Is 2x = sqrt(2)x a Mathematical Paradox?

[mister studebaker]

Here's an odd one I cooked up wandering the streets.

A square (ABCD) has a circumference = 4x. The distance between two given opposed corners (A,C) or (B,D) can be expressed many ways, but two will suffice: doing it in steps (where the distance between AC is 2x) or directly across (where the distance = sqrt2 (x)).

I thought: What would happen if you reduced the size of the steps?

Would you eventually get to the point where 2x = sqrt2)x)?

In terms of physics, you would, as eventually one would reduce the size of the step to <h (planck distance, IIRC, 10^-44 meter) where it becomes immeasurable - at which point the step size practically equals the diagonal, as neither has any useful meaning at that size.

Given the clumpy nature of the universe, you'd probably get tunnelling long before then, but that's a red herring.

What I'm wondering is: where's the mistake? Is dividing by infinity illegal? If so, why? Zero doesn't equal infinity (obviously) but does dividing by infinity = zero?

I'm not a big time math whiz, but I find these subjects interesting.

best to all,

Mr Studebaker

 

[cookiemonster]

So what's the problem again?

If the length is immeasurable, then the whole xSqrt[2] = 2x situation can never happen. How can you measure them to be such?

cookiemonster

 

[Hurkyl]

Don't forget that none of the steps actually is the diagonal; you're merely approaching the diagonal.

This example is basically a proof that length is generally not a property preserved by limiting processes.

 

[mister studebaker]

Don't forget that none of the steps actually is the diagonal; you're merely approaching the diagonal.

This example is basically a proof that length is generally not a property preserved by limiting processes.

True, but if one posits that each step is infinitely small, then it would be indistinguishable from a point on a diagonal, as the distance traveled per step iss infinitely small, i.e., a point, and thusly, 2x = sqrt2(x), which is, logically impossible...

A friend said that dividing the line by infinity is the same as dividing by zero, but I don't see that as necessarily true, as an infinitely divided value is still a value while a zero = nothing - the absence of value.

What do you mean by "length is not a property preserved by limiting processes?

Thanks!

RS

 

[Hurkyl]

But a step can't be infinitely small. You can shrink them to any arbitrary size, but that size is still nonzero.


The problem here is that one's imagination/intuition interferes with one's perception of the facts.

Because we can shrink the steps as small as we like, we can imagine shrinking them so they're infinitessimally long, and we imagine that this entity shares the properties of the real thing (such as having length 2x).

(In some fancy mathematical structures, such as hyperreal analysis, we don't even have to imagine it; stairstep patterns with infinitessimal steps do exist, and, I believe, they will have length 2x)


Now, we look at the problem in a different perspective. When we're shrinking the step sizes, our curve is getting arbitrarily close to the diagonal of the square. Technically, we would say that the diagonal is the limit of the (regular) stairstep pattern as the step size approaches zero.


The irrational leap, here, is to then posit that the diagonal IS our imagined entity with infinitessimal steps!

As the step size approaches zero, we do indeed approach the diagonal, but we don't become the diagonal. Thus, we have no grounds to assume that the diagonal shares all properties with the stairstep patterns. (And in Hyperreal analysis, the infintiessimal stairstep pattern is indeed infinitessimally different from the diagonal)

Part of the point of calculus/analysis is to study when we can and when we can't make these kinds of inferences... but the first trick is to realize that they're not always right! This example, as I mentioned, works as a counterexample that disproves the claim:

The length of the limit of a collection of curves is the limit of the lengths of those curves.

To translate this into the current scenario, you have a collection of curves (your stairstep patterns). These curves approach the diagonal as a limit. The length of each curve is 2x, so the limit of these lengths will be 2x. This statement asserts that the length of the diagonal is 2x, which is wrong, thus this statement is not true in general.


Anyways, the lesson you should take away from this is:

Lengths and limits do not mix, in general.

You need some additional "niceness" properties to justify these types of assertions.

 

[matt grime]

Let us also prove that the limit does not preserve length here. If it did we could make the assumption that it didn't matter what pattern we took for the steps, so let us use the one where we divide the intereval into n equal length parts. The difference between the diagonal and the step is the differencein the lengths, ie 2/n - sqrt(2)/n = (2-sqrt2)/n. Now the total difference between the step lengths and the diagonal is n times this, ie 2-sqrt2, which does not tend to zero as n tends to infinity. So irrespective of how small we take the steps the difference is always 2-sqrt2.
And I hope you see that that argument is independent of the partition.

Putting it more mathematically, if we let d(x,y) be th difference in arc length of two paths x and y, and we let x_n be a sequence of step functions approaching y, the diagonal, then

lim (d(x_n,y) is not the same as d(lim(x_n,y)

(Assuming we have even agreed upon a way to take the limit of the step functions.)

This is an isolated phenomenon.

Here are two more examples:

Let f_n be a function on [0,1] defined as 1-xn for x in [0,1/n] and zero other wise. It is continuous. The limit of f_n isn't.

Let g_n be the funtion on R given by 1 + x/n for x in [-n,0], 1-x/n for x in [0.n] zero otherwise. This converges, in the appropriate sense to the zero function on R, yet the integral of g_n ove R is 1 for all n.