In mathematics there are an infinite number of different numbers. Most of them do not attract attention at all. However, some, at first glance, absolutely uninteresting numbers are so well known that they even have their own names. One of these constants is the irrational number Pi, studied in school and used to calculate the area or perimeter of a circle along a given radius.
From the history of the constant
Interesting facts about the number Pi - history of study. The existence of the constant counts for about 4 thousand years. In other words, it is a little younger than mathematical science itself.
The first evidence that the number Pi was known in ancient Egypt comes from the Ahmes Papyrus, one of the oldest problem books found. The document dates from approximately 1650 BC. e. In papyrus, the constant was taken to be 3.1605. This is a fairly accurate value considering that other peoples used 3 to calculate the circumference of a circle based on its diameter.
The number Pi was calculated a little more accurately by Archimedes, an ancient Greek mathematician. He managed to approximate the value in the form of ordinary fractions 22/7 and 223/71. There is a well-known legend that he was so busy calculating the constant that he did not pay attention to how the Romans captured his city. At that moment, when the warrior approached the scientist, Archimedes shouted to him not to touch his drawings. These words of the mathematician became the last.
The founder of algebra, Al-Khorezmi, who lived in the 8th-9th centuries, worked on calculations of the constant. With a small error, he obtained the number Pi equal to 3.1416.
Eight centuries later, mathematician Ludolf van Zeilen correctly identified 36 decimal places. For this achievement, the number Pi is sometimes called the Ludolf constant (other known names are the Archimedean constant or the circular constant), and the numbers obtained by the scientist were engraved on his tombstone.
Around the same time, the constant began to be used not only for a circle, but also for calculating complex curves - arches and hypocycloids.
It was only at the beginning of the 18th century that the constant began to be called the number Pi. The designation in the form of the letter π was not chosen by chance - it is with it that 2 Greek words begin, meaning circle and perimeter. The name was proposed by the scientist Jones in 1706, and 30 years later the image of this Greek letter was firmly used among other mathematical notations.
In the 19th century, William Shanks worked on calculating the first 707 symbols of the constant. He failed to fully achieve his goal - an error crept into the calculations, and the 527 figure turned out to be incorrect. However, even the result obtained was a good achievement for the science of that time.
At the end of the 19th century, the incorrect value of 3.2 was almost adopted at the state level in Indiana. Fortunately, mathematicians managed to speak out against the bill and prevent the mistake.
In the XX-XXI centuries. With the use of computer technology, the accuracy and speed of calculating the constant has increased thousands of times. By 2002, over 1 trillion digits of the constant had been determined using computers in Japan. After 9 years, the accuracy of the calculation was already 10 trillion decimal places.
In art and marketing
Even though Pi is a mathematical constant, over the years people have tried to use the irrational and mysterious meaning in other areas of life, including works of art.
The very first signs of permanent were found in a monument of architecture in Giza. When determining the dimensions of the Great Pyramid, it turned out that the ratio of the perimeter of its base to its height is equal to π. It is only unknown whether the architect wanted to use his knowledge about this number, or whether this ratio happened by chance.
Currently, the number Pi is also not deprived of attention in creativity. For example, if you designate each note of the minor scale with a number from 0 to 9, and then play the resulting sequence in the form of the number Pi on a musical instrument, you can enjoy an unusual melody with an interesting sound.
The constant has also not spared cinema. The drama film, Pi: Faith in Chaos, won the Best Director award at the Sundance Film Festival. According to the plot, the main character is in search of simple and understandable answers to questions about the constant, which as a result almost drove him to madness. References to the number are also found in other films and TV series.
The number has found its application even in such an unexpected area as marketing. Thus, the Givenchi company released a cologne called “Pi”.
Constant and society
Some features of the number:
- A constant is an irrational quantity. This means that it cannot be represented as a ratio of two numbers. In addition, there is no pattern in his recording.
- Repeating characters in a row in a constant are not uncommon. So, for every 20-30 characters there are usually at least 2 consecutive numbers. Sequences of 3 characters are already more rare; they occur with a frequency of about 1 repetition per 150-300 characters. And at the 763rd sign, a chain of 6 consecutive nines begins. This place in the record even has its own name - the Feynman point.
- If we consider the first million characters, then according to statistics, the rarest numbers in it will be 6 and 1, and the most common – 5 and 4.
- The number 0 appears later in the sequence than the others, only at 31 characters.
