
Why e is base of natural logarithm?
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** e (mathematical constant) **
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"Euler's number" redirects here. For Î³ (gamma), a constant in number
theory, see EulerâMascheroni constant. For other uses, see List of
things named after Leonhard Euler Â§ Euler's numbers.
Part of a series of articles on the
mathematical constant *e*
Euler's formula.svg
Properties
· Natural logarithm
· Exponential function
Applications
· compound interest
· Euler's identity
· Euler's formula
· halflives
· exponential growth and decay
Defining e
· proof that e is irrational
· representations of e
· LindemannâWeierstrass theorem
People
· John Napier
· Leonhard Euler
Related topics
· Schanuel's conjecture
· v
· t
· e
The number *e* is an important mathematical constant that is the base of
the natural logarithm. It is approximately equal to 2.71828,^[1] and is the
limit of (1 + 1//n/)^/n/ as n approaches infinity, an expression that
arises in the study of compound interest. It can also be calculated as the
sum of the infinite series^[2]
e = \displaystyle\sum\limits_{n = 0}^{ \infty} \dfrac{1}{n!} = 1 +
\frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots
The constant can be defined in many ways. For example, e can be defined as
the unique positive number a such that the graph of the function /y/ =
/a/^/x/ has unit slope at /x/ = 0.^[3] The function /f/(/x/) = /e/^/x/ is
called the exponential function, and its inverse is the natural logarithm,
or logarithm to base e. The natural logarithm of a positive number /k/ can
also be defined directly as the area under the curve /y/ = 1//x/ between
/x/ = 1 and /x/
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