So you have a vertical line with an infinite slope. There are no changes in x as you traverse the line, only changes in y. Or said differently, the line is described entirely by a single x-value which corresponds to every possible y-value.
If you think of it in terms of a function, it’s extremely problematic because you no longer have a mapping of a single y-value to each x-value. This violates the requirements of a function. It’s not possible to define the slope value when rise/run is something/zero, therefore we describe the function value as “undefined”.
But even though we can’t calculate a slope or address it with a function, it’s pretty easy to visualize and understand a vertical line. So that’s what dividing by zero represents in concrete terms.
This clicked for me when my teacher explained it in terms of slopes.
The video here breaks it down nicely. https://virtualnerd.com/sat-math/geometry/slope/infinite-slope-definition.
So you have a vertical line with an infinite slope. There are no changes in x as you traverse the line, only changes in y. Or said differently, the line is described entirely by a single x-value which corresponds to every possible y-value.
If you think of it in terms of a function, it’s extremely problematic because you no longer have a mapping of a single y-value to each x-value. This violates the requirements of a function. It’s not possible to define the slope value when rise/run is something/zero, therefore we describe the function value as “undefined”.
But even though we can’t calculate a slope or address it with a function, it’s pretty easy to visualize and understand a vertical line. So that’s what dividing by zero represents in concrete terms.