when you have "x * y = z" you can perform "x = z / y"
ie: "2 * 3 = 6" therefore "2 = 6 / 3"
But it breaks if you try to do division by zero
ie: "2 * 0 = 0" translates into "2 = 0 / 0", we could assume this is good, but if you use any different value for X, you get other nonsensical answers , say "x = 3" ... so "3 * 0 = 0" therefore "3 = 0 / 0"
So as long as you don't divide by zero, the conversion of "x * y = z" into "x = z / y" will work. If "y" is zero, then the conversion breaks down. Hence division by zero is undefined because there is no correct answer.
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The other argument one could do, not mention in the article, is the "limit of division by zero", which is a fancy way, what is the result if we don't divide by zero but get close to it.
Say we have number 2, and we keep dividing it by a number that gets closer and closer to zero
2 / 1 = 2
2 / 0.5 = 4
2 / 0.25 = 8
2 / 0.125 = 16
2 / 0.0625 = 32
So as our divisor gets closer to zero, our result gets closer to infinity.
But the problem is, what happens if we approach the zero from the other side,
2 / -1 = -2
2 / -0.5 = -4
2 / -0.25 = -8
2 / -0.125 = -16
2 / -0.0625 = -32
So as the divisor gets closer from the 'left side', from the negative side, the result is going to negative infinity.
This means that there is no convergence because dividing left of zero, and right of zero, leads to different result.
This is same as studying limit of lim of x->0 for cot(x) where the result leads to undefined value because limit approaching from left side does not converge to the limit approaching from the right side.