Why doesn't the "stair step" method approach the shortest distance as step size approaches 0?

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In summary, The shortest distance between two points is a straight line, but when using a "stair step" method with infinitesimally small step sizes, the total distance does not approach the length of the diagonal line. This is because the limiting scenario of the stair step does not possess all the properties of the diagonal line, as shown in the article "The Limit Paradox." The concept of "infinitely small" is no longer considered meaningful in mathematics, and the stairs and the diagonal line are not the same, even if they may appear similar in a visual representation.
  • #1
climbhi
Alright, I have a question that I know is very dumb, but despite my best efforts I've been unable to come up with an answer that satisfies me. Here it is:

We know the shortest distance b/n two points is a straight line. So, if we have a right triangle with legs of unit length, the shortest distance between the endpoints is the hypotenuse (which will have length [tex]\sqrt{2}[/tex]).

Okay, now let's walk the distance b/n the two points in several different ways:

1.) Walk 1 unit East, then 1 unit North. Total distance = 1 + 1 = 2.

2.) Walk 1/2 unit East, then 1/2 unit North, then 1/2 unit East, then 1/2 unit North. Total Distance = 1/2 + 1/2 + 1/2 + 1/2 = 2.

3.) Walk 1/3 unit East, then 1/3 unit North, 1/3 unit East, then 1/3 unit North, 1/3 unit East, then 1/3 unit North. Total distance = 1/3 + 1/3 + 1/3 + 1/3 + 1/3 + 1/3 = 2.

I'm sure you see where I'm going by now. So, my question is, why doesn't the sum of this "stair step" method approach [tex]\sqrt{2}[/tex] as your step size approaches 0? Is there something fundamentally different about a diagonal line? I just can't see what the difference is b/n a "stair step" of infinitesimal step size and a diagonal line...

This has been bugging me for a while, so someone please set me straight (pun intended).
 
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  • #2
What do you mean "why"? As you have pointed out, the total distance traveled remains 2. For a "stair", if you look at horizontal segments alone, you clearly see that they add up to the horizontal distance between the two bases, independently of how many segments (or stairs, same number) there are. Same goes with the vertical segments.
 
  • #3
Werg22 said:
What do you mean "why"? As you have pointed out, the total distance traveled remains 2. For a "stair", if you look at horizontal segments alone, you clearly see that they add up to the horizontal distance between the two bases, independently of how many segments (or stairs, same number) there are. Same goes with the vertical segments.

Yes, of course I see that, but it doesn't answer my actual question: What is the difference b/n a stair step of infinitesimally small step size and an actual diagonal line? So, to apply the question to the example at hand: Why would a stair step of infinitesimally small step size from start to end yield a total distance of 2, whereas a diagonal line would yield a total distance of [tex]\sqrt{2}[/tex]?
 
  • #5
yenchin said:
You may find this article interesting (very much the same as your question!):

The Limit Paradox
http://www.mathpages.com/home/kmath063.htm

Thanks for the great article. It's nice to realize that I'm not the only one to have been confused by this. I was actually getting a little depressed (I just commented to my wife that I felt like I needed to return my recently-earned physics degree...)

But, despite the great points in the article, it doesn't really satisfy me. Basically, it just says that you have to be careful not to assume that because some limiting case has some of the properties of entity it's approaching that it has all the same properties. That's a great point, but it doesn't actually get into the why of things.

So, why does this limiting scenario (the stair step) not possesses all the properties of the entity it's approaching (the diagonal line)? I'm not seeing, fundamentally, what the difference is.
 
  • #6
climbhi said:
Yes, of course I see that, but it doesn't answer my actual question: What is the difference b/n a stair step of infinitesimally small step size and an actual diagonal line? So, to apply the question to the example at hand: Why would a stair step of infinitesimally small step size from start to end yield a total distance of 2, whereas a diagonal line would yield a total distance of [tex]\sqrt{2}[/tex]?

It seems as though the source of your questions stems from your concept of "infinitely small". The "infinitely small" is a nebulous notion, no longer regarded as meaningful by modern mathematicians. One must speak of the passage to the limit (2), which has a clear definition, instead of obstructing the matter with unclear terms (i.e. "infinitely small step sizes).
 
  • #7
climbhi said:
That's a great point, but it doesn't actually get into the why of things.
You made an assumption about how things behave -- that it didn't matter in which order you applied the steps "compute the length" and "take the limit". You looked at an actual example. The example demonstrated your assumption was flawed. I'm not sure what you find unsatisfactory about this state of affairs.


Maybe simply restating what you have shown will help? In whatever topological space you're using to make sense of that limit, you've proven that the following cannot both be true:
(1) That sequence of stairsteps converges to that line segment
(2) The length function is continuous

In particular, your work says (almost literally) that if (1) holds, then that line segment is a discontinuity of the length function.
 
