What does the number 3 mean? 14. Who discovered the number Pi? History of computing. Is Pi normal?
Introduction
The article contains mathematical formulas, so to read, go to the site to display them correctly. The number \(\pi\) has a rich history. This constant denotes the ratio of the circumference of a circle to its diameter.
In science, the number \(\pi \) is used in any calculations involving circles. Starting from the volume of a can of soda, to the orbits of satellites. And not just circles. Indeed, in the study of curved lines, the number \(\pi \) helps to understand periodic and oscillatory systems. For example, electromagnetic waves and even music.
In 1706, in the book A New Introduction to Mathematics by the British scientist William Jones (1675-1749), the letter of the Greek alphabet \(\pi\) was first used to represent the number 3.141592.... This designation comes from initial letter Greek words περιϕερεια - circle, periphery and περιµετρoς - perimeter. The designation became generally accepted after the work of Leonhard Euler in 1737.
Geometric period
The constancy of the ratio of the length of any circle to its diameter has been noticed for a long time. The inhabitants of Mesopotamia used a rather rough approximation of the number \(\pi\). As follows from ancient problems, they use the value \(\pi ≈ 3\) in their calculations.
A more precise value for \(\pi\) was used by the ancient Egyptians. In London and New York, two pieces of ancient Egyptian papyrus are kept, which are called the “Rinda papyrus”. The papyrus was compiled by the scribe Armes sometime between 2000-1700. BC. Armes wrote in his papyrus that the area of a circle with radius \(r\) is equal to the area of a square with a side equal to \(\frac(8)(9) \) of the diameter of the circle \(\frac(8 )(9) \cdot 2r \), that is, \(\frac(256)(81) \cdot r^2 = \pi r^2 \). Hence \(\pi = 3.16\).
The ancient Greek mathematician Archimedes (287-212 BC) was the first to put the problem of measuring a circle on a scientific basis. He received a score of \(3\frac(10)(71)< \pi < 3\frac{1}{7}\), рассмотрев отношение периметров вписанного и описанного 96-угольника к диаметру окружности. Архимед выразил приближение числа \(\pi \) в виде дроби \(\frac{22}{7}\), которое до сих называется архимедовым числом.
The method is quite simple, but in the absence of ready-made tables trigonometric functions Root extraction will be required. In addition, the approximation converges to \(\pi \) very slowly: with each iteration the error decreases only fourfold.
Analytical period
Despite this, until the mid-17th century, all attempts by European scientists to calculate the number \(\pi\) boiled down to increasing the sides of the polygon. For example, the Dutch mathematician Ludolf van Zeijlen (1540-1610) calculated the approximate value of the number \(\pi\) accurate to 20 decimal digits.
It took him 10 years to calculate. By doubling the number of sides of inscribed and circumscribed polygons using Archimedes' method, he arrived at \(60 \cdot 2^(29) \) - a triangle in order to calculate \(\pi \) with 20 decimal places.
After his death, 15 more exact digits of the number \(\pi\) were discovered in his manuscripts. Ludolf bequeathed that the signs he found be carved on his tombstone. In his honor, the number \(\pi\) was sometimes called the "Ludolf number" or "Ludolf constant".
One of the first to introduce a method different from that of Archimedes was François Viète (1540-1603). He came to the result that a circle whose diameter is equal to one has an area:
\[\frac(1)(2 \sqrt(\frac(1)(2)) \cdot \sqrt(\frac(1)(2) + \frac(1)(2) \sqrt(\frac(1 )(2)) \cdot \sqrt(\frac(1)(2) + \frac(1)(2) \sqrt(\frac(1)(2) + \frac(1)(2) \sqrt (\frac(1)(2) \cdots )))) \]
On the other hand, the area is \(\frac(\pi)(4)\). By substituting and simplifying the expression, we can obtain the following infinite product formula for calculating the approximate value of \(\frac(\pi)(2)\):
\[\frac(\pi)(2) = \frac(2)(\sqrt(2)) \cdot \frac(2)(\sqrt(2 + \sqrt(2))) \cdot \frac(2 )(\sqrt(2+ \sqrt(2 + \sqrt(2)))) \cdots \]
The resulting formula is the first exact analytical expression for the number \(\pi\). In addition to this formula, Viet, using the method of Archimedes, gave, using inscribed and circumscribed polygons, starting with a 6-gon and ending with a polygon with \(2^(16) \cdot 6 \) sides, an approximation of the number \(\pi \) with 9 with the right signs.
