"Euler's Number" e
e is the unique number a, such that the derivative of the
exponential function f(x) = ax (blue curve) at the point x = 0 is equal to 1. For
comparison, functions 2x (dotted
curve) and 4x (dashed
curve) are shown; they are not tangent to the line of slope 1 (red).
The number e is
an important mathematical
constant, approximately
equal to 2.71828, that is the base of the natural logarithms. It is the limit of(1 + 1/n)n as n becomes large, an expression
that arises in the study of compound interest,
and can also be calculated as the sum of the seriese = 2 + 1/2 + 1/(2 × 3) + 1/(2 × 3 ×
4) + 1/(2 × 3 × 4 × 5) + …
The constant can be defined in many ways; for example, e is the unique real number such
that the value of the derivative (slope
of the tangent line) of the function f(x) = ex at
the point x = 0 is
equal to 1. The function ex so
defined 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 betweenx =
1 and x = k, in which case, e is the number whose natural
logarithm is 1. There are also more alternative characterizations.
Sometimes called Euler's number after the Swiss mathematician Leonhard Euler, e is not to be confused with γ—the Euler–Mascheroni
constant, sometimes
called simply Euler's constant.
It is also known as Napier's constant, but Euler's choice of the
symbol e is said to
have been retained in his honor. The number e is of eminent importance in mathematics, alongside 0, 1, π and i. All five of these numbers play important and recurring
roles across mathematics, and are the five constants appearing in one
formulation of Euler's identity.
Like the constant π, e is irrational:
it is not a ratio of integers; and it is transcendental: it is not a root of any non-zero polynomial with
rational coefficients. The numerical value of e truncated to 50decimal places is
