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# Questions tagged [eulers-number]

Euler's number is another name for e, the base of the natural logarithms.

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### Exponential identity ${(x^a)}^b=x^{ab}$

We all know that $e^{\pi i}=\cos \pi + i \sin \pi=-1$, and that ${(x^a)}^b=x^{a \times b}$, also $e^{2 \pi i}=\cos {2\pi} + i \sin {2\pi}=1$. Here's my problem, we have $a \in \mathbb R$, and I ...
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### Is there a “natural tetration function”?

For the natural exponential function, $$f(x)=e^x \to f(x)=f'(x).$$ Is there a natural tetration (tetral?) function? $$f(x)={{^x}b} \to f(x)=f'(x)$$ Is it base $e$?
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### What is the nature of $e^{ix}$, real or complex?

Since, $e^{i\pi}=-1$, right part of the equation ($-1$) is real and left part is seemingly complex as $e^{ix}=\cos(x)+i\sin(x)$ which is a complex number( I am not sure though). I am a freshman, ...
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### Why is Euler's number $2.71828$ and not anything else? [closed]

Why is Euler's number $\mathtt 2.71828$ and not for example $\mathtt 3.7589$? I know that $e$ is the base of natural logarithms. I know about areas on hyperbola ...
638 views

### Non-Numerical proof of $e<\pi$

This is a "coffee-time-style" problem ( to have a taste of this style, you may like to browse the book https://www.amazon.com/Art-Mathematics-Coffee-Time-Memphis/dp/0521693950) interpreted from an ...
### $10-e$ interesting decimal expansion property
The number $e$ has several interesting properties but I noticed something that may just be coincidence; a neat coincidence however! Here is the decimal expansion of $e$. e=2.718281828459045235360\...
### Why is $\lim\limits_{n\to\infty} (1 + \frac{1}{2n})^n = e^{\frac{1}{2}}$
In my textbook it is stated that this is obvious: $\lim\limits_{n\to\infty} (1 + \frac{1}{2n})^n = e^{\frac{1}{2}}$. However I feel stupid for not understanding why? What am I missing?