NUMBER p - the ratio of the circumference of a circle to its diameter, - the value is constant and does not depend on the size of the circle. The number expressing this ratio is usually denoted Greek letter 241 (from "perijereia" - circumference, periphery). This designation became common after the work of Leonhard Euler, referring to 1736, but it 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 relating to circles and round bodies forced us to search for 241 approximations using rational numbers already in ancient times. Information that the circumference is exactly three times longer than the diameter is found in the cuneiform tablets of the Ancient Mesopotamia. Same number value p there is also in the text of the Bible: “And he made a sea of cast copper, from end to end it was ten cubits, completely round, five cubits high, and a string of thirty cubits hugged it around” (1 Kings 7.23). So did the ancient Chinese. But already in 2 thousand BC. the ancient Egyptians used a more accurate value for the number 241, which is obtained from the formula for the area of a circle of diameter d:
This rule from the 50th problem of the Rhind papyrus corresponds to the value 4(8/9) 2 » 3.1605. The Rhinda 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 is only established that the text was created in the second half of the 19th century. BC. Although how the Egyptians got the formula itself is not clear 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, circa 1900 BC, there is another interesting problem about calculating the surface of a basket "with an opening of 4½". 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.
In order to understand how the ancient scientists obtained this or that result, one should try to solve the problem using only the knowledge and methods of calculations of that time. This is exactly what researchers of ancient texts do, but the solutions they manage to find are not necessarily “the same ones”. Very often, several solutions are offered for one task, everyone can choose according to their taste, but no one can say that it was used in antiquity. 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 of 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: in the first approximation, the area of the circle S equal to the difference between the area of a square with a side d and the total area of four small squares A with a party d:
This hypothesis is supported by similar calculations in one of the problems of the Moscow Papyrus, where it is proposed to calculate
From the 6th c. BC. mathematics has developed rapidly 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 is the radius of the circle, l - its length), and the area of a circle is half the product of the circumference and radius:
S = ½ l R = p R 2 .
This evidence is attributed to Eudoxus of Cnidus and Archimedes.
In the 3rd century BC. Archimedes in writing About measuring a circle calculated the perimeters of regular polygons inscribed in a circle and described around it (Fig. 2) - from a 6- to a 96-gon. Thus he established that the number p lies 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, the creator of trigonometry, Claudius Ptolemy (2nd century), but it did not come into use.
Indians and Arabs believed that p= . This value is also given by the Indian mathematician Brahmagupta (598 - ca. 660). In China, scientists in the 3rd century. used the value 3 7/50, which is worse than the approximation of Archimedes, but in the second half of the 5th c. Zu Chun Zhi (c. 430 - c. 501) received for p approximation 355/113 ( p» 3.1415927). It remained unknown to Europeans and was again found by the Dutch mathematician Adrian Antonis only in 1585. This approximation gives an error only in the seventh decimal place.
The search for a more accurate approximation p continued further. For example, al-Kashi (first half of the 15th century) in Treatise on the Circle(1427) computed 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 Ludolph Van Zeilen (1540–1610) found 32 correct decimal places for it (published posthumously in 1615), this approximation is called the Ludolf number.
Number p appears not only in solving geometric problems. Since the time of F. Vieta (1540–1603), the search for the limits of some arithmetic sequences compiled according to simple laws has led to the same number p. For this reason, in determining the number p almost all famous mathematicians took part: F. Viet, H. Huygens, J. Wallis, G. V. Leibniz, L. Euler. They received various expressions for 241 in the form of an infinite product, the 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 not known how he arrived at this result.
In 1665 John Wallis (1616–1703) proved that
This formula bears his name. For the practical determination of the number 241, it is of little use, but is useful in various theoretical reasoning. It entered the history of science as one of the first examples of infinite works.
Gottfried Wilhelm Leibniz (1646–1716) established the following formula in 1673:
expressing 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 calculation 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 characters.
Using the same row for arctg x and formulas
number value p received on a computer with an accuracy of one hundred thousand decimal places. Such calculations are of interest in connection with the concept of random and pseudo random numbers. Statistical processing of an ordered set 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 a number p more precisely than just 3.14. For example, having learned the following quatrain, you can easily name seven decimal places p:
You just need 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 proverb was suggested by Ya.I. Perelman.
