What is the constant e? History of the number e

PERVUSHKIN BORIS NIKOLAEVICH

Private educational institution "St. Petersburg School "Tete-a-Tete"

Mathematics Teacher of the Highest Category

Number e

The number first appeared inmathematicslike something insignificant. This happened in 1618. In the appendix to Napier's work on logarithms, a table of natural logarithms of various numbers was given. However, no one realized that these were logarithms to the base, since the concept of a logarithm at that time did not include such a thing as a base. This is what we now call a logarithm, the power to which the base must be raised to obtain the required number. We'll come back to this later. The table in the appendix was most likely made by Augthred, although the author was not identified. A few years later, in 1624, it appears again in mathematical literature, but again in a veiled manner. This year Briggs gave a numerical approximation decimal logarithm, but the number itself is not mentioned in his work.

The next appearance of the number is again doubtful. In 1647, Saint-Vincent calculated the area of the hyperbola sector. Whether he understood the connection with logarithms can only be guessed at, but even if he understood, it is unlikely that he could come to the number itself. It was not until 1661 that Huygens understood the connection between the equilateral hyperbola and logarithms. He proved that the area under the graph of an equilateral hyperbola of an equilateral hyperbola in the interval from 1 to is equal to 1. This property makes the basis of natural logarithms, but this was not understood by mathematicians of that time, but they were slowly approaching this understanding.

Huygens did next step in 1661. He defined a curve which he called logarithmic (in our terminology we will call it exponential). This is a type curve. And again the decimal logarithm appears, which Huygens finds accurate to 17 decimal digits. However, it arose from Huygens as a kind of constant and was not associated with the logarithm of a number (so, again they came close to , but the number itself remains unrecognized).

IN further work for logarithms, again, the number does not appear explicitly. However, the study of logarithms continues. In 1668, Nicolaus Mercator published a workLogarithmotechnia, which contains a series expansion. In this work, Mercator first uses the name “natural logarithm” for the base logarithm. The number clearly does not appear again, but remains elusive somewhere to the side.

It is surprising that number first appears in explicit form not in connection with logarithms, but in connection with infinite products. In 1683, Jacob Bernoulli tries to find

He uses the binomial theorem to prove that this limit is between 2 and 3, which we can think of as a first approximation of . Although we take this to be the definition of , this is the first time a number has been defined as a limit. Bernoulli, of course, did not understand the connection between his work and the work on logarithms.

It was previously mentioned that logarithms at the beginning of their study were not connected in any way with exponents. Of course, from the equation we find that , but this is a much later way of perceiving. Here we actually mean a function by a logarithm, whereas at first the logarithm was considered only as a number that helped in calculations. Jacob Bernoulli may have been the first to realize that the logarithmic function is the inverse exponential. On the other hand, the first person to connect logarithms and powers may have been James Gregory. In 1684 he certainly recognized the connection between logarithms and powers, but he may not have been the first.

We know that the number appeared in its present form in 1690. Leibniz, in a letter to Huygens, used the designation for it. Finally, a designation appeared (although it did not coincide with the modern one), and this designation was recognized.

In 1697, Johann Bernoulli began studying exponential function and publishesPrincipia calculi exponentialum seu percurrentium. In this work, the sums of various exponential series are calculated, and some results are obtained by their term-by-term integration.

Euler introduced so many mathematical notations that
not surprisingly, the designation also belongs to him. It seems ridiculous to say that he used the letter because it is the first letter of his name. This is probably not even because it is taken from the word “exponential”, but simply because it is the next vowel after “a”, and Euler had already used the notation “a” in his work. Regardless of the reason, the notation first appears in a letter from Euler to Goldbach in 1731. He made many discoveries as he studied further, but not until 1748.Introductio in Analysin infinitorumhe gave full justification for all ideas related to. He showed that

Euler also found the first 18 decimal places of a number:

however, without explaining how he got them. It looks like he calculated this value himself. In fact, if we take about 20 terms of series (1), we get the accuracy that Euler obtained. Among the others interesting results his work shows the connection between the functions sine and cosine and the complex exponential function, which Euler derived from Moivre's formula.

It is interesting that Euler even found the decomposition of a number into continued fractions and gave examples of such decomposition. In particular, he received

Euler did not provide proof that these fractions continue in the same way, but he knew that if there was such a proof, it would prove irrationality. Indeed, if the continued fraction for , continued in the same way as in the given example, 6,10,14,18,22,26, (we add 4 each time), then it would never be interrupted, and (and therefore ) could not be rational. This is obviously the first attempt to prove irrationality.

