Math was never my strong suit, but i don’t think this is correct. The limit as X -> infinity is not 7.07 but rather is 10.
In other words, this series doesn’t converge - it always equals 10, no matter how small a step you take.
That’s what I meant, though I probably wasn’t clear (having left higher math behind eight years ago).
It doesn’t converge, but at infinity the value is different. It approaches 10 (well, rests firmly at 10) as x approaches infinity, but at infinity the value is different (7.07).
I still say the mistake you guys are making is that, yes as the zig zags reach infinity they are still ten, but as the zig-zags apporach infinity they do not approach a line.
The problem is that if you travel from one point either north or east to get to another point that is directly northeast, some of your travel is wasted. Your progress is measured by the cosine of the angle between your travel and your destination, multiplied by the steps you take.
So, no matter how many little jogs you break it into, you are still always walking either north or west. You never walk directly towards your destination, except at the end, by when you must also have spent some of your time walking almost uselessly almost 90 degrees away from your destination.
If you want something that approaches a straight line in a limit, try going from (0,5) to (5,5) to (5,0), then (0,5) to (4,4) to (5,0) then try going through (3,3), then through (2.8,2.8), and so forth. The distance will gradually approach a minimum.
The basic error is the assumption that all functions are continuous. Although the smaller and smaller zig-zags certainly do approach a straight line, their length does not approach that of the line. This is just another way of saying that arc length is not continuous. That doesn’t really help; it just restartes the question. But let me just observe that if you use curves even more sharply curved than the littles zig-zags, you can a series of curves that also approach the straight line, but whose arc length is larger, even much larger, than 10.
Yeah, what Hari Seldon said really captures it. Arclength is extremely sensitive to perturbations, infinitely so. To get any upper bound at all on the length of a curve, you need to know it to full precision; knowing the lengths of a series of approximating curves gives you nothing.
This phenomenon was probably most famous illustrated by Benoit Mandelbrot’s example of measuring the length of the coastline of Britain. You might think a decent approximation to a curve gives you a decent approximation to its length, but it is not at all so; given any wiggle room at all, there is room to make curves of arbitrarily large length.
Another example to illustrate it: Suppose, instead of going directly to the right for one meter, I decide to keep hopping up and down, but never more than a height of X. As X approaches 0, these curves have to approach the direct path; however, with frequent enough hopping, these curves can have arbitrarily large length, no matter what X you choose.
Accidental double post; read the next one.
Or, as perhaps the most simple example: Consider a single jump and fall of height 1 meter. Then consider two jumps of height 1/2 meter. Then consider four jumps of height 1/4 meter. And so on. Each of these has a total distance traveled of 2 meters, even though these curves become better and better approximations (in the same sense as that of the OP) to just staying still, which has a total distance traveled of 0 meters. The length of a path/curve simply isn’t continuous, with respect to this notion of convergence.
Though, I should say, the claim that arclength isn’t continuous depends on what notion of convergence of curves you use. We could come up with a different notion of convergence of curves under which arclength is continuous, but the OP’s series doesn’t approximate a straight line.
I do not follow this reasoning at all. In this particular case, how can Y be 7.071 when X is infinity? If X is infinitely large, Y is infinitely small but inifinitely small does not mean 0. Can you demonstrate this mathematically?
I would also be interested in seeing other examples where as x approaches infinity, the y value approaches some value, but at infinity, the y value is something other than what it was approaching.
Another example.
Nail two small nails into a board. Now tie an end of a thin string to each nail such that the string hangs loosely between them.
Pull the string taught against each nail, running the excess string off to the side:
o---------------------- ------------------------o
| |
| |
| |
| |
| |
-
Now make two offsets of the string so that there are three straight sections between the nails:
o------------ ------------------ -------------o
| | | |
| | | |
- -
Notice that the length of the offsets has shrunk. Now do this with three offsets, four offsets, etc. and each time the size of the offsets will shrink. With an infinite number of offsets, theoretically the size of each offset would be infitesimely small and the string would be straight, all slack in it gone. Yet, quite obviously, a string can’t shrink in length without cutting.
Right, that’s essentially the same as what I was trying to illustrate with my jumping example in #28, though your framing of it is better, I think.
