Size of Infinite SetsDate: 11/20/96 at 21:01:17 From: Gary Zak Subject: Conics and Infinity Which is larger: the number of parabolas, or the number of hyperbolas? For any given double-napped cone, consider a plane surface intersecting the cone, parallel to a generatrix, but not through the vertex of the cone. The points of intersection between the plane surface and the cone form a shape called a parabola. This parabola will have its own vertex, a point on the cone. Through this point there can be one and only one parabola. However, plane surfaces that share this vertex, but are not parallel to that generatrix can generate many hyperbolas - in fact, an infinite number of them. This line of reasoning suggests that there are more hyperbolas than parabolas. However, since there are an infinite number of parabolas on any given double-napped cone (by varying the distance of the plane from the vertex of the cone), is it not possible to map on a one-to-one basis each hyperbola to a parabola? This line of reasoning suggests that there are an equal number of parabolas and hyperbolas, i.e. an infinite number of each. The next consideration is whether the infinite number of possible parabolas is of a different nature than the infinite number of hyperbolas. Since I do not have enough of a high level math background to "horse around" with this myself, I am seeking more opinions and/or facts. I suspect I will need to approach this problem at both a Gallup and a Cantor! Thanks in advance for any replies, Gary Zak Date: 11/21/96 at 14:31:29 From: Doctor Tom Subject: Re: Conics and Infinity Hi Gary, Actually, although "a Gallup and a Cantor" may shed some light on the problem, I think Lebesgue may provide the answer. Lebesgue was one of the founders of the field currently known as "measure theory". A math major will usually run into it in the junior or senior year, and then in a more detailed course in real analysis in graduate school. There are some tricky ideas when you're trying to compare the sizes of infinite sets. Cantor just helps you deal with the cardinality of the sets -- whether they can be put in one-to-one correspondence. In fact, from Cantor's point of view, there are exactly the same number of parabolas as hyperbolas (the same cardinality), but in any reasonable assignment of "measure" to the sets, the measure of the set of parabolas will be zero compared to the set of hyperbolas. Here's a "parable" that may help you see what's going on. Consider a unit square on the plane, and the diagonal line of that square. Both (according to Cantor) have the same cardinality: C, where "C" is the cardinality of the real numbers, and C^2 = C (in transfinite arithmetic). But the area of the square is 1 and the area of the diagonal line is zero. So if you threw darts with an infinitely tiny point at the square, the chance that they would land somewhere in the square is 1.0, and the chance that they would hit the line exactly is 0.0. Consider all conic sections. They are described by a 5 parameter family: A*x^2 + B*x*y + C*y^2 + D*x + E*y + F = 0 There are 6 variables: A, B, C, D, E, and F, but one isn't really there because you can multiply by a constant. Although this isn't exactly right, I'll just divide through by F, so let F=1 and A through E are the 5 parameters. This isn't strictly kosher, since F might have been zero, but let's ignore that for now. So consider any region in any 5-dimensional "block" with the variables A, B, C, D, and E. For some selections of these 5 numbers, the equation represents a parabola, for others a hyperbola, and for still others, an ellipse. The regions corresponding to hyperbolas and ellipses represent full 5-dimensional regions, but the parabolic regions are represented by a 4-dimensional surface in that block. The 5-dimensional volume of a 4-dimensional surface is zero (just as the 2-dimensional "volume" of a line is zero in my parable). The chance of a 5-dimensional "dart" hitting a point corresponding to a parabola is zero. By the way, certain degenerate conics, such as crossed lines, points (x^2 + y^2 = 0), et cetera, have even lower dimensionality than parabolas, and in that sense are "infinitely less" probable than parabolas. If you want to know more about Cantor, look at this web site: http://www.shu.edu/~wachsmut/reals/history/ I hope this helps. -Doctor Tom, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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