The value of p. What is special about Pi? Mathematician answers

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

Ratings

• 510 signs after aim: π ≈ 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 592 307 816 406 286 208 998 628 034 825 342 067 982 148 086 513 282 306 644 69 69 69 550 582 231 725 359 408 128 481 117 450 284 102 701 938 521 105 559 644 622 948 954 930 381 964 428 810 975 665 933 446 128 475 648 233 786 783 165 271 201 909 145 648 566 923 460 213 393 607 260 249 141 273 724 587 006 606 315 588 174 881 520 920 962 829 254 091 715 364 367 892 590 360 011 330 548 820 466 521 384 146 951 941 511 609 433 057 270 365 765 757 59 381 932 611 793 105 118 548 074 462 379 962 749 567 351 885 752 724 891 227 938 183 011 949 129 833 673 362…

Properties

Ratios

There are many formulas with the number π:

• Wallis formula:

• Euler's identity:

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

Transcendence and irrationality

Unresolved issues

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

Calculation history

and Chudnovsky

Mnemonic rules

In order not to 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. So that engage in science, This everyone should know. 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 ( ignoring punctuation marks) and write down these numbers in a row - not forgetting the decimal point after the first digit "3", of course. Get an approximate number of Pi.

This I know and remember perfectly: And many signs are superfluous to me, in vain.

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

So Misha and Anyuta ran to Pi to find out the number they wanted.

(The second mnemonic is correct (with rounding of the last digit) only when using pre-reform orthography: when counting the number of letters in words, hard signs must be taken into account!)

Another version of this mnemonic notation:

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.

If you follow the poetic size, 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

funny facts

Notes

See what "Pi" is in other dictionaries:

number- Reception Source: GOST 111 90: Sheet glass. Specifications original document See also related terms: 109. Number of betatron oscillations … Dictionary-reference book of terms of normative and technical documentation

Ex., s., use. very often Morphology: (no) what? numbers for what? number, (see) what? number than? number about what? about the number; pl. What? numbers, (no) what? numbers for what? numbers, (see) what? numbers than? numbers about what? about mathematics numbers 1. Number ... ... Dictionary Dmitrieva

NUMBER, numbers, pl. numbers, numbers, numbers, cf. 1. A 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 simple1 in 1 value).… … Explanatory Dictionary of Ushakov

An abstract designation, devoid of special content, of any member of a certain series, in which this member is preceded or followed by some other definite member; an 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 one of the manifestations of a more general linguistic category of quantity (see Linguistic Category) along with a lexical manifestation (“lexical ... ... Linguistic Encyclopedic Dictionary

A number approximately equal to 2.718, which is often found in mathematics and science. For example, during the decay of a radioactive substance after time t, a fraction equal to e kt remains from the initial amount of substance, where k is a number, ... ... Collier Encyclopedia

A; pl. numbers, villages, slam; cf. 1. A unit of account expressing one or another quantity. Fractional, integer, simple hours. Even, odd hours. Count as round numbers (approximately, counting as whole units or tens). Natural hours (positive integer ... encyclopedic Dictionary

Wed quantity, count, to the question: how much? and the very sign expressing quantity, the figure. Without number; no number, no count, many many. Put the appliances according to the number of guests. Roman, Arabic or church numbers. Integer, contra. fraction. ... ... Dahl's Explanatory Dictionary

There are a lot of mysteries among the PIs. Rather, these are not even riddles, but a kind of some kind of Truth that no one has yet figured out in the entire history of mankind ...

What is Pi? The PI number is a mathematical "constant" that expresses the ratio of the circumference of a circle to its diameter. At first, due to ignorance, it (this ratio) was considered equal to three, which was roughly approximate, but they were enough. But when prehistoric times gave way to ancient times (that is, already historical), then there was no limit to the surprise of inquisitive minds: it turned out that the number three very inaccurately expresses this ratio. With the passage of time and the development of science, this number began to be considered equal to twenty-two-sevenths.

The English mathematician August de Morgan once called the number PI "... the mysterious number 3.14159... that crawls through the door, through the window and through the roof." Tireless scientists continued and continued to calculate the decimal places of the number Pi, which is actually a wildly non-trivial task, because you can’t just calculate it in a column: the number is not only irrational, but also transcendental (these are just such numbers that are not calculated by simple equations).

