The number π shows how many times the circumference of a circle is greater than its diameter. It doesn't matter what size the circle is, as it was noted at least 4 thousand years ago, the ratio always remains the same. The only question is what does it mean.
To calculate it approximately, an ordinary thread is enough. Greek Archimedes in the 3rd century BC used a more sophisticated method. He drew regular polygons inside and outside the circle. Adding up the lengths of the sides of the polygons, Archimedes more and more accurately determined the fork in which the number π is located, and realized that it was approximately equal to 3.14.
The polygon method was used for almost 2 thousand years after Archimedes, this made it possible to find out the value of the number π up to the 38th digit after the decimal point. One or two more signs - and you can down to the atom calculate the circumference of a circle with a diameter like that of the universe.
While some scientists used the geometric method, others guessed that the number pi can be calculated by adding, subtracting, dividing or multiplying other numbers. Thanks to this, the "tail" has grown to several hundred digits after the decimal point.
With the advent of the first computers and especially modern computers, the accuracy increased by orders of magnitude - in 2016, the Swiss Peter Trub determined the value of the number π up to 22.4 trillion decimal places. If this result is printed on a 14-point line of normal width, the entry will be slightly shorter than the average distance from Earth to Venus.
In principle, nothing prevents achieving even greater accuracy, but for scientific calculations there has long been no need for this - except perhaps for testing computers, algorithms and for research in mathematics. And there is something to explore. Even about the number π itself, not everything is known. Proved that it is written as an infinite non-periodic fraction, that is, there is no limit to the digits after the decimal point, and they do not add up to repeating blocks. But whether numbers and their combinations appear with the same frequency is unclear. Apparently, this is so, but so far no one has provided a rigorous proof.
Further calculations are carried out mainly for sport - and for the same reason people try to remember as many digits after the decimal point as possible. The record belongs to the Indian Rajveer Mina, who in 2015 named 70 thousand characters as a keepsake sitting blindfolded for almost ten hours.
Probably, to surpass his result, you need a special talent. But everyone is capable of simply surprising friends with a good memory. The main thing is to use one of the mnemonic techniques, which can later be useful for something else.
Structure data
The most obvious way is to split the number into identical blocks. For example, you can think of pi as a phone book with ten digit numbers, or you can think of it as a fancy history (and future) textbook that lists years. You won’t remember a lot like that, but to make an impression, a couple of tens of decimal places are enough.
Turn a number into a story
It is believed that the most convenient way to remember numbers is to come up with a story where they will correspond to the number of letters in words (it would be logical to replace zero with a space, but then most words will merge; instead, it is better to use words of ten letters). The phrase "Can I have a large package of coffee beans?" is based on this principle. in English:
May-3,
have-4
large - 5
container - 9
coffee - 6
beans - 5
In pre-revolutionary Russia, they came up with a similar sentence: "Whoever, jokingly and soon wishes (b) Pi to know the number, already knows (b)". Precision - up to the tenth decimal place: 3.1415926536. But it's easier to remember a more modern version: "She was and will be respected at work." There is also a poem: "I know this and remember it perfectly - wee, many signs are superfluous to me, in vain." And the Soviet mathematician Yakov Perelman composed a whole mnemonic dialogue:
What do I know about circles? (3.1415)
So I know the number called pi - well done! (3.1415927)
Learn and know in the number known behind the number the number, how to notice good luck! (3.14159265359)
The American mathematician Michael Keith even wrote a whole book, Not A Wake, the text of which contains information about the first 10 thousand digits of the number π.
Replace numbers with letters
Some people find it easier to remember random letters than random numbers. In this case, the numbers are replaced by the first letters of the alphabet. The first word in the title of the story Cadaeic Cadenza by Michael Keith appeared in this way. In total, 3835 digits of pi are encoded in this work - however, in the same way as in the book Not a Wake.
In Russian, for such purposes, you can use the letters from A to I (the latter will correspond to zero). How convenient it will be to remember the combinations made up of them is an open question.
Come up with images for combinations of numbers
To achieve truly outstanding results, the previous methods are no good. Record breakers use a visualization technique: images are easier to remember than numbers. First you need to match each number with a consonant letter. It turns out that each two-digit number (from 00 to 99) corresponds to a two-letter combination.
Let's say one n- this is "n", four R e - "p", pya T b - "t". Then the number 14 is "nr", and 15 is "nt". Now these pairs should be supplemented with other letters to make words, for example, " n about R a" and " n And T You will need a hundred words in total - it seems like a lot, but there are only ten letters behind them, so remembering is not so difficult.