“So I know the number called Pi. - Well done!" ( p» 3.1415927).
“Learn and know in the number known behind the number the number, how to notice good luck” ( p» 3.14159265359).
The teacher of one of the Moscow schools came up with the line: “I know this and remember it perfectly,” and his student composed a funny continuation: “Many signs are superfluous to me, in vain.” This couplet allows you to define 12 digits.
And this is what 101 digits of a number look like p without 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 value of a number p calculated with millions of correct digits, but such precision is not needed in any calculations. But the possibility of analytical determination of 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- the number is irrational, but only in 1766 the German mathematician Johann Heinrich Lambert (1728–1777), based on the relationship discovered by Euler between the exponential and trigonometric functions, strictly proved this. Number p cannot be represented as a simple fraction, no matter how large the numerator and denominator are.
In 1882, professor at the University of Munich, Carl Louis 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 x n– 1 + … + a 1 x + a 0 = 0 with integer coefficients. This proof put an end to the history of ancient mathematical problem about squaring the circle. For thousands of years, this problem has not yielded to the efforts of mathematicians, the expression "squaring the circle" has become synonymous with an unsolvable problem. And the whole thing turned out to be in 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 of the University of Munich. On the pedestal under his name is a circle crossed by a square of equal area, inside which the letter is inscribed p.
Marina Fedosova
Recently, there is an elegant formula for calculating pi, which was first published in 1995 by David Bailey, Peter Borwein and Simon Pluff:
It would seem: what is special about it - there are a great many formulas for calculating Pi: from the school Monte Carlo method to the incomprehensible Poisson integral and Francois Vieta's formula from the late Middle Ages. But it is on this formula that you should pay special attention - it allows you to calculate nth character pi without finding the previous ones. For information on how it works, as well as for ready-made C code that calculates the 1,000,000th character, I ask for a habrakat.
How does the algorithm for calculating the Nth sign of Pi work?
For example, if we need the 1000th hexadecimal digit of pi, we multiply the whole formula by 16^1000, thereby turning the factor in front of the brackets into 16^(1000-k). When exponentiating, we use the binary exponentiation algorithm or, as will be shown in the example below, exponentiation modulo . After that, we calculate the sum of several terms of the series. Moreover, it is not necessary to calculate a lot: as k increases, 16 ^ (N-k) quickly decreases, so that subsequent terms will not affect the value of the desired digits). That's all magic - ingenious and simple.
The Bailey-Borwein-Pluff formula was found by Simon Pluff using the PSLQ algorithm, which was included in the Top 10 Algorithms of the Century list in 2000. The PSLQ algorithm itself was in turn developed by Bailey. Here is a Mexican series about mathematicians.
By the way, the running time of the algorithm is O(N), the memory usage is O(log N), where N is the ordinal number of the desired character.
I think it would be appropriate to give the C code written directly by the author of the algorithm, David Bailey:
/* This program implements the BBP algorithm to generate a few hexadecimal digits beginning immediately after a given position id, or in other words beginning at position id + 1. On most systems using IEEE 64-bit floating- point arithmetic, this code works correctly so long as d is less than approximately 1.18 x 10^7. If 80-bit arithmetic can be employed, this limit is significantly higher. Whatever arithmetic is used, results for a given position id can be checked by repeating with id-1 or id+1, and verifying that the hex digits perfectly overlap with an offset of one, except possibly for a few trailing digits. The resulting fractions are typically accurate to at least 11 decimal digits, and to at least 9 hex digits. */ /* David H. Bailey 2006-09-08 */ #include
What opportunities does it give? For example: we can create a distributed computing system that calculates the number Pi and set a new record for the calculation accuracy for all Habr (which now, by the way, is 10 trillion decimal places). According to empirical data, the fractional part of the number Pi is a normal numerical sequence (although this has not yet been reliably proven), which means that sequences of digits from it can be used in generating passwords and simply random numbers, or in cryptographic algorithms (for example, in hashing) . You can find a great variety of ways to use it - you just need to turn on your imagination.