In the process of calculating these very signs, many different scientific methods and entire sciences. But the most important thing is that there are no repetitions in the decimal part of pi, as in an ordinary periodic fraction, and the number of decimal places in it is infinite. To date, it has been verified that there really are no repetitions in 500 billion digits of the number pi. There are reasons to believe that they do not exist at all.

Since there are no repetitions in the sequence of signs of the number pi, this means that the sequence of signs of the number pi obeys chaos theory, more precisely, the number pi is chaos written in numbers. Moreover, if desired, this chaos can be represented graphically, and there is an assumption that this Chaos is reasonable.

In 1965, the American mathematician M. Ulam, sitting at a boring meeting, from nothing to do, began to write numbers included in the number pi on checkered paper. Putting 3 in the center and moving in a counterclockwise spiral, he wrote out 1, 4, 1, 5, 9, 2, 6, 5 and other numbers after the decimal point. Along the way, he circled all the prime numbers. What was his surprise and horror when the circles began to line up along the straight lines!

In the decimal tail of pi, you can find any conceived sequence of digits. Any sequence of digits in decimal places of pi will sooner or later be found. Any!

So what? - you ask. And then. Estimate: if your phone is there (and it is), then there is also the phone of the girl who did not want to give you her number. Moreover, there are also credit card numbers, and even all the values winning numbers tomorrow's lottery draw. Why, in general, all lotteries for many millennia to come. The question is how to find them there ...

If you encrypt all the letters in numbers, then in the decimal expansion of the number pi you can find all the world literature and science, and the recipe for making bechamel sauce, and that’s it. holy books all religions. It's strict scientific fact. After all, the sequence is INFINITE and combinations in the number PI are not repeated, therefore it contains ALL combinations of numbers, and this has already been proven. And if everything, then everything. Including those that correspond to the book you have chosen.

And this again means that it contains not only all world literature, which has already been written (in particular, those books that burned down, etc.), but also all the books that WILL be written. Including your articles on the sites. It turns out that this number (the only reasonable number in the Universe!) controls our world. You just need to consider more signs, find the right area and decipher it. This is something akin to a paradox with a herd of chimpanzees hammering on the keyboard. With a long enough (one can even estimate this time) experiment, they will print all of Shakespeare's plays.

This immediately suggests an analogy with periodically appearing reports that the Old Testament allegedly encoded messages to posterity that can be read with the help of ingenious programs. It is not entirely wise to dismiss such an exotic feature of the Bible right off the bat, caballists have been searching for such prophecies for centuries, but I would like to cite the message of one researcher who, using a computer, found in the Old Testament the words that there are no prophecies in the Old Testament. Most likely in very big text, just as in the infinite digits of the number PI, you can not only encode any information, but also “find” phrases that were not originally included there.

For practice, within the Earth, 11 characters after the dot are enough. Then, knowing that the radius of the Earth is 6400 km or 6.4 * 1012 millimeters, it turns out that, having discarded the twelfth digit in the number of PI after the point when calculating the length of the meridian, we will be mistaken by several millimeters. And when calculating the length of the Earth's orbit during rotation around the Sun (as you know, R \u003d 150 * 106 km \u003d 1.5 * 1014 mm), for the same accuracy, it is enough to use the number PI with fourteen digits after the point, but what’s there to trifle - the diameter of our Galaxies are about 100,000 light years (1 light year is approximately equal to 1013 km) or 1018 km or 1030 mm. and them on this moment calculated to 12411 trillion signs!!!

The absence of periodically repeating figures, namely, based on their formula Circumference = Pi * D, the circle does not close, since there is no finite number. This fact can also be closely related to the spiral manifestation in our lives...

There is also a hypothesis that all (or some) universal constants (Planck's constant, Euler number, universal gravitational constant, electron charge, etc.) change their values ​​over time, as the curvature of space changes due to the redistribution of matter or for other reasons unknown to us.

At the risk of incurring the wrath of the enlightened community, we can assume that the number of PI considered today, which reflects the properties of the Universe, may change over time. In any case, no one can forbid us to re-find the value of the number PI, confirming (or not confirming) the existing values.

10 Interesting Facts About Pi

1. The history of number has more than one millennium, almost as long as the science of mathematics exists. Certainly, exact value numbers were not calculated immediately. At first, the ratio of the circumference to the diameter was considered equal to 3. But over time, when architecture began to develop, a more accurate measurement was required. By the way, the number existed, but it received a letter designation only at the beginning of the 18th century (1706) and comes from the initial letters of two Greek words meaning “circumference” and “perimeter”. The mathematician Jones endowed the number with the letter "π", and she firmly entered mathematics already in 1737.

