On Sat, Dec 15, 2012 at 9:04 AM, Robert Hansen wrote:

<< SNIP >>

Even though the student accepted that 0.333... is 1/3, that acceptance was based on nothing more than a presentation (long division). Without being able to produce a similar presentation for 0.999... the student is stuck.

A related demo that may leave a student feeling uneasy has to do with creating a spherical network of more and more triangles.  The algorithm is such that six triangles surround each vertex except in 12 places where five triangles serve.  The triangles are not quite equiangular such that at none of the vertexes do we get a full 360 degrees (perfect flatness), but rather a smidgen less, and this begets convex curvature as viewed from the outside.

In a calculus mindset, looking at this algorithm, we see that the number of degrees v at each vertex differs from 360 by epsilon, and I may always find some number of triangles N such that |360 - v| < e, i.e. lim (N->infinity) |360 - v |, at each vertex, -> 0.

What's wrong with that vision?

Descartes' proved that adding the angular deficits of all such vertexes, no matter their number, yields a constant number, 720 degrees.  Ergo Sigma (360 - v) over all N = 720.  This proves the limit at each vertex is never zero, as every vertex contributes some tiny "tax" or "tithe" to the invariant constant 720.  720/N > 0. |360 - v| > 0 even as N -> infinity.