The number π will appear in the mind as a sequence of images: three integers, a hole, a thread, etc. To better remember this sequence, images can be drawn or printed on a printer and put in front of your eyes. Some people simply lay out the relevant objects around the room and remember the numbers while looking at the interior. Regular training using this method will allow you to remember hundreds and even thousands of decimal places - or any other information, because you can visualize not only numbers.
Marat Kuzaev, Kristina Nedkova
Pi is one of the most popular mathematical concepts. Pictures are written about him, films are made, he is played on musical instruments, poems and holidays are dedicated to him, he is searched for and found in sacred texts.
Who discovered pi?
Who and when first discovered the number π is still a mystery. It is known that the builders of ancient Babylon already used it with might and main when designing. On cuneiform tablets that are thousands of years old, even problems that were proposed to be solved with the help of π have been preserved. True, then it was believed that π is equal to three. This is evidenced by a tablet found in the city of Susa, two hundred kilometers from Babylon, where the number π was indicated as 3 1/8.
In the process of calculating π, the Babylonians discovered that the radius of a circle as a chord enters it six times, and they divided the circle into 360 degrees. And at the same time they did the same with the orbit of the sun. Thus, they decided to consider that there are 360 days in a year.
IN Ancient Egypt pi was 3.16.
IN ancient india – 3,088.
In Italy, at the turn of the epochs, it was believed that π was equal to 3.125.
In Antiquity, the earliest mention of π refers to the famous problem of squaring the circle, that is, the impossibility of constructing a square with a compass and straightedge, the area of \u200b\u200bwhich is equal to the area of a certain circle. Archimedes equated π to the fraction 22/7.
The closest to the exact value of π came in China. It was calculated in the 5th century AD. e. famous Chinese astronomer Zu Chun Zhi. Calculating π is quite simple. It was necessary to write odd numbers twice: 11 33 55, and then, dividing them in half, put the first in the denominator of the fraction, and the second in the numerator: 355/113. The result is consistent with modern calculations of π up to the seventh digit.
Why π - π?
Now even schoolchildren know that the number π is a mathematical constant, equal to the ratio circumference to the length of its diameter and equals π 3.1415926535 ... and further after the decimal point - to infinity.
The number acquired its designation π in a complicated way: at first, the mathematician Outrade called the circumference with this Greek letter in 1647. He took the first letter of the Greek word περιφέρεια - "periphery". In 1706, the English teacher William Jones, in his Review of the Advances of Mathematics, already called the letter π the ratio of the circumference of a circle to its diameter. And the name was fixed by the 18th-century mathematician Leonhard Euler, before whose authority the rest bowed their heads. So pi became pi.
Number uniqueness
Pi is a truly unique number.
1. Scientists believe that the number of characters in the number π is infinite. Their sequence is not repeated. Moreover, no one will ever be able to find repetitions. Since the number is infinite, it can contain absolutely everything, even a Rachmaninov symphony, the Old Testament, your phone number and the year in which the Apocalypse will come.
2. π is related to chaos theory. Scientists came to this conclusion after creating Bailey's computational program, which showed that the sequence of numbers in π is absolutely random, which corresponds to the theory.
3. It is almost impossible to calculate the number to the end - it would take too much time.
4. π is an irrational number, that is, its value cannot be expressed as a fraction.
5. π is a transcendental number. It cannot be obtained by producing any algebraic actions over whole numbers.
6. Thirty-nine decimal places in the number π is enough to calculate the length of a circle encircling known space objects in the Universe, with an error in the radius of a hydrogen atom.
7. The number π is associated with the concept of the "golden section". During measurement Great Pyramid at Giza, archaeologists found that its height is related to the length of its base, just as the radius of a circle is related to its length.
Records related to π
In 2010, Yahoo mathematician Nicholas Zhe was able to calculate two quadrillion decimal places (2x10) in π. It took 23 days, and the mathematician needed a lot of assistants who worked on thousands of computers, united by scattered computing technology. The method allowed making calculations with such a phenomenal speed. It would take more than 500 years to calculate the same on a single computer.
To simply write it all down on paper would require a paper tape over two billion kilometers long. If you expand such a record, its end will go beyond the solar system.
Chinese Liu Chao set a record for memorizing the sequence of digits of the number π. Within 24 hours and 4 minutes, Liu Chao named 67,890 decimal places without making a single mistake.
pi has a lot of fans. It is played on musical instruments, and it turns out that it “sounds” excellently. They remember it and come up with various techniques for this. For the sake of fun, they download it to their computer and brag to each other who downloaded more. Monuments are erected to him. For example, there is such a monument in Seattle. It is located on the steps in front of the Museum of Art.