2. IN different eras and at different peoples pi has different meaning. For example, in Ancient Egypt it was equal to 3.1604, among the Hindus it acquired the value of 3.162, the Chinese used the number equal to 3.1459. Over time, π was calculated more and more accurately, and when computer technology appeared, that is, a computer, it began to have more than 4 billion characters.

3. There is a legend, more precisely, experts believe that the number Pi was used in the construction of the Tower of Babel. However, it was not the wrath of God that caused its collapse, but incorrect calculations during construction. Like, the ancient masters were mistaken. A similar version exists regarding Solomon's temple.

4. It is noteworthy that they tried to introduce the value of the number Pi even at the state level, that is, through the law. In 1897, a bill was drafted in the state of Indiana. Pi was 3.2 according to the document. However, scientists intervened in time and thus prevented an error. In particular, Professor Purdue, who was present at the legislative assembly, spoke out against the bill.

5. Interestingly, several numbers in the infinite sequence Pi have their own name. So, six nines of Pi are named after an American physicist. Once Richard Feynman was giving a lecture and stunned the audience with a remark. He said he wanted to learn the digits of pi up to six nines by heart, only to say "nine" six times at the end of the story, hinting that its meaning was rational. When in fact it is irrational.

6. Mathematicians around the world do not stop doing research related to the number Pi. It is literally shrouded in mystery. Some theorists even believe that it contains a universal truth. To share knowledge and new information about Pi, organized the Pi Club. Entering it is not easy, you need to have an outstanding memory. So, those wishing to become a member of the club are examined: a person must tell as many signs of the number Pi from memory as possible.

7. They even came up with various techniques for remembering the number Pi after the decimal point. For example, they come up with whole texts. In them, words have the same number of letters as the corresponding digit after the decimal point. To further simplify the memorization of such a long number, they compose verses according to the same principle. Members of the Pi Club often have fun in this way, and at the same time train their memory and ingenuity. For example, Mike Keith had such a hobby, who eighteen years ago came up with a story in which each word was equal to almost four thousand (3834) first digits of pi.

8. There are even people who have set records for memorizing Pi signs. So, in Japan, Akira Haraguchi memorized more than eighty-three thousand characters. But the domestic record is not so outstanding. A resident of Chelyabinsk was able to memorize only two and a half thousand numbers after the decimal point of Pi.

9. Pi Day has been celebrated for more than a quarter of a century, since 1988. Once, a physicist from the Popular Science Museum in San Francisco, Larry Shaw, noticed that March 14 was spelled the same as pi. In a date, the month and day form 3.14.

10. There is an interesting coincidence. March 14 was born the great scientist Albert Einstein, who created, as you know, the theory of relativity.

Mathematicians all over the world eat a piece of cake every year on March 14 - after all, this 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 ever since, but does Pi have any secrets left? From ancient origin until an indefinite future, here are some of the most interesting facts about pi.

Memorizing Pi

The record for remembering numbers after the decimal point belongs to Rajveer Meena from India, who managed to remember 70,000 digits - he set the record on March 21, 2015. Before that, the record holder was Chao Lu from China, who managed to memorize 67,890 digits - this record was set in 2005. The unofficial record holder is Akira Haraguchi, who videotaped his repetition of 100,000 digits in 2005 and recently posted a video where he manages to remember 117,000 digits. An official record would only become if this video was recorded in the presence of a representative of the Guinness Book of Records, and without confirmation it remains only an impressive fact, but is not considered an achievement. Mathematics enthusiasts love to memorize the number Pi. Many people use various mnemonic techniques, such as poetry, where the number of letters in each word is the same as pi. Each language has its own variants of such phrases, which help to remember both the first few digits and a whole hundred.

There is a Pi language

Fascinated by literature, mathematicians 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 completely written in the Pi language. Enthusiasts of such creativity write their works in full accordance with the number of letters and the meaning of the numbers. It has no application but is quite common and famous phenomenon in circles of enthusiastic scientists.