The English mathematician William Brounker (1620-1684), using continued fraction, obtained the following results for calculating \(\frac(\pi)(4)\):
\[\frac(4)(\pi) = 1 + \frac(1^2)(2 + \frac(3^2)(2 + \frac(5^2)(2 + \frac(7^2) )(2 + \frac(9^2)(2 + \frac(11^2)(2 + \cdots )))))) \]
This method calculating the approximation of the number \(\frac(4)(\pi)\) requires quite a lot of calculations to get even a small approximation.
The values obtained as a result of substitution are either greater or less than the number \(\pi\), and each time they are closer to the true value, but to obtain the value 3.141592 it will be necessary to perform quite large calculations.
Another English mathematician John Machin (1686-1751) in 1706, to calculate the number \(\pi\) with 100 decimal places, used the formula derived by Leibniz in 1673 and applied it as follows:
\[\frac(\pi)(4) = 4 arctg\frac(1)(5) - arctg\frac(1)(239) \]
The series converges quickly and with its help you can calculate the number \(\pi \) with great accuracy. These types of formulas have been used to set several records during the computer era.
In the 17th century with the beginning of the period of variable-value mathematics, a new stage in the calculation of \(\pi\) began. The German mathematician Gottfried Wilhelm Leibniz (1646-1716) in 1673 found the expansion of the number \(\pi\), in general view it can be written as the following infinite series:
\[ \pi = 1 — 4(\frac(1)(3) + \frac(1)(5) — \frac(1)(7) + \frac(1)(9) — \frac(1) (11) + \cdots) \]
The series is obtained by substituting x = 1 into \(arctg x = x - \frac(x^3)(3) + \frac(x^5)(5) - \frac(x^7)(7) + \frac (x^9)(9) — \cdots\)
Leonhard Euler develops Leibniz's idea in his works on the use of series for arctan x in calculating the number \(\pi\). In the treatise “De variis modis circuli quadraturam numeris proxime exprimendi” (On various methods expressions for squaring a circle by approximate numbers), written in 1738, discusses methods for improving calculations using Leibniz's formula.
Euler writes that the series for the arctangent will converge faster if the argument tends to zero. For \(x = 1\), the convergence of the series is very slow: to calculate with an accuracy of 100 digits it is necessary to add \(10^(50)\) terms of the series. You can speed up calculations by decreasing the value of the argument. If we take \(x = \frac(\sqrt(3))(3)\), then we get the series
\[ \frac(\pi)(6) = artctg\frac(\sqrt(3))(3) = \frac(\sqrt(3))(3)(1 — \frac(1)(3 \cdot 3) + \frac(1)(5 \cdot 3^2) — \frac(1)(7 \cdot 3^3) + \cdots) \]
According to Euler, if we take 210 terms of this series, we will get 100 correct digits of the number. The resulting series is inconvenient because it is necessary to know a fairly accurate value of the irrational number \(\sqrt(3)\). Euler also used in his calculations expansions of arctangents into the sum of arctangents of smaller arguments:
\[where x = n + \frac(n^2-1)(m-n), y = m + p, z = m + \frac(m^2+1)(p) \]
Not all the formulas for calculating \(\pi\) that Euler used in his notebooks were published. In published papers and notebooks, he considered 3 different series for calculating the arctangent, and also made many statements regarding the number of summable terms required to obtain an approximate value of \(\pi\) with a given accuracy.
In subsequent years, refinements to the value of the number \(\pi\) occurred faster and faster. For example, in 1794, Georg Vega (1754-1802) already identified 140 signs, of which only 136 turned out to be correct.
