Mathematical meaning of pi. What is the number PI and what does it mean?

If you compare circles of different sizes, you will notice the following: the sizes of different circles are proportional. This means that when the diameter of a circle increases by a certain number of times, the length of this circle also increases by the same number of times. Mathematically this can be written like this:

where C1 and C2 are the lengths of two different circles, and d1 and d2 are their diameters.
This relationship works in the presence of a coefficient of proportionality - the constant π already familiar to us. From relation (1) we can conclude: the length of a circle C is equal to the product of the diameter of this circle and a proportionality coefficient π independent of the circle:

C = π d.

This formula can also be written in another form, expressing the diameter d through the radius R of a given circle:

С = 2π R.

This formula is precisely the guide to the world of circles for seventh graders.

Since ancient times, people have tried to establish the value of this constant. For example, the inhabitants of Mesopotamia calculated the area of ​​a circle using the formula:

Where does π = 3 come from?

In ancient Egypt, the value for π was more precise. In 2000-1700 BC, a scribe called Ahmes compiled a papyrus in which we find recipes for solving various practical problems. So, for example, to find the area of ​​a circle, he uses the formula:

From what reasons did he arrive at this formula? – Unknown. Probably based on his observations, however, as other ancient philosophers did.

In the footsteps of Archimedes

Which of the two numbers is greater than 22/7 or 3.14?
- They are equal.
- Why?
- Each of them is equal to π.
A. A. Vlasov. From the Examination Card.

Some people believe that the fraction 22/7 and the number π are identically equal. But this is a misconception. In addition to the above incorrect answer in the exam (see epigraph), you can also add one very entertaining puzzle to this group. The task reads: “arrange one match so that the equality becomes true.”

The solution would be this: you need to form a “roof” for the two vertical matches on the left, using one of the vertical matches in the denominator on the right. You will get a visual image of the letter π.

Many people know that the approximation π = 22/7 was determined by the ancient Greek mathematician Archimedes. In honor of this, this approximation is often called the “Archimedean” number. Archimedes managed not only to establish an approximate value for π, but also to find the accuracy of this approximation, namely, to find a narrow numerical interval to which the value π belongs. In one of his works, Archimedes proves a chain of inequalities, which in a modern way would look like this:

can be written more simply: 3,140 909< π < 3,1 428 265...

As we can see from the inequalities, Archimedes found a fairly accurate value with an accuracy of up to 0.002. The most surprising thing is that he found the first two decimal places: 3.14... This is the value we most often use in simple calculations.

Practical use

Two people are traveling on a train:
- Look, the rails are straight, the wheels are round.
Where is the knock coming from?
- Where from? The wheels are round, but the area
circle pi er square, that’s the square that knocks!

As a rule, they become acquainted with this amazing number in the 6th-7th grade, but study it more thoroughly by the end of the 8th grade. In this part of the article we will present the basic and most important formulas that will be useful to you in solving geometric problems, but to begin with we will agree to take π as 3.14 for ease of calculation.

Perhaps the most famous formula among schoolchildren that uses π is the formula for the length and area of ​​a circle. The first, the formula for the area of ​​a circle, is written as follows:

where S is the area of ​​the circle, R is its radius, D is the diameter of the circle.

The circumference of a circle, or, as it is sometimes called, the perimeter of a circle, is calculated by the formula:

C = 2 π R = π d,

where C is the circumference, R is the radius, d is the diameter of the circle.

It is clear that the diameter d is equal to two radii R.

From the formula for circumference, you can easily find the radius of the circle:

where D is the diameter, C is the circumference, R is the radius of the circle.

These are basic formulas that every student should know. Also, sometimes it is necessary to calculate the area not of the entire circle, but only of its part - the sector. Therefore, we present it to you - a formula for calculating the area of ​​a sector of a circle. It looks like this:

where S is the area of ​​the sector, R is the radius of the circle, α is the central angle in degrees.

So mysterious 3.14

Indeed, it is mysterious. Because in honor of these magical numbers they organize holidays, make films, hold public events, write poems and much more.

For example, in 1998, a film by American director Darren Aronofsky called “Pi” was released. The film received many awards.

Every year on March 14 at 1:59:26 a.m., people interested in mathematics celebrate "Pi Day." For the holiday, people prepare a round cake, sit at a round table and discuss the number Pi, solve problems and puzzles related to Pi.

Poets also paid attention to this amazing number; an unknown person wrote:
You just have to try and remember everything as it is - three, fourteen, fifteen, ninety-two and six.

Let's have some fun!

We offer you interesting puzzles with the number Pi. Unravel the words that are encrypted below.

1. π R

2. π L

3. π k

Answers: 1. Feast; 2. File; 3. Squeak.

