Freaky math thing du jour

[essentialsaltes]

                  ...
               (√2)
            (√2)
Consider (√2)


That is, √2 raised to the power of √2 raised to the power of √2 ad infinitum.

At first blush (and maybe second and third blush) it looks freaking infinite. I mean if you go 2^2^2^2... step by step you get 2, 4, 8, 16, 32 ... and it blows up pretty quick.

Ok, root-2 isn't as big as 2, but it's still bigger than 1. And if you raise it to a power bigger than 1 then it's gonna get bigger. And it'll continue to get bigger and bigger each time, over and over again. Bingo, infinite.

But let's just play with it. Since the exponentiation goes on for ever, then if we call that thingy x, then we can say that

x = (√2)^x

O... kay, that equation looks pretty hopeless too.

[a subtle but extremely important point, and a key to understanding the problem correctly (or at least, not wrongly): you can't just break off the 'last' one and say that x = x^(√2), because there is no last one if our beastie goes to infinity.]

Anyway, if we take the log-base-2 of both sides, we get:

log2(x) = x log2 (√2) = x * (1/2)

2 log2(x) = x

That's pretty shitty, too. Let's exponentiate it

x^2 = 2^x

Hey, that's pretty, at least. I still can't solve that fucker without cheating. Watch me closely as I misdirect you. Okay, the right side has to be a power of 2, so on the left side, x must be a power of 2.

If x is 1, then we have 1 = 2. Nope
If x is 2, then we have 4 = 4. Yep
If x is 4, then we have 16 = 16. Yep
If x is 8, then we have 64 = 1,024. Nope
If x is 16, then we have 256 = 65,536. Nope

It is trivial to show (* i.e. I am too lazy to verify, but I have a hunch that) that we're not going to get any more solutions, since the exponential is blowing up way too fast.

Maybe we should have just been satisfied with our original idea that there were zero finite solutions, because now we have two finite answers. That's pretty fucked up. It's probably infinite, but maybe it's 2. Unless it happens to be 4. It certainly can't be both 2 and 4, or the universe would immediately collapse.

So now what?

When analytical tools fail... draw pretty pictures.
If we just go ahead and brute force iterate the successive exponentiations, starting with √2 and then plugging in the new value back in as our new x, we find that indeed, the answer keeps getting bigger. But contrary to my Bingo, infinite conclusion above, we see that there's some diminishing returns, and the value approaches 2 as the limit.

The answer of 4 (although a solution of the derived equation) is basically irrelevant to the original problem, since starting from √2, you get to 2. However, if you examined that same graph near 4, the lines cross the other way round, so that 4 actually drives the pretty pictures away, unlike 2, which acts as an 'attractor'.

So the answer is 2. Weird.

Comments

[ovary.livejournal.com]

...my brain....ow

[essentialsaltes.livejournal.com]

As an additional exercise for the reader, go to the All Wise Google and type:

1.4^1.4

(a close enough approximation to root-2 to the root-2)

Google now does live math, so you can see the result without having to hit enter.

Then add another ^1.4

and another, and another

Knock yourself out.

Watch it rise, but its rise slows.

[my-wits-end.livejournal.com]

Live math? I have to hit enter.

[essentialsaltes.livejournal.com]

Ah, pity. In Mac Safari, when you type the term, it evaluates it. YMM(O)V.

[my-wits-end.livejournal.com]

Oh, you mean in the upper right text field in the browser? Because I have Mac Safari Version 3.1.1 (4525.18).

[essentialsaltes.livejournal.com]

No, from the Google homepage, in the... google text field. When you type in math, it shows the answer in the same manner as it shows google's guesses when it wants to complete what you're typing for you.

Let's see... I'm using Safari Version 4.0.2. Dunno if that makes a difference.

[essentialsaltes.livejournal.com]

ACtually, it works in the upper right text field as well. extra text comes up labelled 'suggestions' with the numerical answer at the top of the list.

Misdirection

[essentialsaltes.livejournal.com]

Who said the answers had to be integers? My solving of that equation was a bunch of crap, but if I cast it as a magic trick, I feel less bad about it.

So, if you go to a good online graphing calculator and plug in:
2^x;
x^2
into the function box (and increase the max y to 20)
Then you can see the three solutions in all their glory.

Wait, did I say three? That's right, there's a negative solution somewhere around -0.75. But if you think the answer can somehow be negative, then I say Good Day to you, sir.

---

If you go back to the previous version of the equation (x = (√2)^x )
you can plug that in as:
2^(x/2);
x
[reset the y max to 5]
and now you can see that 2 and 4 are the only solutions, and this gets us closer to the 'draw pretty pictures' version of the problem.
Note that the only reason there is a solution is because there is that little sliver between the intersections of the graph that sets up the 'attractor'. If we try to infinitely exponentiate the square root of 3, it doesn't work. The answer is infinite. See here for more info on the convergence of a 'hyperpower' sequence.

[my-wits-end.livejournal.com]

Very interesting.

generally speaking

[notjenschiz.livejournal.com]

now prove that if you replace 2 with n, then sqrt(n) infinitely exponentiated in the same way will return n, where n is an positive integer. (4 gives you the same dual weirdness as 2, but presumably with the other answer. For the rest, it looks trivially obvious to me).

Re: generally speaking

[essentialsaltes.livejournal.com]

Sorry, dude, click the "here" link in my comment above.