In order to do any kind of maths, you need definitions and axioms. So, what is your definition of division? The usual definition in number contexts is: assuming a and b are “numbers” (which is already too vague for maths) a/b is “the” number such that
(a/b)×b = a
I put “the” in quotes because, for this to make sense, you need to ensure (by axiom or theorem) that such a “number” exists and is unique, for every “valid” (a,b) combination. How to do so depends on the particular kind of ”numbers” you are considering (integers, rationals, reals, etc.).
*In other words, a/b is defined as the unique solution of equation qb = a (of unknown q), when this equation has a unique solution.*
Now, with usual ”numbers”, including real and complex numbers, you can easily see why dividing by 0 (i.e., applying the definition to (a,b) pairs such that b = 0) is invalid with this definition:
* if a ≠ 0, then there is no number q such that q×0 = a;
* if a = 0, every number q verifies q×0 = a.
In each case, the equation qb = a (where the unknown is q and b is 0) does not meet the criterion of having **one unique solution**.
P.S.: sorry, no cake. :) (Or, beware that this is not maths and is slippery: you pretend that you're going to cut the cake into parts and that, after giving me zero of these parts, I'll have the full cake. No way!)