The first one to calculate quite big number decimal places of the number was Shanks in 1854. Glaisher showed that the first 137 places calculated by Shanks were correct, but then found an error. Shanks corrected it, and 205 decimal places were obtained. In reality, you need about
120 terms of expansion (1) to get 200 correct digits of the number.

In 1864, Benjamin Peirce stood by a board on which was written

In his lectures he might say to his students: “Gentlemen, we have not the slightest idea what this means, but we can be sure that it means something very important.”

Most people believe that Euler proved the irrationality of the number. However, this was done by Hermite in 1873. The question of whether the number is algebraic still remains open. Latest result in this direction is that at least one of the numbers is transcendental.

Next, the next decimal places of the number were calculated. In 1884, Boorman calculated 346 digits, of which the first 187 coincided with Shanks' digits, but the subsequent ones differed. In 1887, Adams calculated the 272 digits of the decimal logarithm.

| Euler number (E)

e - the base of the natural logarithm, a mathematical constant, an irrational and transcendental number. Approximately equal to 2.71828. Sometimes the number is called Euler number or Napier number. Indicated by lowercase Latin letter « e».

Story

Number e first appeared in mathematics as something insignificant. This happened in 1618. In the appendix to John Napier's work on logarithms, a table of natural logarithms of various numbers was given. However, no one realized that these are logarithms to the base e , since the concept of a logarithm of that time did not include such a thing as a base. This is what we now call a logarithm, the power to which the base must be raised to obtain the required number. We'll come back to this later. The table in the appendix was most likely made by Augthred, although the author was not identified. A few years later, in 1624, it appears again in mathematical literature. e , but again in a veiled manner. This year Briggs gave a numerical approximation to the decimal logarithm e , but the number itself e not mentioned in his work.

Next occurrence of the number e again doubtful. In 1647, Saint-Vincent calculated the area of the hyperbola sector. Whether he understood the connection with logarithms can only be guessed at, but even if he did, it is unlikely that he could have arrived at the number itself e . It was not until 1661 that Huygens understood the connection between the equilateral hyperbola and logarithms. He proved that the area under the graph of an equilateral hyperbola xy = 1 equilateral hyperbola on the interval from 1 to e is equal to 1. This property makes e the basis of natural logarithms, but this was not understood by mathematicians of that time, but they were slowly approaching this understanding.

Huygens took the next step in 1661. He defined a curve that he called logarithmic (in our terminology we will call it exponential). This is a curve of the form y = ka x . And the decimal logarithm appears again e , which Huygens finds accurate to 17 decimal digits. However, it arose from Huygens as a kind of constant and was not associated with the logarithm of a number (so, again we came close to e , but the number itself e remains unrecognized).

In further work on logarithms, again the number e does not appear explicitly. However, the study of logarithms continues. In 1668, Nicolaus Mercator published a work Logarithmotechnia, which contains a series expansion log(1 + x) . In this work, Mercator first uses the name “natural logarithm” for the base logarithm e . Number e clearly does not appear again, but remains elusive somewhere to the side.

It's surprising that the number e explicitly appears for the first time not in connection with logarithms, but in connection with infinite products. In 1683, Jacob Bernoulli tries to find

He uses the binomial theorem to prove that this limit is between 2 and 3, which we can think of as a first approximation of the number e . Although we take this as a definition e , this is the first time a number is defined as a limit. Bernoulli, of course, did not understand the connection between his work and the work on logarithms.

It was previously mentioned that logarithms at the beginning of their study were not connected in any way with exponents. Of course, from the equation x = a t we find that t = log a x , but this is a much later way of perceiving. Here we actually mean a function by a logarithm, whereas at first the logarithm was considered only as a number that helped in calculations. Jacob Bernoulli may have been the first to realize that the logarithmic function is the inverse exponential. On the other hand, the first person to connect logarithms and powers may have been James Gregory. In 1684 he certainly recognized the connection between logarithms and powers, but he may not have been the first.

We know that the number e appeared in its current form in 1690. Leibniz, in a letter to Huygens, used the designation for it b . Finally e a designation appeared (although it did not coincide with the modern one), and this designation was recognized.

In 1697, Johann Bernoulli began studying the exponential function and published Principia calculi exponentialum seu percurrentium. In this work, the sums of various exponential series are calculated, and some results are obtained by their term-by-term integration.

