Message about pi interesting facts. Interesting facts about the mystical number pi. Pyramid of Giza

Everyone has come across the number Pi in mathematics lessons at school, but few know how interesting, mysterious and even mystical it is. The maximum that has been deposited in the mass consciousness of the average schoolchild is that it is the result of dividing the length of any circle by its own diameter. In fact, people have been trying to understand its essence and power for thousands of years, using it in both the simplest and most complex mathematical calculations. In the era of antiquity, it interested people from the standpoint of geometry, in the Middle Ages it was actively used in the development of mathematical analysis, and it has not lost its importance in the era of digital computers.

What interesting facts and secrets does this number hide?

It is absolutely endless and irrational, since the sequence of numbers that makes it up has no end and does not repeat with any frequency. Therefore, there is no absolutely exact complete value of π. Essentially, it is an expression of chaos, a digital recording of it.

It was not the same at all times and not among all peoples; serious errors were allowed in determining its meaning. For example, the ancient Egyptians considered Pi to be 3.1604, and the ancient Hindus considered it to be 3.162. The closest to the truth at that time were the Chinese, who accepted its value as 3.1459. According to experts, it was this error, and not God’s wrath and confusion of languages, that caused the failures in the construction of the biblical Tower of Babel.

There is a special Pi Club in the world, organized by mathematicians different countries. Becoming a member is not so easy; for this you need to have not only extraordinary mathematical thinking, but also excellent memory. Memorizing as long a sequence of digits of Pi as possible is your ticket to this elite club.

There are many memory techniques number sequence after the comma. These include specially composed texts, and even poems in which each subsequent word has the required number of letters, and a breakdown into groups with a certain association, and other convenient structuring. Record in in this direction Today it belongs to the Japanese Haraguchi, who was able to reproduce 83 thousand digits from memory.

His record could have easily been broken by a Chinese resident named Liu Chao, who remembered 93 thousand, if he had not made an unfortunate mistake at the 67890 character. It is known that it took him exactly one day and another 4 minutes without breaks for food, sleep and toilet, which is all the more offensive.

There are two dates for celebrating the number Pi in the holiday calendar. The first, as expected, echoes itself and is celebrated on March 14, forming the initial value of the number (3.14). Mathematicians and other scientists in the field exact sciences, of course, do not miss this day, celebrating it cheerfully and creatively. For the holiday, it is customary all over the world to prepare all sorts of goodies with the image of the “birthday boy” or in his form. After the tasting, there are fun and, of course, smart quizzes. By the way, it was on this day that Albert Einstein was born and Stephen Hawking died.

The second date is associated with the European calendar format and is celebrated on July 22 (22/7). It is known that the value of this fraction is an approximate value of the famous constant.

The Bank of Russia issued a coin in denomination “Pi” rubles dedicated to the date. Numismatists, of course, value it much more than 3 rubles 14 kopecks.

The first mention of the number π was found on papyrus in the mathematical calculations of Ahmes in 1650 BC. Now there is an ancient scroll with attempts to calculate a constant using the “squaring of the circle.” It involves measuring the area of ​​a circle by inscribing many squares into it.

Several sequences of numbers in a common chain have their own name. For example, it contains six nines in a row, named after the US physicist Richard Feynman. In one scientific community, he expressed the idea that he would like to learn the sequence up to this point in order to finally pronounce or write down the number “9” six times, which would sound like entering a period, and therefore the rationality of the number. Alas, after the sixth nine (Feyman's point) there is an eight, and irrational infinity continues.

Ancient mathematicians tried to calculate the circumference and area of ​​a circle by inscribing polygons (a geometric equilateral polygon) into it, the perimeter of which was calculated based on the number of angles. The more there were, the more accurate the result. It is known that Archimedes used the 96-gon in his calculations. He was easily overtaken by the Chinese Liu Hui, who managed to fit a 192-gonal polygon into the drawn circle, and then a 3072-gonal polygon. It was his calculation that remained the most accurate in the next millennium.

Today, with the help of computers, the number Pi has been calculated with a sequence of digits of 13.3 trillion digits. This is the limit for now. For further calculations, quantum computers with even higher power and speed are needed.

It has been proven that the famous egyptian pyramid in Giza there is an architectural embodiment of the value of the constant π. The ratio of its height to the sum of the sides of the base (perimeter) exactly repeats this number (golden ratio).

The sequence of the first 6 digits occurs in the first chain of 10 million. characters 6 times, but in reverse order.
If you use the Pi number to calculate the length of the earth's equator, the error will be only 6 mm.