Computing period
The 20th century was marked by a completely new stage in the calculation of the number \(\pi\). Indian mathematician Srinivasa Ramanujan (1887-1920) discovered many new formulas for \(\pi\). In 1910, he obtained a formula for calculating \(\pi\) through the arctangent expansion in a Taylor series:
\[\pi = \frac(9801)(2\sqrt(2) \sum\limits_(k=1)^(\infty) \frac((1103+26390k) \cdot (4k){(4\cdot99)^{4k} (k!)^2}} .\]!}
At k=100, an accuracy of 600 correct digits of the number \(\pi\) is achieved.
The advent of computers made it possible to significantly increase the accuracy of the obtained values over more short time. In 1949, in just 70 hours, using ENIAC, a group of scientists led by John von Neumann (1903-1957) obtained 2037 decimal places for the number \(\pi\). In 1987, David and Gregory Chudnovsky obtained a formula with which they were able to set several records in calculating \(\pi\):
\[\frac(1)(\pi) = \frac(1)(426880\sqrt(10005)) \sum\limits_(k=1)^(\infty) \frac((6k)!(13591409+545140134k ))((3k)!(k!)^3(-640320)^(3k)).\]
Each member of the series gives 14 digits. In 1989, 1,011,196,691 decimal places were obtained. This formula is well suited for calculating \(\pi \) on personal computers. On this moment the brothers are professors at the Polytechnic Institute of New York University.
An important recent development was the discovery of the formula in 1997 by Simon Plouffe. It allows you to extract any hexadecimal digit of the number \(\pi\) without calculating the previous ones. The formula is called the “Bailey-Borwain-Plouffe Formula” in honor of the authors of the article where the formula was first published. It looks like this:
\[\pi = \sum\limits_(k=1)^(\infty) \frac(1)(16^k) (\frac(4)(8k+1) — \frac(2)(8k+4 ) - \frac(1)(8k+5) - \frac(1)(8k+6)) .\]
In 2006, Simon, using PSLQ, came up with some nice formulas for calculating \(\pi\). For example,
\[ \frac(\pi)(24) = \sum\limits_(n=1)^(\infty) \frac(1)(n) (\frac(3)(q^n - 1) - \frac (4)(q^(2n) -1) + \frac(1)(q^(4n) -1)), \]
\[ \frac(\pi^3)(180) = \sum\limits_(n=1)^(\infty) \frac(1)(n^3) (\frac(4)(q^(2n) — 1) — \frac(5)(q^(2n) -1) + \frac(1)(q^(4n) -1)), \]
where \(q = e^(\pi)\). In 2009, Japanese scientists, using the T2K Tsukuba System supercomputer, obtained the number \(\pi\) with 2,576,980,377,524 decimal places. The calculations took 73 hours 36 minutes. The computer was equipped with 640 quad-core AMD Opteron processors, which provided performance of 95 trillion operations per second.
The next achievement in calculating \(\pi\) belongs to the French programmer Fabrice Bellard, who at the end of 2009, on his personal computer running Fedora 10, set a record by calculating 2,699,999,990,000 decimal places of the number \(\pi\). Over the past 14 years, this is the first world record that was set without the use of a supercomputer. For high performance, Fabrice used the Chudnovsky brothers' formula. In total, the calculation took 131 days (103 days of calculations and 13 days of verification of the result). Bellar's achievement showed that such calculations do not require a supercomputer.
Just six months later, Francois's record was broken by engineers Alexander Yi and Singer Kondo. To set a record of 5 trillion decimal places of \(\pi\), a personal computer was also used, but with more impressive characteristics: two Intel Xeon X5680 processors at 3.33 GHz, 96 GB random access memory, 38 TB of disk memory and the Windows Server 2008 R2 Enterprise x64 operating system. For calculations, Alexander and Singer used the formula of the Chudnovsky brothers. The calculation process took 90 days and 22 TB of disk space. In 2011, they set another record by calculating 10 trillion decimal places for the number \(\pi\). The calculations took place on the same computer on which their previous record was set and took a total of 371 days. At the end of 2013, Alexander and Singerou improved the record to 12.1 trillion digits of the number \(\pi\), which took them only 94 days to calculate. This performance improvement is achieved through performance optimization software, increasing the number of processor cores and significantly improving software fault tolerance.