- In trigonometry, a 360 degree angle and a constant are closely related. Oddly enough, the number 360 is located at 358, 359 and 360 positions after the decimal point.
For the purpose of exchanging information about discoveries, the Pi Club was established. Those wishing to join it have to pass a difficult exam: the future member of the mathematical community must correctly name as many symbols of the constant as possible from memory.
Of course, memorizing a long numerical sequence that has no patterns or repetitions is quite a difficult task. To make the task easier, various texts and poems are invented in which the number of letters in a word corresponds to a certain number of the constant. This method of memorization is popular among members of the Pi Club. One of the longest stories contained 3834 first digits.
Monument at the Seattle Art Museum
However, the recognized champions in memorization are, of course, the residents of China and Japan. Thus, the Japanese Akira Haraguchi was able to learn over 83 thousand digits after the decimal point. And the Chinese Liu Chao became famous as the man who was able to name 67,890 symbols of the number Pi in a record time of 24 hours. The average speed was 47 characters per minute. Initially, his goal was to name 93 thousand numbers, but he made a mistake, after which he did not continue.
To emphasize the importance of the constant, a monument in the form of a huge Greek letter π was erected in front of the Seattle Museum of Art.
In addition, since 1988, every March 14th is Pi Day. The date coincides with the first digits of the constant – 3.14. They celebrate it after 1:59. On this day, interested people are treated to cakes and cookies with the Pi symbol, after which various mathematical competitions and quizzes are held. By the way, it was on this day that A. Einstein, astronomer Schiaparelli and cosmonaut Cernan were born.
The number Pi is an amazing constant that has found its application in a variety of fields, from technology and construction to the fields of art. Like any other quantity that is used frequently and which cannot be completely calculated, it will always attract the attention of mathematicians, physicists and other scientists.
Recently, there is an elegant formula for calculating Pi, first published in 1995 by David Bailey, Peter Borwein and Simon Plouffe:
It would seem: what’s special about it - there are a great many formulas for calculating Pi: from the school Monte Carlo method to the incomprehensible Poisson integral and the Francois Vieta formula from the late Middle Ages. But it is this formula that is worth paying special attention to - it allows you to calculate the nth digit of pi without finding the previous ones. For information on how this works, as well as ready-made code in C that calculates the 1,000,000th digit, please subscribe.
How does the algorithm for calculating the Nth digit of Pi work?
For example, if we need the 1000th hexadecimal digit of Pi, we multiply the entire formula by 16^1000, thereby turning the factor in front of the parentheses into 16^(1000-k). When exponentiating, we use the binary exponentiation algorithm or, as the example below will show, modulo exponentiation. After this, we calculate the sum of several terms of the series. Moreover, it is not necessary to calculate a lot: as k increases, 16^(N-k) quickly decreases, so that subsequent terms will not affect the value of the required numbers). That's all magic - brilliant and simple.
The Bailey-Borwine-Plouffe formula was found by Simon Plouffe using the PSLQ algorithm, which was included in the list of Top 10 Algorithms of the Century in 2000. The PSLQ algorithm itself was in turn developed by Bailey. Here's a Mexican series about mathematicians.
By the way, the running time of the algorithm is O(N), memory usage is O(log N), where N is the serial number of the desired sign.
I think it would be appropriate to quote the code in C written directly by the author of the algorithm, David Bailey:
/* This program implements the BBP algorithm to generate a few hexadecimal digits beginning immediately after a given position id, or in other words beginning at position id + 1. On most systems using IEEE 64-bit floating- point arithmetic, this code works correctly so long as d is less than approximately 1.18 x 10^7. If 80-bit arithmetic can be employed, this limit is significantly higher. Whatever arithmetic is used, results for a given position id can be checked by repeating with id-1 or id+1, and verifying that the hex digits perfectly overlap with an offset of one, except possibly for a few trailing digits. The resulting fractions are typically accurate to at least 11 decimal digits, and to at least 9 hex digits. */ /* David H. Bailey 2006-09-08 */ #include
What opportunities does this provide? For example: we can create a distributed computing system that calculates the number Pi and set a new record for the accuracy of calculations for all of Habr (which, by the way, is now 10 trillion decimal places). According to empirical data, the fractional part of the number Pi is a normal number sequence (although this has not yet been proven reliably), which means that sequences of numbers from it can be used in generating passwords and simply random numbers, or in cryptographic algorithms (for example, hashing) . You can find a great variety of ways to use it - you just need to use your imagination.
