| Revision as of 10:12, 26 November 2001 edit165.123.179.xxx (talk) *fixed minor error in infinite series, added sum notation form.← Previous edit | Revision as of 15:51, 25 February 2002 edit undoConversion script (talk | contribs)10 editsm Automated conversionNext edit → | ||
| Line 1: | Line 1: | ||
| 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'' = 2.71828 18284 59045 23536 02874 ..... | : ''e'' = 2.71828 18284 59045 23536 02874 ..... | ||
| 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 ] | ||
| ⚫ | : ''e'' = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ... | ||
| ⚫ | : ''e'' = 1/1! + 1/2! + 1/3! + 1/4! + ... | ||
| <sub>∞</sub> | <sub>∞</sub> | ||
| ''e'' = ∑ (n!)<sup>-1</sup> | ''e'' = ∑ (n!)<sup>-1</sup> | ||
| <sup> n=0</sup> | <sup> n=0</sup> | ||
| Here ''n''! stands for the ] of ''n''. | Here ''n''! stands for the ] of ''n''. | ||
| The number ''e'' is relevant because one can show that the ] exp(''x'') can be written as ''e''<sup>''x''</sup>; the exponential function is important because it is, up to multiplication by a scalar, the unique function which is its own ] and is hence commonly used to model growth or decay processes. | The number ''e'' is relevant because one can show that the ] exp(''x'') can be written as ''e''<sup>''x''</sup>; the exponential function is important because it is, up to multiplication by a scalar, the unique function which is its own ] and is hence commonly used to model growth or decay processes. | ||
| The number ''e'' is known to be ] and even ]. | The number ''e'' is known to be ] and even ]. | ||
| It features (along with a few other fundamental constants) in ]. | It features (along with a few other fundamental constants) in ]. | ||
| ---- | ---- | ||
| ] | |||
| /Talk | |||
Revision as of 15:51, 25 February 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 = 2.71828 18284 59045 23536 02874 .....
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.