Everyone who is interested in mathematics, physics or common philosophy knows the most famous of Zeno’s paradoxes, namely the one with Achilles and the tortoise. Many similar paradoxes exist, and when you understand the first one it is easy to understand them all.
But one day when I was playing with my thoughts, I found a “paradox” that actually requires some extra thinking. We can call it Jensen’s Paradox (don’t mix it up with Jensen’s inequality or The Jensen’s Inequality “Paradox”, a paper on economics by Susan Woodward).
Here it comes:
Consider the figure below which contains the points A, B and C:
If we want to move ourselves from the point A to the point C, we can choose the following path: First we go in a horizontal line to the point B and then we continue in a vertical line upwards to the point C. If the distance is 1 between A and B, and the distance is 1 between B and C, then the total distance (let us call it S) will be 1+1, which is 2, when we choose to go this way.
Now, consider the next figure:
Above, we move ourselves horizontally half the way to B, and then vertically half the way to C. We repeat these two steps once and arrive in the point C. The distance S becomes 2 again.
Let us develop this procedure a bit further:
We have now split every previous “step” from the figure above into two new steps. The distance S is still 2.
We could continue in this way for some time. Below, we have 8 steps:
And now 16 steps:
And 32 steps:
One more time. Here, we have 64 steps:
The distance between A and C have now been split into 128 small distances (64 horizontal and 64 vertical), and it is easy to see that the complete distance, S, is still 2. With a little mathematics, this can be written in the following way:
Let us now imagine that we split the distance into many horizontal and vertical parts (let us assume we then have n steps). Then it looks like this:
The sum of all the parts (the small vertical and horizontal distances) will always be 2, i.e. the total distance is always 2, even if we split the distance into an infinitely large number of steps:
We conclude that the distance below equals 2 (since the limit above is 2):
But according to Pythagoras’ theorem, the hypotenuse of the triangle below is equal to the square root of 2:
So, here is the “paradox”: We have now seen that the distance S between the point A and the point C is always 2, and we do know, simultaneously, that it is always the square root of 2 (according to Pythagoras). Hope you like it!