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 a more precise value for the number 241, which is obtained from the formula for the area of a circle's 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, around 1900 BC, there is another interesting problem about 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 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 digits, but such precision is not needed in any calculation. But the possibility of analytically determining the number ,
In the last formula, the numerator contains all 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 between exponential and trigonometric functions discovered by Euler, 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 of squaring the circle. For millennia, this problem defied the efforts of mathematicians; the expression “squaring the circle” became 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 intersected by a square of equal area, inside which the letter is inscribed p.
Marina Fedosova
Pi (“π”) is a mathematical constant obtained in a rather interesting way. Let us assume that the diameter of the circle is equal to 1 conventional unit. Then the number π is the length of a given circle, which is approximately equal to 3.14 conventional units. In other words, pi expresses the relationship between the circumference of a circle and its diameter. This ratio will always be there.
Pi has a number of properties.
First, pi is an irrational number, which means it cannot be represented as a proper fraction. The value of 3.14 is quite approximate; the decimal places for this constant are not known for certain.
Secondly, the number π is transcendental. This means that it can never be a power of a root of another number. In other words, π is not an algebraic number. Moreover, if any number is raised to the power π, then again we get a transcendental number.
It is worth noting that the ancient mathematicians of Egypt, Greece, Rome, Syria and Iran already knew that the ratio between the diameter of a circle and its length is a constant value. For example, in Babylon this ratio was estimated as 25/8, and in Egypt as 256/81. But the greatest success in calculating the value of the number π was achieved by Archimedes, who, through repeated descriptions and inserting the correct ones into it, achieved fairly accurate results. Archimedes took the perimeter as the minimum value of the number π, and as the maximum. Thus, Archimedes derived the value of the constant π equal to 3.142857142857143.
It's funny to note that there is a "Pi Day" which is celebrated on the 14th. This happens because if you write the day and date as numbers, you get 3.14 - the approximate value of this constant. According to another version, this holiday should be celebrated on July 22, since 22/7 is also one of the first ratios, approximately equal to 3.14
Pi is a mathematical constant that is the ratio of the circumference of a circle to the length of its diameter. This number is usually denoted in mathematics by the Greek letter π.
The final value of pi is still not known. In the process of calculating it, many scientific methods of counting were discovered. Scientists now know more than 500 billion decimal places that separate a decimal fraction from a whole number. The decimal part of the constant pi does not have repetitions, as in a simple periodic fraction, and the number of decimal places is most likely infinite. The infinity of this constant and the absence of periodically repeating digits after the decimal point do not allow the circle to close if, working in reverse order, we multiply pi by the diameter of the circle.
Mathematicians call pi chaos written in numbers. In the decimal fraction of this constant you can find any conceived sequence of numbers: any telephone number, credit card PIN code or historical date. Moreover, if all books were translated into decimal digital code, they could also be found in pi. There are also books there that have not yet been written. Since pi is infinite and the sequence of digits after the decimal point is not repeated, absolutely any information about the Universe can potentially be found in it. This fact allows us to call the constant pi “divine” and “reasonable”.
In school, they usually use the minimum exact value of pi with two decimal places - 3.14. For practice on Earth, the number pi with 11 decimal places is sufficient. To calculate the length of our planet's orbit, you must use a number with 14 decimal places. Accurate calculations within our galaxy are possible using pi to 34 decimal places.
Unsolved pi problems
It is unknown whether pi is algebraically independent. Also, the exact measure of the irrationality of this constant has not been calculated, although it is known that it cannot be greater than 7.6063. It is unknown whether pi to the power n is an integer if n is any positive number.
There is no confirmation whether pi belongs to the period ring. In addition, the question of this number remains unresolved. Normal is any number, when written in the n-ary number system, groups of consecutive digits are formed that occur with the same asymptotic frequency. It is not even known which numbers from 0 to 9 appear an infinite number of times in the decimal representation of pi.
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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 the irrational numbers there is a 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. Practical tasks on finding the numerical value of π were interesting. The number π is one of the most interesting numbers encountered in the study of mathematics. It is found in various school disciplines. There are many interesting facts associated with the number π, 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 the Internet to find out as much information as possible 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 at the present stage of 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 study: 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, science, engineering, and everyday life, few numbers receive as much attention as 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, the 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 the sacred book of Jainism (one of the oldest religions that existed in India and arose in the 6th century BC) there is an indication from which it follows that the number p at that time was taken equal, which gives the fraction 3,162... Ancient Greeks Eudoxus, Hippocrates and others reduced the measurement of a circle to the construction of a segment, and the measurement of a circle to the construction of an equal square. It should be noted that for many centuries, mathematicians from different countries and peoples tried to express the ratio of the circumference to the diameter as a rational number.
