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#### ThePerfectHacker

##### Well-known member

- Jan 26, 2012

- 236

Here is what I tell my students, as far as I know, this is the best explanation I seen because it is essentially a one line explanation that is short and to the point.

**Why**$e$

**is natural**: Let $b>0$, $b\not = 1$ and consider the function $f(x) = b^x$. This is an exponential function with base $b$ and we can show that $ f'(x) = c\cdot b^x$ where $c$ is some constant. In a similar manner we can consider the function $g(x) = \log_b x$, logarithmic function to base $b$, we can show that $g'(x) = k\cdot x^{-1}$ where $k$ is some constant. It would be best and simplest looking derivative formula if those constants, $b$ and $k$, were both equal to one. This happens exactly when $b=e$. Thus, $e$ is

*natural*base for exponent because the derivative formulas for exponential and logarithmic function are as simple as they can be.

**Why radians are natural:**A "degree" is defined by declaring a full revolution to be $360^{\circ}$, a "radian" is defined by declaring a full revolution to be $2\pi$ radians. More generally, we can define a new angle measurement by declaring a full revolution to be $R$ (for instance, $R=400$, results in something known as gradians). Let us denote $\sin_R x$ and $\cos_R x$ to be the sine and cosine functions of $x$ measured along $R$-units of angles. It can be shown that $(\sin_R x)' = C\cdot(\cos_R x)$ and $(\cos_R x)' = -C\cdot(\sin_R x)$ where $C$ is some constant. It would be best if this constant can be made $C=1$ which happens precisely when $R=2\pi$.