Why x*0 = x

[ErGingerbreadMandude]

Also a proof by contradiction:

Let's assume you are correct with you definition of 0 * 1 = 1, and we've already proved that the algebra version of your pattern is correct(granted x*0=0).

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Now let's try to do 1 * 1.

Your pattern states this should be: 1 + (0 * 1) + 0

Hey! That 0 * 1 looks familiar, we have the answer for it as well, so let's substitute: 1 + 1 + 0

Now looks what you've done, 1 * 1 = 2

But we all know 1 * 1 = 1, therefore contradiction, so 0 * 1 cannot equal 1.

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Algebra version is correct if and only if x*0=0 so of course it's going to give a wrong result if x*0=x because the algebraic version is designed to give the correct result only if x*0=0.
That's like going into an open minded argument convinced you're the one that's right.
There can't be any * 's on the R.H.S on the algebra