Editing E (mathematical constant) (section)

== Definitions ==
The number {{mvar|e}} is the [[Limit of a sequence|limit]] 
<math display= block>\lim_{n\to \infty}\left(1+\frac 1n\right)^n,</math>
an expression that arises in the computation of [[compound interest]].

It is the sum of the infinite [[Series (mathematics)|series]]
<math display ="block">e = \sum\limits_{n = 0}^{\infty} \frac{1}{n!} = 1 + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots.</math>

It is the unique positive number {{mvar|a}} such that the graph of the function {{math|1=''y'' = ''a''<sup>''x''</sup>}} has a [[slope]] of 1 at {{math|1=''x'' = 0}}.

One has <math display=block>e=\exp(1),</math> where <math>\exp</math> is the (natural) [[exponential function]], the unique function that equals its own [[derivative]] and satisfies the equation <math>\exp(0)=1.</math> Since the exponential function is commonly denoted as <math>x\mapsto e^x,</math> one has also
<math display=block>e=e^1.</math>

The [[logarithm]] of base {{mvar|b}} can be defined as the [[inverse function]] of the function <math>x\mapsto b^x.</math> Since <math>b=b^1,</math> one has <math>\log_b b= 1.</math> The equation <math>e=e^1</math> implies therefore that {{mvar|e}} is the base of the natural logarithm.

The number {{mvar|e}} can also be characterized in terms of an [[integral]]:<ref>{{dlmf}}</ref>
<math display = block>\int_1^e \frac {dx}x =1.</math>

For other characterizations, see {{slink||Representations}}.