Mathematical number pi. Who discovered the number Pi? History of calculations

(), and it became generally accepted after the work of Euler. This designation comes from the initial letter of the Greek words περιφέρεια - circle, periphery and περίμετρος - perimeter.

Ratings

• 510 decimal places: π ≈ 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 592 307 816 406 286 208 998 628 034 825 342 117 067 982 148 086 513 282 306 647 093 844 609 550 582 231 725 359 408 128 481 117 450 284 102 701 938 521 105 559 644 622 948 954 930 381 964 428 810 975 665 933 446 12 8 475 648 233 786 783 165 271 201 909 145 648 566 923 460 348 610 454 326 648 213 393 607 260 249 141 273 724 587 006 606 315 588 174 881 520 920 962 829 254 091 715 364 367 892 590 360 011 330 530 8 820 466 521 384 146 951 941 511 609 433 057 270 365 759 591 953 092 186 117 381 932 611 793 105 118 548 074 462 379 962 749 567 351 885 752 724 891 227 938 183 011 949 129 833 673 362…

Properties

Ratios

There are many known formulas with the number π:

• Wallis formula:

• Euler's identity:

• T.n. "Poisson integral" or "Gauss integral"

Transcendence and irrationality

Unsolved problems

• It is not known whether the numbers π and e algebraically independent.
• It is unknown whether the numbers π + e , π − e , π e , π / e , π e , π π , e e transcendental.
• Until now, nothing is known about the normality of the number π; it is not even known which of the digits 0-9 appear in the decimal representation of the number π an infinite number of times.

Calculation history

and Chudnovsky

Mnemonic rules

So that we do not make mistakes, We must read 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.”

2. Count the number of letters in each word in the phrases below ( excluding punctuation marks) and write down these numbers in a row - not forgetting about the decimal point after the first digit “3”, of course. The result will be an approximate number of Pi.

This I know and remember perfectly: But many signs are unnecessary for me, in vain.

Whoever, jokingly and soon, wishes Pi to know the number - already knows!

So Misha and Anyuta came running and wanted to find out the number.

(The second mnemonic is correct (with rounding of the last digit) only when using pre-reform spelling: when counting the number of letters in words, it is necessary to take into account hard signs!)

Another version of this mnemonic notation:

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

If you follow the poetic meter, you can quickly remember:

Three, fourteen, fifteen, nine two, six five, three five
Eight nine, seven and nine, three two, three eight, forty six
Two six four, three three eight, three two seven nine, five zero two
Eight eight and four, nineteen, seven, one

Fun facts

Notes

See what “Pi” is in other dictionaries:

number- Receiving source: GOST 111 90: Sheet glass. Specifications original document See also related terms: 109. The number of betatron oscillations ... Dictionary-reference book of terms of normative and technical documentation

Noun, s., used. very often Morphology: (no) what? numbers, what? number, (see) what? number, what? number, about what? about number; pl. What? numbers, (no) what? numbers, why? numbers, (see) what? numbers, what? numbers, about what? about numbers mathematics 1. By number... ... Dictionary Dmitrieva

NUMBER, numbers, plural. numbers, numbers, numbers, cf. 1. The concept that serves as an expression of quantity, something with the help of which objects and phenomena are counted (mat.). Integer. A fractional number. Named number. Prime number. (see simple 1 in 1 value).… … Ushakov's Explanatory Dictionary

An abstract designation devoid of special content for any member of a certain series, in which this member is preceded or followed by some other specific member; abstract individual feature that distinguishes one set from... ... Philosophical Encyclopedia

Number- Number is a grammatical category that expresses the quantitative characteristics of objects of thought. Grammatical number is one of the manifestations of a more general language category quantity (see Category linguistic) along with lexical manifestation (“lexical... ... Linguistic encyclopedic dictionary

A number approximately equal to 2.718, which is often found in mathematics and natural sciences. For example, when a radioactive substance decays after time t, a fraction equal to e kt remains of the initial amount of the substance, where k is a number,... ... Collier's Encyclopedia

A; pl. numbers, sat, slam; Wed 1. A unit of account expressing a particular quantity. Fractional, integer, prime hours. Even, odd hours. Count in round numbers (approximately, counting in whole units or tens). Natural h. (positive integer... encyclopedic Dictionary

Wed. quantity, by count, to the question: how much? and the very sign expressing quantity, number. Without number; there is no number, without counting, many, many. Set up cutlery according to the number of guests. Roman, Arabic or church numbers. Integer, opposite. fraction... ... Dahl's Explanatory Dictionary

Studying Pi numbers starts at primary school, when students study the circle, circumference and the value of Pi is encountered. Since the value of Pi is a constant meaning the ratio of the length of the circle itself to the length of the diameter of a given circle. For example, if we take a circle whose diameter is equal to one, then its length is equal to Pi number. This value of Pi is infinite in mathematical continuation, but there is also a generally accepted designation. It comes from a simplified spelling of the value of Pi, it looks like 3.14.

