Another interpretation of the “infinity times infinitesimal error” statement is just to realize that the staircase only _appears_ to converge to the diagonal; if you zoom in closely enough you see that the staircase never in fact actually converges. So there is no real mystery; it’s just a matter of scale
What do you mean? I don't think you need any prior assumptions, other than the definition of the construction method (which is spelled out in the "proof").
I did spoil the conclusion by giving the answer before we calculate it, but that's because I didn't want to make it too long-winded. The point is that you need to precisely define what you're modeling and not assume that the two effects are the same.
Edit: rephrased the text a bit to move one half of the conclusion down the line.
Uh, actually I'm wrong, the only assumption needed is that n != 2. Since ePath is 2 - n, if n were 2 then the error would always be zero. Since it's not 2 then we can show it doesn't converge.
Assuming n < 2 is pretty reasonable - we are saying that a stair-step is an approximation.
Edit: then again, it would be rather strange to not to allow ourselves to use the Pythagorean theorem. Perhaps all we need to say is that when we replace a right triangle with smaller right triangles, c doesn't converge on a + b, and that's all there is to it. To converge, the angle between a and b would need to increase from 90 towards 180 degrees as we shrink them.
Another interpretation of the “infinity times infinitesimal error” statement is just to realize that the staircase only _appears_ to converge to the diagonal; if you zoom in closely enough you see that the staircase never in fact actually converges. So there is no real mystery; it’s just a matter of scale
To calculate the path error, we need to already know the correct answer. This seems a bit unsatisfying.
What do you mean? I don't think you need any prior assumptions, other than the definition of the construction method (which is spelled out in the "proof").
I did spoil the conclusion by giving the answer before we calculate it, but that's because I didn't want to make it too long-winded. The point is that you need to precisely define what you're modeling and not assume that the two effects are the same.
Edit: rephrased the text a bit to move one half of the conclusion down the line.
Uh, actually I'm wrong, the only assumption needed is that n != 2. Since ePath is 2 - n, if n were 2 then the error would always be zero. Since it's not 2 then we can show it doesn't converge.
Assuming n < 2 is pretty reasonable - we are saying that a stair-step is an approximation.
Edit: then again, it would be rather strange to not to allow ourselves to use the Pythagorean theorem. Perhaps all we need to say is that when we replace a right triangle with smaller right triangles, c doesn't converge on a + b, and that's all there is to it. To converge, the angle between a and b would need to increase from 90 towards 180 degrees as we shrink them.
For some reason, to me, this makes sense intuitively, while on your previous article I mentioned that 0.(9) still bothers me somehow.