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 by heart a fairly large number of its signs.

2. Since 1988, “Pi Day” has been celebrated, which falls on March 14th. They prepare salads, cakes, cookies, and pastries with his image.

3. The number Pi has already been set to music, and it sounds quite good. A monument was even erected to him in Seattle, America, in front of the city Museum of Art.

At that distant time, they tried to calculate the number Pi using geometry. The fact that this number is constant for a wide variety of circles was known by geometers in Ancient Egypt, Babylon, India and Ancient Greece, who stated in their works that it was only a little more than three.

In one of the sacred books of Jainism (an ancient Indian religion that arose in the 6th century BC) it is mentioned that then the number Pi was considered equal to the square root of ten, which ultimately gives 3.162... .

Ancient Greek mathematicians 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 decimal fractions were not yet known, 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 the expansion of trigonometric functions into series. True, only two of his works have survived, and only references and quotes from his students are known for others. The scientific treatise "Mahajyanayana", which is attributed to Madhava, states that the number Pi is 3.14159265359. And in the treatise “Sadratnamala” a number is given with even more exact decimal places: 3.14159265358979324. In the given numbers, the last digits do not correspond to the correct value.

In the 15th century, the Samarkand mathematician and astronomer Al-Kashi calculated the number Pi with sixteen decimal places. His result was considered the most accurate for the next 250 years.

W. Johnson, a mathematician from England, was one of the first to denote the ratio of the circumference of a circle to its diameter by the letter π. Pi is the first letter of the Greek word "περιφέρεια" - circle. But this designation managed to become generally accepted only after it was used in 1736 by the more famous scientist L. Euler.

Conclusion

Modern scientists continue to work on further calculations of the values ​​of Pi. Supercomputers are already used for this. In 2011, a scientist from Shigeru Kondo, collaborating with an American student Alexander Yi, correctly calculated a sequence of 10 trillion digits. But it is still unclear who discovered the number Pi, who first thought about this problem and made the first calculations of this truly mystical number.

March 14, 2012

On March 14, mathematicians celebrate one of the most unusual holidays - International Pi Day. This date was not chosen by chance: the numerical expression π (Pi) is 3.14 (3rd month (March) 14th).

For the first time, schoolchildren encounter this unusual number in the elementary grades when studying circles and circumferences. The number π is a mathematical constant that expresses the ratio of the circumference of a circle to the length of its diameter. That is, if you take a circle with a diameter equal to one, then the circumference will be equal to the number “Pi”. The number π has an infinite mathematical duration, but in everyday calculations a simplified spelling of the number is used, leaving only two decimal places - 3.14.

In 1987, this day was celebrated for the first time. Physicist Larry Shaw from San Francisco noticed that in the American date system (month/day), the date March 14 - 3/14 coincides with the number π (π = 3.1415926...). Typically celebrations begin at 1:59:26 pm (π = 3.14 15926 …).

History of Pi

It is assumed that the history of the number π begins in Ancient Egypt. Egyptian mathematicians determined the area of ​​a circle with diameter D as (D-D/9) 2. From this entry it is clear that at that time the number π was equated to the fraction (16/9) 2, or 256/81, i.e. π 3.160...

In the VI century. BC. in India, in the religious book of Jainism, there are entries indicating that the number π at that time was taken equal to the square root of 10, which gives the fraction 3.162...
In the 3rd century. BC Archimedes in his short work “Measurement of a Circle” substantiated three propositions:

1. 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;
2. The areas of a circle are related to a square built on a diameter as 11 to 14;
3. The ratio of any circle to its diameter is less than 3 1/7 and greater than 3 10/71.

Archimedes justified the last position by sequentially calculating the perimeters of regular inscribed and circumscribed polygons by doubling the number of their sides. According to the exact calculations of Archimedes, the ratio of circumference to diameter is between the numbers 3 * 10 / 71 and 3 * 1/7, which means that the number “pi” is 3.1419... The true value of this ratio is 3.1415922653...
In the 5th century BC. Chinese mathematician Zu Chongzhi found a more accurate value for this number: 3.1415927...
In the first half of the 15th century. The astronomer and mathematician Kashi calculated π with 16 decimal places.

A century and a half later in Europe, F. Viet found the number π with only 9 regular decimal places: he made 16 doublings of the number of sides of polygons. F. Viet was the first to notice that π can be found using the limits of certain series. This discovery was of great importance; it made it possible to calculate π with any accuracy.

In 1706, the English mathematician W. Johnson introduced the notation for the ratio of the circumference of a circle to its diameter and designated it with the modern symbol π, the first letter of the Greek word periferia - circle.

