Pi is one of the most popular mathematical concepts. Pictures are written about him, films are made, he is played on musical instruments, poems and holidays are dedicated to him, they look for him and find him in sacred texts.
Who discovered pi?
Who and when first discovered the number π still remains a mystery. It is known that the builders of ancient Babylon already made full use of it in their design. Cuneiform tablets that are thousands of years old even preserve problems that were proposed to be solved using π. True, then it was believed that π was equal to three. This is evidenced by a tablet found in the city of Susa, two hundred kilometers from Babylon, where the number π was indicated as 3 1/8.
In the process of calculating π, the Babylonians discovered that the radius of a circle as a chord enters it six times, and divided the circle into 360 degrees. And at the same time they did the same with the orbit of the sun. Thus, they decided to consider that there are 360 days in a year.
IN Ancient Egyptπ was equal to 3.16.
In ancient India - 3,088.
In Italy at the turn of the era, it was believed that π was equal to 3.125.
In Antiquity, the earliest mention of π refers to the famous problem of squaring the circle, that is, the impossibility of using a compass and ruler to construct a square whose area is equal to the area of a certain circle. Archimedes equated π to the fraction 22/7.
The closest people to the exact value of π came in China. It was calculated in the 5th century AD. e. famous Chinese astronomer Tzu Chun Zhi. π was calculated quite simply. It was necessary to write odd numbers twice: 11 33 55, and then, dividing them in half, place the first in the denominator of the fraction, and the second in the numerator: 355/113. The result agrees with modern calculations of π up to the seventh digit.
Why π – π?
Now even schoolchildren know that the number π is a mathematical constant equal to the ratio of the circumference of a circle to the length of its diameter and is equal to π 3.1415926535 ... and then after the decimal point - to infinity.
The number acquired its designation π the hard way: first this one Greek letter In 1647, the mathematician Outrade named the circumference. He took the first letter Greek wordπεριφέρεια - “periphery”. In 1706 English teacher William Jones in his work “Review of the Advances of Mathematics” already called the letter π the ratio of the circumference of a circle to its diameter. And the name was cemented by the 18th century mathematician Leonard Euler, before whose authority the rest bowed their heads. So π became π.
Uniqueness of the number
Pi is a truly unique number.
1. Scientists believe that the number of digits in the number π is infinite. Their sequence is not repeated. Moreover, no one will ever be able to find repetitions. Since the number is infinite, it can contain absolutely everything, even a Rachmaninoff symphony, the Old Testament, your phone number and the year in which the Apocalypse will occur.
2. π is associated with chaos theory. Scientists came to this conclusion after creating Bailey's computer program, which showed that the sequence of numbers in π is absolutely random, which is in accordance with the theory.
3. It is almost impossible to calculate the number completely - it would take too much time.
4. π is an irrational number, that is, its value cannot be expressed as a fraction.
5. π – transcendental number. It cannot be obtained by performing any algebraic operations on integers.
6. Thirty-nine decimal places in the number π are enough to calculate the length of the circle encircling known cosmic objects in the Universe, with an error of the radius of a hydrogen atom.
7. The number π is associated with the concept of the “golden ratio”. During the measurement process Great Pyramid At Giza, archaeologists discovered that its height is related to the length of its base, just as the radius of a circle is related to its length.
Records related to π
In 2010, Yahoo mathematician Nicholas Zhe was able to calculate two quadrillion decimal places (2x10) in the number π. This took 23 days, and the mathematician needed many assistants who worked on thousands of computers, united using distributed computing technology. The method made it possible to perform calculations at such a phenomenal speed. To calculate the same thing on one computer would take more than 500 years.
In order to simply write all this down on paper, you would need a paper tape more than two billion kilometers long. If you expand such a record, its end will go beyond the solar system.
Chinese Liu Chao set a record for memorizing the sequence of digits of the number π. Within 24 hours and 4 minutes, Liu Chao said 67,890 decimal places without making a single mistake.
π has many fans. It is played on musical instruments, and it turns out that it “sounds” excellent. It is remembered and invented for this purpose various techniques. For fun, they download it to their computer and brag to each other about who has downloaded the most. Monuments are erected to him. For example, there is such a monument in Seattle. It is located on the steps in front of the Museum of Art.
π is used in decorations and interior design. Poems are dedicated to him, he is looked for in holy books and at excavations. There is even a “Club π”.
