Difference between revisions of "E"

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The number $e$ can be defined in the following way: let $f$ be the unique solution of the initial value problem
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We define the [[real number]] $e$ to be the number such that
$$y'=y;y(0)=1,$$
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$$\displaystyle\int_1^e \dfrac{1}{t} \mathrm{d}t=1.$$
then $e=f(1)$.
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By the definition of the [[logarithm]], we have $\log(e)=1$. The value of $e$ is
 
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$$e=2.71828182846\ldots.$$
 
=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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[[Euler's formula]]<br />
<strong>Theorem:</strong> The real number $e$ is [[irrational]].
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[[e is irrational]]<br />
<div class="mw-collapsible-content">
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[[Log e(z)=log(z)]]<br />
<strong>Proof:</strong> proof goes here █
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[[Log 10(z)=log 10(e)log(z)]]<br />
</div>
 
</div>
 
  
 
=References=
 
=References=
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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?]
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[[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?