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| The constant ''e'' (occasionly called ''Napier's constant'' in honor of the Scottish mathematician ] who introduced logarithms) is the base of the ]. It is approximately equal to | The constant ''e'' (occasionly called ''Napier's constant'' in honor of the Scottish mathematician ] who introduced logarithms) is the base of the ]. It is approximately equal to | ||
| : ''e'' = 2.718281828459045235360287471352662497757247093699959574966 | |||
| : ''e'' = 2.71828 18284 59045 23536 02874 ..... | |||
| 967627724076630353547594571382178525166427427466391932003059 | |||
| 921817413596629043572900334295260595630738132328627943490763 | |||
| 233829880753195251019011573834187930702154089149934884167509 | |||
| 244761460668082264800168477411853742345442437107539077744992 | |||
| 069551702761838606261331384583000752044933826560297606737113 | |||
| 200709328709127443747047230696977209310141692836819025515108 | |||
| 657463772111252389784425056953696770785449969967946864454905 | |||
| 987931636889230098793127736178215424999229576351482208269895 | |||
| 193668033182528869398496465105820939239829488793320362509443 | |||
| 117301238197068416140397019837679320683282376464804295311802 | |||
| 328782509819455815301756717361332069811250996181881593041690 | |||
| 351598888519345807273866738589422879228499892086805825749279 | |||
| 610484198444363463244968487560233624827041978623209002160990 | |||
| 235304369941849146314093431738143640546253152096183690888707 | |||
| 016768396424378140592714563549061303107208510383750510115747 | |||
| 704171898610687396965521267154688957035035402123407849819334 | |||
| 321068170121005627880235193033224745015853904730419957777093 | |||
| 503660416997329725088687696640355570716226844716256079882651 | |||
| 787134195124665201030592123667719432527867539855894489697096 ..... | |||
| It is equal to exp(1) where exp is the ] and therefore it is the ] of (1 + 1/''n'')<sup>''n''</sup> as ''n'' goes to infinity and can | It is equal to exp(1) where exp is the ] and therefore it is the ] of (1 + 1/''n'')<sup>''n''</sup> as ''n'' goes to infinity and can | ||
| also be written as the ] | also be written as the ] | ||
Revision as of 10:01, 30 March 2002
The constant e (occasionly called Napier's constant in honor of the Scottish mathematician John Napier who introduced logarithms) is the base of the natural logarithm. It is approximately equal to
- e = 2.718281828459045235360287471352662497757247093699959574966
967627724076630353547594571382178525166427427466391932003059 921817413596629043572900334295260595630738132328627943490763 233829880753195251019011573834187930702154089149934884167509 244761460668082264800168477411853742345442437107539077744992 069551702761838606261331384583000752044933826560297606737113 200709328709127443747047230696977209310141692836819025515108 657463772111252389784425056953696770785449969967946864454905 987931636889230098793127736178215424999229576351482208269895 193668033182528869398496465105820939239829488793320362509443 117301238197068416140397019837679320683282376464804295311802 328782509819455815301756717361332069811250996181881593041690 351598888519345807273866738589422879228499892086805825749279 610484198444363463244968487560233624827041978623209002160990 235304369941849146314093431738143640546253152096183690888707 016768396424378140592714563549061303107208510383750510115747 704171898610687396965521267154688957035035402123407849819334 321068170121005627880235193033224745015853904730419957777093 503660416997329725088687696640355570716226844716256079882651 787134195124665201030592123667719432527867539855894489697096 .....
It is equal to exp(1) where exp is the exponential function and therefore it is the limit of (1 + 1/n) as n goes to infinity and can also be written as the infinite series
- e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ...
∞ e = ∑ (n!)
Here n! stands for the factorial of n.
The number e is relevant because one can show that the exponential function exp(x) can be written as e; the exponential function is important because it is, up to multiplication by a scalar, the unique function which is its own derivative and is hence commonly used to model growth or decay processes.
The number e is known to be irrational and even transcendental. It features (along with a few other fundamental constants) in the most remarkable formula in the world.