You can find more information on the topic in the article by David Bailey himself, where he talks in detail about the algorithm and its implementation (pdf);
And it looks like you just read the first Russian-language article about this algorithm on the RuNet - I couldn’t find any others.
March 14, 2012
On March 14, mathematicians celebrate one of the most unusual holidays - International Pi Day. This date was not chosen by chance: the numerical expression π (Pi) is 3.14 (3rd month (March) 14th).
For the first time, schoolchildren encounter this unusual number in the elementary grades when studying circles and circumferences. The number π is a mathematical constant that expresses the ratio of the circumference of a circle to the length of its diameter. That is, if you take a circle with a diameter equal to one, then the circumference will be equal to the number “Pi”. The number π has an infinite mathematical duration, but in everyday calculations a simplified spelling of the number is used, leaving only two decimal places - 3.14.
In 1987, this day was celebrated for the first time. Physicist Larry Shaw from San Francisco noticed that in the American date system (month/day), the date March 14 - 3/14 coincides with the number π (π = 3.1415926...). Typically celebrations begin at 1:59:26 pm (π = 3.14 15926 …).
History of Pi
It is assumed that the history of the number π begins in Ancient Egypt. Egyptian mathematicians determined the area of a circle with diameter D as (D-D/9) 2. From this entry it is clear that at that time the number π was equated to the fraction (16/9) 2, or 256/81, i.e. π 3.160...
In the VI century. BC in India, in the religious book of Jainism, there are entries indicating that the number π at that time was taken equal to the square root of 10, which gives the fraction 3.162...
In the 3rd century. BC Archimedes in his short work “Measurement of a Circle” substantiated three propositions:
- Every circle is equal in size to a 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 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 greater than 3 10/71.
Archimedes justified the last position by sequentially calculating the perimeters of regular inscribed and circumscribed polygons by doubling the number of their sides. According to the exact calculations of Archimedes, the ratio of the circumference to the diameter is between the numbers 3 * 10 / 71 and 3 * 1/7, which means that the number “pi” is 3.1419... The true value of this ratio is 3.1415922653...
In the 5th century BC Chinese mathematician Zu Chongzhi found a more accurate value for this number: 3.1415927...
In the first half of the 15th century. The astronomer and mathematician Kashi calculated π with 16 decimal places.
A century and a half later in Europe, F. Viet found the number π with only 9 regular decimal places: he made 16 doublings of the number of sides of polygons. F. Viet was the first to notice that π can be found using the limits of certain series. This discovery was of great importance; it made it possible to calculate π with any accuracy.
In 1706, the English mathematician W. Johnson introduced the notation for the ratio of the circumference of a circle to its diameter and designated it with the modern symbol π, the first letter of the Greek word periferia - circle.
For a long period of time, scientists around the world tried to unravel the mystery of this mysterious number.
What is the difficulty in calculating the value of π?
The number π is irrational: it cannot be expressed as a fraction p/q, where p and q are integers; this number cannot be the root of an algebraic equation. It is impossible to specify an algebraic or differential equation whose root will be π, therefore this number is called transcendental and is calculated by considering a process and is refined by increasing the steps of the process under consideration. Multiple attempts to calculate the maximum number of digits of the number π have led to the fact that today, thanks to modern computing technology, it is possible to calculate the sequence with an accuracy of 10 trillion digits after the decimal point.
The digits of the decimal representation of π are quite random. In the decimal expansion of a number, you can find any sequence of digits. It is assumed that this number contains all written and unwritten books in encrypted form; any information that can be imagined is found in the number π.
You can try to unravel the mystery of this number yourself. Of course, it will not be possible to write down the number “Pi” in full. But for the most curious, I suggest considering the first 1000 digits of the number π = 3,
1415926535 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
Remember the number "Pi"
Currently, with the help of computer technology, ten trillion digits of the number “Pi” have been calculated. The maximum number of numbers that a person could remember is one hundred thousand.
To remember the maximum number of digits of the number “Pi”, various poetic “memories” are used, in which words with a certain number of letters are arranged in the same sequence as the numbers in the number “Pi”: 3.1415926535897932384626433832795…. To restore the number, you need to count the number of characters in each word and write it down in order.
So I know the number called “Pi”. Well done! (7 digits)
So Misha and Anyuta came running
They wanted to know the number Pi. (11 digits)
This I know and remember perfectly:
And many signs are unnecessary for me, in vain.
Let's trust our enormous knowledge
Those who counted the numbers of the armada. (21 digits)
Once at Kolya and Arina's
We ripped the feather beds.
