I'll lead with a question: What is X divided by X?

The immediate and obvious answer is "1", but this is, in fact, incorrect. The answer is, roughly, "1 except where X equals 0". This is both pedantic and important - 0 divided by 0 isn't 1, it's "Undefined". 5 times 0 is -also- 0. "Undefined" in this case really means something like "Every answer simultaneously."

Division, as it is typically defined, is a lossy transformation - you have the potential to lose information in performing the operation. So is multiplication - the equation "5 = 3" can be "made correct" by multiplying by zero, a conceptually valid operation.

Squaring numbers is too. 5^2 is 25 - but once you've done this, you can no longer determine, from the current properties of whatever it is you're working with, whether you started with five or negative five. You've lost information about your starting configuration by performing what we usually consider a perfectly valid operation. Reversing the operation doesn't give you what you started with.

The issue is one of simplification. There isn't one single zero. There are an -infinite- number of 0's. Zero apples isn't the same as zero oranges - they're different zeros.

Squares are similar; a rectangle five feet long and four feet wide has twenty square feet, but it's not the same square feet as from a rectangle ten feet long and two feet wide. A square value doesn't maintain information about its constituent parts - this information is simplified away.

Division, again, is similar; 5 / 5 equals 1. Is it the -same- 1 as provided by 4 / 4? No. They're different 1's, but once we've reduced to a single number, that information is lost to us.

This simplification is great, if you don't need that information, and terrible, if you do.

Every mathematical operation results in a loss of information. Again, this is helpful, if you're looking for a simple result, and worthless, if you end up needing that information. Knowing that the combined length of two walls is 10 doesn't tell you anything about the individual length of the individual walls - you lost that information when you added the two numbers together.

The purity, the cleanness, of mathematics is an illusion, produced by rules which encourage you not to notice the information that goes missing with every step. Mathematics, in truth, is a very messy process, the process of crossing out information until you're left only with the information you think you need. The erasure is intellectually satisfying, but it is wholly the act of hiding complexity to make the complex -seem- simple. The complexity is still there, and knowing the square footage of a room you're tiling tells you next to nothing about the number of tiles you need to cut, and how.

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