The Historical Birth of Pi

The number Pi supposedly got its roots in Ancient Egypt. Since ancient Egyptian scientists calculated the area of ​​a circle using diameter D, which took the value D - D/92. Which corresponded to 16/92, or 256/81, which means Pi is 3.160.
India in the sixth century BC also touched on the number Pi, in the religion of Jainism, records were found that stated that the number Pi is equal to 10 in the square root, which means 3.162.

Archimedes' teachings on the measurement of the circle in the third century BC led him to the following conclusions:

Later, he substantiated his conclusions by a sequence of calculations using examples of correctly inscribed or described polygonal shapes with doubling the number of sides of these figures. In precise calculations, Archimedes concluded the ratio of diameter and circumference in numbers between 3 * 10/71 and 3 * 1/7, therefore the value of Pi is 3.1419... Since we have already talked about the infinite form of this value, it looks like 3, 1415927... And this is not the limit, because the mathematician Kashi in the fifteenth century calculated the value of Pi as a sixteen-digit value.
English mathematician Johnson W. in 1706, began to use the symbol pi for the symbol? (from Greek it is the first letter in the word circle).

Mysterious meaning.

The value of Pi is irrational and cannot be expressed in fraction form because fractions use whole values. It cannot be a root in the equation, which is why it also turns out to be transcendental; it is found by considering any processes, being refined by large quantity steps under consideration this process. There have been many attempts to calculate the largest number of decimal places in Pi, which have resulted in tens of trillions of digits of a given decimal value.

Interesting fact: Oddly enough, the value of Pi has its own holiday. It is called International Pi Day. It is celebrated on March 14th. The date appeared thanks to the very value of Pi 3.14 (mm.yy) and the physicist Larry Shaw, who was the first to celebrate this holiday in 1987.

Note: Legal assistance in obtaining a certificate of absence (presence) of a criminal record for all citizens of the Russian Federation. Follow the link to the state service certificate of no criminal record (http://conviction certificate.rf/) legally, quickly and without queues!

Math enthusiasts around the world eat a piece of pie every year on the fourteenth of March - after all, it is the day of Pi, the most famous irrational number. This date is directly related to the number whose first digits are 3.14. Pi is the ratio of the circumference of a circle to its diameter. Since it is irrational, it is impossible to write it as a fraction. This is an infinitely long number. It was discovered thousands of years ago and has been constantly studied since then, but does Pi still have any secrets? From ancient origin until the uncertain future, here are some of the most interesting facts about Pi.

Memorizing Pi

The record for memorizing decimal numbers belongs to Rajvir Meena from India, who managed to remember 70,000 digits - he set the record on March 21, 2015. Previously, the record holder was Chao Lu from China, who managed to remember 67,890 digits - this record was set in 2005. The unofficial record holder is Akira Haraguchi, who recorded himself on video repeating 100,000 digits in 2005 and recently published a video where he manages to remember 117,000 digits. The record would become official only if this video was recorded in the presence of a representative of the Guinness Book of Records, and without confirmation it remains only impressive fact, but is not considered an achievement. Math enthusiasts love to memorize the number Pi. Many people use various mnemonic techniques, for example poetry, where the number of letters in each word matches the digits of Pi. Each language has its own versions of similar phrases that help you remember both the first few numbers and the whole hundred.

There is a Pi language

Mathematicians, passionate about literature, invented a dialect in which the number of letters in all words corresponds to the digits of Pi in exact order. Writer Mike Keith even wrote a book, Not a Wake, which is entirely written in Pi. Enthusiasts of such creativity write their works in full accordance with the number of letters and the meaning of numbers. This has no practical application, but is quite common and known phenomenon in the circles of enthusiastic scientists.