Archimedes in the 3rd century BC. in his short work “Measuring a Circle” he substantiated three propositions:
Every circle is equal in size to a right triangle, the legs of which are respectively equal to the length of the circle and its radius;
The areas of a circle are related to the square built on the diameter, as 11 to 14;
The ratio of any circle to its diameter is less 3 1/7 and more 3 10/71 .
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 an 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 an alternative version of the 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 a special program, a 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 to remember 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
The first to introduce the modern symbol pi for the ratio of the circumference of a circle to its diameter was an English mathematician W.Johnson in 1706. As a symbol he took the first letter of the Greek word "periphery", which translated means "circle". Entered W.Johnson the designation became commonly used after the publication of the works L. Euler, who used the entered character for the first time in 1736 G.
At the end of the 18th century. A.M.Lagendre based on works I.G. Lambert proved that 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 of an algebraic equation. The search for an exact expression for pi continued after the work F. Vieta. At the beginning of the 17th century. Dutch mathematician from Cologne Ludolf van Zeijlen(1540-1610) (some historians call him L. van Keulen) found 32 correct signs. Since then (year of publication 1615), the value of the number p with 32 decimal places has been called the number Ludolph.
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.
It makes sense for anyone who plans to engage in precise calculations in the future 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 a modern method of 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, mathematicians with the help of 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 techniques for mastering a huge amount of information have been developed. 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 is only important to know special memorization techniques 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 a large number of different formulas.
This and other interdependencies allowed mathematicians to further understand the nature of pi.
The exact value of the number π in the modern world is not only of 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. Summing up the work, I come to the conclusion that this topic is 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. I found out why the 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 the numerical value of the number, 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.
Bibliography.
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/
Recently on Habré, in one article, they mentioned the question “What would happen to the world if the number Pi was equal to 4?” I decided to think a little about this topic, using some (albeit not the most extensive) knowledge in the relevant areas of mathematics. If anyone is interested, please see cat.
To imagine such a world, you need to mathematically realize a space with a different ratio of the circumference of a circle to its diameter. This is what I tried to do.
Attempt No. 1.
Let’s say right away that I will only consider two-dimensional spaces. Why? Because the circle, in fact, is defined in two-dimensional space (if we consider the dimension n>2, then the ratio of the measure of the (n-1)-dimensional circle to its radius will not even be a constant).So, to begin with, I tried to come up with at least some space where Pi is not equal to 3.1415... To do this, I took a metric space with a metric in which the distance between two points is equal to the maximum among the modules of the coordinate difference (i.e., the Chebyshev distance).
What form will the unit circle have in this space? Let's take the point with coordinates (0,0) as the center of this circle. Then the set of points, the distance (in the sense of a given metric) from which to the center is 1, is 4 segments parallel to the coordinate axes, forming a square with side 2 and center at zero.
Yes, in some metric it is a circle!
Let's calculate Pi here. The radius is equal to 1, then the diameter, accordingly, is equal to 2. You can also consider the definition of diameter as the greatest distance between two points, but even so it is equal to 2. It remains to find the length of our “circle” in this metric. This is the sum of the lengths of all four segments, which in this metric have length max(0,2)=2. This means the circumference is 4*2=8. Well, then Pi here is equal to 8/2=4. Happened! But should we be very happy? This result is practically useless, because the space in question is absolutely abstract, angles and turns are not even defined in it. Can you imagine a world where the rotation is not actually defined, and where the circle is a square? I tried, honestly, but I didn't have enough imagination.
The radius is 1, but there are some difficulties in finding the length of this “circle”. After some searching on the Internet, I came to the conclusion that in pseudo-Euclidean space such a concept as “Pi” cannot be defined at all, which is certainly bad.
If someone in the comments tells me how to formally calculate the length of a curve in pseudo-Euclidean space, I will be very glad, because my knowledge of differential geometry, topology (as well as diligent Googling) was not enough for this.
Conclusions:
I don’t know if it’s possible to write about the conclusions after such short-term studies, but something can be said. First, when I tried to imagine space with a different number of pi, I realized that it would be too abstract to be a model of the real world. Secondly, when if you try to come up with a more successful model (similar to our real world), it turns out that the number Pi will remain unchanged. If we take for granted the possibility of a negative squared distance (which for an ordinary person is simply absurd), then Pi will not be defined at all! All this suggests that perhaps a world with a different number Pi could not exist at all? It’s not for nothing that the Universe is exactly the way it is. Or maybe this is real, but ordinary mathematics, physics and human imagination are not enough for this. What do you think?Upd. I found out for sure. The length of a curve in a pseudo-Euclidean space can only be determined on some of its Euclidean subspaces. That is, in particular, for the “circumference” obtained in attempt N3, such a concept as “length” is not at all defined. Accordingly, Pi cannot be calculated there either.
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.
