Difference between revisions of "E"
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| − | We define the [[real number]] $e$ to be the number such that | + | We define the [[irrational]] [[transcendental]] [[real number]] $e$ to be the number such that |
$$\displaystyle\int_1^e \dfrac{1}{t} \mathrm{d}t=1.$$ | $$\displaystyle\int_1^e \dfrac{1}{t} \mathrm{d}t=1.$$ | ||
| − | By the definition of the [[logarithm]], we have $\log(e)=1$. | + | By the definition of the [[logarithm]], we have $\log(e)=1$. The value of $e$ is |
| + | $$e=2.71828182846\ldots.$$ | ||
=Properties= | =Properties= | ||
<div class="toccolours mw-collapsible mw-collapsed"> | <div class="toccolours mw-collapsible mw-collapsed"> | ||
Revision as of 04:06, 7 June 2016
We define the irrational transcendental real number $e$ to be the number such that $$\displaystyle\int_1^e \dfrac{1}{t} \mathrm{d}t=1.$$ By the definition of the logarithm, we have $\log(e)=1$. The value of $e$ is $$e=2.71828182846\ldots.$$
Properties
Theorem: The folllowing formula holds: $$e=\displaystyle\lim_{k \rightarrow \infty} \left( 1 + \dfrac{1}{k} \right)^k,$$ where $e$ denotes Euler's constant.
Proof: █
Theorem: The real number $e$ is irrational.
Proof: proof goes here █
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 4.1.16
