Means pi. What is the number PI? History of discovery, secrets and riddles
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1. Relevance of the work.
In the infinite variety of numbers, just like among the stars of the Universe, individual numbers and their entire “constellations” of amazing beauty stand out, numbers with extraordinary properties and a unique harmony inherent only to them. You just need to be able to see these numbers and notice their properties. Take a closer look at the natural series of numbers - and you will find in it a lot of surprising and outlandish, funny and serious, unexpected and curious. The one who looks sees. After all, people won’t even notice on a starry summer night... the glow. The polar star, if they do not direct their gaze to the cloudless heights.
Moving from class to class, I became acquainted with natural, fractional, decimal, negative, rational. This year I studied irrational. Among ir rational numbers There is special number, the exact calculations of which have been carried out by scientists for many centuries. I came across it back in 6th grade while studying the topic “Circumference and Area of a Circle.” It was emphasized that we would meet with him quite often in classes in high school. Were interesting practical tasks to find numerical value numbers π. The number π is one of the most interesting numbers encountered in the study of mathematics. It is found in various school disciplines. The number pi has a lot to do with interesting facts, so it arouses interest in study.
Having heard a lot of interesting things about this number, I myself decided by studying additional literature and searching on the Internet to find out how to more information about it and answer problematic questions:
How long have people known about the number pi?
Why is it necessary to study it?
What interesting facts are associated with it?
Is it true that the value of pi is approximately 3.14
Therefore, I set myself target: explore the history of the number π and the significance of the number π on modern stage development of mathematics.
Tasks:
Study the literature to obtain information about the history of the number π;
Establish some facts from the “modern biography” of the number π;
Practical calculation of the approximate value of the ratio of circumference to diameter.
Object of study:
Object of study: PI number.
Subject of research: Interesting facts related to the PI number.
2. Main part. Amazing number pi.
No other number is as mysterious as Pi, with its famous never-ending number series. In many areas of mathematics and physics, scientists use this number and its laws.
Of all the numbers used in mathematics, few natural sciences, in engineering and everyday life, is given as much attention as is given to the number pi. One book says, “Pi is captivating the minds of science geniuses and amateur mathematicians around the world” (“Fractals for the Classroom”).
It can be found in probability theory, in solving problems with complex numbers and other unexpected and far from geometry areas of mathematics. The English mathematician Augustus de Morgan once called pi “... the mysterious number 3.14159... that crawls through the door, through the window and through the roof.” This mysterious number, associated with one of the three classical problems of Antiquity - constructing a square whose area is equal to the area of a given circle - entails a trail of dramatic historical and curious entertaining facts.
Some even consider it one of the five most important numbers in mathematics. But as the book Fractals for the Classroom notes, as important as pi is, “it is difficult to find areas in scientific calculations that require more than twenty decimal places of pi.”
3. The concept of pi
The number π is a mathematical constant expressing the ratio of the circumference of a circle to the length of its diameter. The number π (pronounced "pi") is a mathematical constant expressing the ratio of the circumference of a circle to the length of its diameter. Denoted by the letter "pi" of the Greek alphabet.
In numerical terms, π begins as 3.141592 and has an infinite mathematical duration.
4. History of the number "pi"
According to experts, this number was discovered by Babylonian magicians. It was used in the construction of the famous Tower of Babel. However, an insufficiently accurate calculation of the value of Pi led to the collapse of the entire project. It is possible that this mathematical constant underlay the construction of the legendary Temple of King Solomon.
The history of pi, 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. π = 3,160...
IN holy book Jainism (one of ancient religions, which 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, 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 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 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 .
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 π = 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 pi to 16 decimal places. Al-Kashi made unique calculations that were needed to compile a table of sines in steps of 1" . These tables played important role in astronomy.
A century and a half later in Europe F. Viet found pi 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 pi can be found using the limits of certain series. This discovery was of great
value, since it allowed us to calculate pi with any accuracy. Only 250 years after al-Kashi his result was surpassed.
Birthday of the number “”.
The unofficial holiday “PI Day” is celebrated on March 14, which in American format (day/date) is written as 3/14, which corresponds to the approximate value of PI.
There is also alternative option holiday - July 22. It's called Approximate Pi Day. The fact is that representing this date as a fraction (22/7) also gives the number Pi as a result. It is believed that the holiday was invented in 1987 by San Francisco physicist Larry Shaw, who noticed that the date and time coincided with the first digits of the number π.
Interesting facts related to the number “”
Scientists at the University of Tokyo, led by Professor Yasumasa Kanada, managed to set a world record in calculating the number Pi to 12,411 trillion digits. To do this, a group of programmers and mathematicians needed special program, supercomputer and 400 hours of computer time. (Guinness Book of Records).
