If a rational number is a number that can be represented in a fraction that comprises of 2 rational numbers, then shouldn't numbers like 1/0, 2/0, etc, be rational numbers as well?

A rational is a number that can be represented as a quotient with integer numerator and non-zero integer denominator. So x/0 is not rational.

0 * 1 = 0 0 * 2 = 0 0 * 3 = 0

Therefore 0 * 1 = 0 * 2

So if you take that and go backwards, and divide by 0, what is your answer? Is it 1, 2, or 3?

Since there are infinite numbers that, when multiplied by 0 give you 0, it is undefined what the answer is when you go in the reverse direction. If you allow dividing by 0, you can get nonsensical answers like 1 = 2, which is obviously incorrect.

Side note to start: I'm having a weird issue where my instance can't see comments on this post, and I checked and there is no defederation or blocking. Not sure why.

I would, first of all, probably correct the definition of a rational number: A rational number is a number that can be represented as a ratio (fraction, quotient) of two integers, not other rational numbers. This should keep the definitions easier to use, and not self-dependent.

As for the actual meat of the question, I would argue that division by zero results in something that is not a number at all, and it must be a number to be a rational number. Others will (and have) simply define(d) rational numbers to not include division by zero, or to define rational numbers as an integer over a natural number (naturals being 1, 2, 3...).

How you define things in mathematics changes how you use it heavily. If you had a field or branch of mathematics that had a working definition for division by zero, numbers like 1/0 and 2/0 would likely be rational in that context.