I can’t give it a full explanation right now, but let me start with some of the basics.

Math has the notion of a number system, which is basically the name we give to a certain set of numbers and the rules we’re allowed to operate on them with.

The integers are one number system. This is the numbers -3, 0, 5, 1000000, etc. Note that when you add and subtract two integers, you always get another integer. 1000000+5=1000005. Same if you multiply two integers. But if you divide some numbers, you might get something else. 1 divided by 3 isn’t an integer, so we say the result is undefined within the integers.

The “real numbers” are another system. Don’t read too much into the word “real”; it’s just a name. The real numbers include all of the fractions, like 1/3, as well as numbers like pi. It’s a really “big” system.

There are all kinds of other number systems. Mathematicians are inventive! There’s one that sits between the integers and the reals, called the “rational numbers”. Not rational meaning logical, but meaning “ratios”. So 1/3 is a ratio between 1 and 3, and is thus a rational. But pi is not a rational, because there’s no fraction that makes pi. You can get close, like 22/7 or 355/113, but never exactly. You need the real numbers to be exact.

In all of the normal number systems, dividing by 0 is not allowed. It’s just undefined, because it produces a result not in the system. Maybe we say it’s infinity… but infinity isn’t a number in the integers or the reals. So the result doesn’t count and we just say it’s “undefined”.

But again, mathematicians are inventive. And some have invented number systems that contain infinity. Some of them allow dividing by zero. Others allow dividing by infinity for a result that’s not quite zero, but smaller than any real number. 0.0001? Smaller than that. 0.0000001? Smaller than that, too. Add in a million extra zeroes, and it’s still smaller. We call those infinitesimals.

It can be hard to answer questions like this because you can always try adopting a new rule, and seeing what happens. You can allow dividing by 0, but then some other things might break. Like, are 1/0 and 2/0 different? If so, then you have two infinities… or infinity infinities, since then you also have 3/0 and so on. You always lose something when you bend your previous rules.