In the best traditions of π, not one, but two whole days a year are dedicated to the number! The first time π Day is celebrated is March 14th. You need to congratulate each other at exactly 1 hour, 59 minutes, 26 seconds. Thus, the date and time correspond to the first digits of the number - 3.1415926.
For the second time, the π holiday is celebrated on July 22. This day is associated with the so-called “approximate π”, which Archimedes wrote down as a fraction.
Usually on this day, students, schoolchildren and scientists organize funny flash mobs and actions. Mathematicians, having fun, use π to calculate the laws of a falling sandwich and give each other comic rewards.
And by the way, π can actually be found in the holy books. For example, in the Bible. And there the number π is equal to... three.
What is Pi equal to? we know and remember from school. It is equal to 3.1415926 and so on... To an ordinary person it is enough to know that this number is obtained by dividing the circumference of a circle by its diameter. But many people know that the number Pi appears in unexpected areas not only of mathematics and geometry, but also in physics. Well, if you delve into the details of the nature of this number, you will notice many surprising things among the endless series of numbers. Is it possible that Pi is hiding the deepest secrets of the universe?
Infinite number
The number Pi itself appears in our world as the circumference of a circle whose diameter equal to one. But, despite the fact that the segment equal to Pi is quite finite, the number Pi begins as 3.1415926 and goes to infinity in rows of numbers that are never repeated. First amazing fact is that this number, used in geometry, cannot be expressed as a fraction of whole numbers. In other words, you cannot write it as the ratio of two numbers a/b. In addition, the number Pi is transcendental. This means that there is no equation (polynomial) with integer coefficients whose solution would be the number Pi.
The fact that the number Pi is transcendental was proved in 1882 by the German mathematician von Lindemann. It was this proof that became the answer to the question of whether it is possible, using a compass and a ruler, to draw a square whose area is equal to the area of a given circle. This problem is known as the search for squaring a circle, which has worried humanity since ancient times. It seemed that this problem had a simple solution and was about to be solved. But it was precisely the incomprehensible property of the number Pi that showed that there was no solution to the problem of squaring the circle.
For at least four and a half millennia, humanity has been trying to obtain an increasingly accurate value for Pi. For example, in the Bible in the Third Book of Kings (7:23), the number Pi is taken to be 3.
The Pi value of remarkable accuracy can be found in the Giza pyramids: the ratio of the perimeter and height of the pyramids is 22/7. This fraction gives an approximate value of Pi equal to 3.142... Unless, of course, the Egyptians set this ratio by accident. The same value was already obtained in relation to the calculation of the number Pi in the 3rd century BC by the great Archimedes.
In the Papyrus of Ahmes, an ancient Egyptian mathematics textbook that dates back to 1650 BC, Pi is calculated as 3.160493827.
In ancient Indian texts around the 9th century BC, the most accurate value was expressed by the number 339/108, which was equal to 3.1388...
For almost two thousand years after Archimedes, people tried to find ways to calculate Pi. Among them were both famous and unknown mathematicians. For example, the Roman architect Marcus Vitruvius Pollio, the Egyptian astronomer Claudius Ptolemy, the Chinese mathematician Liu Hui, the Indian sage Aryabhata, the medieval mathematician Leonardo of Pisa, known as Fibonacci, the Arab scientist Al-Khwarizmi, from whose name the word “algorithm” appeared. All of them and many other people were looking for the most accurate methods for calculating Pi, but until the 15th century they never got more than 10 decimal places due to the complexity of the calculations.
Finally, in 1400, the Indian mathematician Madhava from Sangamagram calculated Pi with an accuracy of 13 digits (although he was still mistaken in the last two).
Number of characters
In the 17th century, Leibniz and Newton discovered the analysis of infinitesimal quantities, which made it possible to calculate Pi more progressively - through power series and integrals. Newton himself calculated 16 decimal places, but did not mention it in his books - this became known after his death. Newton claimed that he calculated Pi purely out of boredom.
Around the same time, other lesser-known mathematicians also came forward and proposed new formulas for calculating Pi using trigonometric functions.
For example, this is the formula used to calculate Pi by astronomy teacher John Machin in 1706: PI / 4 = 4arctg(1/5) – arctg(1/239). Using analytical methods, Machin derived the number Pi to one hundred decimal places from this formula.
By the way, in the same 1706, the number Pi received an official designation in the form of a Greek letter: William Jones used it in his work on mathematics, taking the first letter of the Greek word “periphery,” which means “circle.” The great Leonhard Euler, born in 1707, popularized this designation, now known to any schoolchild.