For a long period of time, scientists around the world tried to unravel the mystery of this mysterious number.

What is the difficulty in calculating the value of π?

The number π is irrational: it cannot be expressed as a fraction p/q, where p and q are integers; this number cannot be the root of an algebraic equation. It is impossible to specify an algebraic or differential equation whose root will be π, therefore this number is called transcendental and is calculated by considering a process and is refined by increasing the steps of the process under consideration. Multiple attempts to calculate the maximum number of digits of the number π have led to the fact that today, thanks to modern computing technology, it is possible to calculate the sequence with an accuracy of 10 trillion digits after the decimal point.

The digits of the decimal representation of π are quite random. In the decimal expansion of a number, you can find any sequence of digits. It is assumed that this number contains all written and unwritten books in encrypted form; any information that can be imagined is found in the number π.

You can try to unravel the mystery of this number yourself. Of course, it will not be possible to write down the number “Pi” in full. But for the most curious, I suggest considering the first 1000 digits of the number π = 3,
1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989

Remember the number "Pi"

Currently, with the help of computer technology, ten trillion digits of the number “Pi” have been calculated. The maximum number of numbers that a person could remember is one hundred thousand.

To remember the maximum number of digits of the number “Pi”, various poetic “memories” are used, in which words with a certain number of letters are arranged in the same sequence as the numbers in the number “Pi”: 3.1415926535897932384626433832795…. To restore the number, you need to count the number of characters in each word and write it down in order.

So I know the number called “Pi”. Well done! (7 digits)

So Misha and Anyuta came running
They wanted to know the number Pi. (11 digits)

This I know and remember perfectly:
And many signs are unnecessary for me, in vain.
Let's trust our enormous knowledge
Those who counted the numbers of the armada. (21 digits)

Once at Kolya and Arina's
We ripped the feather beds.
The white fluff was flying and spinning,
Showered, froze,
Satisfied
He gave it to us
Old women's headache.
Wow, the spirit of fluff is dangerous! (25 characters)

You can use rhyming lines to help you remember the right number.

So that we don't make mistakes,
You need to read it correctly:
Ninety two and six

If you try really hard,
You can immediately read:
Three, fourteen, fifteen,
Ninety two and six.

Three, fourteen, fifteen,
Nine, two, six, five, three, five.
To do science,
Everyone should know this.

You can just try
And repeat more often:
"Three, fourteen, fifteen,
Nine, twenty-six and five."

Still have questions? Want to know more about Pi?
To get help from a tutor, register.
The first lesson is free!

***

What do a Lada Priora wheel, a wedding ring and your cat's saucer have in common? Of course, you will say beauty and style, but I dare to argue with you. Pi! This is a number that unites all circles, circles and roundness, which in particular include my mother’s ring, the wheel from my father’s favorite car, and even the saucer of my favorite cat Murzik. I'm willing to bet that in the ranking of the most popular physical and mathematical constants, Pi will undoubtedly take first place. But what is hidden behind it? Maybe some terrible curse words from mathematicians? Let's try to understand this issue.

What is the number "Pi" and where did it come from?

Modern number designation π (Pi) appeared thanks to the English mathematician Johnson in 1706. This is the first letter of the Greek word περιφέρεια (periphery, or circle). For those who took mathematics a long time ago, and besides, by no means, let us remind you that the number Pi is the ratio of the circumference of a circle to its diameter. The value is a constant, that is, constant for any circle, regardless of its radius. People knew about this in ancient times. Thus, in ancient Egypt, the number Pi was taken to be equal to the ratio 256/81, and in Vedic texts the value is given as 339/108, while Archimedes proposed the ratio 22/7. But neither these nor many other ways of expressing the number Pi gave an accurate result.

It turned out that the number Pi is transcendental and, accordingly, irrational. This means that it cannot be represented as a simple fraction. If we express it in decimal terms, then the sequence of digits after the decimal point will rush to infinity, and, moreover, without periodically repeating itself. What does all of this mean? Very simple. Do you want to know the phone number of the girl you like? It can probably be found in the sequence of digits after the decimal point of Pi.

You can see the phone number here ↓

Pi number accurate to 10,000 digits.

π= 3,
1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989..

Didn't find it? Then take a look.

In general, this can be not only a phone number, but any information encoded using numbers. For example, if you imagine all the works of Alexander Sergeevich Pushkin in digital form, then they were stored in the number Pi even before he wrote them, even before he was born. In principle, they are still stored there. By the way, the curses of mathematicians in π are also present, and not only mathematicians. In a word, the number Pi contains everything, even thoughts that will visit your bright head tomorrow, the day after tomorrow, in a year, or maybe in two. This is very difficult to believe, but even if we imagine that we believe it, it will be even more difficult to obtain information from it and decipher it. So, instead of delving into these numbers, maybe it’s easier to approach the girl you like and ask her number?.. But for those who are not looking for easy ways, or simply interested in what the number Pi is, I offer several ways calculations. Consider it healthy.