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  • #8
If you draw a picture of your stairs, and make the step size really small, the picture will "look like" a diagonal line, but it is not! Take just one of your stair "steps" and pake the picture bigger - you will see that the diagonal length is still the hypotenuse and its length is still sqrt(x^2 + y^2). No matter how small the step size, this picture of just one step is always the same. When you add up all those little hypotenuses, you get sqrt(2); you do not get 1+1 = 2.

Does that help?
 
  • #9
climbhi said:
Alright, I have a question that I know is very dumb, but despite my best efforts I've been unable to come up with an answer that satisfies me.

Here it is: ...

Climbhi,

Progress, progress, progress. Thanks to the generous people on this forum (generous with their time), you know know that your question is very very far from being a dumb question.

Notice your criteria for an answer, however. You want "an answer that satisfies me."

(Somehow, "You want 'an answer that satisfies you.'" seems like a more natural way to put that last sentence.)

That's not a criteria that anybody can really work to in a logical manner except you. (More accurately, in a "deductive" manner.)

I applaud you for holding out for an answer that satisfies your criteria.

Here's my contribution to the mix.

Let f:[0,1]->[0,1] be the function defined by f(x) = x. the derivative of f(x) is one (f'(x)=1).

Let f_n be the nth staircase function as you describe it.

Then, for f_n converges uniformly to f (as n-->infinity).

("Converges uniformly" means converges pointwise AND for every epsilon in the delta-epsilon definition of a limit, the same N_0 works for all points in the unit interval [1,1].)

(f_n converges pointwise to f as n goes to infinity) iff (for all real numbers x)(for all epsilon greater than zero)(there exists a natural number N_0 such that)(n>N_0 ==> |f_n(x)-f(x)|<epsilon).

However, the limit as n-->infinity of the derivatives of f_n (f_n') do not converge to any function. In fact, loosely speaking, the sequence f_n'(x) is equal to zero an infinite number of times and it is equal to infinity an infinite number of times.

BTW, a question that I have forgotten the answer to, and maybe one of you gurus know off the top of your head is, is the following statement true when (f and f_n are differentiable functions mapping R into R)?

If f_n -> f uniformly and f_n'->g pointwise as n-->infinity, then f = g?

If not, what if f and f_n are continuously differentiable?

What if f_n'->g uniformly?

Are there other cases that are interesting?

DJ
 
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  • #10
Yes, if fn goes to f uniformly then it also goes to f pointwise and the limit is unique.
 
  • #11
HallsofIvy said:
Yes, if fn goes to f uniformly then it also goes to f pointwise and the limit is unique.

Thanks, HallsOfIvy! DJ
 
  • #12
This is an interesting problem, intuitively I would imagine that one can draw a stair case with a blunt pencil, there is a limit where the thickness of the line would cause the staircase to look like a diagonal. This is the point at which definitions of length would be ambiguous because they are the same order of magnitude as the width of the line drawn. No matter how the pencil is sharpened, this point will always be reached. This would indicate that there is not a gradual approximation from a staircase to a diagonal and that the two are fundamentally different objects when viewed at the appropriate resolution. Not too sure how that translates into mathematical terms though.
 
  • #13
Did you notice that this thread was a year old?

In any case, the "mathematical terms" for what you are saying is what I said before: that the convergence is pointwise but not uniform.
 
  • #14
Thanks for the clarification (it helped me understand some of the more technical aspects that were put forward earlier) I did notice the time stamp of a year ago but as I was thinking about the same problem just recently, I decided to venture a comment anyway.
 

1. Why is the "stair step" method not the most accurate method for finding the shortest distance?

The "stair step" method, also known as the Euler method, is a numerical approximation technique that uses a series of small steps to approximate a curve. As the step size approaches 0, the method becomes more accurate but it will never be able to accurately represent the true curve due to the nature of discrete steps.

2. Can the "stair step" method ever reach the shortest distance?

No, the "stair step" method can only approximate the shortest distance but it will never reach it. This is because the method only considers a finite number of small steps rather than a continuous curve.

3. Are there any other methods that can accurately find the shortest distance?

Yes, there are other numerical methods such as the Runge-Kutta method that can provide a more accurate approximation of the shortest distance. However, even these methods will still have a slight error due to the discrete nature of numerical methods.

4. How does the step size affect the accuracy of the "stair step" method?

The smaller the step size, the more accurate the approximation will be. However, as the step size decreases, the number of steps required to reach a certain distance also increases, making the calculation more time-consuming.

5. Is the "stair step" method completely inaccurate?

No, the "stair step" method can still provide a reasonable approximation of the shortest distance, especially for simple and smooth curves. However, for more complex curves, the method may not be as accurate and other numerical methods should be used instead.

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