Exponential growth

Pi is an infinite number, so by definition people will never be able to establish the exact digits of this number. However, the number of decimal places has increased greatly since Pi was first used. The Babylonians also used it, but a fraction of three whole and one eighth was enough for them. Chinese and creators Old Testament and were completely limited to three. By 1665, Sir Isaac Newton had calculated the 16 digits of Pi. By 1719, the French mathematician Tom Fante de Lagny had calculated 127 digits. The advent of computers has radically improved human knowledge of Pi. From 1949 to 1967 the number known to man digits skyrocketed from 2037 to 500,000. Not long ago, Peter Trueb, a scientist from Switzerland, was able to calculate 2.24 trillion digits of Pi! It took 105 days. Of course, this is not the limit. It is likely that with the development of technology it will be possible to establish an even more accurate figure - since Pi is infinite, there is simply no limit to accuracy, and it can only be limited technical features computer technology.

Calculating Pi by hand

If you want to find the number yourself, you can use the old-fashioned technique - you will need a ruler, a jar and some string, or you can use a protractor and a pencil. The downside to using a can is that it needs to be round and accuracy will be determined by how well a person can wrap the rope around it. You can draw a circle with a protractor, but this also requires skill and precision, as an uneven circle can seriously distort your measurements. A more accurate method involves using geometry. Divide a circle into many segments, like a pizza into slices, and then calculate the length of a straight line that would turn each segment into isosceles triangle. The sum of the sides will give the approximate number Pi. The more segments you use, the more accurate the number will be. Of course, in your calculations you will not be able to come close to the results of a computer, nevertheless these simple experiments allow you to understand in more detail what the number Pi actually is and how it is used in mathematics.

Discovery of Pi

The ancient Babylonians knew about the existence of the number Pi already four thousand years ago. Babylonian tablets calculate Pi as 3.125, and an Egyptian mathematical papyrus shows the number 3.1605. In the Bible, Pi is given in the obsolete length of cubits, and the Greek mathematician Archimedes used the Pythagorean theorem, a geometric relationship between the length of the sides of a triangle and the area of ​​the figures inside and outside the circles, to describe Pi. Thus, we can say with confidence that Pi is one of the most ancient mathematical concepts, although the exact name of this number appeared relatively recently.

New look at Pi

Even before the number Pi began to be correlated with circles, mathematicians already had many ways to even name this number. For example, in ancient mathematics textbooks one can find a phrase in Latin that can be roughly translated as “the quantity that shows the length when the diameter is multiplied by it.” The irrational number became famous when the Swiss scientist Leonhard Euler used it in his work on trigonometry in 1737. However, the Greek symbol for Pi was still not used - this only happened in a book by a lesser-known mathematician, William Jones. He used it already in 1706, but it went unnoticed for a long time. Over time, scientists adopted this name, and now it is the most famous version of the name, although it was previously also called the Ludolf number.

Is Pi a normal number?

The number Pi is definitely strange, but how much does it obey the normal ones? mathematical laws? Scientists have already resolved many questions related to this irrational number, but some mysteries remain. For example, it is not known how often all the numbers are used - the numbers 0 to 9 should be used in equal proportion. However, statistics can be traced from the first trillions of digits, but due to the fact that the number is infinite, it is impossible to prove anything for sure. There are other problems that still elude scientists. It is quite possible that further development of science will help shed light on them, but this moment it remains beyond human intellect.

Pi sounds divine

Scientists cannot answer some questions about the number Pi, however, every year they understand its essence better and better. Already in the eighteenth century, the irrationality of this number was proven. In addition, the number has been proven to be transcendental. This means no a certain formula, which would allow us to calculate Pi using rational numbers.

Dissatisfaction with the number Pi

Many mathematicians are simply in love with Pi, but there are also those who believe that these numbers are not particularly significant. In addition, they claim that the Tau number, which is twice as large as Pi, is more convenient to use as an irrational number. Tau shows the relationship between circumference and radius, which some believe represents a more logical method of calculation. However, to unambiguously determine something in this issue impossible, and one and the other number will always have supporters, both methods have the right to life, so it’s simple interesting fact, and not a reason to think that you shouldn’t use Pi.

***

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! This is a number that unites all circles, circles and roundness, which in particular include my mother’s ring, the wheel from my father’s favorite car, and even the saucer of my favorite cat Murzik. I'm willing to bet that in the ranking of the most popular physical and mathematical constants, Pi will undoubtedly take first place. But what is hidden behind it? Maybe some terrible curse words from mathematicians? Let's try to understand this issue.