Before the era of computers, mathematicians worked to calculate as many signs as possible. In this regard, sometimes funny things arose. Amateur mathematician W. Shanks calculated 707 digits of Pi in 1875. These seven hundred signs were immortalized on the wall of the Palais des Discoverys in Paris in 1937. However, nine years later, observant mathematicians discovered that only the first 527 characters were correctly calculated. The museum had to incur significant expenses to correct the error - now all the figures are correct.
When computers appeared, the number of digits of Pi began to be calculated in completely unimaginable orders.
One of the first electronic computers, ENIAC, created in 1946, was enormous in size and generated so much heat that the room warmed up to 50 degrees Celsius, calculated the first 2037 digits of Pi. This calculation took the machine 70 hours.
As computers improved, our knowledge of Pi moved further and further into infinity. In 1958, 10 thousand digits of the number were calculated. In 1987, the Japanese calculated 10,013,395 characters. In 2011, Japanese researcher Shigeru Hondo exceeded the 10 trillion character mark.
Where else can you meet Pi?
So, often our knowledge about the number Pi remains at the school level, and we know for sure that this number is irreplaceable primarily in geometry.
In addition to formulas for the length and area of a circle, the number Pi is used in formulas for ellipses, spheres, cones, cylinders, ellipsoids, and so on: in some places the formulas are simple and easy to remember, and in others they contain very complex integrals.
Then we can meet the number Pi in mathematical formulas, where, at first glance, geometry is not visible. For example, indefinite integral from 1/(1-x^2) is equal to Pi.
Pi is often used in series analysis. As an example, here is a simple series that converges to Pi:
1/1 – 1/3 + 1/5 – 1/7 + 1/9 – …. = PI/4
Among the series, Pi appears most unexpectedly in the famous Riemann zeta function. It’s impossible to talk about it in a nutshell, let’s just say that someday the number Pi will help find a formula for calculating prime numbers.
And absolutely surprisingly: Pi appears in two of the most beautiful “royal” formulas of mathematics - Stirling’s formula (which helps to find the approximate value of the factorial and gamma function) and Euler’s formula (which connects as many as five mathematical constants).
However, the most unexpected discovery awaited mathematicians in probability theory. The number Pi is also there.
For example, the probability that two numbers will be relatively prime is 6/PI^2.
Pi appears in Buffon's needle-throwing problem, formulated in the 18th century: what is the probability that a needle thrown onto a lined piece of paper will cross one of the lines. If the length of the needle is L, and the distance between the lines is L, and r > L, then we can approximately calculate the value of Pi using the probability formula 2L/rPI. Just imagine - we can get Pi from random events. And by the way, Pi is present in the normal probability distribution, appears in the equation of the famous Gaussian curve. Does this mean that Pi is even more fundamental than simply the ratio of circumference to diameter?
We can also meet Pi in physics. Pi appears in Coulomb's law, which describes the force of interaction between two charges, in Kepler's third law, which shows the period of revolution of a planet around the Sun, and even appears in the arrangement of the electron orbitals of the hydrogen atom. And what is again most incredible is that the number Pi is hidden in the formula of the Heisenberg uncertainty principle - the fundamental law of quantum physics.
Secrets of Pi
In Carl Sagan's novel Contact, on which the film of the same name is based, aliens tell the heroine that among the signs of Pi there is a secret message from God. From a certain position, the numbers in the number cease to be random and represent a code in which all the secrets of the Universe are written.
This novel actually reflected a mystery that has occupied the minds of mathematicians around the world: is Pi a normal number in which the digits are scattered with equal frequency, or is there something wrong with this number? And although scientists are inclined to the first option (but cannot prove it), the number Pi looks very mysterious. One Japanese man once calculated how many times the numbers from 0 to 9 occur in the first trillion digits of Pi. And I saw that the numbers 2, 4 and 8 were more common than the others. This may be one of the hints that Pi is not entirely normal, and the numbers in it are indeed not random.
Let's remember everything we read above and ask ourselves, what other irrational and transcendental number is so often found in the real world?
And there are more oddities in store. For example, the sum of the first twenty digits of Pi is 20, and the sum of the first 144 digits is equal to the “number of the beast” 666.
The main character of the American TV series “Suspect,” Professor Finch, told students that due to the infinity of the number Pi, any combination of numbers can be found in it, starting from the numbers of your date of birth to more complex numbers. For example, at position 762 there is a sequence of six nines. This position is called the Feynman point after the famous physicist who noticed this interesting combination.