At the beginning of the new century in Great Britain, on its mysterious fields, known as places of power, circles appeared that were carefully studied by scientists. They managed to decipher a sequence of 10 digits that exactly repeated the beginning of the number π.

In the early 200s, an enterprising Indian took a domain name of the maximum allowed 63 characters, in which he repeated the sequence of the constant. Subsequently, he successfully sold it to the German mathematician Steffens. On his website he posted a proposal to calculate or randomly find a page that displays a million decimal places of Pi. However, it was impossible to find a link to this page.

Another site with the domain name pi.com has nothing but a certain sequence of Pi. You can contact the owner only by transferring 3 dollars according to the specified data.

The famous number was also noted in art. In 1998, the film “Pi: Faith in Chaos” was released. Main character tried to find simple answers to mysterious things, in particular, the “key number” that underlies all natural patterns, as a result of which he went crazy. Darren Aronofosky, who directed the film, was awarded at the Sundance Film Festival for Best Director.

Scientists who decided to simplify the mysterious number and the formulas associated with it introduced an alternative constant, tau, where the diameter of a circle is replaced by its radius. It is proposed, according to calculations, to celebrate Rival Day on June 28. By the way, English phrase“I prefer pi” is read the same from the beginning and from the end (I prefer pi).

Calculating the Pi constant is standard for testing the computing power of another computer device; this is its so-called “digital cardiogram”.

IN mirror image the first three digits resemble English word pie PIE, which is pronounced exactly the same - .

This is just a part of the amazing facts about the number Pi that are understandable to a simple person, far from science. As for mathematical minds, for them the mystical constant opens up truly incredible horizons for research, discoveries and amazing coincidences, which may someday reveal their mysterious patterns to them.

Every person knows about the number Pi since school. In math classes, everyone is immediately told that it is a constant, which is the number obtained by dividing the circumference of a circle by its diameter. The result is an infinite value. There are many interesting facts, as well as myths and legends associated with such a well-known number.

The irrationality of a constant

Until now, it has not been possible to calculate the exact value, since it is irrational. In other words, it is an infinite decimal fraction.

IN different time used by all nations different meaning this indicator. For example, the ancient Egyptians took it for 3,1604 , and the Indians - like 3,162 . In ancient times, the closest to the truth were the Chinese, who used the meaning 3,1459 .

The first person to manually deduce the first 707 digits was W. Shanks. But at the 527th decimal place he made a mistake. But in any case, this is a great achievement for a person who independently calculated the number in the 19th century.

With the advent computer technology It has become much easier to deal with this problem. Thanks to the first computers that appeared, it was possible to output 4 billion decimal places. In 2002, a powerful computer was able to calculate 1.24 trillion. numbers This record was broken in 2011 - it is now known 10 trillion signs.

The modern name appeared only 3 centuries ago

The number Pi as a constant has been known for as long as mathematics itself as a science. Of course, all this time there were often errors in the calculations. Despite this, the name “Pi” itself was adopted in 1706. But there is evidence that such a constant was used as early as 1650 BC in Egypt by a scribe named Ahmes. It is the poet who also called this term the number of Ahmes. There is another name - Rind's number.

Some statistics

It has been established that there is no zero in the first 31 digits of the constant. In addition, scientists were able to calculate that if the first billion digits of such a constant were printed in a standard font, the leaves could be laid out so that they would occupy a route from Kansas to New York.

Number within a number

Pi is also called Archimedes' constant, Ludolph's number, and the circular constant. In the infinity of numbers that come after the decimal point in this number, you can also find several known meanings. For example, if you are careful, you will be able to find six nines. They were named after R. Feynman, a physicist from America. When he lectured to his students, he said that he wanted to memorize the sequence of digits in Pi up to six nines. At the end of his speech, he said the word “nine” six times, implying that it was rational, although in fact, on the contrary, it was irrational.

Ludolf van Zeilen devoted his life to studying the first 36 digits of Pi. As a result of this, the first 36 digits began to be called the Ludolf number. According to legend, they were even written on his tombstone, but it was lost, and it was never discovered.

P club

A club was founded in honor of the number Pi. Its participants are not only mathematicians who have been involved in research on the circular constant. This number is shrouded in a veil of secrets, so many are trying to uncover them. To be able to constantly exchange new knowledge, the Pi Club was founded. It is quite difficult to join it - you must have good memory to take the entrance exam. You need to list in order as many decimal places as possible in the constant. And then the rest will decide whether to accept a new member into this club.

