TITLE: Why is $n$ divided by $n$ equal to $1$?
QUESTION [23 upvotes]: I've understood how to apply this operation for many years but I recently was reminded that:
$n^0 = 1$
Which in high school I accepted and just "solved for", but I'm now curious. Why is that true? So my own cursory research on the internet led me to the most common proof (using recurrence):
$b^1 = b$
$b^{n+1} = b^n \cdot b$
$b^0 = \frac{b^1}{b} = 1$
The most common proof for it relies on the fact that $\frac{n}{n} = 1$ so I started wondering why is that so? There's the rather obvious examples given by teachers using a pie but it feels like there's something else there. Maybe that something else involves Mathematics that is out of my league at the moment (the highest amount I've taught myself is through Algebra).

REPLY [2 votes]: Definition: In mathematics, especially in elementary arithmetic, division ($\div$) is an arithmetic operation. Specifically, if $b$ times $c$
   equals $a$, written:
$\color{red}a = \color{blue}b \times \color{green}c$
where $b$ is not zero, then $a$ divided by $b$ equals $c$, written:
$\color{red}a\div \color{blue}b = \color{green}c$

Using the definition and by letting $\color{green}c=\color{green}1$, $\color{blue}b=\color{blue}{1/n}$ where $n$ is a non zero real number, $\color{blue}b$ times $\color{green}c$ is equal to a number $\color{red}a$: 
$$
\color{red}a=\color{blue}{\dfrac1n}\times\color{green}1=\color{red}{\dfrac1n}
$$
Therefore, from the definition:
$$
\color{red}a\div\color{blue}b=\color{green}c
\\\Downarrow\\
\begin{align}
\color{red}{\dfrac1n}\div\color{blue}{\dfrac1n}&=\color{green}c \tag{1}\\\,\\
\color{red}{\dfrac1n}\cdot\dfrac n1 &=\color{green}c \tag{2} \\\,\\
\color{black}{\dfrac nn} &=\color{green}c\\\,\\&=\color{green}1
\end{align}$$
Thus for any non zero real number $r$ we have that: $$\dfrac rr=1.$$
$\blacksquare$

If you search for why the procedure I applied when going from step $\text{(1)}$ to step $\text{(2)}$ is correct, check this answer: Why is $\frac{1}{\frac{1}{X}}=X$?