so i was thinking about this rationally: anything times anything is whatever amount of whatever you have; ie: 4 x 5 = 20. 4 units of 5 is 20 units. anything divided by anything becomes a fraction of what you had; ie: 20 / 5 = 4. 20 units divided into 5 parts of 4 units. anything times nothing is nothing; 3 x 0 = 0. 3 units of 0 is 0. however, now i get philosophical, when i divide anything by nothing, i'm left with what i had when i started. if i had 4 apples and divided them by nothing, i'm still left with 4 apples. 4 / 0 = 4 this is empirically correct, as far as i've gone to divide an apple by nothing. dividing by zero implies that you are doing nothing to the formula, thereby creating a non-functioning equation. its not that it doesn't work, its that you are not doing anything to the equation, you are not doing anything to x.
however, in thinking of 0 as a unit of measure rather than an absence of measure, it has been shown that any number multiplied by 0 is 0. 5 x 0 = 0, 235 x 0 = 0. 3346356.37839493 x 0 = 0. therefore, 0 divided by 0 should equal every possible outcome. otherwise, any number multiplied by 0 would not equal 0.