[adamzochowski]

The author doesn't explain the argument in detail, but he is trying to say:

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.

[ChainOfFools]

> This means that there is no convergence because dividing left of zero, and right of zero, leads to different result.

Isn't this related to the reason why if a series can be proven to converge absolutely it's considered convergent? And the simpler tests like comparison, limit comparison, integral test for absolute value by just forcing the expression to be absolute notation?

I always just chalk this up to resolving an ambiguous meaning of "less than", in one sense referring to a relative X position from another spot A on the number line, and another sense referring to a relative X position from zero in either direction.