What is Pi equal to? Methods for calculating it:

1. Experimental method. If the number Pi is the ratio of the circumference of a circle to its diameter, then the first, perhaps the most obvious way to find our mysterious constant will be to manually make all the measurements and calculate the number Pi using the formula π=l/d. Where l is the circumference of the circle, and d is its diameter. Everything is very simple, you just need to arm yourself with a thread to determine the circumference, a ruler to find the diameter, and, in fact, the length of the thread itself, and a calculator if you have problems with long division. The role of the sample to be measured can be a saucepan or a jar of cucumbers, it doesn’t matter, the main thing is? so that there is a circle at the base.

The considered method of calculation is the simplest, but, unfortunately, it has two significant drawbacks that affect the accuracy of the resulting Pi number. Firstly, the error of the measuring instruments (in our case, a ruler with a thread), and secondly, there is no guarantee that the circle we are measuring will have the correct shape. Therefore, it is not surprising that mathematics has given us many other methods for calculating π, where there is no need to make precise measurements.

2. Leibniz series. There are several infinite series that allow you to accurately calculate Pi to a large number of decimal places. One of the simplest series is the Leibniz series. π = (4/1) - (4/3) + (4/5) - (4/7) + (4/9) - (4/11) + (4/13) - (4/15) ...
It’s simple: we take fractions with 4 in the numerator (this is what’s on top) and one number from the sequence of odd numbers in the denominator (this is what’s below), sequentially add and subtract them with each other and get the number Pi. The more iterations or repetitions of our simple actions, the more accurate the result. Simple, but not effective; by the way, it takes 500,000 iterations to get the exact value of Pi to ten decimal places. That is, we will have to divide the unfortunate four as many as 500,000 times, and in addition to this, we will have to subtract and add the obtained results 500,000 times. Want to try?

3. Nilakanta series. Don't have time to tinker with the Leibniz series? There is an alternative. The Nilakanta series, although it is a little more complicated, allows us to quickly get the desired result. π = 3 + 4/(2*3*4) — 4/(4*5*6) + 4/(6*7*8) — 4/(8*9*10) + 4/(10*11 *12) - (4/(12*13*14) ... I think if you look carefully at the given initial fragment of the series, everything becomes clear, and comments are unnecessary. Let's move on with this.

4. Monte Carlo method A rather interesting method for calculating Pi is the Monte Carlo method. It got such an extravagant name in honor of the city of the same name in the kingdom of Monaco. And the reason for this is coincidence. No, it was not named by chance, the method is simply based on random numbers, and what could be more random than the numbers that appear on the roulette tables of the Monte Carlo casino? Calculating Pi is not the only application of this method; in the fifties it was used in calculations of the hydrogen bomb. But let's not get distracted.

Take a square with a side equal to 2r, and inscribe a circle with radius r. Now if you put dots in a square at random, then the probability P The fact that a point falls into a circle is the ratio of the areas of the circle and the square. P=S kr /S kv =2πr 2 /(2r) 2 =π/4.

Now let's express the number Pi from here π=4P. All that remains is to obtain experimental data and find the probability P as the ratio of hits in the circle N cr to hitting the square N sq.. In general, the calculation formula will look like this: π=4N cr / N square.

I would like to note that in order to implement this method, it is not necessary to go to a casino; it is enough to use any more or less decent programming language. Well, the accuracy of the results obtained will depend on the number of points placed; accordingly, the more, the more accurate. I wish you good luck 😉

Tau number (Instead of a conclusion).

People who are far from mathematics most likely do not know, but it so happens that the number Pi has a brother who is twice its size. This is the number Tau(τ), and if Pi is the ratio of the circumference to the diameter, then Tau is the ratio of this length to the radius. And today there are proposals from some mathematicians to abandon the number Pi and replace it with Tau, since this is in many ways more convenient. But for now these are only proposals, and as Lev Davidovich Landau said: “The new theory begins to dominate when the supporters of the old one die out.”

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 a 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 that gives the ratio of the circumference to the diameter of the circle goes back to ancient times,” 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 the legend, Archimedes was so carried away by calculations that he did not notice how Roman soldiers took his hometown of 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.

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

“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

At South Ural State University, the birthday of the constant is celebrated by all teachers and mathematics students. This has always been the case - it cannot be said that interest has appeared only in recent years. The number 3.14 is even welcomed with a special holiday concert!