The white fluff was flying and spinning,
Showered, froze,
Satisfied
He gave it to us
Old women's headache.
Wow, the spirit of fluff is dangerous! (25 characters)
You can use rhyming lines to help you remember the right number.
So that we don't make mistakes,
You need to read it correctly:
Ninety two and six
If you try really hard,
You can immediately read:
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."
Still have questions? Want to know more about Pi?
To get help from a tutor, register.
The first lesson is free!
For many centuries and even, oddly enough, millennia, people have understood the importance and value for science of a mathematical constant equal to the ratio of the circumference of a circle to its diameter. the number Pi is still unknown, but the best mathematicians throughout our history have been involved with it. Most of them wanted to express it as a rational number.
1. Researchers and true fans of the number Pi have organized a club, to join which you need to know by heart a fairly large number of its signs.
2. Since 1988, “Pi Day” has been celebrated, which falls on March 14th. They prepare salads, cakes, cookies, and pastries with his image.
3. The number Pi has already been set to music, and it sounds quite good. They even erected a monument to him in American Seattle in front of the city Museum of Art.
At that distant time, they tried to calculate the number Pi using geometry. The fact that this number is constant for a wide variety of circles was known by geometers in Ancient Egypt, Babylon, India and Ancient Greece, who stated in their works that it was only a little more than three.
In one of the sacred books of Jainism (an ancient Indian religion that arose in the 6th century BC) it is mentioned that then the number Pi was considered equal to the square root of ten, which ultimately gives 3.162... .
Ancient Greek mathematicians measured a circle by constructing a segment, but in order to measure a circle, they had to construct an equal square, that is, a figure equal in area to it.
When decimal fractions were not yet known, the great Archimedes found the value of Pi with an accuracy of 99.9%. He discovered a method that became the basis for many subsequent calculations, inscribing regular polygons in a circle and describing it around it. As a result, Archimedes calculated the value of Pi as the ratio 22 / 7 ≈ 3.142857142857143.
In China, mathematician and court astronomer, Zu Chongzhi in the 5th century BC. e. designated a more precise value for Pi, calculating it to seven decimal places and determined its value between the numbers 3, 1415926 and 3.1415927. It took scientists more than 900 years to continue this digital series.
Middle Ages
The famous Indian scientist Madhava, who lived at the turn of the 14th - 15th centuries and became the founder of the Kerala school of astronomy and mathematics, for the first time in history began to work on the expansion of trigonometric functions into series. True, only two of his works have survived, and only references and quotes from his students are known for others. The scientific treatise "Mahajyanayana", which is attributed to Madhava, states that the number Pi is 3.14159265359. And in the treatise “Sadratnamala” a number is given with even more exact decimal places: 3.14159265358979324. In the given numbers, the last digits do not correspond to the correct value.
In the 15th century, the Samarkand mathematician and astronomer Al-Kashi calculated the number Pi with sixteen decimal places. His result was considered the most accurate for the next 250 years.
W. Johnson, a mathematician from England, was one of the first to denote the ratio of the circumference of a circle to its diameter by the letter π. Pi is the first letter of the Greek word "περιφέρεια" - circle. But this designation managed to become generally accepted only after it was used in 1736 by the more famous scientist L. Euler.
Conclusion
Modern scientists continue to work on further calculations of the values of Pi. Supercomputers are already used for this. In 2011, a scientist from Shigeru Kondo, collaborating with American student Alexander Yee, correctly calculated a sequence of 10 trillion digits. But it is still unclear who discovered the number Pi, who first thought about this problem and made the first calculations of this truly mystical number.
The text of the work is posted without images and formulas.
The full version of the work is available in the "Work Files" tab in PDF format
1. Relevance of the work.
In the infinite variety of 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 the 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 the Internet to find out as much information as possible 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 research: 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, few numbers receive as much attention as pi. One book says, “Pi is captivating the minds of science 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 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 sacred book of Jainism (one of the oldest religions that 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 substantiated three propositions in his short work “Measuring a Circle”:
Every circle is equal in size to a 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 an 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 an alternative version of the 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 a special program, a 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 to remember 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. As a symbol he took the first letter of the 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 an infinite decimal fraction. 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 a modern method of 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 the modern world is not only of 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 dates back more than twenty-two centuries.
The work I did was interesting. I wanted to learn about the history of pi, practical applications, 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. I learned some aspects of its 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 in various 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.
References.
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/
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