Leonhard Euler introduced so much mathematical notation that it is not surprising that the notation e also belongs to him. It seems ridiculous to say that he used the letter e due to the fact that it is the first letter of his name. It's probably not even because e taken from the word “exponential”, it is simply the next vowel after “a”, and Euler had already used the notation “a” in his work. Regardless of the reason, the notation first appears in a letter from Euler to Goldbach in 1731. He made many discoveries while studying e later, but only in 1748 Introductio in Analysin infinitorum he gave full justification for all ideas related to e . He showed that

Euler also found the first 18 decimal places of the number e :

True, without explaining how he got them. It looks like he calculated this value himself. In fact, if we take about 20 terms of series (1), we get the accuracy that Euler obtained. Among other interesting results in his work is the connection between the functions sine and cosine and the complex exponential function, which Euler derived from De Moivre's formula.

Interestingly, Euler even found a decomposition of the number e into continued fractions and gave examples of such decomposition. In particular, he received

Euler did not provide proof that these fractions continue in the same way, but he knew that if there was such a proof, it would prove irrationality e . Indeed, if the continued fraction for (e - 1) / 2 , continued in the same way as in the above example, 6,10,14,18,22,26, (we add 4 each time), then it would never have been interrupted, and (e -1) / 2 (and therefore e ) could not be rational. Obviously, this is the first attempt to prove irrationality e .

The first to calculate a fairly large number of decimal places of a number e , was Shanks in 1854. Glaisher showed that the first 137 characters calculated by Shanks were correct, but then found an error. Shanks corrected it, and 205 decimal places of the number were obtained e . In fact, about 120 terms of expansion (1) are needed to get 200 correct digits of the number e .

In 1864, Benjamin Peirce stood by a board on which was written

In his lectures he might say to his students: “Gentlemen, we have not the slightest idea what this means, but we can be sure that it means something very important.”

Most believe that Euler proved the irrationality of the number e . However, this was done by Hermite in 1873. The question still remains open whether the number is e e algebraic. The final result in this direction is that at least one of the numbers e e And e e 2 is transcendental.

Next, the following decimal places of the number were calculated e . In 1884, Boorman calculated 346 digits e , of which the first 187 coincided with Shanks's signs, but the subsequent ones differed. In 1887, Adams calculated the 272 digits of the decimal logarithm e .

J. J. Connor, E. F. Robertson. The number e.

Like something insignificant. This happened in 1618. In the appendix to Napier's work on logarithms, a table of natural logarithms of various numbers was given. However, no one realized that these were logarithms to the base, since the concept of a logarithm at that time did not include such a thing as a base. This is what we now call a logarithm, the power to which the base must be raised to obtain the required number. We'll come back to this later. The table in the appendix was most likely made by Augthred, although the author was not identified. A few years later, in 1624, it appears again in mathematical literature, but again in a veiled manner. This year Briggs gave a numerical approximation of the decimal logarithm, but the number itself is not mentioned in his work.

The next appearance of the number is again doubtful. In 1647, Saint-Vincent calculated the area of the hyperbola sector. Whether he understood the connection with logarithms can only be guessed at, but even if he understood, it is unlikely that he could come to the number itself. It was not until 1661 that Huygens understood the connection between the equilateral hyperbola and logarithms. He proved that the area under the graph of an equilateral hyperbola of an equilateral hyperbola on the interval from to is equal to . This property makes the basis of natural logarithms, but this was not understood by mathematicians of that time, but they were slowly approaching this understanding.

Huygens took the next step in 1661. He defined a curve that he called logarithmic (in our terminology we will call it exponential). This is a type curve. And again the decimal logarithm appears, which Huygens finds accurate to 17 decimal digits. However, it arose from Huygens as a kind of constant and was not associated with the logarithm of a number (so, again they came close to , but the number itself remains unrecognized).

In further work on logarithms, the number again does not appear explicitly. However, the study of logarithms continues. In 1668, Nicolaus Mercator published a work Logarithmotechnia, which contains a series expansion. In this work, Mercator first uses the name “natural logarithm” for the base logarithm. The number clearly does not appear again, but remains elusive somewhere to the side.

It is surprising that number first appears in explicit form not in connection with logarithms, but in connection with infinite products. In 1683, Jacob Bernoulli tries to find

He uses the binomial theorem to prove that this limit is between and , which we can think of as a first approximation of . Although we take this to be the definition of , this is the first time a number has been defined as a limit. Bernoulli, of course, did not understand the connection between his work and the work on logarithms.

Euler introduced so many mathematical notations that
not surprisingly, the designation also belongs to him. It seems ridiculous to say that he used the letter because it is the first letter of his name. This is probably not even because it is taken from the word “exponential”, but simply because it is the next vowel after “a”, and Euler had already used the notation “a” in his work. Regardless of the reason, the notation first appears in a letter from Euler to Goldbach in 1731. He made many discoveries as he studied further, but not until 1748. Introductio in Analysin infinitorum he gave full justification for all ideas related to. He showed that

Euler also found the first 18 decimal places of a number:

however, without explaining how he got them. It looks like he calculated this value himself. In fact, if we take about 20 terms of series (1), we get the accuracy that Euler obtained. Among other interesting results in his work is the connection between the functions sine and cosine and the complex exponential function, which Euler derived from De Moivre's formula.