Exponential Growth

Pi is an infinite number, so people, by definition, will never be able to figure out the exact numbers of this number. However, the number of digits after the decimal point has increased greatly since the first use of the Pi. Even the Babylonians used it, but a fraction of three and one eighth was enough for them. The Chinese and the creators of the Old Testament were completely limited to the three. By 1665, Sir Isaac Newton had calculated 16 digits of pi. By 1719, French mathematician Tom Fante de Lagny had calculated 127 digits. The advent of computers has radically improved man's knowledge of Pi. From 1949 to 1967 the number known to man numbers skyrocketed from 2037 to 500,000. Not so long ago, Peter Trueb, a scientist from Switzerland, was able to calculate 2.24 trillion digits of Pi! This 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 only the technical features of computer technology can limit it.

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 string, you can also use a protractor and a pencil. The downside to using a jar is that it has to be round, and accuracy will be determined by how well the person can wrap the rope around it. It is possible to 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 the use of geometry. Divide the circle into many segments, like pizza slices, and then calculate the length of a straight line that would turn each segment into an isosceles triangle. The sum of the sides will give an approximate number of pi. The more segments you use, the more accurate the number will be. Of course, in your calculations you will not be able to approach the results of a computer, nevertheless these simple experiments allow you to understand in more detail what the number pi is in general 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. The Babylonian tablets calculate Pi as 3.125, and the Egyptian mathematical papyrus contains the number 3.1605. In the Bible, the number Pi is given in an obsolete length - in cubits, and the Greek mathematician Archimedes used the Pythagorean theorem to describe Pi, the geometric ratio of the length of the sides of a triangle and the area of ​​\u200b\u200bthe figures inside and outside the circles. Thus, it is safe to say that Pi is one of the most ancient mathematical concepts, although the exact name given number and appeared relatively recently.

A new take on Pi

Even before pi was related to circles, mathematicians already had many ways to even name this number. For example, in old mathematics textbooks one can find a phrase in Latin, which 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 - it only happened in a book less famous mathematician William Jones. He used it as early as 1706, but it was long neglected. Over time, scientists adopted this name, and now this is the most famous version of the name, although before it was also called the Ludolf number.

Is pi normal?

The number pi is definitely strange, but how does it obey the normal 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 digits are used - the numbers from 0 to 9 should be used in equal proportion. However, statistics can be traced for the first trillion 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 science will help to shed light on them, but for the moment this remains beyond the scope of human intelligence.

Pi sounds divine

Scientists cannot answer some questions about the number Pi, however, every year they understand its essence better. Already in the eighteenth century, the irrationality of this number was proved. In addition, it has been proved that the number is transcendental. This means that there is no definite formula that would allow you to calculate pi using rational numbers.

Dissatisfaction with Pi

Many mathematicians are simply in love with Pi, but there are those who believe that these numbers have no special significance. In addition, they claim that the number Tau, which is twice the size of Pi, is more convenient to use as an irrational one. Tau shows the relationship between the circumference and the radius, which, according to some, represents a more logical method of calculation. However, to unambiguously define something in this issue impossible, and one and the other number will always have supporters, both methods have the right to life, so this is just an interesting fact, and not a reason to think that you should not use Pi.

One of the most mysterious numbers known to mankind, of course, is the number Π (read - pi). In algebra, this number reflects the ratio of the circumference of a circle to its diameter. Previously, this quantity was called the Ludolf number. How and where the number Pi came from is not known for certain, but mathematicians divide the entire history of the number Π into 3 stages, into the ancient, classical and era digital computers.

The number P is irrational, that is, it cannot be represented as a simple fraction, where the numerator and denominator are integers. Therefore, such a number has no end and is periodic. For the first time, the irrationality of P was proved by I. Lambert in 1761.

In addition to this property, the number P cannot also be the root of any polynomial, and therefore is a number property, when it was proved in 1882, it put an end to the almost sacred dispute of mathematicians “about the squaring of the circle”, which lasted for 2,500 years.

It is known that the first to introduce the designation of this number was the Briton Jones in 1706. After Euler's work appeared, the use of such a designation became generally accepted.

To understand in detail what Pi is, it should be said that its use is so widespread that it is difficult to even name a field of science in which it would be dispensed with. One of the simplest and most familiar school curriculum values ​​is the designation of the geometric period. The ratio of the length of a circle to the length of its diameter is constant and equal to 3.14. This value was known even to the most ancient mathematicians in India, Greece, Babylon, Egypt. The earliest version of calculating the ratio dates back to 1900 BC. e. More close to contemporary meaning P was calculated by the Chinese scientist Liu Hui, in addition, he invented and fast way such a calculation. Its value remained generally accepted for almost 900 years.

