 # Understanding e and e to the power of imaginary number How did the fee of e come from compound hobby equation. What does the cost of e honestly mean... Capacitors and inductors charge and discharge exponentially, radioactive elements decay exponentially and even bacterial growth follows exponential i.e., (2.71)x ,why can't it be 2x or something. Felix Cohen 2022-09-11 Answered
Understanding e and e to the power of imaginary number
How did the fee of e come from compound hobby equation. What does the cost of e honestly mean
Capacitors and inductors charge and discharge exponentially, radioactive elements decay exponentially and even bacterial growth follows exponential i.e., $\left(2.71{\right)}^{x}$ ,why can't it be ${2}^{x}$ or something.
Also ${e}^{2}$ means $e\ast e$ ,${e}^{3}$ means $e\ast e\ast e$ But what exactly ${e}^{ix}$ mean
I want to know how to visualise ${e}^{i\pi }=-1$ in graphs
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There are many homes that make the exponential function unique; one which I discover specially instructive is:
There's exactly one feature f such that ${f}^{\prime }\left(x\right)=f\left(x\right)$ for all x and $f\left(0\right)=1$. This feature is the exponential function, and it seems that there's a specific real variety e such that $f\left(a\right)={e}^{a}$ each time $a\in \mathbb{Q}$. therefore it makes experience to apply the notation ${e}^{x}$ for $f\left(x\right)$ for all x.
The property ${f}^{\prime }\left(x\right)=f\left(x\right)$ is what makes this particular exponential function useful for describing exponential growth and decay, because it makes it easy to relate the instantaneous rate of change to the current size of the thing that is growing or decaying.
In the complex plane it so happens that $f\left(ix\right)$ will be a point x radians counterclockwise along the unit circle. This is forced by the relation ${f}^{\prime }\left(x\right)=f\left(x\right)$, though it doesn't have any particular intuitive relation to repeated multiplication. One just has to get used to the fact that the unique function that obeys the nice rules we know from the real exponential happens to behave that way for complex arguments.

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