Probably a stupid question but where did the constant $e$ come from? How did it come about? How is it derived mathematically other than $e^{i\cdot \pi} = -1$? What exactly does natural growth mean? Or is Euler's constant arbitrary?
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3"Where does it come from?" It comes from incredibly many places. That's the reason it's an important constant. – Arthur Jul 16 '20 at 19:33
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What would it mean for $e$ to be arbitrary? Why wouldn't we just pick $e$ to be $1$ then? – pancini Jul 16 '20 at 19:34
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1See https://math.stackexchange.com/questions/26037/intuitive-understanding-of-the-constant-e?noredirect=1&lq=1 – A. Goodier Jul 16 '20 at 19:35
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All exponents are the same, they just have different factors in the exponent to convert from one to another, meaning you can directly relate an exponent with base $a$ to an exponent with base $b$. $e$ has a special role as a result of other properties which are calculus based. – Cameron Williams Jul 16 '20 at 19:42
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If $f(x)=a^x$ is an exponential growth function, then $f'(x)=a^x\ln a$, so it is natural to take $a=e$, and then $f'(x)=f(x)$. Also, I changed the tag to [tag:eulers-number] from [tag:eulers-constant], which is different – J. W. Tanner Jul 16 '20 at 19:47
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Have you read the Wikipedia article Euler's number yet? – Somos Jul 16 '20 at 20:33
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@Somos Do you want the better answer or the honest answer? – Jul 16 '20 at 21:23
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Honesty is the best policy in general. However, MSE is supposed to be a forum to help answer people's questions. You decide if you were helped. – Somos Jul 16 '20 at 21:49
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Some definitions of $e$:
- The number such that $\int_1^e \frac{\mathrm d x}{x}=1$.
- $\sum_{n=0}^\infty \frac{1}{n!}$.
- $\lim\limits_{n \to \infty} \left(1+ \frac{1}{n}\right)^n$
Interesting topic is to prove that all those definitions lead to the same real number.
mathcounterexamples.net
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The number $e$ is the unique $a>0$ such that the function $f(x)=a^x$ has derivative $f'(x)=a^x=f(x).$
md2perpe
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