# What's so special about $e$?

• Intuitive Understanding of the constant “$e$”
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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\$.