It is interesting that Euler even found the decomposition of a number into continued fractions and gave examples of such decomposition. In particular, he received
And
Euler did not provide proof that these fractions continue in the same way, but he knew that if there was such a proof, it would prove irrationality. Indeed, if the continued fraction for continued in the same way as in the above example (we add each time), then it would never be interrupted, and (and therefore) could not be rational. This is obviously the first attempt to prove irrationality.

The first to calculate a fairly large number of decimal places was Shanks in 1854. Glaisher showed that the first 137 digits calculated by Shanks were correct, but then found an error. Shanks corrected it, and 205 decimal places were obtained. In reality, you need about
120 terms of expansion (1) to get 200 correct digits of the number.

In 1864, Benjamin Peirce stood by a board on which was written

In his lectures he might say to his students: “Gentlemen, we have not the slightest idea what this means, but we can be sure that it means something very important.”

Most people believe that Euler proved the irrationality of the number. However, this was done by Hermite in 1873. The question of whether the number is algebraic still remains open. The final result in this direction is that at least one of the numbers is transcendental.

y (x) = e x, the derivative of which is equal to the function itself.

The exponent is denoted as , or .

Number e

The basis of the exponent degree is number e. This is an irrational number. It is approximately equal
e ≈ 2,718281828459045...

The number e is determined through the limit of the sequence. This is the so-called second wonderful limit:
.

The number e can also be represented as a series:
.

Exponential graph

Exponential graph, y = e x .

The graph shows the exponent e to a degree X.
y (x) = e x
The graph shows that the exponent increases monotonically.

Formulas

The basic formulas are the same as for the exponential function with a base of degree e.

;
;
;

Expression of an exponential function with an arbitrary base of degree a through an exponential:
.

Private values

Let y (x) = e x.
.

Then

Exponent Properties e > 1 .

The exponent has the properties of an exponential function with a power base

Domain, set of values (x) = e x Exponent y
defined for all x.
- ∞ < x + ∞ .
Its domain of definition:
0 < y < + ∞ .

Its many meanings:

Extremes, increasing, decreasing

The exponential is a monotonically increasing function, so it has no extrema. Its main properties are presented in the table.

The inverse of the exponent is the natural logarithm.
;
.

Derivative of the exponent

Derivative e to a degree X equal to e to a degree X :
.
Derivative of nth order:
.
Deriving formulas > > >

Integral

Complex numbers

Operations with complex numbers are carried out using Euler's formulas:
,
where is the imaginary unit:
.

Expressions through hyperbolic functions

; ;
.

Expressions using trigonometric functions

; ;
;
.

Power series expansion

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.

Doctor of Geological and Mineralogical Sciences, Candidate of Physical and Mathematical Sciences B. GOROBETS.

Graphs of the functions y = arcsin x, the inverse function y = sin x

Graph of the function y = arctan x, the inverse of the function y = tan x.

Normal distribution function (Gaussian distribution). The maximum of its graph corresponds to the most probable value of a random variable (for example, the length of an object measured with a ruler), and the degree of “spreading” of the curve depends on the parameters a and sigma.

The priests of Ancient Babylon calculated that the solar disk fits in the sky 180 times from dawn to sunset and introduced a new unit of measurement - a degree equal to its angular size.

Dimensions natural formations- sand dunes, hills and mountains - increase with each step by an average of 3.14 times.

Science and life // Illustrations

The pendulum, swinging without friction or resistance, maintains a constant amplitude of oscillation. The appearance of resistance leads to exponential attenuation of oscillations.

In a very viscous medium, a deflected pendulum moves exponentially toward its equilibrium position.

Scales pine cones and the curls of the shells of many mollusks are arranged in logarithmic spirals.

Science and life // Illustrations

A logarithmic spiral intersects all rays emanating from point O at the same angles.

Probably, any applicant or student, when asked what numbers and e are, will answer: - this is a number equal to the ratio of the circumference to its diameter, and e is the base of natural logarithms. If asked to define these numbers more strictly and calculate them, students will give formulas:

e = 1 + 1/1! + 1/2! + 1/3! + ... 2.7183…

(remember that factorial n! =1 x 2x 3x… x n);

3(1+ 1/3x 2 3 + 1x 3/4x 5x 2 5 + .....) 3,14159…

(Newton's series is the last one, there are other series).