The current record is that of Alexander Yee and Singer Kondo, which is 12.1 trillion decimal places \(\pi\).
Thus, we looked at methods for calculating the number \(\pi\) used in ancient times, analytical methods, and also looked at modern methods and records for calculating the number \(\pi \) on computers.
List of sources
- Zhukov A.V. The ubiquitous number Pi - M.: Publishing house LKI, 2007 - 216 p.
- F.Rudio. On the squaring of the circle, with the application of a history of the issue compiled by F. Rudio. / Rudio F. – M.: ONTI NKTP USSR, 1936. – 235c.
- Arndt, J. Pi Unleashed / J. Arndt, C. Haenel. – Springer, 2001. – 270p.
- Shukhman, E.V. Approximate calculation of Pi using the series for arctan x in published and unpublished works of Leonhard Euler / E.V. Shukhman. — History of science and technology, 2008 – No. 4. – P. 2-17.
- Euler, L. De variis modis circuli quadraturam numeris proxime exprimendi/ Commentarii academiae scientiarum Petropolitanae. 1744 – Vol.9 – 222-236p.
- Shumikhin, S. Number Pi. A history of 4000 years / S. Shumikhin, A. Shumikhina. - M.: Eksmo, 2011. - 192 p.
- Borwein, J.M. Ramanujan and the number Pi. / Borwein, J.M., Borwein P.B. In the world of science. 1988 – No. 4. – pp. 58-66.
- Alex Yee. Number world. Access mode: numberworld.org
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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. Were interesting practical tasks to find the numerical value of the number π. 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 search 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 " modern biography» numbers 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, associated with the number PI.
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 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 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 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 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 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 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
He was the first to introduce the notation for the ratio of circumference to diameter modern symbol pi 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.
IN late XVIII V. 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 the 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 big number various 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. Summarizing the work, I come to the conclusion that this topic 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. Found out why 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/
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. number Pi is still unknown, but the most related to it best mathematicians throughout our history. 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. A monument was even erected to him in Seattle, America, 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 claimed in their works that it is 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 they measured a circle by constructing a segment, but in order to measure a circle, they had to build an equal square, that is, a figure equal to it in area.
When they didn't know yet decimals, 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 in a circle and describing around it regular polygons. 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 an American student Alexander Yi, 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.
Number meaning(pronounced "pi") is a mathematical constant equal to the ratio
Denoted by the letter "pi" of the Greek alphabet. Old name - Ludolph number.
What is pi equal to? In simple cases, it is enough to know the first 3 signs (3.14). But for more
complex cases and where greater accuracy is needed, you need to know more than 3 digits.
What is pi? First 1000 decimal places of pi:
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...
Under normal conditions, the approximate value of pi can be calculated following the steps,
given below:
- Let's take it circle, wrap the thread around its edge once.
- We measure the length of the thread.
- We measure the diameter of the circle.
- Divide the length of the thread by the length of the diameter. We got the number pi.
Properties of Pi.
- pi —irrational number, i.e. the value of pi cannot be accurately expressed in the form
fractions m/n, Where m And n are integers. From this it is clear that the decimal representation
pi never ends and it is not periodic.
- pi- transcendental number, i.e. it cannot be the root of any polynomial with whole
coefficients. In 1882, Professor Koenigsbergsky proved the transcendence pi numbers, A
later, professor at the University of Munich Lindemann. The proof has been simplified
Felix Klein in 1894.
- since in Euclidean geometry area of a circle And circumference- these are functions of pi,
that proof of the transcendence of pi put an end to the dispute about the squaring of the circle, which lasted more than
2.5 thousand years.
- pi is an element of the period ring (that is, a computable and arithmetic number).
But no one knows whether it belongs to the ring of periods.
Pi number formula.
- Francois Viet:
- Wallis formula:
- Leibniz series:
- Other rows: