Number e

Number e . It is one of the most important real numbers next to 0, 1 and pi. Obtained by a new resource in the mathematics of motion: limit of a sequence. It is an irrational and transcendent number , the basis of the system of natural or neperian logarithms , whose first reference is due to the Scottish mathematician and theologian John Napier in his work Mirifici Logarithmorum Canonis Descriptio from 1614 , although there was no approximation of the numerical value of and until Jacob Bernoulli obtained the first value by solving a compound interest problem on a fixed initial quantity with continuous variation of time.

The functions that start from their value: the natural logarithm and the exponential function have well-known properties within mathematics and simple numerical calculation expressions that allow other traditional functions to be supported computationally. For example, the less typical values of sine and cosine are calculated by taking advantage of the relationship between the trigonometric and exponential representations of complex numbers .

Table of Contents

Summary

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1 Equivalent names
2 History
3 Definitions
4 Importance
5 Presence and peculiarities
- 1 Features
6 Sources
7 References and notes

Equivalent names

Among its many equivalent names are:

Constant e.
Euler’snumber or constant .
Napier number or constant (or Neper, as it is usually Latinized).

History

The first indirect reference to e is given in Mirifici Logarithmorum Canonis Descriptio , the famous 1614 text by John Napier where his ideas about logarithms, antilogarithms, results and tables of calculations of them were first exposed; However, the first approximation would be obtained by Jacob Bernoulli in solving the problem of long-term interest of an initial fixed quantity that led him, after successive iterations, to the now well-known limit:

whose value set at 2.7182818 . Then the mathematician and philosopher Gottfried Leibniz in the letters of 1690 and 1691 to Christiaan Huygens made use of the value by representing it with the letter b . It was Leonard Euler who began in 1727 to identify the number with its current symbol: the letter e , but it is not until ten years later that he introduces it to the mathematical community in his book Mechanica .

Later the specialists would use a , b , c and e , until the latter finally triumphed to represent the irrational number. Charles Hermite demonstrated in 1873 that this was a momentous number. Their approximations began with the work of Bernoulli, then Euler made an approximation of 18 places after the comma and thus they have occurred as in the determination of the places of pi a kind of race that had its most recent edition in 2010 when Shigeru Kondo and Alexander J. Yee determined the exact 1 trillion decimals of e .

Definitions

Euler’s number, along with Archimedes’ pi number, has occupied a prominent place in mathematics, since Euler’s introduction tothe work Introductio in Analyssin Infinotorum appeared in 1748 . It provides a wonderful method in which the principle of monotonous sequences can be used to define a new real number. Using the notation

n! = 1 · 2.3.4 … n

for the first n positive integers, let be the sequence a ₁ , a ₂ , a ₃ in which

. a _n = 1 + 1/1! + 1/2! + … + 1 / n!

The terms a _n form an increasing sequence, since the following is obtained by adding a positive value 1 / (i + 1)! They are also bounded superior to _n <C = 3, which is proved comparing with a geometric progression of ratio 1/2.

According to the principle of monotonous sequences, when n tends to ∞, a must approach a limit to _n , and we will call this limit e . To denote e = lim a _n , we can write e in series form

e = 1 + 1/1! + 1/2! + … + 1 / n! + … ^[1]

The real number e is the result of the following expressions:

( Algebraic fundamental limit).
Continuous fractions:
1. (discovered by Leonard Euler).
2. .
ln x = 1.

Importance

Euler’s constant is of radical importance first in mathematics and then in many other sectors of production, science, and everyday life. Although the number pi is important in the area of circular trigonometry ; e plays a major role in the analysis , as it is part of many of the fundamental results of limits , derivatives , integrals, differential equations, hyperbolic trigonometry, series, etc. This constant has a series of properties that allow its use in the definition of expressions of great application in many areas of human knowledge: the Gaussian bell.

Firstly, e is an irrational number whose demonstration is carried out by the method of reduction to the absurd . Charles Hermite showed that it is a transcendent number and argues that it is normal .

Then, its value constitutes the result of the algebraic fundamental limit:

It is the basis of the defined natural logarithm function:

It has the following decomposition into power series according to Taylor:

for x> 0

What makes it a potential candidate for the calculation of other logarithms in view of the fundamental property of the logarithm:

which is the inverse function of the exponential:

as it is known, it retains its property before the integral and the derivative, that is:

who can be computationally calculated, for example, by developing Taylor series of derivatives:

(one)

This development is very advantageous for the computational calculation and therefore of exponentiations in other bases from the transformation only as a function of e ^x and ln (x) :

a ^x= e ^x

The exponential function can be expanded to the body of complex numbers thanks to the relationship between the exponential and trigonometric representations of those.

and since the argument is any real (angle in radians) this allows obtaining versions of (1) as:

that once the powers are developed and the real and imaginary parts such as cosines and sines, respectively, are grouped, they are expressed:

or better
or

The definition of the hyperbolic functions senh and cosh can be easily described by the relation:

e ^x= senh (x) + cosh (x)

since:

It is also used in the modeling of phenomena in finance and economics , the growth behavior of populations of biological species, the mean time of disintegration of isotopes and subatomic particles , climate models, etc.

Presence and peculiarities

It serves as the basis of the system of natural or neperian logarithms; it is denoted by lnx = t, where x is a positive real number and t is positive for x> 1 and negative for x <1. ln 1 = 0. The equivalence is lnx = t if, only if x = e ^t.
It is present in the definition of the function y (x) = e ^x, or y (x) = exp (x), its set of admissible values CVA being the set R of all real numbers.
When defining a complex number of modulo 1, we have the expression e ^{ix}= cosx + isenx. Geometrically represents a circumference of radius1, center: O
From the above it results and ^iπ+ 1 = 0

Features

e is an irrational real number, that is, it cannot express as the ratio of two integers; well it’s not a rational number ..
e is a transcendent number, which means that it cannot be the root of an algebraic equation, of those that have rational coefficients.
It is presented in the solution of ordinary differential equations, among others, of the type dx / dt = kx
Its polynomial development serves to define the exponential of a matrix, used in solving a system of ordinary differential equations.