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Amb
Amb the Hitched.

 Posted: Sat Jan 12, 2013 7:47 pm    Post subject: 1 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
Tired of his old title

 Posted: Sat Jan 12, 2013 9:27 pm    Post subject: 2 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
No offense, but....

 Posted: Sun Jan 13, 2013 12:57 am    Post subject: 3 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
Daedalian Member

Posted: Sun Jan 13, 2013 6:45 pm    Post subject: 4

 Amb wrote: 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)

This sounds like two separate questions.
1) Does this imply pi is incorrect.
A. No. Pi as the limit of an infinite series (4/1 - 4/3 + 4/5 - 4/7 . . .) or the as the twice the definite integral of sqrt(1-x^2) does not depend on a physical construction.

2a) For all practical purposes, can you round-off pi?
A. This is trickier and probably requires you to define your "practical purposes". If all your practical purposes are architecture and civil engineering, then probably yes. In many cases, you can round pi off to 3 or 22/7ths without any ill effects.

2b) Is knowing the decimal value of pi to lots of decimal places useful?
A. Not particularly, unless you are using it for cryptography, or as an example for pseudo-randomness tests. Mathimatically, the symbol for pi (and the idea it represents) usually suffices.
L'lanmal
Daedalian Member

Posted: Sun Jan 13, 2013 6:52 pm    Post subject: 5

 Amb wrote: And if not, does this change the 0.9999.... = 1 argument, because the 9's must stop after x digits.

Funny how you phrase this. It does not change the argument in the slightest. The Greeks, not knowing about convergent series but believing that Achilles must catch the tortoise (that is, they intuitively believed that 0.999999... eventually equaled 1), used this concept to argue that there must be some indivisible unit of space/stuff - in their language an "atom".

So this is perhaps the oldest known argument on the topic, revisited.
DejMar
(Possibly a robot)

 Posted: Sun Jan 13, 2013 8:18 pm    Post subject: 6 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*
Guest

 Posted: Sun Jan 13, 2013 8:22 pm    Post subject: 7 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*
Guest

Posted: Sun Jan 13, 2013 8:26 pm    Post subject: 8

 Quote: A planck length is not simply 'a point of smallness, beyond which you cannot go further',

I thought that the point of smallness beyond which you cannot go further was the definition of said length? Which is in keeping with the bizarre world of quantum physics where things can 'spin' without returning back to original position after one 'rotation' etc

Is there a reference that backs you up, because that point would nullify my question, and thus bring resolution to the enigma.
The Potter
Feat of Clay

 Posted: Mon Jan 14, 2013 1:39 am    Post subject: 9 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
Paragon in Training

Posted: Mon Jan 14, 2013 3:08 am    Post subject: 10

 The Potter wrote: The engineer said, "The math universe is different then the real universe."

So, either he was attempting a play on words, or he failed his English classes. Maybe that's why he's an engineer.
_________________
Paragon Tally: 18 mafia, 3 SKs (1 twice), 1 cultist, numerous chat scum...and counting.
Jack_Ian
Big Endian

 Posted: Mon Jan 14, 2013 5:28 pm    Post subject: 11 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
Amb the Hitched.

 Posted: Mon Jan 14, 2013 7:11 pm    Post subject: 12 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
Big Endian

 Posted: Mon Jan 14, 2013 8:54 pm    Post subject: 13 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
Daedalian Member

Posted: Tue Jan 15, 2013 12:10 am    Post subject: 14

 Amb wrote: 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.

Let us take a "simpler" irrational number, such as the square root of 2.

Which of the following steps would you use planck length to dispute?

1. No positive integer is both odd and even.
2. Any integer greater than 1 has a unique prime factorization.
3. For any positive integer, its square has an even number (possibly zero) of 2s in its prime factorization.
4. For any positive integer, twice its square has an odd number of 2s in its prime factorization.
5. Therefore, a-squared does not equal twice b-squared for any integer a and b. (Because then a-squared would have both an even and odd number of 2s in its prime factorization.)
6. Therefore, a^2/b^2 is not 2.
7. Therefore the square root of 2 is not a/b for any integer a and b.
(Note that any terminating decimal can be written as a/b, where b is the "place" (tenths place, hundredths place, thousandths place, etc.) where it terminates.)

I imagine you will object at step 6. In this case, it is not that you have a problem with accepting irrational numbers in particular... it is the ability to divide anything exactly that you can't accept.

If you object at step 5 instead, it is that you believe that an integer can be "close enough" to a different integer to be considered equal. Which would be a rather intriguing position to take, as it removes the number of places after the decimal from the discussion.
extro...*
Guest

Posted: Tue Jan 15, 2013 2:26 pm    Post subject: 15

 Amb wrote: 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.

It doesn't, since planck length applies to physical reality, and not to mathematical abstractions such as the circle (all points on a plane at a fixed distance r>0 from some central point).
Amb
Amb the Hitched.

 Posted: Thu Feb 14, 2013 3:11 am    Post subject: 16 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
No offense, but....

 Posted: Thu Feb 14, 2013 9:51 am    Post subject: 17 If pi were redefined in terms of physically constructable circles, yes.
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