Before talking about history of pi , we note that the number Pi is one of the most mysterious quantities in mathematics. You will now see this for yourself, my dear reader...
Let's start our story with a definition. So, the number Pi is abstract number , denoting the ratio of the circumference of a circle to the length of its diameter. This definition has been familiar to us since school. But then the mysteries begin...
It is impossible to fully calculate this value; it is equal to 3,1415926535 , then after the decimal point - to infinity. Scientists believe that the sequence of numbers is not repeated, and this sequence is absolutely random...
The mystery of Pi It doesn't end there. Astronomers are confident that thirty-nine decimal places in this number are enough to calculate the circumference that encircles known cosmic objects in the Universe, with an error of the radius of a hydrogen atom...
irrational , i.e. it cannot be expressed as a fraction. This value transcendental – i.e. it cannot be obtained by performing any operations on integers….The number Pi is closely related to the concept of the golden ratio. Archaeologists have found that the height of the Great Pyramid of Giza is related to the length of its base, just like the radius of a circle is to its length...
History of the number P also remains a mystery. It is known that builders also used this value for design. Preserved, several thousand years old, which contained problems whose solution involved the use of the number Pi. However, the opinion about the exact value of this value among scientists different countries was ambiguous. So in the city of Susa, located two hundred kilometers from Babylon, a tablet was found where the number Pi was indicated as 3¹/8 . In Ancient Babylon it was discovered that the radius of a circle as a chord enters it six times, and it was there that it was first proposed to divide the circle into 360 degrees. Let us note by the way that a similar geometric action was made with the orbit of the Sun, which led ancient scientists to the idea that there should be approximately 360 days in a year. However, in Egypt the number Pi was equal to 3,16 , and in ancient India - 3, 088 , in ancient Italy - 3,125 . believed that this quantity is equal to the fraction 22/7 .
The number Pi was most accurately calculated by a Chinese astronomer Zu Chun Zhi in the 5th century AD. To do this, he wrote odd numbers twice 11 33 55, then he divided them in half, placed the first part in the denominator of the fraction, and the second part in the numerator, thus obtaining a fraction 355/113 . Surprisingly, the value coincides with modern calculations up to the seventh digit...
Who gave the first official name to this quantity?
It is believed that in 1647 mathematician Outtrade named the Greek letter π for the circumference of a circle, taking the first letter of the Greek word for this περιφέρεια - “periphery” . But in 1706 work came out English teacher William Jones “Review of the Achievements of Mathematics,” in which he denoted by the letter Pi the ratio of the circumference of a circle to its diameter. This symbol was finally fixed in the 20th century mathematician Leonhard Euler .
History of pi
The history of the number p, which expresses the ratio of the circumference of a circle to its diameter, began in Ancient Egypt. Area of 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. p = 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 to , which gives the fraction 3,162...
Ancient Greeks Eudoxus, Hippocrates and others reduced the measurement of a circle to the construction of a segment, and the measurement of a circle to the construction of an equal square. It should be noted that for many centuries, mathematicians from different countries and peoples tried to express the ratio of the circumference to the diameter as a rational number.
Archimedes in the 3rd century BC in his short work “Measuring a Circle” he substantiated three propositions:
Every circle is equal in size 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 .
Last offer Archimedes justified by the sequential calculation of the perimeters of regular inscribed and circumscribed polygons by doubling the number of their sides. First, he doubled the number of sides of regular inscribed and inscribed hexagons, then dodecagons, etc., bringing the calculations to the perimeters of regular inscribed and inscribed polygons with 96 sides. 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 p = 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 p with 16 decimal places. He doubled the number of sides of polygons 27 times and arrived at a polygon with 3*2 28 angles. 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 a number p 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 p can be found using the limits of certain series. This discovery was of great importance, as it made it possible to calculate p with any accuracy. Only 250 years after al-Kashi his result was surpassed.
The first to introduce the notation for the ratio of circumference to diameter with the modern symbol p 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 the number p 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. From the latter it follows that using only a compass and a ruler, construct a segment equal in circumference to impossible, and therefore there is no solution to the problem of squaring the circle.
The search for the exact expression for p 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.
Towards the end of the 19th century, after 20 years of hard work, the Englishman William Shanks found 707 digits of the number p. However, in 1945 it was discovered with the help of a computer that Shanks in his calculations he made an error in the 520th digit and his further calculations turned out to be incorrect.