Record breakers

There are people who have set records for memorizing digits in Pi. For example, one of the most famous is the Japanese Harakuchi Akira, who was able to voice more than 83 thousand characters. The Chinese planned to write more than 93 thousand characters in one day, but he made a mistake, so he could only write 67,890 numbers.

Pi Day

Since 1988, International Pi Day has been introduced. I celebrate it on March 14th. And this is not surprising, since the date is written as 3 and 14 - these are the first digits in the constant. Mathematicians from all over the world try not to miss such a celebration, although the holiday itself is not the most fun.

Myths

There are suggestions that the number Pi was used in the construction of the Tower of Babel. And it collapsed because the calculations were incorrect (inaccurate). By the way, there is the same story about the Temple of Solomon.

Other facts

There are many other interesting facts:

1. If you write the number Pi as a fraction, then it will have no end, as well as repetitions.
2. The indicator is used in weather forecasting.
3. In 2008, mysterious circles suddenly appeared in the UK. Scientists were able to decode the first 10 digits from a known constant.
4. If you write the first 3 digits in a mirror image, you will get the English word “pie”, which translates as “pie”.
5. If you number all the keys on the piano and start playing according to the numbers from Pi, you get a beautiful melody.
6. Givenchy named one of its men's perfume collections after the famous mathematical constant.

Pi is the most popular and indispensable constant in calculations. Although it has been used for several millennia, its modern name appeared only 300 years ago, and full meaning has not been established, because humanity will never be able to calculate it accurately. That's why this number is one of the most interesting in mathematics.

International Pi Day is celebrated annually on March 14th. At first glance, this event is quite insignificant. After all, what is this number “Pi”? Simply the ratio of the circumference of a circle to its diameter. However, this mysterious number has worried the minds of many mathematicians since ancient times. Therefore, several decades ago, scientists agreed to celebrate the annual holiday of the number “Pi”. Why exactly March 14? It's also very simple. In American calculus, this day is written as 3.14 - that is, the first three digits of Pi.

International Pi Day is celebrated annually on March 14th. At first glance, this event is quite insignificant. After all, what is this number “Pi”? Simply the ratio of the circumference of a circle to its diameter. However this mysterious the number has been troubling the minds of many since ancient times mathematicians. Therefore, several decades ago, scientists agreed to celebrate the annual holiday of the number “Pi”. Why exactly March 14? It's also very simple. In American calculus, this day is written as 3.14 - that is, the first three digits of Pi.

Pi is the most famous constant in the mathematical world.

In the Star Trek episode "The Wolf in the Fold," Spock commands the tinfoil computer to "compute to the last digit the value of Pi."

Comedian John Evans once quipped, “What do you get if you divide the circumference of a jack-o-lantern with eye, nose and mouth holes cut into it by its diameter? Pumpkin π!

Scientists in Carl Sagan's novel "The Bound" tried to unravel the fairly precise value of Pi in order to find hidden messages from the creators human race and give people access to "deeper levels of universal knowledge."

The symbol Pi (π) has been used in mathematical formulas for 250 years.

During the famous trial of OJ Simpson, a dispute arose between lawyer Robert Blasier and an FBI agent about the actual meaning of Pi. This was all intended to reveal shortcomings in the level of knowledge of a civil service agent.

Men's cologne from Givenci, called "Pi", is intended for attractive and forward-thinking people.

We will never be able to accurately measure the circumference or area of ​​a circle, since we do not know the full value of Pi. This “magic number” is irrational, that is, its numbers are forever changing in a random sequence.

In the Greek (“π” (piwas)) and English (“p”) alphabets, this symbol is located in position 16.

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, 1/2π

In mathematics, π is defined as the ratio of the circumference of a circle to its diameter. In other words, π the number of times the diameter of a circle is equal to its perimeter.

The first 144 decimal places of Pi end with 666, which is referred to in the Bible as the “number of the beast.”

In 1995, Hiryuki Goto was able to reproduce memory 42,195 decimal places of Pi, and is still considered the true champion in this field.

Ludolf van Zeijlen (b. 1540 – d. 1610) spent most of his life calculating the first 36 decimal places of Pi (which were called “Ludolf's digits”). According to legend, these numbers were engraved on his tombstone after his death.

William Shanks (b.1812-d.1882) worked for many years to find the first 707 digits of Pi. As it turned out later, he made an error in the 527th bit.

