# Jensen’s Paradox

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!

Great stuff thanks ))

I do not know why this is called Jensen’s Paradox. It has been known and discussed for years, possibly centuries. I believe I first sawit in one of the books of Martin Gardner

Hi David. Which book? I haven’t seen this before … nevertheless, this text was simply posted here (on this page) and I was more or less joking about the name … I am certainly not claiming official credit or anything. However, I have read quite a few books and never seen it before. My text was posted several years ago, so until I see its main idea presented/published/discussed before that, I don’t have a problem referring to the content of this blog post as my own. If I see evidence that someone else presented the idea earlier than I did, then I will gladly change the title of this post.

The straight line is the sum of all the little hypotenuses of all the little right angled triangles. There is no paradox because there is no limit that is any different for each hypotenuse.

That is correct; it is not a mathematical paradox. I used the word paradox in the same way that Xeno did so long ago (the paradox with Achilles of course isn’t really a paradox). The interesting thing with this fractal is that it will always look like a straight line, no matter how much we “zoom in”. So, when we see a straight line … could it in fact be this fractal (or a similar one) we are seeing? The straight line has a length of √2 and the fractal has a length of 2, that is indisputable. Still we cannot draw the fractal in a way that makes it distinguishable from the straight line. That’s amusing (but it still does not qualify as a “real” paradox, of course). Thanks for reading and commenting!

Great piece

thanks

Drew Hempel writes about this too

can google it out

sum(1/n) does not converge

Well, reader zajo, that is correct. But that has nothing to do with the mathematical expression above. There are no

sums here, only the limit ofinfinitesums.finiteI understand that the figure we get when we apply the “split into steps”-algorithm, and thus split the distance into an “infinitely” large number of steps, is, in fact, a

fractal. This fractal has the length 2. Furthermore it has no derivatives (not even in one single point).What is so fascinating about this present figure is that it has the same appearance as a normal straight line.

So one question arises: When we see a line, how do we know it is really a line and not a fractal? If you have to drive a car from point A to point C, you might want to know how much fuel you need. Is the distance 2 or almost 30% less?

I think in reality it is 2 since it is composed of knowable distinct parts. The straight line just cuts through the perpendicular lines… cheating in the way 🙂

Well, Ian. The shortest distance between two points is a straight line, no doubt! Or? 🙂

The phrase “The shortest distance between two points is a straight line.” is a mathematical piece of nonsense. Distance is a number. A line is a geometric object. The distance cannot be a line. And “shortest distance” is a phrase that does not make sense. What is probably meant is that if we look at all paths between two points, and we look at the length of each path–and we must distinguish between the geometric object “the path” and its length (a number)–then the path which is the straight line containing those two points is a straight line, and its length is the distance (by definition) between those two points. The correct statement should be “The distance between two points is the length of the shortest path between the points,” or more accurately, the infimum of the lengths of all paths between the two points. While there are multiple problems with the standard wording, the two main ones are that (1) the phrase “shortest distance” is an oxymoron, and (2) there appears to be confusion between a path or line and its length, which are two different things. The standard statement is an example of fuzzy thinking. Many students do poorly in math because they use fuzzy thinking and unclear definitions and do not know how to distinguish properly between connected, but different ideas. The use of “fuzzy” speech by teachers is often a major contributing factor to students using fuzzy thinking and doing poorly in math. If this is a professional site, let us not use fuzzy speech or thinking!

David, I hear you, and I agree with everything you’re saying!

First of all, I was commenting on reader Ian Walker’s comment, which I think was some kind of joke (hence my smiley), so I didn’t start my internal deep-thought-engine (which requires both time and energy to work). I normally start that whenever necessary.

Secondly, English is my third language. In my first language, Danish, there is a common proverb stating “den korteste afstand mellem to punkter er en ret linie” (it could, fuzzily, be translated to “the shortest distance between two points is a straight line”), and in Danish the word “short” (i.e. “kort”) can be used as a synonym to “small”, but the rest of this “everyday truism” is, in fact, just as meaningless as you’ve just pointed out. I might have simply been using that phrase … well, I don’t know why, but strange things sometimes happen, when I write in English.

Furthermore, I am a supporter of precision, certainly. For some years, I used this example, https://blog.offcircle.com/a-small-exercise/, to point out that one needs to read the mathematical formulas thoroughly before applying them (it is even possible to find this kind of imprecision in books used officially within the Swedish school system). I fight imprecise sentences whenever I discover them (errare humanum est, I might add). Of course, a distance between two points is a number (non-negative), and the same goes for the distance between to numbers for that matter (numbers can be regarded as one-dimensional points and thus |a-b| can be calculated as sqrt((a-b)²) which is, per definition, the

positivesolution for x in the equation x²=(a-b)², which brings me to the fact that for any real number, x, one should always define |x| as sqrt(x²), which I believe would eliminate confusion among first-year students and increase understanding of something as simple as square roots and the symbol √). In fundamental mechanics, I’ve seen students mix up the terms speed and velocity, another example of confusion between scalar quantities and vectors (not saying that a geometric straight line necessarily always should be (or can be) treated as a vector).Finally, this is not a professional site (but of course, I agree with you that fuzzy thinking should be avoided). I do neither refer to myself as a mathematician nor a math teacher (I teach programming part-time). I am writing a novel (in Swedish) at the moment, but sometimes “things from the past” bubble up, interfering with my writing, and I have to get them out before I can continue, which was the case with the text above. The reactions when I posted it gave me reason to read a few more books (I knew very little about fractals before), and that is always a good thing.

Thanks for reading and commenting!

David, the point behind this kind of mathematics is the enjoyment of it. I thoroughly understood all of the mathematics involved and I enjoyed the problem. I had never heard of Jensens Paradox, but now I have and I feel richer for it. I didn’t figure it out until I read the post by Geoffrey Campbell. For me it was an aha! moment. Any criticism of how unprecise any of this is, including how we define the distance between two points truly stands out as nothing but ugly.

Yes yes !! Cheers 🙂