After the development of methods of differential and integral calculus, many formulas were found that contain the number "pi". Some of these formulas allow you to calculate pi using techniques other than the method Archimedes and more rational. For example, you can arrive at the number pi by looking for the limits of certain series. So, G. Leibniz(1646-1716) received a row in 1674
1-1/3+1/5-1/7+1/9-1/11+... =p /4,
which made it possible to calculate p in a shorter way than Archimedes. However, this series converges very slowly and therefore requires quite lengthy calculations. To calculate "pi" it is more convenient to use the series obtained from the expansion arctg x at value x=1/ , in which the expansion of the function arctan 1/=p /6 in a series gives equality
p /6 = 1/,
those.
p= 2
Partial sums of this series can be calculated using the formula
S n+1 = S n + (2)/(2n+1) * (-1/3) n,
in this case, “pi” will be limited by the double inequality:
S 2n< p < S 2n+1
An even more convenient formula for calculating p received J. Machin. Using this formula, he calculated p(in 1706) with an accuracy of 100 correct characters. A good approximation for pi is given by
p = +
However, it should be remembered that this equality must be considered as approximate, because its right side is an algebraic number, and its left side is a transcendental one, therefore, these numbers cannot be equal.
As indicated in her articles E.Ya.Bakhmutskaya(60s of the XX century), back in the XV-XVI centuries. South Indian scientists, including Nilakanta, using methods of approximate calculations of the number p, we found a way to decompose arctan x into a power series similar to the series found Leibniz. Indian mathematicians gave a verbal formulation of the rules for expansion in series sine And cosine. By this they anticipated the discovery of European mathematicians of the 17th century. Nevertheless, their computational work, isolated and limited by practical needs, had no impact on the further development of science.
In our time, the work of computers has been replaced by computers. With their help, the number "pi" was calculated with an accuracy of more than a million decimal places, and these calculations lasted only a few hours.
In modern mathematics, the number p is not only the ratio of the circumference to the diameter; it is included in a large number of different formulas, including the formulas of non-Euclidean geometry, and the formula L. Euler, which establishes a connection between the number p and the number e
as follows:
e 2 p i = 1 , Where i = .
This and other interdependencies allowed mathematicians to further understand the nature of the number p.
On March 14, a very unusual holiday is celebrated all over the world - Pi Day. Everyone has known it since school. Students are immediately explained that the number Pi is a mathematical constant, the ratio of the circumference of a circle to its diameter, which has an infinite value. It turns out that there are many interesting facts associated with this number.
1. The history of numbers goes back more than one thousand years, almost as long as the science of mathematics has existed. Of course, the exact value of the number was not immediately calculated. At first, the ratio of the circumference to the diameter was considered equal to 3. But over time, when architecture began to develop, a more accurate measurement was required. By the way, the number existed, but it received a letter designation only at the beginning of the 18th century (1706) and comes from initial letters two Greek words, meaning “circle” and “perimeter”. The letter “π” was given to the number by the mathematician Jones, and it became firmly established in mathematics already in 1737.
2. In different eras and among different peoples, the number Pi had different meanings. For example, in Ancient Egypt it was equal to 3.1604, among the Hindus it acquired a value of 3.162, and the Chinese used a number equal to 3.1459. Over time, π was calculated more and more accurately, and when computing technology, that is, a computer, appeared, it began to number more than 4 billion characters.
3. There is a legend, or rather experts believe, that the number Pi was used in the construction of the Tower of Babel. However, it was not the wrath of God that caused its collapse, but incorrect calculations during construction. Like, the ancient masters were wrong. A similar version exists regarding the Temple of Solomon.
4. It is noteworthy that they tried to introduce the value of Pi even at the state level, that is, through law. In 1897, the state of Indiana prepared a bill. According to the document, Pi was 3.2. However, scientists intervened in time and thus prevented the mistake. In particular, Professor Perdue, who was present at the legislative meeting, spoke out against the bill.
5. It is interesting that several numbers in the infinite sequence Pi have their own name. So, six nines of Pi are named after the American physicist. Richard Feynman once gave a lecture and stunned the audience with a remark. He said he wanted to memorize the digits of Pi up to six nines, only to say "nine" six times at the end of the story, implying that its meaning was rational. When in fact it is irrational.
6. Mathematicians around the world do not stop conducting research related to the number Pi. It is literally shrouded in some mystery. Some theorists even believe that it contains universal truth. To exchange knowledge and new information about Pi, a Pi Club was organized. It’s not easy to join; you need to have an extraordinary memory. Thus, those wishing to become a member of the club are examined: a person must recite from memory as many signs of the number Pi as possible.
7. They even came up with various techniques for remembering the number Pi after the decimal point. For example, they come up with entire texts. In them, words have the same number of letters as the corresponding number after the decimal point. To make it even easier to remember such a long number, they compose poems according to the same principle. Members of the Pi Club often have fun in this way, and at the same time train their memory and intelligence. For example, Mike Keith had such a hobby, who eighteen years ago came up with a story in which each word was equal to almost four thousand (3834) of the first digits of Pi.
