as constrained by the Plank Length. Any ideas on how to solve this
Eh ? Pi is a mathematical constant, not a physical measure, so I don't see how you can "measure" Pi.
Or do you mean: how many digits of Pi can we "write down" in the current universe or something ?
I guess what the questioner means is "what is the ratio of the circumference of the largest possible circle in the Universe, measured as an integer number of Planck lengths, to its radius, measured as an integer number of Planck lengths".
If not, what did you mean?
Unfortunately, my formulation of the question contains so many false assumptions, it's impossible to answer. For a start, it assumes that fractions of a Planck length are impossible, which isn't true. It begs all sorts of questions about the geometry and topology of the Universe. Is the Universe flat? Do you want to use the observable Universe or the whole Universe? When are you going to measure an expanding Universe? Finally even if you could get a ratio of two integers it could be a recurring decimal and go on forever.
As vanesch indicated, a circle is abstract mathematical concept and the value of π is decided by mathematical formula, not physical experiment. The number of decimal places is limited only by the size of your computer and how long you're prepared to wait for an answer.
Now, the computability of pi using the entire sum of states of the Universe to produce a turing program or something similar... that is in theory answerable, but then that would be a vast busy beaver number... so it isn't answerable again. T.T
I think he meant what is the finest physical measurement of pi possible... which runs into the uncertainty principle I would think.
Yeah I meant the smallest physical measurement. Would this work?
Two times the interval from -1 to 1 of the square root of 1-x^2 dx with
Where h is the Plank length?
Again, Pi is not some physical constant, so it is not something that you can measure. You could eventually write down an operational procedure to measure a physical quantity which might be close to Pi in a certain model universe, but 1) what guarantees you that our universe satisfies that theory with such a precision and 2) how would you know whether the number you've measured is equal to Pi ?
You see, what you are trying to do is similar to, say, try to find the sum of 2 and 2 to very high precision by measuring the volume of 2 liters of water, and adding 2 liters of water to it. While a mathematician might tell you that 2 + 2 = 4, you might want to "measure it" to 60 decimal places. Well, guess what ? You will probably not find 4.0000000000000000000000000000... but rather 3.99999999999999999999999345923424252... or something of the kind.
Because there are effects that will play a role, such as the self-gravitation of the water, and its compression and lots of things. And we're only to our 60th decimal. Does that mean that 2 + 2 is actually 3.999.... , and that 4 was just a mathematician's approximation ? While mathematically, 2 + 2 really equals 4, it is in fact your theory that 2 liters of water + 2 liters of water make 4 liters of water is simply not accurate to 60 decimal places. Same with your cosmic measurement of Pi.
I think it was a much simpler question. What's the largest real number (how many digits), in the universe, the estimated size of the universe in terms of planks constant. PI has been calculated to billions of digits, but I doubt there are many "real numbers" of the order 10^(10^9)). Most of our "constants" or "units" have less than 12 digits of accuracy.
Note about pi, you can measure it, its how many times the diameter of a circle fits onto its circumference.
And pi is calculated to about a trillion decimal places now (1,000,000,000,000) and because of computers can keep getting calculated to more.
and its something in the region of 10^39 decimal places means that uses pi and is measurable is accurate to around the size of a hydrogen atom.
You cannot answer this question. Depending on how you build the machine to measure pi (you might even use mathematical results to evaluate series by a machine), this machine will stops sooner or later or even never.
It is 39 decimal places, not 10^39. With PI given to the 39 decimal place you can compute the circumference of the known universe (from its diameter) with an accuracy to the size of a hydrogen atom. Is that what you tired to say here?
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