3 14 complete. What does pi hide?

Of course, these are not the most efficient ways to calculate pi. There are still a huge number of formulas. For example, the Chudnovsky formula, variations of which are used in Maple. However, in normal programming practice, the Gaussian formula is quite sufficient, so these methods will not be described in the article. It is unlikely that anyone wants to calculate billions of digits of pi, for which a complex formula gives a large increase in speed.

There are a lot of mysteries among the PIs. Or rather, these are not even riddles, but a kind of Truth that no one has yet solved in the entire history of mankind...

What is Pi? The PI number is a mathematical “constant” that expresses the ratio of the circumference of a circle to its diameter. At first, out of ignorance, it (this ratio) was considered equal to three, which was a rough approximation, but it was enough for them. But when prehistoric times gave way to ancient times (i.e., already historical), the surprise of inquisitive minds knew no bounds: it turned out that the number three very inaccurately expresses this ratio. With the passage of time and the development of science, this number began to be considered equal to twenty-two sevenths.

The English mathematician Augustus de Morgan once called the number PI “... the mysterious number 3.14159... that crawls through the door, through the window and through the roof.” Tireless scientists continued and continued to calculate the decimal places of the number Pi, which is actually a wildly non-trivial task, because you can’t just calculate it in a column: the number is not only irrational, but also transcendental (these are just such numbers that cannot be calculated by simple equations).

In the process of calculating these same signs, many different scientific methods and entire sciences were discovered. But the most important thing is that there are no repetitions in the decimal part of pi, as in an ordinary periodic fraction, and the number of decimal places is infinite. Today it has been verified that there are indeed no repetitions in 500 billion digits of pi. There is reason to believe that there are none at all.

Since there are no repetitions in the sequence of pi signs, this means that the sequence of pi signs obeys the theory of chaos, or more precisely, the number pi is chaos written in numbers. Moreover, if desired, this chaos can be represented graphically, and there is an assumption that this Chaos is intelligent.

In 1965, the American mathematician M. Ulam, sitting at one boring meeting, with nothing to do, began to write the numbers included in pi on checkered paper. Putting 3 in the center and moving counterclockwise in a spiral, he wrote out 1, 4, 1, 5, 9, 2, 6, 5 and other numbers after the decimal point. Along the way, he circled all the prime numbers. Imagine his surprise and horror when the circles began to line up along straight lines!

In the decimal tail of pi you can find any desired sequence of digits. Any sequence of digits in the decimal places of pi will be found sooner or later. Any!

So what? - you ask. Otherwise... Think about it: if your phone is there (and it is), then there is also the phone number of the girl who didn’t want to give you her number. Moreover, there are credit card numbers, and even all the values ​​of the winning numbers for tomorrow's lottery draw. What is there, in general, all lotteries for many millennia to come. The question is how to find them there...

If you encrypt all the letters with numbers, then in the decimal expansion of the number pi you can find all the world literature and science, and the recipe for making bechamel sauce, and all the holy books of all religions. This is a strict scientific fact. After all, the sequence is INFINITE and the combinations in the number PI are not repeated, therefore it contains ALL combinations of numbers, and this has already been proven. And if everything, then ALL. Including those that correspond to the book you have chosen.

And this again means that it contains not only all the world literature that has already been written (in particular, those books that burned, etc.), but also all the books that WILL yet be written. Including your articles on websites. It turns out that this number (the only reasonable number in the Universe!) governs our world. You just need to look at more signs, find the right area and decipher it. This is somewhat akin to the paradox of a herd of chimpanzees hammering away at a keyboard. Given a long enough experiment (you can even estimate the time) they will print all of Shakespeare's plays.

This immediately suggests an analogy with periodically appearing messages that the Old Testament supposedly contains encoded messages to descendants that can be read using clever programs. It is not entirely wise to immediately dismiss such an exotic feature of the Bible; cabalists have been searching for such prophecies for centuries, but I would like to cite the message of one researcher who, using a computer, found words in the Old Testament that there are no prophecies in the Old Testament. Most likely, in a very large text, as well as in the infinite digits of the PI number, it is possible not only to encode any information, but also to “find” phrases that were not originally included there.