All this is true, but, as you know, numbers and e are included in many formulas in mathematics, physics, chemistry, biology, and also in economics. This means they reflect some general laws nature. Which ones exactly? The definitions of these numbers through series, despite their correctness and rigor, still leave a feeling of dissatisfaction. They are abstract and do not convey the connection of the numbers in question with the outside world through everyday experience. It is not possible to find answers to the question posed in the educational literature.

Meanwhile, it can be argued that the constant e is directly related to the homogeneity of space and time, and to the isotropy of space. Thus, they reflect the laws of conservation: the number e - energy and momentum (momentum), and the number - torque (momentum). Usually such unexpected statements cause surprise, although essentially, from the point of view of theoretical physics, there is nothing new in them. The deep meaning of these world constants remains terra incognita for schoolchildren, students and, apparently, even for the majority of teachers of mathematics and general physics, not to mention other areas of natural science and economics.

In the first year of university, students can be baffled by, for example, a question: why does the arctangent appear when integrating functions of type 1/(x 2 +1), and circular trigonometric functions of the arcsine type, expressing the magnitude of the arc of a circle? In other words, where do the circles “come from” during integration and where do they then disappear during the inverse action - differentiating the arctangent and arcsine? It is unlikely that the derivation of the corresponding formulas for differentiation and integration will answer the question posed by itself.

Further, in the second year of university, when studying probability theory, the number appears in the formula for the law of normal distribution random variables(see "Science and Life" No. 2, 1995); from it you can, for example, calculate the probability with which a coin will fall on the coat of arms any number of times with, say, 100 tosses. Where are the circles here? Does the shape of the coin really matter? No, the formula for probability is the same for a square coin. Indeed, these are not easy questions.

But the nature of the number e is useful for students of chemistry and materials science, biologists and economists to know more deeply. This will help them understand the kinetics of the decay of radioactive elements, saturation of solutions, wear and tear of materials, proliferation of microbes, the impact of signals on the senses, processes of capital accumulation, etc. - an infinite number of phenomena in living and inanimate nature and human activities.

Number and spherical symmetry of space

First, we formulate the first main thesis, and then explain its meaning and consequences.

1. The number reflects the isotropy of the properties of the empty space of our Universe, their sameness in any direction. The law of conservation of torque is associated with the isotropy of space.

This leads to well-known consequences that are studied in high school.

Corollary 1. The length of the arc of a circle along which its radius fits is the natural arc and angular unit radian.

This unit is dimensionless. To find the number of radians in an arc of a circle, you need to measure its length and divide by the length of the radius of this circle. As we know, along any full circle its radius is approximately 6.28 times. More precisely, the length of a full arc of a circle is 2 radians, and in any number systems and units of length. When the wheel was invented, it turned out to be the same among the Indians of America, the nomads of Asia, and the blacks of Africa. Only the units of arc measurement were different and conventional. Thus, our angular and arc degrees were introduced by the Babylonian priests, who considered that the disk of the Sun, located almost at the zenith, fits 180 times in the sky from dawn to sunset. 1 degree is 0.0175 rad or 1 rad is 57.3°. It can be argued that hypothetical alien civilizations would easily understand each other by exchanging a message in which the circle is divided into six parts “with a tail”; this would mean that the “negotiating partner” has already at least passed the stage of reinventing the wheel and knows what the number is.

Corollary 2. Purpose trigonometric functions- express the relationship between the arc and linear dimensions of objects, as well as between the spatial parameters of processes occurring in a spherically symmetrical space.

From the above it is clear that the arguments of trigonometric functions are, in principle, dimensionless, like those of other types of functions, i.e. these are real numbers - points on the number axis that do not need degree notation.

Experience shows that schoolchildren, college and university students have difficulty getting used to dimensionless arguments for sine, tangent, etc. Not every applicant will be able to answer the question without a calculator what cos1 (approximately 0.5) or arctg / 3. The last example is especially confusing. It is often said that this is nonsense: “an arc whose arctangent is 60 o.” If you say exactly that, then the error will be in unauthorized use degree measure to the function argument. And the correct answer is: arctg(3.14/3) arctg1 /4 3/4. Unfortunately, quite often applicants and students say that = 180 0, after which they have to correct them: in the decimal number system = 3.14…. But, of course, we can say that a radian is equal to 180 0.

Let us examine another non-trivial situation encountered in probability theory. It concerns the important formula for the probability of a random error (or normal law probability distribution), which includes the number . Using this formula, you can, for example, calculate the probability of a coin falling on the coat of arms 50 times with 100 tosses. So, where did the number in it come from? After all, no circles or circles seem to be visible there. But the point is that the coin falls randomly in a spherically symmetrical space, in all directions of which random fluctuations should be equally taken into account. Mathematicians do this by integrating around a circle and calculating the so-called Poisson integral, which is equal to and included in the specified probability formula. A clear illustration of such fluctuations is the example of shooting at a target under constant conditions. The holes on the target are scattered in a circle (!) with the highest density near the center of the target, and the probability of a hit can be calculated using the same formula containing the number .

