[br][img width=412,height=304]https://mathworld.wolfram.com/images/eps-gif/EAreaPlot_1000.gif[/img][br][br]The [url=https://mathworld.wolfram.com/Constant.html]constant[/url] [img width=6,height=15]https://mathworld.wolfram.com/images/equations/e/Inline1.gif[/img] is base of the [url=https://mathworld.wolfram.com/NaturalLogarithm.html]natural logarithm[/url]. [img width=6,height=15]https://mathworld.wolfram.com/images/equations/e/Inline2.gif[/img] is sometimes known as Napier's constant, although its symbol ([img width=6,height=15]https://mathworld.wolfram.com/images/equations/e/Inline3.gif[/img]) honors Euler.[img width=6,height=15]https://mathworld.wolfram.com/images/equations/e/Inline4.gif[/img] is the unique number with the property that the area of the region bounded by the [url=https://mathworld.wolfram.com/Hyperbola.html]hyperbola[/url] [img width=46,height=15]https://mathworld.wolfram.com/images/equations/e/Inline5.gif[/img], the [url=https://mathworld.wolfram.com/x-Axis.html][i]x[/i]-axis[/url], and the vertical lines [img width=31,height=15]https://mathworld.wolfram.com/images/equations/e/Inline6.gif[/img] and [img width=30,height=15]https://mathworld.wolfram.com/images/equations/e/Inline7.gif[/img] is 1. In other words,[table][tr][td][img width=107,height=37]https://mathworld.wolfram.com/images/equations/e/NumberedEquation1.gif[/img][/td][td][br][/td][/tr][/table]With the possible exception of [img width=7,height=15]https://mathworld.wolfram.com/images/equations/e/Inline8.gif[/img], [img width=6,height=15]https://mathworld.wolfram.com/images/equations/e/Inline9.gif[/img] is the most important constant in mathematics since it appears in myriad mathematical contexts involving [url=https://mathworld.wolfram.com/Limit.html]limits[/url] and [url=https://mathworld.wolfram.com/Derivative.html]derivatives[/url]. The numerical value of [img width=6,height=15]https://mathworld.wolfram.com/images/equations/e/Inline10.gif[/img] is[table][tr][td][img width=324,height=15]https://mathworld.wolfram.com/images/equations/e/NumberedEquation2.gif[/img][/td][td][br][/td][/tr][/table][img width=366,height=226]https://mathworld.wolfram.com/images/eps-gif/ELimit_1001.gif[/img][br][br][img width=6,height=15]https://mathworld.wolfram.com/images/equations/e/Inline11.gif[/img] can be defined by the [url=https://mathworld.wolfram.com/Limit.html]limit[/url][table][tr][td][img width=95,height=35]https://mathworld.wolfram.com/images/equations/e/NumberedEquation3.gif[/img][/td][td][br][/td][/tr][/table](illustrated above), or by the infinite series[table][tr][td][img width=61,height=44]https://mathworld.wolfram.com/images/equations/e/NumberedEquation4.gif[/img][/td][td][br][/td][/tr][/table]as first published by Newton (1669; reprinted in Whiteside 1968, p. 225).[img width=6,height=15]https://mathworld.wolfram.com/images/equations/e/Inline12.gif[/img] is given by the unusual limit[table][tr][td][img width=188,height=43]https://mathworld.wolfram.com/images/equations/e/NumberedEquation5.gif[/img][br][br][br][/td][/tr][/table]