Difference between revisions of "E"

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We define the [[irrational]] [[transcendental]] [[real number]] $e$ to be the number such that
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We define the [[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 $\log(e)=1$. The value of $e$ is
 
By the definition of the [[logarithm]], we have $\log(e)=1$. The value of $e$ is

Revision as of 04:06, 7 June 2016

We define the 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

Who proved $e$ is irrational?