Update: There is more subtlety to the infinite case, which I’ve now addressed in the post!
Update 2: Learning so much messing with this! Added a bit more discussion near the end.
So on twitter yesterday, the following mathematical identity was making the rounds…
Basically, it says that the infinite series of powers on the left — “the square root of 2, raised to the power of [the square root of 2, raised to the power of (the square root of 2, raised to the power of…)],” is simply equal to 2!
I love simple but baffling little mathematical puzzles like this, and I thought I would show you how to demonstrate this, and whether it can be generalized to other cases!
To make it clear, we use a little bit of algebraic manipulation. First of all, let’s solve the following mathematical identity:
This expression isn’t too hard to solve! The answer is x = 2, because squaring the square root of 2 is equal to 2.
So let’s work under the condition that x = 2. But, if we reverse the equation above, we can also say that
Now we can plug this value of x into our first equation, to get
Now we flip this expression around,
and we can again plug it into our original equation!
Hopefully you can see the trend. If we keep putting the new value of x into our original equation, and do it an infinite number of times, we will get an infinite chain of powers which must evidently approach 2.
To check our work, at any stage of the calculation we can simply input x = 2 and see that it falls right back into place.
This isn’t a proof of the original infinite power identity, but is rather a plausibility argument. There are mathematical subtleties to proving the infinite sequence of powers in fact does properly converge to 2 and doesn’t do something weird — because infinity — but this gives you the basic idea!
In fact, now that we’ve seen how this basic process works, we can ask: can this be done for any number? For example, we can apparently write a similar expression for the cube root of 3,
Again, the trick is to note that we may write, algebraically,
where now x = 3 is the solution.
In general, then, for any integer N = 1,2,3,4,…, we may also apparently write
The process for demonstrating this result is exactly the same as for 2 or 3, but you can horrify your non-mathematical friends by presenting them with equations like
But does this really hold for an infinite number of iterations? As I hinted above, the answer isn’t as simple. We note that the chain of calculations we’ve done so far, to any finite number of powers, all end with an “N” in the top exponent,
However, that final red N doesn’t appear at all in our infinite expression: there is no “final” power in the infinite case! In order to check whether the infinite chain of powers converges to the expected result, one must go instead from the ground up, so to speak, and calculate a large number of values in the sequence
Note the difference with the expression above it: we don’t end with N, but with the Nth root of N, the only thing that appears in our infinite expression. One can calculate this sequence in order numerically, and see if it approaches the expected value N or not.
It so happens that a paper just appeared on the arXiv this week by Luca Moroni about such “infinite power towers” (h/t to Greg Egan on twitter for the link) and it turns out that the sequence does not always converge to N! For the case N = 2 it does, but it does not for N = 3, and for some values of N it does not even converge to a single value, instead oscillating between several as one continues the sequence! So we can’t use our simple plausibility proof to argue that the infinite case works for any N.
Thanks to Danny Kodicek on twitter, we can actually say a bit more about where that non-N value comes from! We started our argument on the assumption that there is a single solution for the following equation,
and that the solution is x = N. However, for a large range of N values, there are in fact two solutions to this expression, which can be readily seen by plotting both the left and right sides of the above equation as a function of x. Below, we plot the case N = 3.
It is a little difficult to see in this plot, but there are two intersection points between the linear (red) curve and the exponential (blue) curve: one at x = 3, but also one at x = 2.478!
If we go ahead and look at how our value of the infinite power tower of cubes depends on the number of iterations we do, we get the following picture:
The value of the power tower approaches the lower value x = 2.478, not the value at x =3!
So, in fact, our infinite power tower does approach a value dictated by our simple equation,
but it usually approaches the non-obvious solution, not the integer one that we would think. In the case of the square root, with N = 2, the infinite power tower takes on the simple value we expected.
This just goes again to show something that I’ve said a lot on this blog: infinity is weird! Strange and unexpected things can happen when one takes a finite expression to its infinite limit.
You might ask: is there any use or purpose to expressions like this? In this particular case, I don’t think so, but formulas that involve an infinite number of mathematical operations have a lot of historic importance in mathematics. For example, in 1656 the mathematician John Wallis published what is now called the Wallis product, of the form
A formula like this gives one a way to approximate the value of π as accurately as one likes. The more one continues the chain of multiplications, the closer one gets to the true value of π/2. So if you are ever on a plane that crashes in the wilderness, and you need to calculate the value of π by hand, now you know what to do!
I originally started this blog post to show a simple way to derive the equation for the square root of 2, and it turned into a curious exploration of the non-intuitive nature of infinite mathematical expressions! This is why I love blogging.
(Work has been crazy busy lately, which is why my blogging has been less frequent. Hoping to get back to a bunch of new posts on physics and everything else in the near future!)