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Geometrical paradox?

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Okay, it's probably not really a paradox at all. My kids and I were talking the other day about geometry and we arrived at a question I couldn't answer.

A line segment is usually defined to be the set of all points between two other points. But a line has one dimension; a point has none. No matter how many points you add up, infinity times zero is still zero. If this definition is true, the length of all line segments is zero.

So what's the problem here? The definition itself?

(Not an Earth-shaking question but the flame wars this afternoon are getting rather depressing.)
 
The problem here is evaluating infinity times zero.

Think of it this way:

Imagine a line segment broken into 10 smaller segments. Then imagine breaking it down further into 100 line segments 1/10th the size of the original segments. This can go further and further, ever larger numbers of ever smaller segments, until it becomes more or less clear that what you are approaching is an infinite number of infinitesimally small segments.

The infinities involved cancel each other out, and by noticing this you've performed a bit of calculus.

You can't really throw around infinities and use them in direct arithmetic, but you instead do a study of limits (what happens as the numbers increase without end or approach a certain finite value)... and that gives you the result.

I hope this helps...
 
If the length of all line segments was 0, then we could never leave a room.
 
infinity times zero is still zero
Click to expand...
You are not multiplying, therefore this statement is not relevant. The points may have zero dimensions, but in Geometry the points are located in 2D or 3D space. The length of a line is the distance between the point locations, which is not zero.
 
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A line segment is usually defined to be the set of all points between two other points. But a line has one dimension; a point has none. No matter how many points you add up, infinity times zero is still zero. If this definition is true, the length of all line segments is zero.
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To extend this to a higher dimension, say that instead of putting poitns in a line to get a line, say we are putting lines in a stack to get a rectangle. Surely we will get a rectangle, even though lines are 1-dimensional, we are stacking them in a 2-dimensional manner.

Also, with the infinity*0 = 0 thing; this isn't always the case. The idea is that we are wanting to find:

number of points*"width" of a point = length of the line, or

n*w=L

Really we are taking limits as n->oo (infinity), and as w->0.

So w never equals 0, it just approaches 0 mathematically. -So we will always get some positive length for the line.

There are many examples of functions in mathematics, say f and g, where the limit

f(x)*g(x) does not equal 0, as f(x) approaches infinity and g(x) approaches 0.
 
sundog said:
infinity times zero is still zero.
Click to expand...
To be pedantic, infinity times zero is undefined.

To be even more pendatic, infinity as such is not a number, and therefore cannot take part in arithmetical operations.
 
Well, actually,

The correct reply would be that the Hausdorff measure (or the Lebesgue, depending of the space you work in) is only countably additive, i.e. only over sets which have at most the size of the set of natural numbers.

There are set's with just as much points in it as the line has, but has length 0 (cantor sets)
(OK, it hasn't got the same dimension as a line or a point, but somewhere inbetween;
But who's nitpicking now.)
 
The definiton of a line is: The location of all points between the endpoints (all points where the sum of the distances to the endpoints are equal).

The line does not consist of those points, they are merely located on the line. And since a point has zero size, an infinite number of points can be located on any line.

Hans
 
MRC_Hans said:
The definiton of a line is: The location of all points between the endpoints (all points where the sum of the distances to the endpoints are equal).

The line does not consist of those points, they are merely located on the line. And since a point has zero size, an infinite number of points can be located on any line.
Click to expand...

Hmm, I'm not so happy with this. I definitely recall a line(segment) being defined as the set of all points between to points (between meaning at the shortest possible distance from the two points).

Besides, the location of a point and the point itself are synonymous in mathematics.
 
Betweeness itself is defined in terms of the coordinate system and the distance formula, in most versions of Euclidian geometry. You have the triangle inequality: Given three points A, B, C, the following is assumed true (it's an axiom of Euclidean geometry).

d(A,B) + d(B,C) >= d(A,C)

i.e. the distance from A to B plus the distance from B to C is greater than or equal to the distance from B to C. Note that, in an earlier axiom, distances between distinct points are always positive numbers.

Definition: Between(B, A, C) iff d(A,B)+d(B,C) = d(A,C)
In other words, B is between A and C when the distances add up exactly.

Dimensional measure of a set of points is literally definied in terms of line segments.

Suppose you had a line segment, e.g. the unit segment from 0 to 1. It's fairly easy to show that there are more points there than there are integers (i.e. if you try to pair points in [0,1] up with integers, you'll end up with points left over). If there's interest, I'll go into this in another message.

Now suppose you take away the open-interval middle third of that, so you have two line segments: [0,1/3] and [2/3,1].

Similarly, take away the middle thirds of each of those.

Repeat this process as many times as there are integers.

When you get done, there will be no line segments left (any such line segment would have a minimum length l, but the process has at least chopped l in thirds, therefore such a line segment cannot be present any more). But there will still be a point at 0, a point at 1, a point at 1/3, a point at 2/3, etc... By an interesting theorem, it turns out that there are still too many points left for you to pair them up with the integers. This set is called "the Cantor Set" or sometimes "The Cantor Dust", after the mathematician who first noted it.
 
The message is, don't mess with infinity unless you know what you're going up against. Most apparent mathematical paradoxes involve the use of infinity in arithmetic. The first thing to look for in a bit of arithmetical flim-flam is the divide-by-zero.
 
arSenic said:


Hmm, I'm not so happy with this. I definitely recall a line(segment) being defined as the set of all points between to points (between meaning at the shortest possible distance from the two points).

Besides, the location of a point and the point itself are synonymous in mathematics.
Click to expand...
But isnt that the same? The key words: "Between two points". The definition contains the end points, and thus the length of the line. It includes all points on the line, or the quality that any point between the end points are part of the line.

--- Yeah, Cantor dust is another intriguing way to view it.

Hans
 
The reaction of my kids, after I tried to convey some of this to them, was silence as they tried to absorb it, then my son said, "I guess it wasn't such a dumb question after all." :D
 
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