Re: Calculus XOR Probability

Tony Orlow wrote:
Matt Gutting said:
Tony Orlow wrote:
Virgil said:
<snip>

The length of a broken or polygonal line depends only on the location of of the "joints" and not at all on the direction of the segments between joints.
What is the slope of the staircase at x=0. It's infinite. Does that ever change as the staircase approaches the limit? No. That point at x=0 always has slope infinite, as opposed to the slope of the diagonal. The direction between 0,0 and 0,1/n never changes.
But you're taking into account multiple points (2 points). The sequence of
staircases converges pointwise; you don't *have* a staircase as the limit.


Uh, yeah, you do have an infinite sequence of infinitesimal steps, and the slope at x=0 remains infinite.

I hate to repeat myself, but in the absence of a precise, non-circular
definition of "infinite," the truth value of the above statement can't be
evaluated.


Virgil:
That may be a TO requirement, but it is not a mathematical one. The only mathematical requirement for a curve to have a length is the the lengths of its polygonal approximations have a finite upper bound, and then that upper bound is then defined to be the length.

Tony:
If the endpoints are on the curve.
Whether the endpoints are on the curve or not, as long as the series of
lengths is convergent.

Gee, the series of your lengths on the staircase "converges" to the same value it has in all finite cases. Some point on the segments must be parallel to the curve at a point on the curve perpendicular to a point in that segment, or your measure is off.


How can a point be parallel, or perpendicular, to anything?

Matt

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