You can find more information on the topic in the article by David Bailey himself, where he talks in detail about the algorithm and its implementation (pdf);
And it seems that you have just read the first Russian-language article about this algorithm in RuNet - I could not find others.
Today is the birthday of the number Pi, which, at the initiative of American mathematicians, is celebrated on March 14 at 1 hour and 59 minutes in the afternoon. This is due to a more accurate value of Pi: we are all used to counting this constant as 3.14, but the number can be continued like this: 3, 14159... Translating this into a calendar date, we get 03.14, 1:59.
Photo: AIF / Nadezhda Uvarova
Vladimir Zalyapin, professor at the Department of Mathematical and Functional Analysis at South Ural State University, says that July 22 should still be considered “pi day”, because in the European date format this day is written as 22/7, and the value of this fraction is approximately equal to the value of Pi .
“The history of the number that gives the ratio of the circumference of a circle to the diameter of a circle goes back to ancient times,” says Zalyapin. — The Sumerians and Babylonians already knew that this ratio does not depend on the diameter of the circle and is constant. One of the first mentions of the number Pi can be found in the texts Egyptian scribe Ahmes(about 1650 BC). The ancient Greeks, who borrowed a lot from the Egyptians, contributed to the development of this mysterious quantity. According to the legend, Archimedes was so carried away by the calculations that he did not notice how the Roman soldiers took his hometown of Syracuse. When a Roman soldier approached him, Archimedes shouted in Greek, "Don't touch my circles!" In response, the soldier stabbed him with a sword.
Plato received a fairly accurate value of pi for his time - 3.146. Ludolf van Zeilen spent most of his life on the calculations of the first 36 digits after the decimal point of pi, and they were engraved on his tombstone after death.
Irrational and abnormal
According to the professor, at all times the pursuit of calculating new decimal places was determined by the desire to get the exact value of this number. It was assumed that the number Pi is rational and, therefore, can be expressed as a simple fraction. And this is fundamentally wrong!
Pi is also popular because it is mystical. Since ancient times, there has been a religion of worshipers of the constant. In addition to the traditional value of Pi - a mathematical constant (3.1415 ...), expressing the ratio of the circumference of a circle to its diameter, there are many other values \u200b\u200bof the number. Such facts are curious. In the process of measuring the dimensions of the Great Pyramid of Giza, it turned out that it has the same ratio of height to the perimeter of its base as the radius of a circle to its length, that is, ½ Pi.
If we calculate the length of the Earth's equator using Pi to the ninth decimal place, the calculation error is only about 6 mm. Thirty-nine decimal places in the number Pi is enough to calculate the circumference of a circle encircling known space objects in the Universe, with an error no greater than the radius of a hydrogen atom!
Mathematical analysis is also involved in the study of Pi. Photo: AIF / Nadezhda Uvarova
Chaos in numbers
According to a professor of mathematics, in 1767 Lambert established the irrationality of the number Pi, that is, the impossibility of representing it as a ratio of two integers. This means that the sequence of decimal digits of pi is chaos embodied in numbers. In other words, the "tail" of decimal places contains any number, any sequence of numbers, any texts that were, are and will be, but it is not possible to extract this information!
“It is impossible to know the exact value of Pi,” continues Vladimir Ilyich. But these attempts are not abandoned. In 1991 Chudnovsky achieved new 2260000000 decimal digits of the constant, and in 1994 - 4044000000. After that, the number of correct digits of the number Pi increased like an avalanche.
Chinese man holds world record for memorizing pi Liu Chao, who managed to memorize 67890 decimal places without error and reproduce them within 24 hours and 4 minutes.
About the "golden section"
By the way, the connection between "pi" and another amazing quantity - the golden ratio - has not actually been proven. People have long noticed that the "golden" proportion - it is also the Phi number - and the number Pi divided by two differ from each other by less than 3% (1.61803398... and 1.57079632...). However, for mathematics, these three percent are too significant a difference to consider these values identical. In the same way, we can say that the number Pi and the number Phi are relatives of another well-known constant - the Euler number, since the root of it is close to half the number of Pi. One second of Pi is 1.5708, Phi is 1.6180, the root of E is 1.6487.
