I've been really interested in tetration lately. So I came up with a seemingly simple problem to solve, which is 2 tetrated 0.5 times, which I'll write as the following.
2^^0.5
To make sense of this notation, consider the following where A represents a real number:
A^A = A^^2
A^A^A = A^^3
etc.
Here's where my problem is. The answer I got and the one on Wikipedia are different. I'm assuming the answer on Wikipedia is the correct one, but I would like to know what I did wrong.
So here's how I tried to solve this problem:
First I say 2^^0.5 is the same as the "super square root" of 2 (I don't exactly know how to format this), which is equal to X.
Next I tetrate or "super square" both sides by 2, so the "super square root" of 2 becomes 2, and X becomes X tetrated 2 times, which looks like the following:
X^^2 = 2
Then I rewrite X tetrated 2 times as X to the power of X.
X^X = 2
Finally I graphed Y = X^X and Y = 2 on my calculator and found the intersection point in the first quadrant, which should be the answer of 2^^0.5. And I got the following:
X = 1.559610469 (approximately)
However, the answer to 2^^0.5 on Wikpedia is approximately 1.45933.
Does anyone know what I did wrong when trying to solve this problem? Any answers would be appreciated. Also, if you have any questions of what I did or what I'm asking, feel free to ask.
Best Answer
The disparity is because you're starting from a different interpretation of the expression "2^^0.5" than Wikipedia is. You're saying that (essentially) it ought to be the case that for any $y$, ($y$^^0.5)^^2$=y$. This is a natural extension of the related law of exponentiation - but why would laws of exponentiation apply to tetration? Notice that it doesn't work for other numbers in place of $0.5$ and $2$:
$$(x\text{^^}2)\text{^^}2 = (x^x)\text{^^}2 = (x^x)^{x^x} = x^{x\cdot x^x} \neq x^{x^{x^x}} = x\text{^^}4$$
I can't think of a good, natural way to define $x$^^$0.5$, and the Wikipedia article on tetration agrees. It's not an easy task to extend things that are naturally defined only on integers into the reals - we should really do a better job of expressing to students how weird it is that it worked out just fine for multiplication and exponentiation.