The German king Frederick II was so fascinated by this number that he dedicated to it... the entire palace of Castel del Monte, in the proportions of which PI can be calculated. Now the magical palace is under the protection of UNESCO.
How to remember the first digits of the number “”.
The first three digits of the number = 3.14... are not difficult to remember. And for remembering more signs there are funny sayings and poems. For example, these:
You just have to try
And remember everything as it is:
Ninety two and six.
S. Bobrov. "Magic bicorn"
Anyone who learns this quatrain will always be able to name 8 signs of the number :
In the following phrases, the number signs can be determined by the number of letters in each word:
“What do I know about circles?” (3.1416);
“So I know the number called Pi. - Well done!"
(3,1415927);
“Learn and know the number behind the number, how to notice good luck.”
(3,14159265359)
5. Notation for pi
He was the first to introduce the notation for the ratio of circumference to diameter modern symbol pi 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.
IN late XVIII V. A.M.Lagendre based on works I.G. Lambert proved that pi 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 algebraic equation. Search exact expression continued even 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.
6. How to remember the number "Pi" accurate to eleven digits
The number "Pi" is the ratio of the circumference of a circle to its diameter, it is expressed as an infinite decimal fraction. In everyday life, it is enough for us to know three signs (3.14). However, some calculations require greater accuracy.
Our ancestors did not have computers, calculators or reference books, but since the time of Peter I they have been engaged in geometric calculations in astronomy, mechanical engineering, and shipbuilding. Subsequently, electrical engineering was added here - there is the concept of “circular frequency of alternating current”. To remember the number “Pi,” a couplet was invented (unfortunately, we do not know the author or the place of its first publication; but back in the late 40s of the twentieth century, Moscow schoolchildren studied Kiselev’s geometry textbook, where it was given).
The couplet is written according to the rules of old Russian orthography, according to which after consonant must be placed at the end of the word "soft" or "solid" sign. Here it is, this wonderful historical couplet:
Who, jokingly, will soon wish
“Pi” knows the number - he already knows.
For those who plan to study in the future accurate calculations, it makes sense to remember this. So what is the number "Pi" accurate to eleven digits? Count the number of letters in each word and write these numbers in a row (separate the first number with a comma).
This accuracy is already quite sufficient for engineering calculations. In addition to the ancient one, there is also modern way memorization, which was pointed out by a reader who identified himself as Georgiy:
So that we don't make mistakes,
You need to read it correctly:
Three, fourteen, fifteen,
Ninety two and six.
You just have to try
And remember everything as it is:
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."
Well, with the help of mathematics modern computers can calculate almost any number of digits of Pi.
7. Pi memory record
Humanity has been trying to remember the signs of pi for a long time. But how to put infinity into memory? A favorite question of professional mnemonists. Many unique theories and methods of mastering have been developed huge amount information. Many of them have been tested on pi.
The world record set in the last century in Germany is 40,000 characters. The Russian record for pi values was set on December 1, 2003 in Chelyabinsk by Alexander Belyaev. In an hour and a half with short breaks, Alexander wrote 2500 digits of pi on the blackboard.
Before this, listing 2,000 characters was considered a record in Russia, which was achieved in 1999 in Yekaterinburg. According to Alexander Belyaev, head of the center for the development of figurative memory, any of us can conduct such an experiment with our memory. It's only important to know special techniques memorize and practice periodically.
Conclusion.
The number pi appears in formulas used in many fields. Physics, electrical engineering, electronics, probability theory, construction and navigation are just a few. And it seems that just as there is no end to the signs of the number pi, there is no end to the possibilities for the practical application of this useful, elusive number pi.
In modern mathematics, the number pi is not only the ratio of the circumference to the diameter, it is included in large number various formulas.
This and other interdependencies allowed mathematicians to further understand the nature of pi.
The exact value of the number π in modern world represents not only its own scientific value, but is also used for very precise calculations (for example, the orbit of a satellite, the construction of giant bridges), as well as assessing the speed and power of modern computers.
Currently, the number π is associated with a difficult-to-see set of formulas, mathematical and physical facts. Their number continues to grow rapidly. All this speaks of a growing interest in the most important mathematical constant, the study of which has spanned more than twenty-two centuries.