We also know that the number Pi contains the sequence 0123456789, but it is located at the 17,387,594,880th digit.
All this means that in the infinity of the number Pi you can find not only interesting combinations of numbers, but also the encoded text of “War and Peace”, the Bible and even The Main Secret The universe, if such a thing exists.
By the way, about the Bible. The famous popularizer of mathematics, Martin Gardner, stated in 1966 that the millionth digit of Pi (at that time still unknown) would be the number 5. He explained his calculations by the fact that in the English version of the Bible, in the 3rd book, 14th chapter, 16 verse (3-14-16) the seventh word contains five letters. The millionth figure was reached eight years later. It was the number five.
Is it worth asserting after this that the number Pi is random?
Today is the birthday of Pi, which, on the initiative of American mathematicians, is celebrated on March 14 at 1 hour and 59 minutes in the afternoon. This is connected with a more precise value of Pi: we are all accustomed to considering this constant as 3.14, but the number can be continued as follows: 3.14159... Translating this into calendar date, we get 03.14, 1:59.
Photo: AiF/ Nadezhda Uvarova
Professor of the Department of Mathematical and Functional Analysis of South Ural State University Vladimir Zalyapin says that July 22 should still be considered “Pi day”, because in the European date format this day is written as 22/7, and the value of this fraction is approximately equal to the value of Pi .
“The history of the number giving the ratio of the circumference of a circle to the diameter of a circle goes back to distant antiquity, says Zalyapin. - Already the Sumerians and Babylonians knew that this ratio does not depend on the diameter of the circle and is constant. One of the first mentions of the number Pi can be found in the texts Egyptian scribe Ahmes(circa 1650 BC). The ancient Greeks, who borrowed a lot from the Egyptians, contributed to the development of this mysterious quantity. According to legend, Archimedes was so carried away by calculations that he did not notice how the Roman soldiers took him hometown Syracuse. When the Roman soldier approached him, Archimedes shouted in Greek: “Don’t touch my circles!” In response, the soldier stabbed him with a sword.
Plato received a fairly accurate value of Pi for his time - 3.146. Ludolf van Zeilen spent most of his life working on the calculations of the first 36 decimal places of Pi, and they were engraved on his tombstone after his death."Irrational and abnormal
According to the professor, at all times the pursuit of calculating new decimal places was determined by the desire to obtain the exact value of this number. It was assumed that Pi was rational and could therefore be expressed as a simple fraction. And this is fundamentally wrong!
The number Pi is also popular because it is mystical. Since ancient times, there has been a religion of worshipers of the constant. In addition to the traditional value of Pi - a mathematical constant (3.1415...), expressing the ratio of the circumference of a circle to its diameter, there are many other meanings of the number. Such facts are interesting. In the process of measuring the dimensions of the Great Pyramid of Giza, it turned out that it has the same ratio of height to the perimeter of its base as the radius of a circle to its length, that is, ½ Pi.
If you calculate the length of the Earth's equator using Pi to the ninth decimal place, the error in the calculations will be only about 6 mm. Thirty-nine decimal places in Pi are enough to calculate the circumference of the circle surrounding known cosmic objects in the Universe, with an error no greater than the radius of a hydrogen atom!
The study of Pi also includes mathematical analysis. Photo: AiF/ Nadezhda Uvarova
Chaos in numbers
According to a mathematics professor, in 1767 Lambert established the irrationality of the number Pi, that is, the impossibility of representing it as a ratio of two integers. This means that the sequence of decimal places of Pi is chaos embodied in numbers. In other words, the “tail” of decimal places contains any number, any sequence of numbers, any texts that were, are and will be, but it’s just not possible to extract this information!“It is impossible to know the exact value of Pi,” continues Vladimir Ilyich. - But these attempts are not abandoned. In 1991 Chudnovsky achieved a new 2260000000 decimal places of the constant, and in 1994 - 4044000000. After that, the number of correct digits of Pi increased like an avalanche.”
Chinese holds world record for memorizing Pi Liu Chao, who was able to remember 67,890 decimal places without error and reproduce them within 24 hours and 4 minutes.
