Walk the Planck with a pi

[Amb]

According to quantam physics, there is a point of smallness, beyond which you cannot go any further. This is known as the planck length, and is mind bogglingly small.

Does this imply that an eternal number like 'pi' is actually incorrect? That is is to say, that for all practical purposes, pi simply stops after a certain number of digits. (This would imply that the formula for pi delves into noise after this point, or should be adjusted to stop)

Or am I just making up stuff again?

And if not, does this change the 0.9999.... = 1 argument, because the 9's must stop after x digits. (Im not sure how to convert 1.616199(97)×10 ⁻³⁵ meters into the right number of digits)

Im assuming this would also affect Phi and other constants.

[Zag]

You have a pretty good argument for pi, since it is defined as a ratio of actual distances. If you could measure, accurate to planck lengths, the diameter and circumference of a particular perfect circle, then divide, you'd have a decent argument to say that it has a finite number of digits (and is, therefore, a rational number).

As for 0.999..., you have to argue with Achilles as he is sitting on the tortoise. But let's consider another repeating decimal. If you have a piece of something which you measure to be exactly 100 planck lengths, and you want to cut it into thirds. You pull out your handy planck-length-marked ruler, count out 33 units, and say, well, I'm not allowed to cut in between the markers. I'll have to round down. However, once you have three of those and put them end-to-end, you don't have the same length as your full piece.

My point is that even though it is impossible to measure a fraction of a planck length, that doesn't mean that one doesn't exist, conceptually.

[extropalopakettle]

A perfect circle does not require physical existence, and it's diameter and circumference aren't constrained to being measured as a number of planck lengths.

Heck, if space is curved, there is no constant P (like pi) such that a "real" circle in real 3 space has circumference equal to pi times diameter. But that's not the sort of circle we mean when we define pi. (nor do we mean one constrained by planck lengths)

[L'lanmal]

[L'lanmal]

[DejMar]

L'lanmal responded to most of your questions with answers I agree to. Still I will add to it by commenting on your understanding of planck length. A planck length is not simply 'a point of smallness, beyond which you cannot go further', but the shortest measurable length in which one can measure beyond the quantum effects that dominate at that and below that length. As our current scientific understanding has no way to measure beyond the limit as we approach lengths less than the planck length. The data becomes...how you said, "noise", and with no practical tools to decipher it. One of the values used in defining planck length is the gravitational constant. It is theoretically possible that this value changes as the universe expands, stretches, shrinks, cools and heats up. Yet the difference is practically immeasurable. Just as an infinitismal is immeasurably close to zero or some rational point on a virtual number line, in calculations it can be treated as if it were correspondingly zero or zero distance from the rational point, though it is not actually equal to zero.

[Amb*]

I confess I only mentioned the .999...=1 because of the recurrent nature of it on the GL.

I guess what I am getting at, is: Does knowing that there is a planck length change any of the assumptions around "irrational" numbers? If we were to change our system to measure in PL (Planck Lengths) would we still need to calculate pi beyond PL digits? My guess is that it would work like an imaginary number. It means nothing in reality (just like 0 or -1, or SQRT(-2) etc) but very useful for maths regardless.

Also regarding diving by 3: If I have one apple and I divide it into 3, none of those chunks will be exactly 1/3 of the original apple. This is because there will be some of the apple left on the blade of the knife - even if only juice

[Amb*]

[The Potter]

The engineer said, "The math universe is different then the real universe."
_________________
Artwork | Fractals | Don't ignore your dreams; don't work too much; say what you think; cultivate friendships; be happy.

[Jedo the Jedi]

[Jack_Ian]

In the real world it is impossible to draw a perfect circle or divide something perfectly.
As for calculating things with Pi, we already truncate our answers. A computer or calculator does not waste time trying to get the perfect answer, rather it stops when an acceptable degree of accuracy is reached.
In fact calculating the +/- of your result is a very important part of Computer Science.
Some apparently simple calculations can give wildly different results based on the number of decimal places you keep in your calculations.
One example of this phenomenon is the Hilbert Matrix. If you keep the elements as fractions, you can get the correct result, but once you move to decimal, the results are way off.
A more simple example is just to put a number into a calculator and hit the square root about 50 times followed by the x ² 50 times. You will not get your original number. Does that mean you should throw out your calculator? Of course not. But you need to be aware that there are limitations to what we can measure and calculate.

[Amb]

Be that as it may, I'd actually rather focus on whether or not the planck length puts a limit on an irrational number such as pi. Ie is it possible that Pi is actually a constant with x number of digits, and any digits that occur after X are imaginary (still useful), noise/irrelevant, or technically incorrect - thus exposing a minor/near irrelevant fault in the formula for pi.

Admittedly that X number of digits is a lot of digits, and far exceeds practical purposes anyway.

Dej Mar: From what I understand, anything smaller than the planck length loses the property of locality - and is therefore everywhere at once. Whether that can actually occur or not I have no idea. I'm not far enough through my book "In Search of Schrodingers Cat" yet

[Jack_Ian]

π is what it is by definition, not by the decimal representation of 3.141592653589793238462643383279502884197169399375105820974944…

As for restricting π to a physically representable value, regardless of the size of ℓ _P , if you want more precision from a physical object then just use an arbitrarily large circle. If the universe is infinite, then you can have infinite precision.

[L'lanmal]

[extro...*]

[Amb]

The implication to me, is that because of the planck length, the most perfect circle that you can possibly construct is still a polygon - and thus the actual ratio is approximately close to pi.

[extropalopakettle]

If pi were redefined in terms of physically constructable circles, yes.