In 2002, a Japanese scientist calculated 1.24 trillion digits in the number Pi using a powerful Hitachi SR 8000 computer. In October 2011, the number π was calculated with an accuracy of 10,000,000,000,000 decimal places

Since 360 ​​degrees in a full circle and Pi are closely related, some mathematicians were delighted to learn that the numbers 3, 6 and 0 are at the three hundred and fifty-ninth decimal place in Pi.

One of the first mentions of the number Pi can be found in the texts of an Egyptian scribe named Ahmes (circa 1650 BC), now known as the Ahmes Papyrus (Rinda).

People have been studying the number pi for 4,000 years.

The Ahmes papyrus records the first attempt to calculate Pi using the “squaring of the circle,” which involved measuring the diameter of a circle using squares created inside.

In 1888, a doctor named Edwin Goodwin claimed to have the "supernatural value" of accurately measuring a circle. Soon a bill was proposed in parliament, according to which Edwin could publish copyright on his mathematical results. But this never happened - the bill did not become law, thanks to a mathematics professor in the legislature who proved that Edwin's method led to another incorrect value for Pi.

The first million decimal places in Pi consist of: 99959 zeros, 99758 ones, 100026 twos, 100229 threes, 100230 fours, 100359 fives, 99548 sixes, 99800 sevens, 99985 eights and 100106 nines.

Pi Day is celebrated on March 14 (chosen because it is similar to 3.14). The official celebration begins at 1:59 pm in order to comply with 3/14|1:59.

The meaning of the first numbers in Pi was first correctly calculated by some of the greatest mathematicians ancient world, Archimedes from Syracuse (b.287 - d.212 BC). He represented this number as several fractions. According to legend, Archimedes was so carried away by calculations that he did not notice how the Roman soldiers took it hometown Syracuse. When the Roman soldier approached him, Archimedes shouted in Greek: “Don’t touch my circles!” In response to this, the soldier stabbed him with a sword.

The exact value of Pi was obtained Chinese civilization much earlier than the Western one. The Chinese had two advantages over most other countries in the world: they used decimal system notation and zero symbol. European mathematicians, on the contrary, did not use the symbolic designation of zero in counting systems until late Middle Ages, until they came into contact with Indian and Arab mathematicians.

Al-Khwarizmi (the founder of algebra) worked hard to calculate Pi and achieved the first four numbers: 3.1416. The term “algorithm” comes from the name of this great Central Asian scientist, and from his text Kitab al-Jaber wal-Muqabala the word “algebra” appeared.

Ancient mathematicians tried to calculate Pi, each time entering polygons with big amount sides that fit much more closely into the area of ​​the circle. Archimedes used the 96-gon. Chinese mathematician Liu Hui inscribed the 192-gon, and then the 3072-gon. Tsu Chun and his son managed to fit a polygon with 24576 sides

William Jones (b.1675–d.1749) introduced the symbol “π” in 1706, which was later popularized in the mathematical community by Leonardo Euler (b.1707–d.1783).

The Pi symbol "π" came into use in mathematics only in the 1700s, the Arabs invented the decimal system in 1000, and the equal sign "=" appeared in 1557.

Leonardo da Vinci (b. 1452 – d. 1519) and the artist Albrecht Durer (b. 1471 – d. 1528) had small developments in the “squaring of the circle,” that is, they knew the approximate value of the number Pi.

Isaac Newton calculated Pi to 16 decimal places.

Some scientists argue that humans are programmed to find patterns in everything because that is the only way we can make sense of the world and ourselves. And that is why we are so attracted to the “irregular” number Pi))

Pi may also be referred to as the "circular constant", "Archimedean constant" or "Ludolf number".

In the seventeenth century, Pi expanded beyond the circle and began to be used in mathematical curves such as the arc and hypocycloid. This happened after the discovery that in these areas some quantities can be expressed through the number Pi itself. In the twentieth century, Pi was already used in many mathematical fields, such as number theory, probability and chaos.

The first six digits of Pi (314159) are reversed at least six times among the first 10 million decimal places.

Many mathematicians argue that the correct formulation would be: “a circle is a figure with an infinite number of angles.”

Thirty-nine decimal places in Pi are enough to calculate the circumference of a circle encircling known space objects in the Universe, with an error of no more than the radius of a hydrogen atom.

Plato (b. 427 - d. 348 BC) obtained a fairly accurate value for the number Pi for his time: √ 2 + √ 3 = 3.146.

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 length of a circle whose diameter is 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. The first surprising 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 signs

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 the number Pi through 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 surpassed 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, but 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. For 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 amazing: Pi appears in two of the most beautiful “royal” formulas of mathematics - the Stirling formula (which helps to find approximate value 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 all over 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. A Japanese man once calculated how many times the numbers 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, ranging 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?