The classical period in the development of mathematics was marked by the fact that in order to establish exactly what the number Pi is, scientists began to use the methods of mathematical analysis. In the 1400s, the Indian mathematician Madhava used the theory of series to calculate and determined the period of the number P with an accuracy of 11 digits after the decimal point. The first European, after Archimedes, who investigated the number P and made a significant contribution to its justification, was the Dutchman Ludolf van Zeulen, who already determined 15 digits after the decimal point, and wrote very entertaining words in his will: "... whoever is interested - let him go further." It was in honor of this scientist that the number P received its first and only nominal name in history.

The era of computer computing brought new details to the understanding of the essence of the number P. So, in order to find out what the number Pi is, in 1949 the ENIAC computer was used for the first time, one of the developers of which was the future "father" of the theory of modern computers J. The first measurement was carried out on for 70 hours and gave 2037 digits after the decimal point in the period of the number P. The mark of a million characters was reached in 1973. In addition, during this period, other formulas were established that reflect the number P. So, the Chudnovsky brothers were able to find one that made it possible to calculate 1,011,196,691 digits of the period.

In general, it should be noted that in order to answer the question: "What is the number Pi?", Many studies began to resemble competitions. Today, supercomputers are already dealing with the question of what it really is, the number Pi. Interesting Facts associated with these studies permeate almost the entire history of mathematics.

Today, for example, world championships in memorizing the number P are held and world records are set, the latter belongs to the Chinese Liu Chao, who named 67,890 characters in a little over a day. In the world there is even a holiday of the number P, which is celebrated as "Pi Day".

As of 2011, 10 trillion digits of the number period have already been established.

Table of values trigonometric functions

Note. This table of values ​​of trigonometric functions uses the sign √ to denote square root. To denote a fraction - the symbol "/".

see also useful materials:

For determining the value of a trigonometric function, find it at the intersection of the line indicating the trigonometric function. For example, a sine of 30 degrees - we are looking for a column with the heading sin (sine) and we find the intersection of this column of the table with the line "30 degrees", at their intersection we read the result - one second. Similarly, we find cosine 60 degrees, sine 60 degrees (once again, at the intersection of the sin (sine) column and the 60 degree row, we find the value sin 60 = √3/2), etc. In the same way, the values ​​of sines, cosines and tangents of other "popular" angles are found.

Sine of pi, cosine of pi, tangent of pi and other angles in radians

The table of cosines, sines and tangents below is also suitable for finding the value of trigonometric functions whose argument is given in radians. To do this, use the second column of angle values. Thanks to this, you can convert the value of popular angles from degrees to radians. For example, let's find the 60 degree angle in the first line and read its value in radians under it. 60 degrees is equal to π/3 radians.

The number pi uniquely expresses the dependence of the circumference on degree measure angle. So pi radians equals 180 degrees.

Any number expressed in terms of pi (radian) can be easily converted to degrees by replacing the number pi (π) with 180.

Examples:
1. sine pi.
sin π = sin 180 = 0
thus, the sine of pi is the same as the sine of 180 degrees and is equal to zero.

2. cosine pi.
cos π = cos 180 = -1
thus, the cosine of pi is the same as the cosine of 180 degrees and is equal to minus one.

3. Tangent pi
tg π = tg 180 = 0
thus, the tangent of pi is the same as the tangent of 180 degrees and is equal to zero.

Table of sine, cosine, tangent values ​​for angles 0 - 360 degrees (frequent values)

If in the table of values ​​​​of trigonometric functions, instead of the value of the function, a dash is indicated (tangent (tg) 90 degrees, cotangent (ctg) 180 degrees), then when given value the degree measure of the angle function has no definite meaning. If there is no dash, the cell is empty, so we have not yet entered the desired value. We are interested in what requests users come to us for and supplement the table with new values, despite the fact that the current data on the values ​​of cosines, sines and tangents of the most common angle values ​​is enough to solve most problems.

Table of values ​​of trigonometric functions sin, cos, tg for the most popular angles
0, 15, 30, 45, 60, 90 ... 360 degrees
(numerical values ​​"as per Bradis tables")