Is number involved in natural structures?

Let's try to understand the phenomena, the causes of which are far from clear, but which, perhaps, were also not without number.

Domestic geographer V.V. Piotrovsky compared the average characteristic sizes natural reliefs in the next row: sand riffle on shallows, dunes, hills, mountain systems Caucasus, Himalayas, etc. It turned out that the average increase in size is 3.14. A similar pattern seems to have been recently discovered in the topography of the Moon and Mars. Piotrovsky writes: “Tectonic structural forms formed in earth's crust and expressed on its surface in the form of relief forms, develop as a result of some general processes occurring in the body of the Earth, they are proportional to the size of the Earth." To be more precise, they are proportional to the ratio of its linear and arc dimensions.

The basis of these phenomena may be the so-called law of distribution of maxima of random series, or the “law of triplets”, formulated back in 1927 by E. E. Slutsky.

Statistically, according to the law of threes, sea coastal waves are formed, which the ancient Greeks knew. Every third wave is on average slightly higher than its neighbors. And in the series of these third maxima, every third one, in turn, is higher than its neighbors. This is how the famous ninth wave is formed. He is the peak of the "second rank period". Some scientists suggest that according to the law of triplets, fluctuations in solar, comet and meteorite activities also occur. The intervals between their maxima are nine to twelve years, or approximately 3 2 . What does the doctor think? biological sciences G. Rosenberg, we can continue constructing time sequences as follows. The period of the third rank 3 3 corresponds to the interval between severe droughts, which averages 27-36 years; period 3 4 - secular cycle solar activity(81-108 years old); period 3 5 - glaciation cycles (243-324 years). The coincidences will become even better if we depart from the law of “pure” triplets and move on to powers of number. By the way, they are very easy to calculate, since 2 is almost equal to 10 (once in India the number was even defined as the root of 10). You can continue to adjust the cycles of geological epochs, periods and eras to whole powers of three (which is what G. Rosenberg does, in particular, in the collection “Eureka-88”, 1988) or the numbers 3.14. And you can always take wishful thinking with varying degrees of accuracy. (In connection with adjustments, a mathematical joke comes to mind. Let us prove that

odd numbers

The essence of the numbers is simple. We take: 1, 3, 5, 7, 9, 11, 13, etc., and 9 here is an experimental error.) And yet the idea of the unobvious role of the number p in many geological and biological phenomena, it seems, is not completely empty , and perhaps it will show itself in the future. The number e and the homogeneity of time and space Now let's move on to the second great world constant - the number e. The mathematically flawless determination of the number e using the series given above, in essence, does not in any way clarify its connection with physical or other

Everyone knows that a continuous wave in time can be described by a sine wave or the sum of sine and cosine waves. In mathematics, physics, and electrical engineering, such a wave (with an amplitude equal to 1) is described by the exponential function e iβt =cos βt + isin βt, where β is the frequency of harmonic oscillations. One of the most famous mathematical formulas is written here - Euler's formula. It was in honor of the great Leonhard Euler (1707-1783) that the number e was named after the first letter of his last name.

This formula is well known to students, but it needs to be explained to students of non-mathematical schools, because in our time, from ordinary school programs Complex numbers are excluded. The complex number z = x+iy consists of two terms - the real number (x) and the imaginary number, which is the real number y multiplied by the imaginary unit. Real numbers are counted along the real axis O x, and imaginary numbers are counted on the same scale along the imaginary axis O y, the unit of which is i, and the length of this unit segment is the modulus | i | =1. That's why complex number corresponds to a point on the plane with coordinates (x, y). So, unusual look a number e with an exponent containing only imaginary units i means the presence of only undamped oscillations described by a cosine and sine wave.

It is clear that an undamped wave demonstrates compliance with the law of conservation of energy for electromagnetic wave in a vacuum. This situation occurs during the “elastic” interaction of a wave with a medium without loss of its energy. Formally, this can be expressed as follows: if you move the reference point along the time axis, the energy of the wave will be preserved, since the harmonic wave will retain the same amplitude and frequency, that is, energy units, and only its phase will change, the part of the period distant from the new reference point. But the phase does not affect the energy precisely because of the uniformity of time when the reference point is shifted. So, parallel transfer of the coordinate system (it is called translation) is legal due to the homogeneity of time t. Now, it is probably clear in principle why homogeneity in time leads to the law of conservation of energy.

