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In order to understand why $0! = 1$, we need to understand a little bit about how multiplication over series of numbers works in general.  I'll try to limit this description to using basic algebra and will explain (briefly) anything that I think might be beyond basic algebra.

The first thing we need to understand is Set Theory, or at the very least, what a set is.  A set is a grouping of numbers (or other things) without duplicates.  For example:

$$\begin{align}S &= \{1, 2, 3, 4\}\end{align}$$

We use $\{$ and $\}$ to indicate a set in mathematics.  The example above $(1)$ is the set of the four numbers $1$, $2$, $3$, and $4$.

Next we need to understand big-Pi notation (and you may have already seen big-Sigma notation):

$$\begin{align}\displaystyle\prod\{1, 2, 3, 4\} &= 1\times2\times3\times4 = 24\\ \displaystyle\sum\{1, 2, 3, 4\} &= 1+2+3+4 = 10\end{align}$$

$\prod$ means to multiply everything in the set together.  $\sum$ means to add everything in the set together.  In this case, we're mostly interested in $\prod$.

Now, because multiplication is associative, we can re-arrange these in any groupings we care to:

$$\begin{align}\displaystyle\prod\{2, 3, 4, 5\} &= 120\\ \displaystyle\prod\{2, 3\}\times\displaystyle\prod\{4, 5\} &= 120\\ \displaystyle\prod\{2, 4\}\times\displaystyle\prod\{3, 5\} &= 120\\ \displaystyle\prod\{5, 3\}\times\displaystyle\prod\{4, 2\} &= 120\\ \displaystyle\prod\{2, 3, 4\}\times\displaystyle\prod\{5\} &= 120\\ \displaystyle\prod\{2, 3, 4, 5\}\times\displaystyle\prod\{\} &= 120\end{align}$$

Those last two are interesting.  Specifically $\prod\{5\}$ and $\prod\{\}$.  $\prod\{5\}$ is fairly easy.  If you only have one thing to multiply together, you get it, so $\prod\{5\} = 5$.  However, $\prod\{\}$ may seem more complicated.  You have no things to multiply together.  Because it makes sense that if you have no more things to multiply together, it shouldn't change the result, we define this to be $1$.  So $\prod\{\} = 1$.

Next, we need to look into set-builder notation.  Set-builder notation is where instead of actually listing everything in the set, you give a rule that things in the set follow.  It looks like this:

$$\begin{align}S &= \{2x \mid x \in \mathbb{Z} \}\end{align}$$

That looks more complicated than it is.  Let's break it down.  $\{ a \mid b \}$ is the basic form for set-builder notation.  $a$ is the expression for each element in the set, and $b$ is the condition or limitation on $a$.  So, the set above is the set where every element is $2x$, and where $x$ is constrained by $x \in \mathbb{Z}$.

So what does $\in$ mean? $\in$ is the "is an element of" operator.  On the left is an element and the right is a set.  $\notin$ is the "is not an element of" operator.  So:

$$\begin{align}2 &\in \{1, 2, 3, 4\}\\3 &\in \{1, 3, 5, 7\}\\9 &\notin \{5, 6, 7, 8\}\end{align}$$

And $\mathbb{Z}$ is a named constant: The Integers.  This is the set of all integers and looks like this:

$$\begin{align}\mathbb{Z} &= \{..., -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, ...\}\end{align}$$

And it continues in both directions forever.  So, what does our $\{2x \mid x \in \mathbb{Z}\}$ set above $(10)$ look like?  Well, it's the even integers:

$$\begin{align}\{2x \mid x \in \mathbb{Z}\} &= \{..., -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10, ...\}\end{align}$$

Next, we need to understand what $\land$ means in math.  $\land$ is the "and" symbol.  We can use it in the set-builder constraint to add more constraints.  (And for those curious, $\lor$ is the "or" symbol.)  For example:

$$\begin{align}\{2x \mid x \in \mathbb{Z} \land 1 \le x \le 4\} &= \{2, 4, 6, 8\}\end{align}$$

Now, we can move on to ranged big-Pi notation:

$$\begin{align}\displaystyle\prod_{k=a}^{b}c &= \prod\{c \mid k \in \mathbb{Z} \land a \le k \le b \}\end{align}$$

For example:

$$\begin{align}\displaystyle\prod_{k=1}^{4}2k &= \prod\{2k \mid k \in \mathbb{Z} \land 1 \le k \le 4\}\\&= \prod\{2, 4, 6, 8\}\\&= 2 \times 4 \times 6 \times 8\\&= 384\end{align}$$

Finally we get to factorial.  Factorial is just a short-hand for when you don't want to write something out in big-Pi notation:

$$\begin{align}x! &= \displaystyle\prod_{k=1}^x k\end{align}$$

For example:

$$\begin{align}5! &= \displaystyle\prod_{k=1}^5 k\\&= \prod\{k \mid k \in \mathbb{Z} \land 1 \le k \le 5\}\\&= \prod\{1, 2, 3, 4, 5\}\\&= 1 \times 2 \times 3 \times 4 \times 5\\&= 120\end{align}$$

So, finally, what does $0!$ mean?  Well, let's look:

$$\begin{align}0! &= \displaystyle\prod_{k=1}^0 k\\&= \prod\{k \mid k \in \mathbb{Z} \land 1 \le k \le 0\}\\&\text{Well, there are no integers that $1$ is less than or equal to,}\\&\text{and $0$ is greater than or equal to.}\\&= \prod\{\}\\&\text{We saw this above in $(9)$.}\\&= 1\end{align}$$

Sun, 18 Apr 2021 17:00 GMT