For x⋅0=a (i.e. 1/0) no number fits. It's like a database search that finds nothing. It's a null. A point that is not in the set.

For x⋅0=0 (i.e. 0/0) any number fits. Is there such a thing as an anti-null? In the infinite set, 0/0 is infinite.

If a/0 isn't any number if you exclude 0, then a/0 should be zero. Tada!

...of course zero wasn't in the set, which makes it a null