The work I did was interesting. I wanted to learn about the history of pi, practical applications, and I think I achieved my goal. Summarizing the work, I come to the conclusion that this topic relevant. There are many interesting facts associated with the number π, so it arouses interest in study. In my work, I became more familiar with number - one of the eternal values that humanity has been using for many centuries. I learned some aspects of its rich history. Found out why ancient world did not know the correct ratio of circumference to diameter. I looked clearly at the ways in which the number can be obtained. Based on experiments, I calculated the approximate value of the number in various ways. Processed and analyzed the experimental results.
Any schoolchild today should know what a number means and approximately equals. After all, everyone’s first acquaintance with a number, its use in calculating the circumference of a circle, the area of a circle, occurs in the 6th grade. But, unfortunately, this knowledge remains formal for many and after a year or two, few people remember not only that the ratio of the length of a circle to its diameter is the same for all circles, but they even have difficulty remembering numerical value numbers equal to 3.14.
I tried to lift the veil of the rich history of the number that humanity has been using for many centuries. I made a presentation for my work myself.
The history of numbers is fascinating and mysterious. I would like to continue researching other amazing numbers in mathematics. This will be the subject of my next research studies.
References.
1. Glazer G.I. History of mathematics in school, grades IV-VI. - M.: Education, 1982.
2. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook - M.: Prosveshchenie, 1989.
3. Zhukov A.V. The ubiquitous number “pi”. - M.: Editorial URSS, 2004.
4. Kympan F. History of the number “pi”. - M.: Nauka, 1971.
5. Svechnikov A.A. a journey into the history of mathematics - M.: Pedagogika - Press, 1995.
6. Encyclopedia for children. T.11.Mathematics - M.: Avanta +, 1998.
Internet resources:
- http:// crow.academy.ru/materials_/pi/history.htm
Http://hab/kp.ru// daily/24123/344634/
The meaning of the number "Pi", as well as its symbolism, is known all over the world. This term denotes irrational numbers (that is, their value cannot be accurately expressed as a fraction y/x, where y and x are integers) and is borrowed from the ancient Greek phraseology "perepheria", which can be translated into Russian as "circle".The number "Pi" in mathematics denotes the ratio of the circumference of a circle to the length of its diameter. The history of the origin of the number "Pi" goes back to the distant past. Many historians have tried to establish when and by whom this symbol was invented, but they were never able to find out.
Pi is a transcendental number, or saying in simple words it cannot be the root of some polynomial with integer coefficients. It can be designated as a real number or as an indirect number that is not algebraic.
The number "Pi" is 3.1415926535 8979323846 2643383279 5028841971 6939937510...
Pi can be not only an irrational number that cannot be expressed using several different numbers. The number "Pi" can be represented by a certain decimal, which has an infinite number of digits after the decimal point. More interesting point- all these numbers cannot be repeated.
Pi can be correlated with fractional number 22/7, the so-called "triple octave" symbol. The ancient Greek priests knew this number. In addition, even ordinary residents could use it to solve any everyday problems, and also use it for design, such the most complex buildings like tombs.
According to scientist and researcher Hayens, a similar number can be traced among the ruins of Stonehenge, and also found in the Mexican pyramids.
Pi Ahmes, a famous engineer at that time, mentioned in his writings. He tried to calculate it as accurately as possible by measuring the diameter of the circle using the squares drawn inside it. Probably in some sense this number has some mystical, sacred meaning for the ancients.
Pi is essentially the most mysterious mathematical symbol. It can be classified as delta, omega, etc. It represents a relationship that will turn out to be exactly the same, regardless of what point in the universe the observer will be located. In addition, it will be unchanged from the object of measurement.
Most likely, the first person who decided to calculate the number "Pi" using a mathematical method is Archimedes. He decided to draw regular polygons in a circle. Considering the diameter of a circle to be one, the scientist designated the perimeter of a polygon drawn in a circle, considering the perimeter of an inscribed polygon as an upper estimate, and as a lower estimate of the circumference
What is the number "Pi"
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:
| C 1 | C 2 | ||
| = | |||
| d 1 | d 2 | (1) |
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 accurate. In 2000-1700 BC, a scribe called Ahmes compiled a papyrus in which we find recipes for resolving various practical problems. So, for example, to find the area of a circle, he uses the formula:
| 8 | 2 | |||||
| S | = | ( | d | ) | ||
| 9 |
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:
| 10 | 6336 | 14688 | 1 | |||||||||
| 3 | < | < | π | < | < | 3 | ||||||
| 71 | 1 | 1 | 7 | |||||||||
| 2017 | 4673 | |||||||||||
| 4 | 2 | |||||||||||
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 Application
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:
| π D 2 | |
| S=π R 2 = | |
| 4 |
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:
| α | |||
| S | = | π R 2 | |
| 360 ˚ |
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.