This is just the same thing in a little different shape. This reasoning leaps to the conclusion that infinitesimally small equals zero, which is not the case. I still think this is a false paradox. I was looking for a mathematical description of this. Can you express this as a function which you can solve for X=infinity? My guess is that such a function would be undefined at infinity, similar to how division by zero is undefined*, but IANAM.
*Division by zero is not infinity, it’s undefined.
Let me elaborate. Suppose we have a rectangle of area 1 square unit. We divide that rectangle into x equal rectangles. The function that gives the total area of all the rectangles is
y = x * 1/x
where x is the number of equal-sized rectangles.
y is 1 for arbitrarily large values of x. As x approaches infinity, y remains 1. However, if were to follow the reasoning shown in earlier posts, at x=infinity, 1/x becomes 0 and therefore y is 0. However, 1/infinity is *not * zero. You cannot have a total area of zero after adding up the area of an infinite number of non-zero sized rectangles.
I think it depends on which of the two is correct:
∞ * [sup]1[/sup]/[sub]∞[/sub] = 0
or
∞ * [sup]1[/sup]/[sub]∞[/sub] = 1
By normal calculus, once something hits [sup]1[/sup]/[sub]∞[/sub] you take it to mean zero, and zero times anything (including infinity, I would assume) is zero. In that case, once you have an infinite number of infinitesimally small offsets, they all disappear leaving the straight line. But I don’t know how calculus views that rule in relation to the n * [sup]1[/sup]/[sub]n[/sub] = 1 rule and how they go in terms of precedence.
Hm, well Dr. Math seems to hold that 0 * infinity is indeterminate. So there appears to be no answer
I have no idea what you guys are doing starting with post 33. Can you explain the connection to the OP’s question a little more?
I believe the problem here is in taking the idea of a limit from one to two dimensions. In order for a limit to exist in two dimensions, the function needs to approach the limit from all directions, not just one or two, else it doesn’t exist. Also, some ‘approaches’ are not mathematically valid in higher dimensional limits, but I’ll let somebody else explain that.
The problem makes you think we’re only dealing with a normal single variable function, y=f(x), but really we are dealing with the distance function s=f(x,y)*.
So the OP just showed us that the distance function we are used to is not the limit of a “taxicab function” broken up into smaller parts. We just have to find another way of defining it. Such as dS=sqrt((dx/dt)^2 + (dy/dt)^2)dt.
*Actually, it would be s=f(a,b), where a=a(x,y) and b=b(x,y). Distance is more complicated than you thought!
Man, I really wish I could type math better.
Well, since I posted #33 I’ll respond.
The OP suggests a function that gives the length of the zig-zag line as a function of the number of zigs (or is zags?). The OP does not actually give us the function, but one way to write it is
f(z) = z * d/(z * sqrt(2))
where d is the straight-line distance and z is the number of zigs (or triangle peaks). The distance for a single peak is the straight-line distance divided by the square root of 2 (which is the sum of the two equal sides of an isosceles right triangle). When you have two zigs, you have two times as many sides that are each half the size. When you have a zillion zigs, you have a zillion times as many sides that are each a zillionth the size. For any arbitrarily large number, the z terms cancel out, and the answer is indeed d.
The OP claims that
f(infinity) = d
However, when z becomes infinity, we have to resolve the term
d/infinity
I suggested that a real number divided by infinity is non-zero, although the cite to Dr. Math above says that it is equal to zero, in which case I’m mistaken.
If you determine that this number is zero, then we still have to resolve
infinity * 0
And the good Dr. Math says this is an indeterminate form, so the overall answer to the zig-zag problem is that the total length of the zig-zag line when the number of zigs *becomes * infinity is indeterminate.
As the number of zigzags grows, the crooked path approaches a straight line. All that’s happening is that the limit of the length fails to be the length of the limit. That just means that arc length is discontinuous, as Hari Seldon mentioned.
There’s another example of this sort of thing that might be brought up in an introductory analysis class. Define the function f[sub]n/sub as n when 0 < x < 1/n and 0 otherwise. The area under the curve of f[sub]n[/sub] is equal to 1 for any value of n, but the area under the limiting curve is zero. In that case, you’re getting bizarre behavior because the sequence of curves only converges in a weak sense.