When looking about proofs that you cannot divide by 0, the most common argument why you couldn't was because of an example like this... 8/0=x is the same as x*O=8 which doesn't make sense because you can't multiply a number by 0 and get a number other then 0.
Well I said why did 8/0=x equal to x*8=0. Oh that's right, because the rule a/b=c is b*c=a. Well whoever discovered that rule didn't create it because they wanted to, they had to prove it works.
if you multiply on both sides of the equation a/b=c by the denominator (b*a/b=c*b) the b's cancel on the left and you get a=c*b, or b*c=a.
for an example, 8/4=2. 4*8/4=2*4. the 4's on the left cancel and you get 8=2*4, or 4*2=8. so a/b=c does equal b*c=a.
so lets see if 8/0=x equals x*0=8.
8=a, 0=b, x=c,
8/0=x, Times both sides by denominator such as how the rule is possible. 0*8/0=x*0
well on the left (0*8/0) equals 0 because any number multiplied by 0 equals zero. so you result in the expression 0=x*0, or x*0=0 and not x*0=8, which it's possible.
so the original argument that 8/0=x is equivalent to x*0=8 was wrongfully assumed that the a/b=c rule applied to an expression with 0 as the denominator.
another common argument was set up in a long division form, which is difficult to type on the computer, but I have a reason why their math was off with that equation also