E (mathematical constant)

The Euler number = 2.718281828459045235 ... (named after the Swiss mathematician Leonhard Euler ) is an irrational and even transcendental real number.

It is the base of natural logarithms and the ( natural ) exponential function. These (special ) exponential function is the basis of this relationship to the number often abbreviated function.

The Euler number of plays throughout the analysis and all related areas of mathematics, particularly in differential and integral calculus a central role. It is one of the most important constants of mathematics.

• 9.1 Evolution of the number of known digits of e
• 9.2 The first 200 digits of e

Definition

The number may be defined, inter alia, by limiting formation. The two most popular representations are:

As a series, with, by definition, the empty product.

With this, the faculty of, so the product of the natural numbers to designated. Both images corresponding to the function value of the exponential function (or " function" ) at the point 1; the series notation also corresponds to the Taylor expansion of the exponential function around the point zero evaluated at 1.

Properties

The Euler number is a transcendent (Proof by Charles Hermite, 1873) and thus irrational number ( proof). You can also ( as well as the circle number after Ferdinand von Lindemann 1882) are neither as a fraction of two integers or as a solution of an algebraic equation of finite degree and therefore has an infinite non-periodic decimal expansion. The Irrationalitätsmaß of 2 and thus as small as possible for an irrational number, in particular, is not liouvillesch. It is not known whether any base is normal.

In the Eulerian identity

Be put in context with fundamental mathematical constants: the integer 1, Euler's number, the imaginary unit of the complex numbers and the county number.

The Euler number is the only positive real number, which is valid for:

It occurs also in the asymptotic estimate of the Faculty (see Stirling formula ):

Origin of the symbol e

The character for this figure was first used in his work Mechanica Euler in 1736. There is no evidence that this was in line with its name also is unclear whether he did this based on the exponential function or, for practical reasons the determination of the much-used letters a, b, c or d. Although other names are in use, were about c in d' Alembert's Histoire de l' Académie, has prevailed.

Other representations for the Euler number

The Euler number can be achieved by also

Or described by the limit of the quotient of faculty and Subfakultät:

A connection to the distribution of prime numbers is via the formulas

Significantly, with the prime function and the symbol represents the primorial number.

Even more of an exotic appeal as of practical importance is the representation catalansche

Continued fractions

In connection with the number there at least since the publication of Leonhard Euler's Introductio in Analysin Infinitorum in 1748 a large number of continued fractions for and from derivable quantities.

So Euler found the following classical identity for:

The identity ( 1) apparently has a regular pattern, which continues to infinity. You are a regular continued fraction again, which was derived from Euler from the following:

The latter continued fraction is in turn the following with a special case:

Another classical continued fraction expansion, which is not, however, regularly, is also by Euler:

On Euler and Ernesto Cesàro another continued fraction expansion of Euler's number is declining, which is from another pattern as in (1):

In connection with the Euler's number, a large number of general chain breakage theoretical functional equations exists beyond. So Oskar Perron identifies one of several following the general representation of the function:

Another example of this is derived from Johann Heinrich Lambert development of the hyperbolic tangent, which is expected to Lambertian continued fractions:

Illustrative interpretations of Euler's number

Compound interest

The following example makes the calculation of Euler's number is not only descriptive, but it also describes the history of the discovery of Euler's number: your first posts were found by Jakob Bernoulli in the investigation of compound interest.

The limit of the first formula can be interpreted as follows: Somebody charged on January 1, a € a on the bench. The bank guarantee him a momentary interest at a rate per annum. How big is his balance on January 1 of next year, when he applies the interest on the same terms?

After the compound interest formula from the start capital after interest rates with interest rate, the capital

In this example, and if the interest is payable annually, or when the interest premium - times takes place in the year, so at under -standing interest.

If annual surcharge would be

In semi- annual award has,

So a bit more. With daily interest rate ( ), we obtain

If you currently earn interest, is infinitely great, and you get the above mentioned first formula.

Probability Theory

Is also frequently encountered in probability theory: For example, assume that a baker for each roll is a raisin in the dough and knead this well. After that, statistically speaking, each containing - te bun no raisin. The probability that none of the rolls with raisins in a tightly selected, results in the limit of (37 % rule ):

Characterization of the Euler's number after Steiner

In the fortieth volume of Crelle's Journal in 1850 the Swiss mathematician Jacob Steiner gives a characterization of the Euler's number, which can be understood as a solution of an extreme value problem. Steiner showed namely that the number can be characterized as the one uniquely determined positive real number, which by itself gives the square root, the largest root. Literally Steiner writes: If any number radicirt by itself, so the number e granted the greatest root.

Steiner addressed the question whether the function

The global maximum exists and how it is to be determined. His statement is that it exists and that it is believed in, and only in.

In his book Triumph of mathematics Heinrich Dorrie is an elementary solution of these extreme value problem. His approach is based on the following always valid for the real exponential inequality:

This is true for all positive real numbers always

By means of simple transformations follows immediately

And finally

Fraction approximations

For the number and quantities derived from it, there are various approximate representations by fractures. So Charles Hermite found the following fraction approximations:

This gives way to the former fraction by less than 0.0003 percent of ab.

The optimal Bruchnäherung in the three-digit number range, ie the optimal Bruchnäherung with is,

However, this approximation is not the best Bruchnäherung within the meaning of the requirement that the denominator should be no more than three digits. The best in this sense Bruchnäherung calculated as the 9th approximate fraction of the continued fraction expansion of Euler's number:

From the proximity of the quarries belonging to continued fractions (see above) to break approximations result for arbitrary precision and quantities derived from it. These are found very effective best break approximations of Euler's number in any number of areas. So receives about five-digit number range the best Bruchnäherung

Which shows that the Bruchnäherung found by Charles Hermite for the Euler number in the five-digit number range was not optimal.

In the same manner as CD Olds has shown that the approximation

For the Euler number, a further improvement, namely

Can be achieved.

Overall, the result of the best convergents of Euler's number, which result from their regular continued fraction begins as follows:

Others

Change in number of known digits of e

The first 200 digits of e

The decimal expansion of including the name of the first 200 decimal places is: