Equals a constant e. Math I like

Everybody knows geometric meaning numbers π is the circumference of a circle with a unit diameter:

And here is the meaning of another important constant, e, tends to be quickly forgotten. That is, I don’t know about you, but each time it’s worth the effort for me to remember why this number equal to 2.7182818284590 is so remarkable ... (however, I wrote down the value from memory). Therefore, I decided to write a note so that more does not fly out of memory.

Number e by definition - the limit of a function y = (1 + 1 / x) x at x → ∞:

x	y
1	(1 + 1 / 1) 1	= 2
2	(1 + 1 / 2) 2	= 2,25
3	(1 + 1 / 3) 3	= 2,3703703702...
10	(1 + 1 / 10) 10	= 2,5937424601...
100	(1 + 1 / 100) 100	= 2,7048138294...
1000	(1 + 1 / 1000) 1000	= 2,7169239322...
∞	lim × → ∞	= 2,7182818284590...

This definition, unfortunately, is not clear. It is not clear why this limit is remarkable (despite the fact that it is called the "second remarkable"). Just think, they took some clumsy function, calculated the limit. Another function will have another.

But the number e for some reason pops up in a whole bunch of the most different situations in mathematics.

For me main point numbers e is revealed in the behavior of another, much more interesting function, y = k x. This feature has unique property at k = e, which can be shown graphically as follows:

At point 0, the function takes on the value e 0 = 1. If we draw a tangent at the point x= 0, then it will pass to the x-axis at an angle with tangent 1 (in yellow triangle the ratio of the opposite leg 1 to the adjacent 1 is 1). At point 1, the function takes on the value e 1 = e. If we draw a tangent at a point x= 1, then it will pass at an angle with tangent e(in green triangle opposite leg ratio e to adjacent 1 is equal to e). At point 2 the value e 2 function again coincides with the tangent of the slope of the tangent to it. Because of this, at the same time, the tangents themselves intersect the x-axis exactly at the points −1, 0, 1, 2, etc.

Among all features y = k x(e.g. 2 x , 10 x , π x etc.), function e x- the only one has such beauty that the tangent of its slope at each of its points coincides with the value of the function itself. So, by definition, the value of this function at each point coincides with the value of its derivative at this point: ( e x)´ = e x. For some reason the number e= 2.7182818284590... must be raised to different degrees to get this picture.

That, in my opinion, is its meaning.

Numbers π and e are included in my favorite formula - Euler's formula, which connects the 5 most important constants - zero, one, imaginary one i and actually numbers π and e:

eip + 1 = 0

Why is the number 2.7182818284590... in complex degree 3,1415926535...i suddenly equal to minus one? The answer to this question is beyond the scope of a note and could form the content of a small book that would require some initial understanding of trigonometry, limits and series.

I have always been amazed by the beauty of this formula. Perhaps in mathematics there are more amazing facts, but for my level (three in the Physics and Mathematics Lyceum and five for complex analysis at the university) is the most important miracle.

NUMBER e
A number approximately equal to 2.718, which is often found in mathematics and natural sciences. For example, when breaking radioactive substance after time t, a fraction equal to e-kt remains from the initial amount of the substance, where k is a number characterizing the decay rate given substance. The reciprocal value of 1/k is called the average lifetime of an atom of a given substance, since on average an atom exists for the time 1/k before decaying. The value 0.693/k is called the half-life of the radioactive substance, i.e. the time it takes for half of the original amount of the substance to decay; the number 0.693 is approximately equal to loge 2, i.e. logarithm of 2 to base e. Similarly, if bacteria in a nutrient medium multiply at a rate proportional to their number in this moment, then after time t initial quantity bacteria N turns into Nekt. attenuation electric current I in a simple circuit with serial connection, resistance R and inductance L occurs according to the law I = I0e-kt, where k = R/L, I0 is the current strength at time t = 0. Similar formulas describe stress relaxation in a viscous fluid and attenuation magnetic field. The number 1/k is often called the relaxation time. In statistics, the value of e-kt occurs as the probability that during the time t there were no events occurring randomly with an average frequency of k events per unit time. If S is the amount of money invested at r percent with continuous accrual instead of accrual at discrete intervals, then by time t the initial amount will increase to Setr/100. The reason for the "omnipresence" of the number e is that the formulas mathematical analysis, containing exponential functions or logarithms, are written more simply if the logarithms are taken to the base e, rather than 10 or some other base. For example, the derivative of log10 x is (1/x)log10 e, while the derivative of loge x is simply 1/x. Similarly, the derivative of 2x is 2xloge 2, while the derivative of ex is simply ex. This means that the number e can be defined as the base b, for which the graph of the function y = logb x has a tangent at the point x = 1 with slope factor equal to 1, or for which the curve y = bx has a tangent at x = 0 with slope equal to 1. Logarithms in base e are called "natural" and are denoted by ln x. Sometimes they are also called "non-Perean", which is incorrect, since in reality J. Napier (1550-1617) invented logarithms with a different base: the non-Perian logarithm of the number x is 107 log1 / e (x / 107) (see also LOGARITHM). Various combinations of powers of e are so common in mathematics that they have special names. These are, for example, hyperbolic functions

The graph of the function y = ch x is called a catenary; a heavy inextensible thread or chain suspended by the ends has such a shape. Euler formulas

where i2 = -1, associate the number e with trigonometry. special case x = p leads to the famous relation eip + 1 = 0, linking the 5 most famous numbers in mathematics. When calculating the value of e, some other formulas can also be used (the first of them is most often used):

The value of e with 15 decimal places is 2.718281828459045. In 1953, the value of e was calculated with 3333 decimal places. The symbol e for this number was introduced in 1731 by L. Euler (1707-1783). The decimal expansion of the number e is non-periodic (e is an irrational number). In addition, e, like p, is a transcendental number (it is not the root of any algebraic equation with rational coefficients). This was proved in 1873 by Sh. Hermit. It was shown for the first time that a number that arises in such a natural way in mathematics is transcendental.
see also
MATHEMATICAL ANALYSIS ;
CONTINUED FRACTIONS ;
NUMBERS THEORY;
NUMBER p;
ROWS.

Collier Encyclopedia. - Open society. 2000 .

See what "NUMBER e" 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. Concept, serving as an expression quantities, that with which the counting of objects and phenomena is made (mat.). Integer. 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 grammatical category expressing quantitative characteristics objects of thought. grammatical number one of the manifestations of a more general language category quantity (see Language category) along with lexical manifestation("lexical ... ... Linguistic Encyclopedic Dictionary

BUT; 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 (integer positive number). Simple h. ( natural number, not… … Explanatory dictionary of Ozhegov

NUMBER "E" (EXP), an irrational number that serves as the basis of natural LOGARITHMS. It's valid decimal number, an infinite fraction equal to 2.7182818284590...., is the limit of the expression (1/) as n tends to infinity. In fact,… … Scientific and technical encyclopedic dictionary

Quantity, cash, composition, strength, contingent, amount, figure; day.. Wed. . See day, quantity. not big number, there is 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. A textbook on extrasensory perception (number of volumes: 3)
Name number. A new look at numbers. Numerology - the way of knowledge (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…

Describing e as “a constant approximately equal to 2.71828…” is like calling the number pi “ irrational number, approximately equal to 3.1415 ... ". No doubt it is, but the essence still eludes us.

The number pi is the ratio of the circumference of a circle to its diameter, the same for all circles.. This is a fundamental proportion common to all circles, and therefore, it is involved in calculating the circumference, area, volume and surface area for circles, spheres, cylinders, etc. Pi shows that all circles are connected, not to mention trigonometric functions derived from circles (sine, cosine, tangent).