8. There are even people who have set records for memorizing Pi signs. So, in Japan, Akira Haraguchi memorized more than eighty-three thousand characters. But the domestic record is not so outstanding. A resident of Chelyabinsk managed to recite by heart only two and a half thousand numbers after the decimal point of Pi.
"Pi" in perspective
9. Pi Day has been celebrated for more than a quarter of a century, since 1988. One day, a physicist from the popular science museum in San Francisco, Larry Shaw, noticed that March 14, when written, coincides with the number Pi. In the date, the month and day form 3.14.
10. Pi Day is celebrated not exactly in an original way, but in a fun way. Of course, scientists involved in exact sciences do not miss it. For them, this is a way not to break away from what they love, but at the same time relax. On this day, people gather and prepare various delicacies with the image of Pi. There is especially room for pastry chefs to roam. They can make cakes with pi written on them and cookies with similar shapes. After tasting the delicacies, mathematicians arrange various quizzes.
11. There is an interesting coincidence. On March 14, the great scientist Albert Einstein, who, as we know, created the theory of relativity, was born. Be that as it may, physicists can also join in the celebration of Pi Day.
Pi- a mathematical constant equal to the ratio of the circumference of a circle to its diameter. The number pi is , the digital representation of which is an infinite non-periodic decimal fraction - 3.141592653589793238462643... and so on ad infinitum.
100 decimal places: 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34 211 70679.
The history of refining the value of pi
In every book on entertaining mathematics you will certainly find a story about clarifying the value of pi. At first, in ancient China, Egypt, Babylon and Greece, fractions were used for calculations, for example, 22/7 or 49/16. In the Middle Ages and the Renaissance, European, Indian and Arab mathematicians refined the value of pi to 40 digits after the decimal point, and by the beginning of the computer age, through the efforts of many enthusiasts, the number of pi was increased to 500.
Such accuracy is of purely academic interest (more on this below), but for practical needs within the Earth, 10 decimal places are sufficient. With the radius of the Earth 6400 km or 6.4·10 9 mm, it turns out that, discarding the twelfth digit of pi after the decimal point, when calculating the length of the meridian, we will be mistaken by several millimeters. And when calculating the length of the Earth’s orbit around the Sun (its radius is 150 million km = 1.5 10 14 mm), for the same accuracy it is enough to use the number pi with fourteen decimal places. The average distance from the Sun to Pluto, the farthest planet in the solar system, is 40 times greater than the average distance from Earth to the Sun. To calculate the length of Pluto's orbit with an error of several millimeters, sixteen digits of pi are enough. Why bother about trifles, the diameter of our Galaxy is about 100 thousand light years (1 light year is approximately equal to 10 13 km) or 10 19 mm, and yet in the 17th century 35 signs of pi were obtained, excessive even for such distances.
What is the difficulty in calculating the value of pi? The fact is that it is not only irrational, that is, it cannot be expressed as a fraction p/q, where p and q are integers. Such numbers cannot be written down exactly; they can only be calculated by successive approximations, increasing the number of steps to obtain greater accuracy. The simplest way is to consider regular polygons inscribed in a circle with an ever-increasing number of sides and calculate the ratio of the perimeter of the polygon to its diameter. As the number of sides increases, this ratio tends to pi. This is how, in 1593, Adrian van Romen calculated the perimeter of an inscribed regular polygon with 1073741824 (i.e. 2 30) sides and determined 15 digits of pi. In 1596, Ludolf van Zeijlen obtained 20 signs by calculating an inscribed polygon with 60 2 33 sides. Subsequently, he brought the calculations to 35 characters.
Another way to calculate pi is to use formulas with an infinite number of terms. For example:
π = 2 2/1 (2/3 4/3) (4/5 6/5) (6/7 8/7) ...
π = 4 (1/1 - 1/3) + (1/5 - 1/7) +(1/9 - 1/11) + ...
Similar formulas can be obtained by expanding, for example, the arctangent in the Maclaurin series, knowing that
arctan(1) = π/4(since tg(45°) = 1)
or expanding the arcsine in a series, knowing that
arcsin(1/2) = π/6(side lying opposite an angle of 30°).
Modern calculations use even more effective methods. With their help for today.
Pi day
Pi Day is celebrated by some mathematicians on March 14 at 1:59 (in the American date system - 3/14; the first digits of the number π = 3.14159). It is usually celebrated at 1:59 pm (on the 12-hour system), but those who adhere to the 24-hour light time system consider it to be 1:59 pm and prefer to celebrate at night. At this time, they read laudatory speeches in honor of the number pi, its role in the life of humanity, draw dystopian pictures of a world without pi, and eat pie pie ( pie), drink drinks and play games starting with pi.