Next, let's imagine a wave not in time, but in space. A clear example it can be a standing wave (oscillations of a string motionless at several points-nodes) or coastal sand ripples. Mathematically, this wave along the O x axis will be written as e ix = cos x + isin x. It is clear that in this case, translation along x will not change either the cosine or sinusoid if the space is homogeneous along this axis. Again, only their phase will change. It is known from theoretical physics that the homogeneity of space leads to the law of conservation of momentum (momentum), that is, mass multiplied by speed. Let now space be homogeneous in time (and the law of conservation of energy is satisfied), but inhomogeneous in coordinate. Then at different points of inhomogeneous space the speed would also be unequal, since per unit of homogeneous time there would be different meanings the length of the segments covered per second by a particle with a given mass (or a wave with a given momentum).

So, we can formulate the second main thesis:

2. The number e as the basis of a function of a complex variable reflects two basic laws of conservation: energy - through the homogeneity of time, momentum - through the homogeneity of space.

And yet, why exactly the number e, and not some other, was included in Euler’s formula and turned out to be at the base of the wave function? Staying within limits school courses mathematics and physics, answering this question is not easy. The author discussed this problem with the theorist, Doctor of Physical and Mathematical Sciences V.D. Efros, and we tried to explain the situation as follows.

The most important class of processes - linear and linearized processes - retains its linearity precisely due to the homogeneity of space and time. Mathematically, a linear process is described by a function that serves as a solution to a differential equation with constant coefficients(this type of equations is studied in the first and second years of universities and colleges). And its core is the above Euler formula. So the solution contains a complex function with base e, just like the wave equation. Moreover, it is e, and not another number in the base of the degree! Because only the function ex does not change for any number of differentiations and integrations. And therefore, after substitution into the original equation, only the solution with the base e will give an identity, as a correct solution should.

Now let’s write down the solution to a differential equation with constant coefficients that describes the propagation of a harmonic wave in a medium, taking into account the inelastic interaction with it, leading to energy dissipation or the acquisition of energy from external sources:

f(t) = e (α+ib)t = e αt (cos βt + isin βt).

We see that Euler's formula is multiplied by a real variable e αt, which is the amplitude of the wave changing over time. Above, for simplicity, we assumed it to be constant and equal to 1. This can be done in the case of undamped harmonic oscillations, with α = 0. In the general case of any wave, the behavior of the amplitude depends on the sign of the coefficient a with the variable t (time): if α > 0, the amplitude of oscillations increases if α< 0, затухает по экспоненте.

Perhaps the last paragraph is difficult for graduates of many ordinary schools. It, however, should be understandable to students of universities and colleges who thoroughly study differential equations with constant coefficients.

Now let’s set β = 0, that is, we will destroy the oscillatory factor with number i in the solution containing Euler’s formula. From the former oscillations, only the “amplitude” that decays (or grows) exponentially will remain.

To illustrate both cases, imagine a pendulum. In empty space it oscillates without damping. In space with a resistive medium, oscillations occur with exponential decay in amplitude. If you deflect a not too massive pendulum in a sufficiently viscous medium, then it will smoothly move towards the equilibrium position, slowing down more and more.

So, from thesis 2 we can deduce the following corollary:

Corollary 1. In the absence of an imaginary, purely vibrational part of the function f(t), at β = 0 (that is, at zero frequency), the real part exponential function describes many natural processes that proceed in accordance with the fundamental principle: the increase in value is proportional to the value itself .

The formulated principle mathematically looks like this: ∆I ~ I∆t, where, let’s say, I is a signal, and ∆t is a small time interval during which the signal ∆I increases. Dividing both sides of the equality by I and integrating, we obtain lnI ~ kt. Or: I ~ e kt - the law of exponential increase or decrease of the signal (depending on the sign of k). Thus, the law of proportionality of the increase in value to the value itself leads to natural logarithm and thus to the number e. (And here this is shown in a form accessible to high school students who know the elements of integration.)

Many processes in physics, chemistry, biology, ecology, economics, etc., proceed exponentially with a real argument, without hesitation. We especially note the universal psychophysical law of Weber - Fechner (for some reason ignored in educational programs schools and universities). It reads: “The strength of sensation is proportional to the logarithm of the strength of stimulation.”

Vision, hearing, smell, touch, taste, emotions, and memory are subject to this law (naturally, until physiological processes abruptly turn into pathological ones, when the receptors have undergone modification or destruction). According to the law: 1) a small increase in the irritation signal in any interval corresponds to a linear increase (with a plus or minus) in the strength of sensation; 2) in the area of weak irritation signals, the increase in the strength of sensation is much steeper than in the area of strong signals. Let's take tea as an example: a glass of tea with two pieces of sugar is perceived as twice as sweet as tea with one piece of sugar; but tea with 20 pieces of sugar is unlikely to seem noticeably sweeter than with 10 pieces. The dynamic range of biological receptors is colossal: signals received by the eye can vary in strength by ~ 10 10 , and by the ear - by ~ 10 12 times. Live nature adapted to such ranges. It protects itself by taking a logarithm (by biological limitation) of incoming stimuli, otherwise the receptors would die. The widely used logarithmic (decibel) scale of sound intensity is based on the Weber-Fechner law, in accordance with which the volume controls of audio equipment operate: their displacement is proportional to the perceived volume, but not to the sound intensity! (The sensation is proportional to lg/ 0. The threshold of audibility is taken to be p 0 = 10 -12 J/m 2 s. At the threshold we have lg1 = 0. An increase in the strength (pressure) of sound by 10 times corresponds approximately to the sensation of a whisper, which is 1 bel above the threshold on a logarithmic scale. Sound amplification a million times from a whisper to a scream (up to 10 -5 J/m 2 s) on a logarithmic scale is an increase of 6 orders of magnitude or 6 Bel.)

Probably, such a principle is optimally economical for the development of many organisms. This can be clearly observed in the formation of logarithmic spirals in mollusk shells, rows of seeds in a sunflower basket, and scales in cones. The distance from the center increases according to the law r = ae kj. At each moment, the growth rate is linearly proportional to this distance itself (which is easy to see if we take the derivative of the written function). The profiles of rotating knives and cutters are made in a logarithmic spiral.

Corollary 2. The presence of only the imaginary part of the function at α = 0, β 0 in the solution of differential equations with constant coefficients describes a variety of linear and linearized processes in which undamped harmonic oscillations take place.

This corollary brings us back to the model already discussed above.

Corollary 3. When implementing Corollary 2, there is a “closing” in a single formula of numbers and e through Euler’s historical formula in its original form e i = -1.

In this form, Euler first published his exponent with an imaginary exponent. It is not difficult to express it through the cosine and sine on the left side. Then the geometric model of this formula will be motion in a circle with a speed constant in absolute value, which is the sum of two harmonic oscillations. According to the physical essence, the formula and its model reflect all three fundamental properties of space-time - their homogeneity and isotropy, and thereby all three conservation laws.

Conclusion

The statement about the connection of conservation laws with the homogeneity of time and space is undoubtedly correct for Euclidean space in classical physics and for the pseudo-Euclidean Minkowski space in the General Theory of Relativity (GR, where time is the fourth coordinate). But within the framework of general relativity, a natural question arises: what is the situation in regions of huge gravitational fields, near singularities, in particular, near black holes? The opinions of physicists here differ: the majority believe that the indicated fundamental provisions are preserved in these extreme conditions. However, there are other points of view of authoritative researchers. Both are working on creating a new theory of quantum gravity.

To briefly imagine what problems arise here, let us quote the words of theoretical physicist Academician A. A. Logunov: “It (Minkowski space. - Auto.) reflects properties common to all forms of matter. This ensures the existence of unified physical characteristics- energy, momentum, angular momentum, laws of conservation of energy, momentum. But Einstein argued that this is possible only under one condition - in the absence of gravity<...>. From this statement of Einstein it followed that space-time becomes not pseudo-Euclidean, but much more complex in its geometry - Riemannian. The latter is no longer homogeneous. It changes from point to point. The property of space curvature appears. The exact formulation of conservation laws, as they were accepted in classical physics, also disappears in it.<...>Strictly speaking, in general relativity, in principle, it is impossible to introduce the laws of conservation of energy-momentum; they cannot be formulated" (see "Science and Life" No. 2, 3, 1987).

The fundamental constants of our world, the nature of which we talked about, are known not only to physicists, but also to lyricists. Thus, the irrational number equal to 3.14159265358979323846... inspired the outstanding Polish poet of the twentieth century, laureate Nobel Prize 1996 to Wisław Szymborska for the creation of the poem “Pi,” with a quote from which we will end these notes:

A number worthy of admiration:
Three comma one four one.
Each number gives a feeling
start - five nine two,
because you will never reach the end.
You can’t grasp all the numbers at a glance -
six five three five.
Arithmetic operations -
eight nine -
is no longer enough, and it’s hard to believe -
seven nine -
that you can’t get away with it - three two three
eight -
nor an equation that does not exist,
not a joking comparison -
you can't count them.
Let's move on: four six...
(Translation from Polish - B. G.)