π is used in decorations and interiors. Poems are dedicated to him, he is searched for in holy books and in excavations. There is even a "Club π".
In the best traditions of π, not one, but two whole days a year are devoted to the number! The first time Pi Day is celebrated on March 14th. It is necessary to congratulate each other at exactly 1 hour, 59 minutes, 26 seconds. Thus, the date and time correspond to the first digits of the number - 3.1415926.
The second time π is celebrated on July 22. This day is associated with the so-called "approximate π", which Archimedes wrote down as a fraction.
Usually on this day π students, schoolchildren and scientists arrange funny flash mobs and actions. Mathematicians, having fun, use π to calculate the laws of a falling sandwich and give each other comic awards.
And by the way, pi can actually be found in holy books. For example, in the Bible. And there the number pi is… three.
14 Mar 2012
On March 14, mathematicians celebrate one of the most unusual holidays - International Pi Day. This date was not chosen by chance: numeric expressionπ (Pi) - 3.14 (3rd month (March) 14th day).
For the first time, schoolchildren are faced with this unusual number already in lower grades in the study of circles and circles. 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 we 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 they use a simplified spelling of the number, 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 system of writing dates (month / day), the date March 14 - 3/14 coincides with the number π (π \u003d 3.1415926 ...). Celebrations usually start at 1:59:26 p.m. (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 a diameter D as (D-D/9) 2 . From this entry it can be seen 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 records indicating that the number π at that time was taken equal to square root out of 10, which gives the fraction 3.162...
In the III century. BC Archimedes in his short work "Measurement of the circle" substantiated three positions:
- Every circle is equal right triangle, whose legs are respectively equal to the circumference and its radius;
- The areas of a circle are related to a square built on a diameter as 11 to 14;
- The ratio of any circle to its diameter is less than 3 1/7 and greater than 3 10/71.
Archimedes substantiated the latter position by sequentially calculating the perimeters of regular inscribed and circumscribed polygons with doubling the number of their sides. According to the exact calculations of Archimedes, the ratio of circumference to diameter is between 3*10/71 and 3*1/7, which means that the number "pi" is equal to 3.1419... true value 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 XV century. 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 correct decimal places: he made 16 doublings of the number of sides of polygons. F. Wiet was the first to notice that π can be found using the limits of some series. This discovery had great importance, it allowed us to calculate π with any accuracy.
In 1706, the English mathematician W. Johnson introduced the notation for the ratio of the circumference to the diameter and designated it modern symbolπ is the first letter of the Greek word periferia, circumference.
For a long period of time, scientists around the world have been trying 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 a root algebraic equation. You cannot specify an algebraic or differential equation, whose root is π, so this number is called transcendental and is calculated by considering a process and 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 a sequence with an accuracy of 10 trillion digits after the decimal point.
The digits of the decimal representation of the number π are quite random. In the decimal expansion of a number, you can find any sequence of digits. It is assumed that in this number in encrypted form there are all written and unwritten books, any information that can only be represented is in the number π.
You can try to solve the mystery of this number yourself. Writing down the number "Pi" in full, of course, will not work. But I propose to the most curious to consider 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 using computer science calculated in ten trillion digits of the number "Pi". The maximum number of digits that a person could remember is one hundred thousand.
To memorize the maximum number of characters of the number "Pi", various poetic "memory" 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 of the words and write it down in order.
So I know the number called "Pi". Well done! (7 digits)
So Misha and Anyuta came running
Pi to know the number they wanted. (11 digits)
This I know and remember very well:
Pi many signs are superfluous to me, in vain.
Let's trust the vast knowledge
Those who have counted, numbers armada. (21 digits)
Once at Kolya and Arina
We ripped the feather beds.
White fluff flew, circled,
Courageous, froze,
blissed out
He gave us
Headache of old women.
Wow, dangerous fluff spirit! (25 characters)
You can use rhyming lines that help you remember the right number.
So that we don't make mistakes
It needs to be read correctly:
ninety two and six
If you try 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 keep repeating:
"Three, fourteen, fifteen,
Nine, twenty-six and five."
Do you have any questions? Want to know more about Pi?
To get help from a tutor, register.
The first lesson is free!
They mentioned the question “What would happen to the world if the number Pi was 4?” I decided to reflect a little on this topic, using some (albeit not the most extensive) knowledge in the relevant areas of mathematics. To whom it is interesting - I ask under cat.