- Pi (number) - Wikipedia
Ever since humans were able to count and began exploring the properties of abstract objects called numbers, generations of inquisitive minds have made fascinating discoveries. As our knowledge of numbers has increased, some of them have attracted special attention, and some were even given mystical meanings. Was, which stands for nothing, and which when multiplied by any number gives itself. There was, the beginning of everything, also possessing rare properties, prime numbers. Then they discovered that there are numbers that are not integers, but are sometimes obtained by dividing two integers - rational numbers. Irrational numbers that cannot be obtained as a ratio of whole numbers, etc. But if there is a number that has fascinated and caused a lot of writing to be done, it is (pi). A number that, despite a long history, was not called what we call it today until the eighteenth century.
Start
The number pi is obtained by dividing the circumference of a circle by its diameter. In this case, the size of the circle is not important. Big or small, the ratio of length to diameter is the same. Although it is likely that this property was known earlier, the earliest evidence of this knowledge is the Moscow Mathematical Papyrus of 1850 BC. and the Ahmes papyrus 1650 BC. (although this is a copy of an older document). It contains a large number of mathematical problems, some of which come close to , which is slightly more than 0.6\% different from the exact value. Around this time, the Babylonians considered equals. In the Old Testament, written more than ten centuries later, Yahweh keeps things simple and establishes by divine decree what exactly equals .
However, the great explorers of this number were the ancient Greeks such as Anaxagoras, Hippocrates of Chios and Antiphon of Athens. Previously, the value was determined almost certainly by experimental measurements. Archimedes was the first to understand how to theoretically evaluate its significance. The use of circumscribed and inscribed polygons (the larger one is circumscribed around the circle in which the smaller one is inscribed) made it possible to determine what is greater and less. Using Archimedes' method, other mathematicians obtained better approximations, and already in 480 Zu Chongzhi determined that the values were between and . However, the polygon method requires a lot of calculations (remember that everything was done by hand and not in a modern number system), so it had no future.
Submissions
It was necessary to wait until the 17th century, when a revolution in calculation took place with the discovery of the infinite series, although the first result was not close, it was a product. Infinite series are the sums of an infinite number of terms that form a certain sequence (for example, all numbers of the form , where takes values from to infinity). In many cases the sum is finite and can be found by various methods. It turns out that some of these series converge to or some quantity related to . In order for a series to converge, it is necessary (but not sufficient) that the summed quantities tend to zero as they grow. Thus, than more numbers we add, the more accurately we get the value. Now we have two options for getting a more accurate value. Either add more numbers, or find another series that converges faster, so that you can add fewer numbers.
Thanks to this new approach, the accuracy of the calculation increased dramatically, and in 1873, William Shanks published the result of many years of work, giving a value with 707 decimal places. Fortunately, he did not live until 1945, when it was discovered that he had made a mistake and all the numbers, starting with , were incorrect. However, his approach was most accurate before the advent of computers. This was the penultimate revolution in computing. Mathematical operations that would take several minutes to perform manually are now completed in fractions of a second, with virtually no errors. John Wrench and L. R. Smith managed to calculate 2,000 digits in 70 hours on the first electronic computer. The million-digit barrier was reached in 1973.
The latest (currently) advance in computing is the discovery of iterative algorithms that converge to faster than infinite series, so that much higher accuracy can be achieved with the same computing power. The current record is just over 10 trillion correct digits. Why calculate so accurately? Considering that, knowing the 39 digits of this number, you can calculate the volume of the known Universe to the nearest atom, there is no need... yet.
Some interesting facts
However, calculating the value is only a small part of its story. This number has properties that make this constant so interesting.
Perhaps the biggest problem associated with , is the famous squaring of the circle problem, the problem of using a compass and straightedge to construct a square whose area is equal to the area of a given circle. The squaring of the circle tormented generations of mathematicians for twenty-four centuries until von Lindemann proved that it is a transcendental number (it is not a solution to any polynomial equation with rational coefficients) and, therefore, impossible to grasp the immensity. Until 1761, it was not proven that the number is irrational, that is, that there are no two natural numbers such that . Transcendence was not proven until 1882, but it is not yet known whether the numbers or ( is another irrational transcendental number) are irrational. Many relationships appear that are not related to circles. This is part of the normalization factor of the normal function, probably the most widely used function in statistics. As mentioned earlier, a number appears as the sum of many series and is equal to infinite products, it is also important in the study of complex numbers. In physics, it can be found (depending on the system of units used) in the cosmological constant (Albert Einstein's biggest mistake) or the constant magnetic field constant. In a number system with any base (decimal, binary...), the numbers pass all tests of randomness, there is no observed order or sequence. The Riemann zeta function closely relates number to prime numbers. This number has a long history and probably still holds many surprises.