January 13, 2017***
What does a wheel from a Lada Priora have in common? wedding ring and your cat's saucer? Of course, you will say beauty and style, but I dare to argue with you. Pi number! 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, 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 it is expressed in decimal terms, then the sequence of digits after the decimal point will rush to infinity, and, moreover, without periodically repeating itself. What does this all 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 Pi is the ratio of the circumference of a circle to its diameter, then the first, perhaps most obvious, way to find our mysterious constant will be to manually make all the measurements and calculate 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 up to large quantity 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 precisely 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 results obtained 500,000 times. Want to try it?
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 view 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 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: “ New theory begins to dominate when the supporters of the old die out.”
NUMBER p – the ratio of the circumference of a circle to its diameter, is a constant value and does not depend on the size of the circle. The number expressing this relationship is usually denoted by the Greek letter 241 (from “perijereia” - circle, periphery). This notation came into use with the work of Leonhard Euler in 1736, but was first used by William Jones (1675–1749) in 1706. Like any irrational number, it is represented by an infinite non-periodic decimal fraction:
p= 3.141592653589793238462643... The needs of practical calculations related to circles and round bodies forced us to look for 241 approximations using rational numbers already in ancient times. Information that the circle is exactly three times longer than the diameter is found in the cuneiform tablets of Ancient Mesopotamia. Same number value p is also in the text of the Bible: “And he made a sea cast of copper, ten cubits from end to end, completely round, five cubits high, and a string of thirty cubits encircled it” (1 Kings 7:23). The ancient Chinese believed the same. But already in 2 thousand BC. the ancient Egyptians used more exact value number 241, which is obtained from the formula for the area of a circle with diameter d:
This rule from the 50th problem of the Rhind papyrus corresponds to the value 4(8/9) 2 » 3.1605. The Rhind Papyrus, found in 1858, is named after its first owner, it was copied by the scribe Ahmes around 1650 BC, the author of the original is unknown, it has only been established that the text was created in the second half of the 19th century. BC Although how the Egyptians received the formula itself is unclear from the context. In the so-called Moscow papyrus, which was copied by a certain student between 1800 and 1600 BC. from an older text, approximately 1900 BC, there is another interesting task on calculating the surface of a basket “with a 4½ hole”. It is not known what shape the basket was, but all researchers agree that here for the number p the same approximate value 4(8/9) 2 is taken.
To understand how ancient scientists obtained this or that result, you need to try to solve the problem using only the knowledge and calculation techniques of that time. This is exactly what researchers of ancient texts do, but the solutions they manage to find are not necessarily “the same.” Very often, several solution options are offered for one problem; everyone can choose to their liking, but no one can claim that this was the solution that was used in ancient times. Regarding the area of a circle, the hypothesis of A.E. Raik, the author of numerous books on the history of mathematics, seems plausible: the area of a circle is the diameter d is compared with the area of the square described around it, from which small squares with sides and are removed in turn (Fig. 1). In our notation, the calculations will look like this: to a first approximation, the area of a circle S equal to the difference between the area of a square and its side d and the total area of four small squares A with the side d:
This hypothesis is supported by similar calculations in one of the problems of the Moscow papyrus, where it is proposed to count
From the 6th century BC mathematics has developed rapidly in Ancient Greece. It was the ancient Greek geometers who strictly proved that the circumference of a circle is proportional to its diameter ( l = 2p R; R– radius of the circle, l – its length), and the area of the circle is equal to half the product of the circumference and radius:
S = ½ l R = p R 2 .
These proofs are attributed to Eudoxus of Cnidus and Archimedes.
In the 3rd century. BC Archimedes in his essay About measuring a circle calculated the perimeters of regular polygons inscribed in a circle and circumscribed around it (Fig. 2) - from a 6- to a 96-gon. Thus he established that the number p is between 3 10/71 and 3 1/7, i.e. 3.14084< p < 3,14285. Последнее значение до сих пор используется при расчетах, не требующих особой точности. Более точное приближение 3 17/120 (p"3.14166) was found by the famous astronomer, creator of trigonometry Claudius Ptolemy (2nd century), but it did not come into use.
Indians and Arabs believed that p= . This meaning is also given by the Indian mathematician Brahmagupta (598 - ca. 660). In China, scientists in the 3rd century. used a value of 3 7/50, which is worse than the Archimedes approximation, but in the second half of the 5th century. Zu Chun Zhi (c. 430 – c. 501) received for p approximation 355/113 ( p"3.1415927). It remained unknown to Europeans and was rediscovered by the Dutch mathematician Adrian Antonis only in 1585. This approximation produces an error of only the seventh decimal place.