This is only part of the meaning of Pi. Photo: Screenshot
Pi's birthday
In the South Ural state university Constant's birthday is celebrated by all teachers and mathematics students. It has always been like this - it cannot be said that interest appeared only in last years. The number 3.14 is even welcomed with a special holiday concert!
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 sought 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 a circle, that is, the impossibility of constructing a square with a compass and straightedge, the area of which 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 of the circumference of a circle 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 Rachmaninoff symphony, 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 performing any algebraic operations on integers.
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". In the process of measuring the Great Pyramid of 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.
PI
The symbol PI stands for the ratio of the circumference of a circle to its diameter. For the first time in this sense, the symbol p was used by W. Jones in 1707, and L. Euler, having accepted this designation, introduced it into scientific use. Even in ancient times, mathematicians knew that calculating the value of p and the area of a circle are closely related tasks. The ancient Chinese and ancient Jews considered the number p equal to 3. The value of p, equal to 3.1605, is contained in the ancient Egyptian papyrus of the scribe Ahmes (c. 1650 BC). Around 225 BC e. Archimedes, using regular 96-gons inscribed and circumscribed, approximated the area of a circle using a method that resulted in a PI value between 31/7 and 310/71. Another approximate value of p, equivalent to the usual decimal representation of this number 3.1416, has been known since the 2nd century. L. van Zeulen (1540-1610) calculated the value of PI with 32 decimal places. By the end of the 17th century. new methods of mathematical analysis made it possible to calculate the value of p in many different ways. In 1593 F. Viet (1540-1603) derived the formula
In 1665 J. Wallis (1616-1703) proved that
In 1658, W. Brounker found a representation of the number p in the form of a continued fraction
G. Leibniz in 1673 published a series
Series allow you to calculate the value of p with any number of decimal places. In recent years, with the advent of electronic computers, the value of p has been found with more than 10,000 digits. With ten digits, the value of PI is 3.1415926536. As a number, PI has some interesting properties. For example, it cannot be represented as a ratio of two integers or as a periodic decimal; the number PI is transcendental, i.e. cannot be represented as a root of an algebraic equation with rational coefficients. The PI number is included in many mathematical, physical and technical formulas, including those not directly related to the area of a circle or the length of an arc of a circle. For example, the area of an ellipse A is given by A = pab, where a and b are the lengths of the major and minor semiaxes.
Collier Encyclopedia. - Open society. 2000 .
See what "PI NUMBER" 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 of Dmitriev
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. The grammatical number is one of the manifestations of a more general linguistic category of quantity (see the 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
NUMBER, a, pl. numbers, villages, slam, cf. 1. The basic concept of mathematics is the value, with the help of which the swarm is calculated. Integer hours Fractional hours Real hours Complex hours Natural hours (positive integer). Simple hours (natural number, not ... ... Explanatory dictionary of Ozhegov
NUMBER "E" (EXP), an irrational number that serves as the basis of natural LOGARITHMS. This real decimal number, an infinite fraction equal to 2.7182818284590...., is the limit of the expression (1/) as n goes to infinity. In fact,… … Scientific and technical encyclopedic dictionary
Quantity, cash, composition, strength, contingent, amount, figure; day.. Wed. . See day, quantity. a small number, no number, grow in number... Dictionary of Russian synonyms and expressions similar in meaning. under. ed. N. Abramova, M .: Russians ... ... Synonym dictionary
Books
- Name number. Secrets of numerology. Exit from the body for the lazy. ESP Primer (number of volumes: 3), Lawrence Shirley. Name number. Secrets of numerology. Shirley B. Lawrence's book is a comprehensive study of the ancient esoteric system - numerology. To learn how to use number vibrations to…
- Name number. The sacred meaning of numbers. Symbolism of the Tarot (number of volumes: 3), Uspensky Petr. Name number. Secrets of numerology. Shirley B. Lawrence's book is a comprehensive study of the ancient esoteric system - numerology. To learn how to use number vibrations to…
