considered a record in Russia, which was achieved in 1999 in Yekaterinburg. According to Alexander Belyaev, head of the center for the development of figurative memory, any of us can conduct such an experiment with our memory. It is only important to know special memorization techniques and practice periodically.

Conclusion.

The number pi appears in formulas used in many fields. Physics, electrical engineering, electronics, probability theory, construction and navigation are just a few. And it seems that just as there is no end to the signs of the number pi, there is no end to the possibilities for the practical application of this useful, elusive number pi.

In modern mathematics, the number pi is not only the ratio of the circumference to the diameter; it is included in a large number of different formulas.

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

The exact value of the number π in modern world represents not only its own scientific value, but is also used for very precise calculations (for example, the orbit of a satellite, the construction of giant bridges), as well as assessing the speed and power of modern computers.

Currently, the number π is associated with a difficult-to-see set of formulas, mathematical and physical facts. Their number continues to grow rapidly. All this speaks of a growing interest in the most important mathematical constant, the study of which has spanned more than twenty-two centuries.

The work I did was interesting. I wanted to know about the history of the number pi, practical application and I think I achieved my goal. Summing up the work, I come to the conclusion that this topic is relevant. There are many interesting facts associated with the number π, so it arouses interest in study. In my work, I became more familiar with number - one of the eternal values ​​that humanity has been using for many centuries. Learned some aspects of it rich history. I found out why the ancient world did not know the correct ratio of circumference to diameter. I looked clearly at the ways in which the number can be obtained. Based on experiments, I calculated the approximate value of the number different ways. Processed and analyzed the experimental results.

Any schoolchild today should know what a number means and approximately equals. After all, everyone’s first acquaintance with a number, its use in calculating the circumference of a circle, the area of ​​a circle, occurs in the 6th grade. But, unfortunately, this knowledge remains formal for many and after a year or two, few people remember not only that the ratio of the length of a circle to its diameter is the same for all circles, but they even have difficulty remembering the numerical value of the number, equal to 3 ,14.

I tried to lift the veil of the rich history of the number that humanity has been using for many centuries. I made a presentation for my work myself.

The history of numbers is fascinating and mysterious. I would like to continue researching other amazing numbers in mathematics. This will be the subject of my next research studies.

Bibliography.

1. Glazer G.I. History of mathematics in school, grades IV-VI. - M.: Education, 1982.

2. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook - M.: Prosveshchenie, 1989.

3. Zhukov A.V. The ubiquitous number “pi”. - M.: Editorial URSS, 2004.

4. Kympan F. History of the number “pi”. - M.: Nauka, 1971.

5. Svechnikov A.A. a journey into the history of mathematics - M.: Pedagogika - Press, 1995.

6. Encyclopedia for children. T.11.Mathematics - M.: Avanta +, 1998.

Internet resources:

- http:// crow.academy.ru/materials_/pi/history.htm

Http://hab/kp.ru// daily/24123/344634/