Just two things:
1. I need someone to prove to me that 0*0=0
2. If I were to (hypothetically) try and introduce a new mathematical concept, how would I go about introducing it to mathematicians?
Mathematicians (as opposed to teachers of mathematics in primary and secondary school) are quite open to looking at alternative kinds of mathematics. They insist only on careful definitions, clear identification of axioms, logical proof of theorems, and the lack of any inconsistency among theorems.
There are for example three different types of plane geometry that have been thoroughly developed.
Whether anyone other than pure mathematicians will be interested in a new branch of mathematics depends on whether it produces results that can be observed in some actual physical or social context of interest outside of mathematics.
To get a formal proof that 0 * 0 = 0, you will have to go to the axioms of ring theory or to Peano's Postulates and work from there. I can, however, give you an informal argument.
We define 0 as follows:
\(\displaystyle x = 0 \iff a + x = a = x + a \text { for any number } a.\)
That is, we define the number 0 as the additive identity element, meaning that when you add zero to a number, the sum is the number. But let's change the definition so the definition is silent on 0 * 0. No mathematician will blink an eye at that. We have left open what 0 * 0 is at the stage of definition.
Finally, we accept as an axiom the following distributive property:
\(\displaystyle \text {If } p,\ q, \text { and } r \text { are numbers, then } p * (q + r) \equiv (p * q) + (p * r).\)
Now if you accept those, then
\(\displaystyle n \ne 0 \implies 0 * (n + 0) = 0 * n = 0.\)
\(\displaystyle n\ne 0 \implies 0 * ( n + 0) = (0 * n) + (0 * 0) = 0 + (0 * 0) = (0 * 0).= 0 * 0.\)
\(\displaystyle \text {But } 0 * (n + 0) = 0 \text { and } 0 * (n + 0) = 0 * 0 \implies 0 * 0 = 0.\)
So if you build a consistent arithmetic in which, by definition, 0 * 0 does not equal 0, you cannot have the distributive property in that arithmetic as a universal rule. Other weird things may happen. But a universal distributive property and a very general definition of zero as the additive identity lead to the conclusion that 0 * 0 = 0. That in turn says that we can define 0 as the universal identity element without any harm to the rest of normal arithmetic.