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

Modern number designation π (Pi) appeared thanks to the English mathematician Johnson in 1706. This is the first letter of the Greek word περιφέρεια (periphery, or circle). For those who took mathematics a long time ago, and besides, by no means, let us remind you that the number Pi is the ratio of the circumference of a circle to its diameter. The value is a constant, that is, constant for any circle, regardless of its radius. People knew about this in ancient times. So in ancient Egypt Pi number was taken equal to the ratio 256/81, and in Vedic texts the value is given as 339/108, while Archimedes proposed a ratio of 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 all of this mean? Very simple. Do you want to know the phone number of the girl you like? It can probably be found in the sequence of digits after the decimal point of Pi.

You can see the phone number here ↓

Pi number accurate to 10,000 digits.

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

Didn't find it? Then take a look.

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

What is Pi equal to? Methods for calculating it:

1. Experimental method. If 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 shortcomings, affecting the accuracy of the resulting number Pi. Firstly, the error measuring instruments(in our case, this is a ruler with a thread), and secondly, there is no guarantee that the circumference we are measuring will have correct form. Therefore, it is not surprising that mathematics has given us many other methods for calculating π, where there is no need to make precise measurements.

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

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 Enough interesting method 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 it be more random than 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 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 calculation formula will look like this: π=4N cr / N square.

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

Tau number (Instead of a conclusion).

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

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 was noticed at least 4 thousand years ago, the ratio always remains the same. The only question is what it equals.

To calculate it approximately, an ordinary thread is enough. Greek Archimedes in the 3rd century BC. used more tricky way. He drew circles inside and outside regular polygons. By adding the lengths of the sides of the polygons, Archimedes more and more precisely 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 decimal place. One or two more signs - and you can calculate with atomic precision the length of a circle with a diameter like the Universe.

While some scientists used the geometric method, others realized that the number π could be calculated by adding, subtracting, dividing or multiplying other numbers. Thanks to this, the “tail” grew to several hundred decimal places.

With the advent of the first computers and especially modern computers, the accuracy has increased by orders of magnitude - in 2016, the Swiss Peter Trüb determined the value of the number π to 22.4 trillion decimal places. If you print this result in 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 us from achieving even greater accuracy, but for scientific calculations there is no need for this for a long time - except for testing computers, algorithms and for research in mathematics. And there is a lot to explore. Not everything is known even about the number π itself. It has been proven that it is written as an infinite non-periodic fraction, that is, there is no limit to the numbers after the decimal point, and they do not add up to repeating blocks. But it is unclear whether numbers and their combinations appear with the same frequency. Apparently this is true, but no one has yet provided rigorous proof.

Further calculations are carried out mainly for sport - and for the same reason people try to remember as many decimal places as possible. The record belongs to the Indian Rajvir Meena, who in 2015 named 70 thousand characters from memory while sitting blindfolded for almost ten hours.

Probably, to surpass his result, you need a special talent. But everyone can simply surprise their friends with a good memory. The main thing is to use one of the mnemonic techniques, which can then be useful for something else.

Structure data

The most obvious way is to split the number into equal blocks. For example, you can think of π as a phone book with ten-digit numbers, or you can think of it as a fancy history (and future) textbook listing the years. You won’t remember much, but a couple of dozen decimal places are enough to make an impression.

Turn a number into a story

It is believed that the most convenient way remember numbers - 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 came up with a similar sentence: “Whoever, jokingly and soon, wishes (b) Pi to know the number, already knows (b).” Accuracy - up to the tenth decimal place: 3.1415926536. But it's easier to remember more modern version: “She was and will be respected at work.” There is also a poem: “I know this and remember it perfectly - no, many signs are unnecessary for me, in vain.” And the Soviet mathematician Yakov Perelman composed an entire mnemonic dialogue:

What do I know about circles? (3.1415)

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)

American mathematician Michael Keith even wrote an entire 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 Michael Keith's story Cadaeic Cadenza 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 similar purposes, you can use letters from A to I (the latter will correspond to zero). How convenient it will be to remember the combinations made from them is an open question.

Come up with images for combinations of numbers

To achieve truly outstanding results, previous methods will not work. Record holders use visualization techniques: 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", fours R e - "r", pya T b - "t". Then the number 14 is “nr”, and 15 is “nt”. Now these pairs should be supplemented with other letters to form words, for example, " n O R a" and " n And T b". In total, you will need a hundred words - it seems like a lot, but there are only ten letters behind them, so it’s not that difficult to remember.

The number π will appear in the mind as a sequence of images: three whole numbers, a hole, a thread, etc. To better remember this sequence, the images can be drawn or printed and placed before your eyes. Some people simply place the corresponding items 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