For practice, 11 characters after the dot are enough within the Earth. Then, knowing that the radius of the Earth is 6400 km or 6.4 * 1012 millimeters, it turns out that if we discard the twelfth digit in the PI number after the 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 when rotating around the Sun (as is known, R = 150 * 106 km = 1.5 * 1014 mm), for the same accuracy it is enough to use the number PI with fourteen digits after the dot, and what’s there to waste - the diameter of our Galaxies are about 100,000 light years away (1 light year is approximately equal to 1013 km) or 1018 km or 1030 mm, and back in the 17th century, 34 digits of the PI number were obtained, excessive for such distances, and they are currently calculated to 12411 trillionth sign!!!

The absence of periodically repeating numbers, namely, based on their formula Circumference = Pi * D, the circle does not close, since there is no finite number. This fact can also be closely related to the spiral manifestation in our lives...

There is also a hypothesis that all (or some) universal constants (Planck’s constant, Euler’s number, universal gravitational constant, electron charge, etc.) change their values ​​over time, as the curvature of space changes due to the redistribution of matter or for other reasons unknown to us.

At the risk of incurring the wrath of the enlightened community, we can assume that the PI number considered today, reflecting the properties of the Universe, may change over time. In any case, no one can forbid us to re-find the value of the number PI, confirming (or not confirming) the existing values.

10 interesting facts about PI 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 circumference to 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 the initial letters of 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. Interestingly, 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.

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. There is an interesting coincidence. On March 14, the great scientist Albert Einstein, who, as we know, created the theory of relativity, was born.

Ever since humans were able to count and began exploring the properties of abstract objects called numbers, generations of inquisitive minds have made fascinating discoveries. As our knowledge of numbers has increased, some of them have attracted special attention, and some have even been given mystical meanings. Was, which stands for nothing, and which when multiplied by any number gives itself. 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 written, it is (pi). A number that, despite a long history, was not called what we call it today until the eighteenth century.

Start

The number pi is obtained by dividing the circumference of a circle by its diameter. In this case, the size of the circle is not important. Big or small, the ratio of length to diameter is the same. Although it is likely that this property was known earlier, the earliest evidence of this knowledge is the Moscow Mathematical Papyrus of 1850 BC. and the Ahmes papyrus 1650 BC. (although this is a copy of an older document). It contains a large number of mathematical problems, 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.

Representation

It was necessary to wait until the 17th century, when a revolution in calculation took place with the discovery of the infinite series, although the first result was not close, it was a product. Infinite series are the sums of an infinite number of terms that form a certain sequence (for example, all numbers of the form , where takes values ​​from to infinity). In many cases the sum is finite and can be found by various methods. It turns out that some of these series converge to or some quantity related to . In order for a series to converge, it is necessary (but not sufficient) that the summed quantities tend to zero as they grow. Thus, the more numbers we add, the more accurate we get the value. Now we have two options for getting a more accurate value. Either add more numbers, or find another series that converges faster, so that you can add fewer numbers.

Thanks to this new approach, the accuracy of the calculation increased dramatically, and in 1873, William Shanks published the result of many years of work, giving a value with 707 decimal places. Fortunately, he did not live until 1945, when it was discovered that he had made a mistake and all the numbers, starting with , were incorrect. However, his approach was most accurate before the advent of computers. This was the penultimate revolution in computing. Mathematical operations that would take several minutes to perform manually are now completed in fractions of a second, with virtually no errors. John Wrench and L. R. Smith managed to calculate 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 reason... yet.

Some interesting facts

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

Perhaps the biggest problem associated with , is the famous squaring of the circle problem, the problem of constructing, using a compass and ruler, a square whose area is equal to the area of ​​a given circle. The squaring of the circle tormented generations of mathematicians for twenty-four centuries until von Lindemann proved that it is a transcendental number (it is not a solution to any polynomial equation with rational coefficients) and, therefore, impossible to grasp the immensity. Until 1761, it was not proven that the number is irrational, that is, that there are no two natural numbers 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, apparently the most widely used in statistics. As mentioned earlier, a number appears as the sum of many series and is equal to infinite products, it is also important in the study of complex numbers. In physics, it can be found (depending on the system of units used) in the cosmological constant (Albert Einstein's biggest mistake) or the constant magnetic field constant. In a number system with any base (decimal, binary...), the numbers pass all tests of randomness, there is no order or sequence. The Riemann zeta function closely relates number to prime numbers. This number has a long history and probably still holds many surprises.