= A Basic Concept First
Multiplication is defined as follows:
$ a * b = sum_(c=1)^a b $

Such that $a * b$ is equivilent to the following:
$ underbrace(b + b + ... + b, a "times") $

= Explaining Why $n/0$ is not $infinity$
The inverse of division is multiplication, such as the following:
$ 6 / 2 = 3 $
$ 3 * 2 = 6 $

So if we pretend for a moment that $n/0 = infinity$, then the following is true (where $n$ is any number and $n eq.not 0$):
$ n / 0 = infinity $
$ infinity * 0 = n $

However, a fundemental rule in multiplication is that anything times $0$ is $0$, due to the following:
$ a * 0 = sum_(c=1)^a 0 = underbrace(0 + 0 + ... + 0, a "times") = 0 $

And $n eq.not 0$, so $infinity * 0 eq.not n$, thus proving that anything divided by $0$ is not infinity.

= Explaining Why $0/0$ is undefined
Let $n$ and $k$ be numbers such that $n eq.not k$, and let $alpha = n * 0$ and $beta = k * 0$. As anything times $0$ is $0$, $alpha = beta$. Now let's set up the following equation:
$ alpha = beta $

Expanded, this would be:
$ n * 0 = k * 0 $

So far so good. Now let's try dividing $0$ on both sides:
$ (n * 0) / 0 = (k * 0) / 0 $

If we pretend for a moment that $0/0$ is defined, then the following occurs:
$ (n * cancel(0)) / cancel(0) = (k * cancel(0)) / cancel(0) $

Leaving:
$ n = k $

However, $n$ and $k$ are defined such that $n eq.not k$, so division by $0$ is undefined.

