College Mathematics Physical Science Biology: The number e mathematical constant is the base of the natural logarithm.

The number $e$ is an important mathematical constant that is the base of the natural logarithm. It is approximately equal to 2.71828,^[and is the limit of

(1 + 1/ n) n

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]

$e = 1 + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \frac{1}{1\cdot 2\cdot 3\cdot 4}+\cdots$

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) = e x

at the point

x = 0

is equal to 1.^[3] The function

e x

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

between

x = 1

and

x = k

, in which case,

e

is the number whose natural logarithm is 1. There are also more alternative characterizations.

The area between the

x

-axis and the graph

y = 1/ x

, between

x = 1

and

x = e

is 1.

College Mathematics Physical Science Biology