To imagine such a world, it is necessary 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 #1.
We will stipulate at once that I will consider only 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 for starters, 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 kind will it have unit circle in this space? Let's take a point with coordinates (0,0) as the center of this circle. Then the set of points, the distance (in the sense of the given metric) from which to the center is equal to 1, is 4 segments parallel to the coordinate axes, forming a square with side 2 and centered at zero.
Yes, in some metric it is a circle!
Let's calculate Pi here. The radius is 1, so the diameter is 2, respectively. You can also consider the definition of the diameter as the largest distance between two points, but even so it is 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 the length max(0,2)=2. So the circumference is 4*2=8. Well, then Pi here is equal to 8/2=4. Happened! But is it really necessary to rejoice? This result is practically useless, because the space in question is absolutely abstract, it does not even define angles and turns. Can you imagine a world where no turn is actually defined and where the circle is a square? I tried, honestly, but I didn't have the imagination.
The radius is 1, but there are some difficulties with finding the length of this “circle”. After some searching for information on the Internet, I came to the conclusion that in a pseudo-Euclidean space, such a concept as “Pi number” 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 happy, because my knowledge of differential geometry, topology (as well as hard googling) was not enough for this.
Conclusions:
I don’t know if it’s possible to write about the conclusions after such not very long studies, but something can be said. First, when I tried to imagine a 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 better model (similar to ours, 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 ordinary person- just absurd), then Pi will not be defined at all! All this suggests that, perhaps, a world with a different Pi number could not exist at all? After all, it is not for nothing that the Universe is exactly the way it is. Or maybe this is real, only ordinary mathematics, physics and human imagination are not enough for this. What do you think?Upd. I knew 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 “circle” obtained in the attempt N3, such a concept as “length” is not defined at all. Accordingly, Pi cannot be calculated there either.
To calculate any large number of signs of pi, the previous method is no longer suitable. But there is a large number of sequences that converge to Pi much faster. Let's use, for example, the Gauss formula:
| p | = 12 arctan | 1 | + 8 arctan | 1 | - 5 arctan | 1 |
| 4 | 18 | 57 | 239 |
The proof of this formula is simple, so we will omit it.
Program source, including "long arithmetic"
The program calculates NbDigits of the first digits of Pi. The arctan calculation function is named arccot, since arctan(1/p) = arccot(p), but the calculation is carried out according to the Taylor formula for the arctangent, namely arctan(x) = x - x 3 /3 + x 5 /5 - . .. x=1/p, so arccot(x) = 1/p - 1 / p 3 / 3 + ... Calculations are recursive: the previous element of the sum is divided and gives the next one.
/* ** Pascal Sebah: September 1999 ** ** Subject: ** ** A very easy program to compute Pi with many digits. ** No optimisations, no tricks, just a basic program to learn how ** to compute in multiprecision. ** ** Formulae: ** ** Pi/4 = arctan(1/2)+arctan(1/3) (Hutton 1) ** Pi/4 = 2*arctan(1/3)+arctan(1/ 7) (Hutton 2) ** Pi/4 = 4*arctan(1/5)-arctan(1/239) (Machin) ** Pi/4 = 12*arctan(1/18)+8*arctan(1 /57)-5*arctan(1/239) (Gauss) ** ** with arctan(x) = x - x^3/3 + x^5/5 - ... ** ** The Lehmer"s measure is the sum of the inverse of the decimal ** logarithm of the pk in the arctan(1/pk). The more the measure ** is small, the more the formula is efficient. ** For example, with Machin"s formula: ** ** E = 1/log10(5)+1/log10(239) = 1.852 ** ** Data: ** ** A big real (or multiprecision real) is defined in base B as: ** X = x(0) + x(1)/B^1 + ... + x(n-1)/B^(n-1) ** where 0<=x(i)Work with double instead of long and the base B can ** be choosen as 10^8 ** => During the iterations the numbers you add are smaller ** and smaller, take this in account in the +, *, / ** => In the division of y=x/d, you may precompute 1/d and ** avoid multiplications in the loop (only with doubles) ** => MaxDiv may be increased to more than 3000 with doubles ** => . .. */#includeOf course, these are not the most efficient ways to calculate pi. There are many more formulas. For example, Chudnovsky's formula, variations of which are used in Maple. However, in normal programming practice, the Gauss formula is enough, so these methods will not be described in the article. It is unlikely that anyone wants to calculate billions of digits of pi, for which a complex formula gives a large increase in speed.