About the “golden ratio”
By the way, the connection between “pi” and another amazing quantity - the golden ratio - has never actually been proven. People have long noticed that the “golden” proportion - also known as the number Phi - and the number Pi divided by two differ from each other by less than 3% (1.61803398... and 1.57079632...). However, for mathematics, these three percent are too significant a difference to consider these values identical. In the same way, we can say that the Pi number and the Phi number are relatives of another well-known constant - the Euler number, since the root of it is close to half the Pi number. One half of Pi is 1.5708, Phi is 1.6180, the root of E is 1.6487.
This is only part of the value of Pi. Photo: Screenshot
Pi's birthday
In South Ural state university Constant's birthday is celebrated by all mathematics teachers and students. It has always been this way - it cannot be said that interest appeared only in recent years. The number 3.14 is even welcomed with a special holiday concert!
The ratio of the circumference of a circle to its diameter is the same for all circles. This ratio is usually denoted by the Greek letter (“pi” - the initial letter of the Greek word 
, which meant “circle”).
Archimedes, in his work “Measurement of a Circle,” calculated the ratio of the circumference to the diameter (number) and found that it was between 3 10/71 and 3 1/7.
For a long time, the number 22/7 was used as an approximate value, although already in the 5th century in China the approximation 355/113 = 3.1415929... was found, which was rediscovered in Europe only in the 16th century.
IN Ancient India considered equal to = 3.1622….
The French mathematician F. Viète calculated in 1579 with 9 digits.
The Dutch mathematician Ludolf Van Zeijlen in 1596 published the result of his ten-year work - the number calculated with 32 digits.
But all these clarifications of the value of the number were carried out using methods indicated by Archimedes: the circle was replaced by a polygon with all a large number sides The perimeter of the inscribed polygon was less than the circumference of the circle, and the perimeter of the circumscribed polygon was greater. But at the same time, it remained unclear whether the number was rational, that is, the ratio of two integers, or irrational.
Only in 1767 did the German mathematician I.G. Lambert proved that the number is irrational.
And more than a hundred years later, in 1882, another German mathematician, F. Lindemann, proved its transcendence, which meant the impossibility of constructing a square equal in size to a given circle using a compass and a ruler.
The simplest measurement
Draw a diameter circle on thick cardboard d(=15 cm), cut out the resulting circle and wrap a thin thread around it. Measuring the length l(=46.5 cm) one full turn of the thread, divide l per diameter length d circles. The resulting quotient will be an approximate value of the number, i.e. = l/ d= 46.5 cm / 15 cm = 3.1. This rather crude method gives, under normal conditions, an approximate value of the number with an accuracy of 1.
Measuring by weighing
Draw a square on a piece of cardboard. Let's write a circle in it. Let's cut out a square. Let's determine the mass of a cardboard square using school scales. Let's cut a circle out of the square. Let's weigh him too. Knowing the masses of the square m sq. (=10 g) and the circle inscribed in it m cr (=7.8 g) let's use the formulas
where p and h– density and thickness of cardboard, respectively, S– area of the figure. Let's consider the equalities:
Naturally, in in this case the approximate value depends on the weighing accuracy. If the cardboard figures being weighed are quite large, then even on ordinary scales it is possible to obtain such mass values that will ensure the approximation of the number with an accuracy of 0.1.
Summing the areas of rectangles inscribed in a semicircle
Figure 1
Let A (a; 0), B (b; 0). Let us describe the semicircle on AB as a diameter. Divide the segment AB into n equal parts by points x 1, x 2, ..., x n-1 and restore perpendiculars from them to the intersection with the semicircle. The length of each such perpendicular is the value of the function f(x)=. From Figure 1 it is clear that the area S of a semicircle can be calculated using the formula
S = (b – a) ((f(x 0) + f(x 1) + … + f(x n-1)) / n.
In our case b=1, a=-1. Then = 2 S.
The more division points there are on segment AB, the more accurate the values will be. To facilitate monotonous computing work, a computer will help, for which program 1, compiled in BASIC, is given below.
Program 1REM "Pi Calculation"
REM "Rectangle Method"
INPUT "Enter the number of rectangles", n
dx = 1/n
FOR i = 0 TO n - 1
f = SQR(1 - x^2)
x = x + dx
a = a + f
NEXT i
p = 4 * dx * a
PRINT "The value of pi is ", p
END
The program was typed and launched with different parameter values n. The resulting number values are written in the table:
Monte Carlo method
This is actually a statistical testing method. It got its exotic name from the city of Monte Carlo in the Principality of Monaco, famous for its gambling houses. The fact is that the method requires the use of random numbers, and one of the simplest devices that generates random numbers is a roulette. However, you can get random numbers using...rain.