Usually our knowledge about the number Pi ends here: 3.14159. Not everyone even remembers that this number shows the ratio of the circumference of a circle and its diameter.

Pi - ir rational number, that is, it cannot be written as a simple fraction.

Moreover, it is infinite and non-periodic decimal, making it one of the most mysterious numbers known to man.

First calculation

Archimedes was the first to talk about the existence of the number Pi

It is believed that Archimedes was the first to talk about the number Pi. Around 220 BC. he derived the formula S = Pi R2 by approximating the area of ​​a circle based on the area of ​​the polygon inscribed in the circle and the area of ​​the polygon around which the circle was circumscribed. Both polygons outlined the lower and upper boundaries of the circle, thereby allowing Archimedes to realize that the missing piece (Pi) was somewhere between 3 1/7 and 3 10/71.

The famous Chinese mathematician and astronomer Zu Chongzhi (429–501) calculated Pi a little later, dividing 355 by 113, but it is still unknown how he came to this conclusion, since no records of his work have survived.

The area of ​​the circle is actually unknown

Pi is an irrational number

In the 18th century, Johann Heinrich Lambert proved the irrationality of Pi. Irrational numbers cannot be expressed as a whole fraction. Any rational number can always be written as a fraction, where the numerator and denominator are expressed as a whole number. You can, of course, imagine Pi as a simple ratio of the circumference and diameter (Pi = C/D), and it will always turn out that if the diameter is represented by an integer, then the circumference will be expressed by an integer, and vice versa.

The irrationality of the number Pi is expressed in the fact that we never know the real circumference (and subsequently the zone) of the circle. This fact seemed inevitable to scientists, but some mathematicians insisted that it would be more accurate to imagine that a circle had an infinite number of tiny angles, rather than to assume that the circle itself was straight.

Using Buffon's problem you can calculate Pi without using a circle

Scientists first paid attention to Buffon's needle problem in 1777. This problem has been recognized as one of the most intriguing in the history of geometric probability. Here's how it works.
If you were faced with the task of throwing a needle of a certain length onto a piece of paper on which lines of the same length were drawn, then the probability that the needle would cross one of the lines would be equal to the number Pi.

There are two variables in throwing a needle: 1. the angle of incidence and 2. the distance from the center of the needle to the nearest line. The angle can vary from 0 to 180 degrees and is measured from a line parallel to the lines on the paper.

It turns out that the probability of the needle landing this way is 2/Pi, or about 64%. Accordingly, the number Pi can theoretically be calculated using this technique, if there is someone who has the patience to carry out this dreary experiment. Please note that there is no circle involved here.

It may be difficult to imagine all this, but if you have the desire, you can try.

Pi and the tape problem

The circumference of a circle increases strictly in relation to Pi

Imagine taking a ribbon and wrapping it around globe. (To simplify the experiment, we suggest taking it as a truth that the Earth is a flat sphere with a circumference of 40,000 km). Now try to determine the required length of tape that can be wrapped around the Earth at a distance of 2.54 cm above its surface. If you think that the second tape should be longer, then you are not alone in your guesses. But in fact, this is not at all true: the second tape will be only 2Pi longer, which is approximately 16 cm.

And here is the solution: let’s say that the Earth is a perfect sphere, a huge circle, the length of which is 40,000 km (along the equator). Therefore, its radius will be equal to 40000/2Pi, or 6.37 km. Now the second ribbon, which passes at a distance of 2.54 cm above the surface of the Earth: its radius will increase by only 2.54 cm relative to the radius of the Earth. We get the equation C = 2 Pi(r+1), which is equivalent to C = 2 Pi(r) + 2 Pi. Based on this, we can say that the circumference of the second ribbon will increase by only 2 Pi. It doesn't really matter which radius you take into account (the Earth and the hoops of a basketball hoop), increasing this radius by 2.54 cm will only increase the circumference by 2Pi (about 16 cm).

Navigation

Pi is used in navigation calculations

The number Pi plays very important role in navigation, especially when it comes to determining location on large territory. The size of a person is very small relative to the Earth, so it seems to us that we are always moving in a straight line, but this is not so. For example, airplanes fly in a circle and their path must be calculated in order to calculate the flight time, the amount of fuel and take into account all the nuances.

Additionally, when you determine your location on Earth using GPS, Pi plays an important role in these calculations.

