Euler's constant e

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Number

$e$

2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003059921817413596629043572900334295260595630738132328627943490763233829880753195251019011573834187930702154089149934884167509244761460668082264800168477411853742345442437107539077744992069

Definition

The number $e$ is the unique positive real number $a$ such that the function $x \mapsto a^x$ is its own derivative.

Formulas

(1)

$e = \sum_{n\geq 0} 1/n!$.

(2)

$e = \lim_{n\to\infty} (1+1/n)^n$.

(3)

$\int_1^e 1/x\, dx = 1$.

Programs

(P1)

Sage

from sage.rings.real_arb import RealBallField

numbers = [RealBallField(1024)(1).exp()]

Links

[1]

Wikipedia: e

[2]

OEIS: A001113

Data properties

Entries are of type: real number

Table is complete: yes (it holds the single number named by the definition)

How well the digits are known: proven

How they were obtained:

The entry is the decimal truncation of $\exp(1)$. Recomputing $\exp(1)$ in ball arithmetic at 4000 bits confirms every digit written; the stored decimal is within one unit in the last place.