The number e is the basic growth ratio for all continuously growing processes. The number e allows you to take a simple growth rate (where the difference is visible only at the end of the year) and calculate the components of this indicator, normal growth, in which every nanosecond (or even faster) everything grows by a little more.

The number e participates in both exponential and constant growth systems: population, radioactive decay, interest calculation, and many, many others. Even stepped systems that do not grow uniformly can be approximated by the number e.

Just as any number can be viewed as a "scaled" version of 1 (the base unit), any circle can be viewed as a "scaled" version of unit circle(with radius 1). And any growth factor can be considered as a "scaled" version of e (a "single" growth factor).

So the number e is not a random number taken at random. The number e embodies the idea that all continuously growing systems are scaled versions of the same metric.

The concept of exponential growth

Let's start by looking at the basic system that doubles behind certain period time. For example:

Bacteria divide and "doubling" in numbers every 24 hours
We get twice as many noodles if we break them in half
Your money doubles every year if you get 100% profit (lucky!)

And it looks something like this:

Dividing by two or doubling is a very simple progression. Of course, we can triple or quadruple, but doubling is more convenient for explanation.

Mathematically, if we have x divisions, we get 2^x times more good than we had at the beginning. If only 1 partition is made, we get 2^1 times more. If there are 4 partitions, we get 2^4=16 parts. General formula looks like that:

growth= 2 x

In other words, a doubling is a 100% increase. We can rewrite this formula like this:

growth= (1+100%) x

This is the same equality, we just divided "2" into its constituent parts, which in essence this number is: initial value(1) plus 100%. Smart, right?

Of course, we can substitute any other number (50%, 25%, 200%) instead of 100% and get the growth formula for this new ratio. The general formula for x periods of the time series will look like:

growth = (1+growth) x

This simply means that we use the rate of return, (1 + growth), "x" times in a row.

Let's take a closer look

Our formula assumes that growth occurs in discrete steps. Our bacteria wait, wait, and then bam!, and in last minute they double in number. Our profit on interest from the deposit magically appears exactly after 1 year. Based on the formula written above, profits grow in steps. Green dots appear suddenly.

But the world is not always like this. If we zoom in, we can see that our bacteria friends are constantly dividing:

The green kid doesn't come out of nothing: it slowly grows out of the blue parent. After 1 period of time (24 hours in our case), the green friend is already fully ripe. Having matured, he becomes a full-fledged blue member of the herd and can create new green cells himself.

Will this information somehow change our equation?

Nope. In the case of bacteria, the half-formed green cells still can't do anything until they grow up and completely separate from their blue parents. So the equation is correct.

NUMBER e. 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 a time t from the initial amount of the substance remains a fraction equal to e–kt, where k- a number characterizing the rate of decay of a given substance. Reciprocal 1/ k is called the average lifetime of an atom of a given substance, since on average an atom exists for the time 1/ k. Value 0.693/ k is called the half-life of a radioactive substance, i.e. the time it takes for half of the original amount of the substance to decay; the number 0.693 is approximately equal to log e 2, i.e. base logarithm of 2 e. Similarly, if bacteria in a nutrient medium multiply at a rate proportional to their number at the moment, then after time t initial number of bacteria N turns into Ne kt. Attenuation of electric current I in a simple circuit with a series connection, resistance R and inductance L happens according to law I = I 0 e–kt, where k = R/L, I 0 - current strength at the time t= 0. Similar formulas describe stress relaxation in a viscous fluid and magnetic field damping. Number 1/ k often referred to as relaxation time. In statistics, the value e–kt occurs as the probability that over time t there were no events occurring randomly with an average frequency k events per unit of time. If a S- amount of money invested r interest with continuous accrual instead of accrual at discrete intervals, then by the time t the initial amount will increase to Setr/100.

