This little geometric riddle is something that I have thought about way too much, and it's driving me crazy. I know this is incorrect but I can't figure out why:
The diagonal of a square is twice the length of its sides.
Crazy, right? We all know it's not true. The Pythagorean Theorem states that for a triangle with sides
a,
b, and hypotenuse
c,
a2 + b2 = c2
which means that for triangles formed by a diagonally bisected square with sides of length
n,
n2 + n2 = c2
where c is the length of the diagonal. Simple algebra gives us
c = √2n
That's what we all learned in geometry, right? The length of the diagonal of a square is √2 times the length of its sides. Physical reality backs up this fact. Grab a ruler and see for yourself. Now then, why does the following "proof" make sense?
Assuming a square with sides of length
a, the distance between opposite corners of the square traveling along the border of the square is
2a.
By dividing both sides into two equal segments and alternating directions, we create a new path of the same distance
2a.
2(a/2) + 2(a/2) = a + a = 2a
By continually dividing each side into
n equal segments of length
a/n and rearranging them in this manner, we create a path that is arbitrarily close to the diagonal without changing the total length of the path:
n(a/n) + n(a/n) = a + a = 2a
As the number of segments
n approaches infinity, the total length remains unchanged.
lim(n → ∞) 2an/n = lim(n → ∞) 2a = 2a
Thus the length of the diagonal is
2a or twice the length of a side.
Please, someone tell me why I'm wrong.