What about navigation, which requires even more precise positioning than flying from New York to Tokyo? Susan Gomez, a NASA scientist, says that most of NASA's calculations are made using the numbers 15 or 16, especially when it comes to very precise calculations for a program that controls and stabilizes spaceships During the flight.

Signal Processing and Fourier Transform

Pi plays an important role in signal transmission

Most often, the number Pi is used in such geometric problems, as a measurement of a circle, however, its role is also important in signal processing, mainly in the process known as the Fourier transform, which transforms the signal into a spectrum of frequencies. The Fourier transform is called the “map frequency domain» of the original signal, where it relates to both the frequency domain and mathematical operations, which combine the frequency domain and the time function.

People and technology use this phenomenon when basic signal conversion is needed, such as when your iPhone receives a message from a cell phone tower, or when your ear distinguishes sounds different frequencies. Pi, which appears in the Fourier transform formula, plays a decisive and, at the same time, strange role in the transformation process, since it lies in the exponent of the Euler number (the well-known mathematical constant 2.71828...)

Therefore, you can thank Pi every time you make a cell phone call or listen to a broadcast signal.

Normal probability distribution

Using Pi, you can calculate the vibration force of a large structure

And if the use of Pi is expected in operations such as the Fourier transform, which relates directly to signals (and, accordingly, waves), then its appearance in the normal probability distribution formula is surprising. You've no doubt encountered this notorious distribution before—it's involved in a wide range of phenomena we observe on a regular basis, from dice rolls to test scores.

Every time you discover that Pi is hidden in an equation, imagine that there is a circle hidden somewhere among the mathematical formulas. In the case of a normal probability distribution, Pi is expressed in terms of the Gaussian integral (also known as the Euler-Poisson integral), which is Square root from Pi. In fact, all that is required is small changes in the variables in the Gaussian integral to calculate the normalization constant of the normal distribution.

One common but counterintuitive application of the Gaussian integral involves "white noise" - a normally distributed random variable, used to predict everything from the effect of wind on an aircraft to the vibration force of a beam in a large-scale structure.

Rivers make their winding path in accordance with the number Pi

A completely unexpected fact is that the number Pi is related to meandering rivers. The floodplain of a river most often looks like a sinusoid, which bends in one place and then in another, crossing the plain. In mathematical terms, it can be described as the length of a winding path divided by the length of the river from source to mouth. It turns out that regardless of the length of the river and the number of its bends, its sinuosity is approximately equal to the number Pi.

Albert Einstein made several suggestions as to why rivers behave this way. He noticed that water flowed faster on the outside of the bend, causing more erosion of the shoreline and making the bend worse. Then these bends “meet” each other and sections of the river are connected. This back and forth motion seems to constantly correct itself as the river continues to bend in accordance with Pi.

Pi and the Fibonacci sequence

Pi can be calculated using the Febonacci sequence

Usually, 2 methods have always been used to calculate Pi: the first was invented by Archimedes, the second was developed by the Scottish mathematician James Gregory.

Each subsequent number in the Fibonacci sequence is equal to the sum of the previous two numbers. The sequence looks like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... It is endless.

And since the arctangent of 1 is equal to Pi/4, this means that Pi can be expressed in terms of the Fibonacci sequence via the following equation: arctan(1)*4=pi.

In addition to being just a beautiful collection of numbers, the Febonacci sequence plays an important role in some natural phenomena. It can be used to model and describe a large number of phenomena in mathematics, science, art and nature. The mathematical ideas that the Febonacci sequence leads to, such as the golden ratio, spirals, curves, are highly valued for their aesthetic appearance, but mathematicians are still trying to explain the depth of the connection.

Pi and quantum mechanics

Pi is also closely related to Einstein's theory of relativity.

Pi is, without a doubt, the inevitable and complex basis of our world, but what about our endless universe? Pi works throughout the universe and is directly involved in explaining the nature of the cosmos. It is a fact that many formulas used in the field of quantum mechanics, which governs the world of atoms and nuclei, contain Pi.

Some of the most famous equations in this field are the Einstein equations of gravitational field (also known simply as the Einstein equations). These are 10 equations compiled within the theory of relativity that describe the fundamental interaction of gravity as a result of the curvature of space-time by mass and energy. The amount of gravity present in a system is proportional to the amount of energy and momentum, with the constant of proportionality associated with G being a numerical constant.

We hope that our article has helped you better understand the nature and purpose of the number Pi. Who would have thought that it is an integral part of our Everyday life and even natural processes occur in accordance with its meaning.