Yes and no. It depends on how you use the numbers. I just took the square root of 37 on my calculator, then I ran esup/2[/sup] which gave the same number, as one would expect. Subtracting the two yielded 2x10[sup]-15[/sup] which is pretty good but not perfect.
What I am saying is that when you work with represented values in any medium, you can expect inaccuracies. Numbers on a slipstick are perfectly accurate, as long as you line up the indexes and the hairline where they need to be, but at some point, you have to take the number off it, at which point you lose accuracy.
You’re no longer talking about numbers but about the calculations used to approximate the numbers. Two different things.
Wait, the numbers are perfectly accurate until you have to use them?
Slide rules were almost immediately replaced when calculators were invented because using two-dimensional lines to represent points is inherently fantastically inaccurate. Again, you are confusing calculating a number for the number itself.
OTOH, your confusion inadvertently does help to drive home what we’ve been arguing: the obvious fact that our measurements and calculations can never represent reality to the last decimal place.
Because you can’t make a perfect circle or whatever shape from particles or finite points, but you can make one from arc or line segments, which a probability field can make.
Which is why this is possible
Electrons arrange themselves to a field which is much flatter than an atoms width variance.
If a helium atom were a basketball this mirror would be a court with a few holes and frisbees on it.
All this really blows your claim of not being able to measure to that level completely out of the water and cues us into the fact youre using outdated information, but to your credit information you seem to have learned rather thoroughly at the time.
The deposit may be single-atom, but the underlying surface is not: though it may be exceptionally smooth by normal standards it will be bumpy by atomic standards. It is not perfect in any sense of the word.
And neither is the atomically smooth coating. They’re measuring to 30 nm resolution, but that just means they know nothing about the smoothness at 0.00000000030 nm resolution or a billion billion billion times that.
That says, again, that real-world objects fail at being mathematically perfect by essentially an infinite order of magnitudes. And we can’t even begin to measure the extent of their imperfection because are tools are so crude.
Mathematically “perfect” is not a possible attribute of real-world things, not even layers of single atoms.
Visualising droplet dynamics at 30nm resolution doesn’t mean measuring the surface at 30nm resolution.
Atomically flar surfaces vary by no more than an atoms width .1-.5nm
And their electron fields can arrange to be flatter than that.
You’re still thinking in terms of finite particles, not fields.
That mirror can reflect the majority of the helium atoms sent at it with a predictable exit trajectory because it’s surface is near perfectly flat, even from the perspective of a helium atom.
So while a circle of basketballs can’t make a perfect circle, their motion can transcribe one, just as a quantum field could in theory. Meaning you’d have an effectively solid object that formed a perfect shape.
Certainly within a fraction of .1nm in any case.
Which is definitely less than varying by an entire atom.
Building a shape with more precision than the OP referred to when using single atoms is not only possible, it’s been done.
The whole idea of making the depositions so thin is that their surface qualities are dominated by quantum properties .rather than perfect arrangement of the matter.
Right. I don’t think the OP is saying anything about pi that isn’t also true about the square root of two (and that bothered people a lot earlier, historically).