What's so special about $e$?

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  • Intuitive Understanding of the constant “$e$”

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If you put 1 dollar in a deposit, and the yearly interest rate is 100%, after a year you’ll end up with 2 dollars — that is, if interest is accrued yearly.

If it’s accrued twice a year, you’ll get 50 cents after 6 months, because yearly percentage is 100%, and half a year has passed, so it’s 50% rate for 6 months. After second half a year, you’ll actually get 75 cents, as you had 1.50 dollar after a 6 months, and 50% of 1.50 is 0.75, so now you have 2.25 dollars.

Similarly, if the interest is accrued monthly, you’ll end up with around $2.63. And if the interest is accrued daily, it’s around 2.71 dollars.

Sadly, no matter how often the interest is accrued, you’ll never end up with more then e dollars, because the more often the interest is accrued, the closer you balance will be to e at the and of the year.

$$\dfrac d{dx}\left(e^x\right)=e^x$$
$$\int e^xdx=e^x+C$$
It is the only function that does this

Also, $e^x$ can be expressed like this:
$$e^x = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}……$$

I believe that
e^{i\theta} = \cos \theta + i \sin \theta
should be enough!
Anyway if you’re interested in other things, like differential equations which appear in physics or biology or economics and so on, $e$ is quite important.

Here is an application of $e$ in economics, which may be accessible to someone without any special education in mathematics.

The thing that is special is the exponential function $\mathrm{exp}$, which satisfies
$$\mathrm{exp}(0)=1,\quad \mathrm{exp’}=\mathrm{exp}.$$
Then of course $e:=\mathrm{exp}(1)$, and because of $\mathrm{exp}(x+y)=\mathrm{exp}(x)\mathrm{exp}(y)$ it makes sense to write $\exp(x)=e^x$.