For the experiment, let's prepare a piece of cardboard, draw a square on it and inscribe a quarter of a circle in the square. If such a drawing is kept in the rain for some time, then traces of drops will remain on its surface. Let's count the number of tracks inside the square and inside the quarter circle. Obviously, their ratio will be approximately equal to the ratio of the areas of these figures, since drops will fall into different places in the drawing with equal probability. Let N cr– number of drops in a circle, N sq. is the number of drops squared, then
4 N cr / N sq.
Figure 2
Rain can be replaced with a table of random numbers, which is compiled using a computer using a special program. Let us assign two random numbers to each trace of a drop, characterizing its position along the axes Oh And Oh. Random numbers can be selected from the table in any order, for example, in a row. Let the first four-digit number in the table 3265 . From it you can prepare a pair of numbers, each of which is greater than zero and less than one: x=0.32, y=0.65. We will consider these numbers to be the coordinates of the drop, i.e. the drop seems to have hit the point (0.32; 0.65). We do the same with all selected random numbers. If it turns out that for the point (x;y) If the inequality holds, then it lies outside the circle. If x + y = 1, then the point lies inside the circle.
To calculate the value, we again use formula (1). The calculation error using this method is usually proportional to , where D is a constant and N is the number of tests. In our case N = N sq. From this formula it is clear: in order to reduce the error by 10 times (in other words, to get another correct decimal place in the answer), you need to increase N, i.e. the amount of work, by 100 times. It is clear that the use of the Monte Carlo method was made possible only thanks to computers. Program 2 implements the described method on a computer.
Program 2REM "Pi Calculation"
REM "Monte Carlo Method"
INPUT "Enter the number of drops", n
m = 0
FOR i = 1 TO n
t = INT(RND(1) * 10000)
x = INT(t\100)
y = t - x * 100
IF x^2 + y^2< 10000 THEN m = m + 1
NEXT i
p=4*m/n
END
The program was typed and launched with different values of the parameter n. The resulting number values are written in the table:
| n | ||||||
| n | ||||||
Dropping needle method
Let's take an ordinary sewing needle and a sheet of paper. We will draw several parallel lines on the sheet so that the distances between them are equal and exceed the length of the needle. The drawing must be large enough so that an accidentally thrown needle does not fall outside its boundaries. Let us introduce the following notation: A- distance between lines, l– needle length.
Figure 3
The position of a needle randomly thrown onto the drawing (see Fig. 3) is determined by the distance X from its middle to the nearest straight line and the angle j that the needle makes with the perpendicular lowered from the middle of the needle to the nearest straight line (see Fig. 4). It's clear that
Figure 4
In Fig. 5 let's graphically represent the function y=0.5cos. All possible needle locations are characterized by points with coordinates (; y ), located on section ABCD. The shaded area of the AED is the points that correspond to the case where the needle intersects a straight line. Probability of event a– “the needle has crossed a straight line” – is calculated using the formula:
Figure 5
Probability p(a) can be approximately determined by repeatedly throwing the needle. Let the needle be thrown onto the drawing c once and p since it fell while crossing one of the straight lines, then with a sufficiently large c we have p(a) = p/c. From here = 2 l s / a k.
Comment. The presented method is a variation of the statistical test method. It is interesting from a didactic point of view, as it helps to combine simple experience with the creation of a rather complex mathematical model.
Calculation using Taylor series
Let us turn to the consideration of an arbitrary function f(x). Let us assume that for her at the point x 0 there are derivatives of all orders up to n th inclusive. Then for the function f(x) we can write the Taylor series:
Calculations using this series will be more accurate the more members of the series are involved. It is, of course, best to implement this method on a computer, for which you can use program 3.
Program 3REM "Pi Calculation"
REM "Taylor series expansion"
INPUT n
a = 1
FOR i = 1 TO n
d = 1 / (i + 2)
f = (-1)^i * d
a = a + f
NEXT i
p = 4 * a
PRINT "value of pi equals"; p
END
The program was typed and run with different values of the parameter n. The resulting number values are written in the table:
There are very simple mnemonic rules for remembering the meaning of a number: |
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 enough by heart large number his 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 claimed in their works that it is only a little more than three.
In one of holy books Jainism (an ancient Indian religion that arose in the 6th century BC) mentions 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 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 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 decomposition trigonometric functions into the ranks. 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.