Difference between revisions of "E"

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=Properties=
 
=Properties=
 
[[Euler's formula]]<br />
 
[[Euler's formula]]<br />
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[[e is irrational]]<br />
<strong>Theorem:</strong> The folllowing formula holds:
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[[Log e(z)=log(z)]]<br />
$$e=\displaystyle\lim_{k \rightarrow \infty} \left( 1 + \dfrac{1}{k} \right)^k,$$
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[[Log 10(z)=log 10(e)log(z)]]<br />
where $e$ denotes [[E|Euler's constant]].
 
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<strong>Proof:</strong>
 
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<strong>Theorem:</strong> The real number $e$ is [[irrational]].
 
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<strong>Proof:</strong> proof goes here █
 
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=References=
 
=References=
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Logarithm at -i|next=e is limit of (1+1/n)^n}}: 4.1.16
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Logarithm at -i|next=e is limit of (1+1/n)^n}}: $4.1.16$
 
[http://eulerarchive.maa.org/hedi/HEDI-2006-02.pdf Who proved $e$ is irrational?]
 
[http://eulerarchive.maa.org/hedi/HEDI-2006-02.pdf Who proved $e$ is irrational?]
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Latest revision as of 19:35, 25 June 2017

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

Euler's formula
e is irrational
Log e(z)=log(z)
Log 10(z)=log 10(e)log(z)

References

Who proved $e$ is irrational?