The search for a more accurate approximation p continued in the future. For example, al-Kashi (first half of the 15th century) in Treatise on the Circle(1427) calculated 17 decimal places p. In Europe, the same meaning was found in 1597. To do this, he had to calculate the side of a regular 800 335 168-gon. The Dutch scientist Ludolf Van Zeijlen (1540–1610) found 32 correct decimal places for it (published posthumously in 1615), an approximation called the Ludolf number.
Number p appears not only when solving geometric problems. Since the time of F. Vieta (1540–1603), the search for the limits of certain arithmetic sequences compiled according to simple laws, led to the same number p. In this regard, in determining the number p Almost all famous mathematicians took part: F. Viet, H. Huygens, J. Wallis, G. W. Leibniz, L. Euler. They received various expressions for 241 in the form of an infinite product, a sum of a series, an infinite fraction.
For example, in 1593 F. Viet (1540–1603) derived the formula
In 1658, the Englishman William Brounker (1620–1684) found a representation of the number p as an infinite continued fraction
however, it is unknown how he arrived at this result.
In 1665 John Wallis (1616–1703) proved that
This formula bears his name. It is of little use for the practical determination of the number 241, but is useful in various theoretical discussions. It went down in the history of science as one of the first examples of endless works.
Gottfried Wilhelm Leibniz (1646–1716) in 1673 established the following formula:
expressing a number p/4 as the sum of the series. However, this series converges very slowly. To calculate p accurate to ten digits, it would be necessary, as Isaac Newton showed, to find the sum of 5 billion numbers and spend about a thousand years of continuous work on this.
London mathematician John Machin (1680–1751) in 1706, applying the formula
got the expression
which is still considered one of the best for approximate calculations p. It only takes a few hours of manual counting to find the same ten exact decimal places. John Machin himself calculated p with 100 correct signs.
Using the same series for arctg x and formulas
number value p was obtained on a computer with an accuracy of one hundred thousand decimal places. This kind of calculation is of interest in connection with the concept of random and pseudorandom numbers. Statistical processing of an ordered collection of a specified number of characters p shows that it has many of the features of a random sequence.
There are some fun ways to remember numbers p more accurate than just 3.14. For example, having learned the following quatrain, you can easily name seven decimal places p:
You just have to try
And remember everything as it is:
Three, fourteen, fifteen,
Ninety two and six.
(S. Bobrov Magic bicorn)
Counting the number of letters in each word of the following phrases also gives the value of the number p:
“What do I know about circles?” ( p"3.1416). This saying was proposed by Ya.I. Perelman.
“So I know the number called Pi. - Well done!" ( p"3.1415927).
“Learn and know the number behind the number, how to notice luck” ( p"3.14159265359).
A teacher at one of the Moscow schools came up with the line: “I know this and remember it perfectly,” and his student composed a funny continuation: “And many signs are unnecessary for me, in vain.” This couplet allows you to define 12 digits.
This is what 101 numbers look like p no rounding
3,14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679.
Nowadays, with the help of a computer, the meaning of a number p calculated with millions of correct signs, but such precision is not needed in any calculations. But the possibility of analytically determining the number ,
In the last formula, the numerator includes everyone prime numbers, and the denominators differ from them by one, and the denominator is greater than the numerator if it has the form 4 n+ 1, and less otherwise.
Although since the end of the 16th century, i.e. Since the very concepts of rational and irrational numbers were formed, many scientists have been convinced that p- an irrational number, but only in 1766 the German mathematician Johann Heinrich Lambert (1728–1777), based on the relationship discovered by Euler between exponential and trigonometric functions, strictly proved this. Number p cannot be represented as a simple fraction, no matter how large the numerator and denominator are.
In 1882, professor at the University of Munich Carl Louise Ferdinand Lindemann (1852–1939), using the results obtained by the French mathematician C. Hermite, proved that p– a transcendental number, i.e. it is not the root of any algebraic equation a n x n + a n– 1 xn– 1 + … + a 1 x+a 0 = 0 with integer coefficients. This proof put an end to the history of the ancient mathematical problem about squaring the circle. For millennia, this problem has defied the efforts of mathematicians; the expression “squaring the circle” has become synonymous with an unsolvable problem. And the whole point turned out to be the transcendental nature of the number p.
In memory of this discovery, a bust of Lindemann was erected in the hall in front of the mathematical auditorium at the University of Munich. On the pedestal under his name there is a circle crossed by a square equal area, inside of which a letter is inscribed p.
Marina Fedosova
