Wrong method. Try geometry:
1. Draw a right triangle on a grid, with
each leg being 1 unit long and following a grid line. The length of the
hypotenuse is the square root of 2. Note that.
2. Measure the shortest
distance between one end of the hypotenuse and the other, following only the
grid. This length will be 2.
3. Divide the grid in half along both axes,
giving you 4 squares where there was only one. Measure the shortest distance
between the endpoints of the hypotenuse again, following only the new
finer-grained grid. This distance is still 2. But now the error in the
approximation, the area between the actual hypotenuse and the path you followed,
is smaller. This means that our path is a better approximation to the actual
hypotenuse.
4. Repeat the previous division an arbitrary number of times.
Each time the error in the approximation will be smaller, as the grid gets
finer-grained and the area between your path and the hypotenuse is less and
less.
5. Define an acceptable error E such that if the error in the
approximation is less than E you can consider the approximation "close enough"
to the actual hypotenuse that it can be considered equivalent to the hypotenuse.
Continue repeating step 3 until the error is less than E. You can now argue that
we can treat the approximation as equivalent to the hypotenuse.
6. But
the length of our approximate path still follows the grid, and is still 2. Since
our path is equivalent to the hypotenuse, the length of the hypotenuse is 2 as
well. And since the length of the hypotenuse is the square root of 2, as we
noted in step 1, the square root of 2 must be 2.
The flaw here of course
lies in step 5, because no matter how small the difference is you still can't
make that jump from the sums of the lengths of the sides along the path to the
length of the diagonal. But how often have we seen legal arguments that amount
to "Since we've gotten our approximation by a valid path, we have to ignore the
actual diagonal."? [ Reply to This | Parent | # ]
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