Difference between revisions of "E"
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(Created page with "The number $e$ can be defined in the following way: let $f$ be the unique solution of the initial value problem $$y'=y;y(0)=1,$$ then $e=f(1)$.") |
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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.$$ |
| − | + | By the definition of the [[logarithm]], we have $\log(e)=1$. The value of $e$ is | |
| + | $$e=2.71828182846\ldots.$$ | ||
| + | =Properties= | ||
| + | [[Euler's formula]]<br /> | ||
| + | [[e is irrational]]<br /> | ||
| + | [[Log e(z)=log(z)]]<br /> | ||
| + | [[Log 10(z)=log 10(e)log(z)]]<br /> | ||
| + | |||
| + | =References= | ||
| + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Logarithm at -i|next=e is limit of (1+1/n)^n}}: $4.1.16$ | ||
| + | [http://eulerarchive.maa.org/hedi/HEDI-2006-02.pdf Who proved $e$ is irrational?] | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 19:35, 25 June 2017
We define the real number $e$ to be the number such that $$\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
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.1.16$
