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In english, the square root of two is 2.

Proof:
take two lines of equal length placed end to end at right angles to each other. Let's pretend that these are two streets you need to walk down to get to, say, the market from your house. Each street is 1 mile long, so you walk a total of two miles.

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Now, one day, you decide to take a road halfway down the first one, .5 miles from your starting point, and after walking half a mile down that road you decide to turn again so that you will meet the second leg of your original path 1/2 way along that second path, so that your walking route now looks like:

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There is no difference in the length of this path -- you are just travelling four 1/2 mile lengths instead of two 1 mile lengths.

One day you decide to halve this again, making you walk eight 1/4 mile lengths:

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Once again, Same length.

If you continue to follow this process of going shorter and shorter distances while taking more and more turns, your path, as you approach an infinite number of turns, will look like this:

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(more or less). In other words, the hypotenuse of the right triangle formed with the two legs of your original path. AND, using the pythagorean theorum, we know the length of that hypotenuse to be (1^2 + 1^2)^(.5) or, the square root of two. AND, seeing as each iteration of our walk did not change the length of the path walked, our path should be equal to the length of the two original sides, i.e., 2. SO, 2 = 2^(1/2).

The problem with Sand Jack's `proof' is that it improperly uses mathematical induction. We have
  1. The length of figure 0 (Sand Jack's first figure) is 2.
  2. If the length of figure k is 2, then the length of figure k+1 is 2.
From this, mathematical induction allows us to establish that the length of figure n is 2, for any natural number n. However, induction does not allow us to make such a claim for the limit. Sand Jack's proof is parallel to the following, also incorrect, one:
Consider the sequence 1, 1/2, . . ., 1/n, . . . . Clearly, each element of this sequence is greater than zero. Hence the limit of this sequence (that is, zero) is greater than zero. Therefore, zero is greater than itself.

I have heard Sand Jack's proof attacked with ``the limit of the sequence of figures is not a straight line''. However, the limit is a straight line, for a reasonable definition of `limit': for each figure n, let f_n be the natural parametrisation of that figure. Then the sequence { f_n } converges (uniformly) to the natural parametrisation of the diagonal line.


To reiterate: after any finite number of steps, the length is indeed 2, but the figure is not a straight line. In the limit, the figure does become a straight line (not a fractal as Dhericean supposes), but the length is not 2. Dhericean and evan927 state the first part of this, but the second is just as important.

Well lets try and make neil's proof a little more rigorous. First define a function called l with its domain as the set of all real valued continuous functions on R, and its range as R. l(f) tells you the length of the graph of f between two points a and b.
We use the so called infinite norm on the domain of l i.e we define the distance between two functions f and g as:
d(f,g) = supx in R |f(x)-g(x)|
The set of all functions is of course isomorphic to a subset of the set of all curves in R2. Thus we can speak of the length of a curve using this function l(f) provided this curve is representable by a continuous function. So now I'll use the word curve with this understanding.

Now for the proof. We have above a sequence of curves, and the limit of this sequence(under the metric defined above) is the straight line between a and b(the two points involved).
The fallacy in the proof, of course, is that the proof uses the fact that:
lim(l(fn)) = l(lim(fn))
This would be true only if l was a continuous function and what this tells you is that l is not one. So mathematically speaking thats all there is here. We cant take the limit inside the function sign unless the function is continuous.

Physically speaking this is far more interesting. We would expect the length of two very similar curves to be almost equal. So this is actually an instance of a fractal. Its difficult to fit this into the usual definition of a fractal because the object is not really self-similar, but the idea is the same - If you examine an object in detail its length is not what it seems to be if you examine the same object from a distance.

First, draw two line segments, each one unit in length, perpendicular to each other at an endpoint:

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The sum of their lengths is clearly two units. According to the Pythagorean theorem, the distance between the two endpoints equals the square root of the sum of each segment--in this case, the square root of two. Simple geometry, right?

But what if we divide each of those line segments in half, and shuffle them around a bit?

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The length of the entire path is still two units, four times one-half. No surprises there. Even if you do it again:

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...you'd still have a two-unit-long path. But if you continued this to infinity, you'd have an infinite number of infinitely-short line segments, which must be the same as a straight line between the two points, right? And this proves that the square root of 2 equals 2.

Well, no, it doesn't, for the simple but oft-ignored reason that infinity is not a number. But let's try this from a purely geometric approach, instead.

If you look at the third "zigzag" above, you can imagine two parallel straight lines that pass through all of the corners. As the line segments grow smaller and smaller, these parallel lines will draw closer and closer. But they will never actually touch each other. No matter how small you make those line segments, they will still have finite length. They will never form a single diagonal line because to do that, each line would need to be a single point--in other words, it has zero length. But you can't shrink it to zero length by chopping it in half repeatedly. No matter how many times you divide a number by two, it is still greater than zero.

The length of the "zigzag" will always be exactly two, even if you take the limit to infinity. It will never equal the square root of two, because you will never reach infinity.

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