Re: Calculus XOR Probability

In article <MPG.1ec14bf59cdb4d3a98aca3@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:


And, if you think the direction of the approximating elements doesn't
matter in measuring the curve, then you're simply wrong, and need to
look back at the basics of how calculus works.

The measure of the length of a curve is standardly defined to be the
least upper bound, if there is one, of the polygonal lengths, with
polygons or broken lines having a finite number of segments.

In the case of an actual staircase curve, the longest polygonal
approximation is the staircase itself.

But for the limit function all polygons lie strictly along the diagonal.

Thus the lengths of the staircases are strictly greater that the length
of the limit diagonal as the step size goes to zero.
.

Relevant Pages

  • Re: Calculus XOR Probability
    ... According to TO's infinite induction principle it would necessarily be ... The length of the staircase in the limit IS 2, ... The segments have directions, which alternate between vertical and horizontal, ... We're not defining the curve by the endpoints any more. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... What is the slope of the staircase at x=0. ... infinite, as opposed to the slope of the diagonal. ... As long as those locations are ON the curve, ... The mathematics involved is quite straightforward and pedestrian. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... the locations of the staircase in the limit are indistinguishable ... mathematical requirement for a curve to have a length is the the lengths ... it cannot be used as a measure of a curve. ... The mathematics involved is quite straightforward and pedestrian. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... your concept of the diagonal line being the staircase in the limit is ... segments which constitute the staircase. ... curve, which is precisely what causes the problem here. ... infinite case, and that is why the proof fails. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... your concept of the diagonal line being the staircase in the limit is ... segments which constitute the staircase. ... According to the definitions of arc length in calculus, ... parallel to the curve they're approximating. ...
    (sci.math)