Hey, this is Presh Talwalkar.
In a previous video, I explained a trick to solve the following problem:
Imagine you have X to the power of X to the power of X to the power of X, and so on,
repeated infinitely many times, and you want all of that to be equal to 2.
How can you solve this problem?
Now by "Solve," I meant "How would you find an answer in a high school math competition as fast as possible?"
I was asked this problem in a high school math competition, and the first person to get the answer got the points.
Nearly everyone loved this 1 minute video, which never claimed to be a rigorous proof presented in a graduate level real analysis course.
So, what was the trick?
You could look at this equation, and there's a pattern.
Consider all of the exponents above the base.
You have X to the power of X to the power of X and so on repeated infinitely many times.
Now, all of that is equal to 2.
Therefore, we can substitute 2 for the exponent of this base of X.
So we then have X to the power of 2 is equal to 2.
Now we're looking for a positive root, so we get X is equal to the square root of 2.
And that IS correct,
That square root of 2 to the power of square root of 2 to the power of square root of 2 repeated infinitely many times is equal to 2.
Now some of the people who watched this video actually had some interesting follow-up emails to me,
Including Andrew, George P., Diviad, Dolev, Oliver, and Vineet.
They tried applying the method I presented to another problem:
So imagine you have this infinite tower of exponents set equal to 4.
Well, using the same trick as beFOUR,
We would then have FOUR being equal to X raised to all these infinite powers
So we end up with X to the power of FOUR is equal to four.
Now if we take the fourth root of four, and we're looking for a positive root,
we end up with X is equal to square root of 2.
So we SEEMINGLY have that square root 2 raised to the power square root of 2 raised to the power square root of 2 infinitely many times is also equal to 4.
And this raises a potential paradox.
So we've shown that 4 is equal to 2...
...but this is clearly absurd!
So where is the mistake in this supposed proof?
Can you figure it out?
Give it a try, and when you're ready, keep watching the video for an explanation.
*Ding*
*Tick*
So here's a SHORT explanation:
This equation, where X raised to the power of X infinitely many times is equal to 4,
This is the mistake: this equation actually has NO solution, so any subsequent steps are not true.
It's like starting out with the equation X plus 4 is equal to X plus 2.
You could subtract X from both sides, and you would seemingly prove that 4 is equal to 2.
But we have not really proved anything. Our original equation has no solution.
And similarly, this original equation, where we set this infinite tower of exponentials equal to 4, this has no solution.
So anything that we deduce from that will not be true.
So for a little bit of a longer explanation,
imagine we have this infinite tower of exponents: X to the power of X all the way infinitely many times equal to Y
(This is known as "tetration." Or it's also sometimes referred to as "power tower," "super exponentiation," "iterated exponentials," or "hyper power.")
So in order to solve for X, we must first check that a solution exists.
In other words, you want to check for values of x that converge to a value.
For a proof, I refer you to the award-winning paper by Arthur Knoebel,
"Exponentials Reiterated" in The American Mathematical Monthly in 1981.
I provided a link in the video description.
So we can plot out this function
It actually has the following curve:
the minimum value is X is equal to 1 divided by e to the power of e,
and that has a value of y equals 1 over e.
Now, when X is equal to 1, y is also equal to 1.
And then the maximum value for which this converges
is X is equal to e to the power 1 divided by e
And this gives the maximum value of y, which is e.
So the range of this function goes between 1 over e and e,
and the domain is between 1 over e to the power of e and e to the power of 1.
So if you have values outside of these ranges,
you're not going to get a solution
So to bring it back,
For y equals 2, this IS within the range of this function, so we can find a solution:
X is equal to square root of 2
But for y equals 4, there is no solution, as this value exceeds the maximum value of the infinite power tower.
So this problem reminds us of a very important point:
Whenever you have an infinite number of items,
you generally need to check for convergence before you simply start making substitutions and doing usual arithmetic to solve for an answer.
Thanks for watching this video, please subscribe to my channel, I make videos on math.
You can catch me on my blog, Mind Your Decisions.
If you like this video,
You can check out my books, which are listed in the video description, and you can support me on Patreon.
If you have a math topic, you can email me presh [at] mindyourdescisions [dot] com.