The reason for the "omnipresence" of the number e is that the formulas of mathematical analysis containing exponential functions or logarithms are written easier if the logarithms are taken in base e, not 10 or some other base. For example, the derivative of log 10 x equals (1/ x)log 10 e, while the derivative of log ex is just 1/ x. Similarly, the derivative of 2 x equals 2 x log e 2, while the derivative of e x equals just ex. This means that the number e can be defined as the basis b, for which the graph of the function y= log b x has at the point x= 1 tangent with slope equal to 1, or at which the curve y = b x has in x= 0 tangent with slope equal to 1. Base logarithms e are called "natural" and denoted by ln x. Sometimes they are also called "non-Perian", which is not true, since in reality J. Napier (1550–1617) invented logarithms with a different base: the non-Perian logarithm of a number x equals 10 7 log 1/ e (x/10 7) .

Various degree combinations e are so common in mathematics that they have special names. These are, for example, the hyperbolic functions

Function Graph y=ch x called a catenary; a heavy inextensible thread or chain suspended by the ends has such a shape. Euler formulas

where i 2 = -1, bind number e with trigonometry. special case x = p leads to the famous relation ip+ 1 = 0, linking the 5 most famous numbers in mathematics.

| Euler number (E)

e - base of natural logarithm, mathematical constant, 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 an appendix to John Napier's work on logarithms, a table of natural logarithms was given various numbers. However, no one understood that these are base logarithms e , since such a thing as a base was not included in the concept of the logarithm of that time. This is now what we call the 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 Ougthred, although the author was not credited. A few years later, in 1624, the mathematical literature reappears e , but again veiled. This year, Briggs gave a numerical approximation of the base 10 logarithm e , but the number itself e not mentioned in his work.

The next occurrence of the number e again doubtful. In 1647, Saint-Vincent calculated the area of a hyperbolic sector. Whether he understood the connection with logarithms, one can only guess, but even if he understood, it is unlikely that he could come to the number itself e . It was not until 1661 that Huygens understood the connection between the isosceles hyperbola and logarithms. He proved that the area under the graph of an isosceles hyperbola xy = 1 isosceles hyperbola on the interval from 1 to e is 1. This property makes e the base of natural logarithms, but the mathematicians of that time did not understand this, but they slowly approached this understanding.

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

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 the series expansion log(1 + x) . In this work, Mercator first uses the name “ natural logarithm” for base logarithm e . Number e obviously does not appear again, but remains elusive somewhere in the distance.

Surprisingly, the number e explicitly arises 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, and this 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 that 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 associated with exponents in any way. Of course, from the equation x = a t we find that t = log x , but this is a much later way of perceiving. Here we really mean by logarithm a function, whereas at first the logarithm was considered only as a number that helped in calculations. Perhaps Jacob Bernoulli was the first to realize that logarithmic function is inversely exponential. On the other hand, the first to link logarithms and powers could be James Gregory. In 1684 he definitely recognized the connection between logarithms and powers, but he may not have been the first.

We know that the number e appeared in the form as it is now, 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 begins to study the exponential function and publishes Principia calculi exponentialum seu percurrentium. In this paper, the sums of various exponential series are calculated, and some results are obtained by integrating them term by term.

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

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

True, without explaining how he got them. It looks like he calculated this value himself. In fact, if you take about 20 terms of the series (1), you get the accuracy that Euler got. Among the others interesting results in his work, the relationship 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 expansions. In particular, he received

Euler did not provide proof that these fractions continue in the same way, but he knew that if there were such a proof, then it would prove irrationality e . Indeed, if the continued fraction for (e - 1) / 2 , continued in the same way as in the above sample, 6,10,14,18,22,26, (we add 4 each time), then it would never be 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 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 were received e . In fact, it takes about 120 terms of the expansion (1) to get 200 correct digits of the number e .

In 1864, Benjamin Pierce (Peirce) stood at the blackboard on which was written

In his lectures, he might say to his students, "Gentlemen, we have no 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. It still remains open question whether the number e e algebraic. The final result in this direction is that at least one of the numbers e e and e e 2 is transcendent.

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

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