Exploring the Smallest Measurable Units: Planck Length & Pi

[slipperyfish10]

Hi all. This is my first time posting so forgive me If I am doing something wrong. I am a year 7 student interested in all types of physics and my question is, if nothing can be smaller than Planck length then wouldn't past a certain point the digits of pi become obsolete? Simply because the changes would become smaller than Planck length? Could someone explain this to me please?

 

[A.T.]

Pi is a ratio, not a length. Make the radius bigger, and you need more digits for certain precission of the arc length.

 

[willoughby]

There is no realistic applied use of Pi that has more than just 5 significant figures or so (3.14159 - rarely is more than this necessary for ANY application). As far as realistic applications go, anything beyond this is pretty meaningless. As A.T. points out, it's not a length, but a ratio. Pi continues to be calculated for the purposes of applied mathematics. I'm not a mathematician, but I do believe that fundamental applications and methods in math were developed through the process of coming up with all sorts of different algorithms to calculate Pi. We do it because we can and because it is fun, and we learn more about mathematics in the process. Not for any realistic physically applicable reason.

Just for reference, a calculation of a circle's area using Pi to 5 digits gives a result that is only off by 0.0000844664% from the same calculation using Pi to 12 digits (adding more digits to Pi is not going to move that '8' any closer to that decimal point). The only field where this would equate to a significant distance would be astronomy. When was the last time someone told you that the nearest star to Earth was 4.3266854126885933575 light years away (I made up every digit after 4.3!)? They don't! That's not for your sake. It's just the nature of astro physics. It's a science of estimations, so there is never a need for Pi to be accurate to more than just a few digits - even on the grandest scale.

 

[slipperyfish10]

Ok thanks all I wasn't asking this question to find an answer with an actual practical use.

 

[craigi]

Hi all. This is my first time posting so forgive me If I am doing something wrong. I am a year 7 student interested in all types of physics and my question is, if nothing can be smaller than Planck length then wouldn't past a certain point the digits of pi become obsolete? Simply because the changes would become smaller than Planck length? Could someone explain this to me please?

In Physics, it's important to consider the unit of a quantity.

The Planck length is a measure of distance.

We could use it in metres, killometres or whatever unit we want. All of which give us a different number.

We can't measure distances smaller than the plank length, but that has nothing to say about the accuracy of other measurements and nothing to do with the precision with which we can quote dimensionless numbers such as Pi.

 

[DrClaude]

There is no realistic applied use of Pi that has more than just 5 significant figures or so.

Some fundamental constants are known (measured / determined) to much higher precision than that. For instance, the electron g-factor is known to 14 decimal places. Using π with more digits can be necessary.

Also, in numerical simulations, it would be foolish to introduce additional errors deliberately. Using π up to machine precision makes sense then.

if nothing can be smaller than Planck length

That is a common misconception. There is no reason why something couldn't be smaller than Planck's length.

 

[sophiecentaur]

Pi is a transcendental number, which cannot be written as a ratio of two integers. The Maths which involves Pi is used to describe Physical (Scientific) relationships concerning the real world. But all these mathematical models of the world are only models and run out of accuracy at some stage - e.g. for very small / very large values of the quantities involved. The formulae you can see in textbooks 'work' within certain limits - in the same way that a map of your town doesn't show the blades of grass or the curvature of the Earth. The best Science can do is to get the map good enough to within a certain accuracy and it uses Maths.

You mention the Planck length, which can only be understood in the context of a lot more Scientific concepts that you will have come across by Year 7. It is a terrific idea to read around about advanced Science but do not be surprised or disappointed when you find that you cannot tie it all together - until you are well int Post Graduate level work.

 

[willoughby]

Some fundamental constants are known (measured / determined) to much higher precision than that. For instance, the electron g-factor is known to 14 decimal places. Using π with more digits can be necessary.

I was specifically talking about Pi. The point is, Pi is not calculated to trillions of digits because it is necessary. The vast majority of calculations involving Pi are precise enough to 5 significant figures - give or take a few.

 

[sophiecentaur]

Pi is calculated by Mathematicians, to many decimal places, because they CAN. Mathematicians are nice chaps and they are essential for Scientists. But they do tend to be a bit nutty (doesn't mean you're a bad person!) and it is their quirkiness that produces all those powerful mathematical tools that we, near mortals use so gratefully.

 

[Meir Achuz]

Pi is not just a ratio of physical quantities.
Its value is given by a power series, and does not depend on any physical measurement.
There are many calculations that require pi to more than five significant figures.

 

[Meir Achuz]

"nothing can be smaller than Planck length"
The Planck length is not the smallest possible length. It is just used as a unit for very small distances.

 

[dauto]

I was specifically talking about Pi. The point is, Pi is not calculated to trillions of digits because it is necessary. The vast majority of calculations involving Pi are precise enough to 5 significant figures - give or take a few.

DrClaude's point is that some very important applications do make use of PI and require more than 5 digits. Though you're correct that trillions of digits are not needed, 5 is sometimes not enough. High level physics is often done to 9 significant digits or more.

 

[slipperyfish10]

Every I am very sorry I would not have asked this question if I knew that Planck was just a measurement. I guess I'm just another little kid trying to understand big things.

 

[DrClaude]

Every I am very sorry I would not have asked this question if I knew that Planck was just a measurement. I guess I'm just another little kid trying to understand big things.

Don't be sorry. If you can't ask such a question here, where can you ask it? PF is a place for learning.

 

[jbunniii]

There is no realistic applied use of Pi that has more than just 5 significant figures or so (3.14159 - rarely is more than this necessary for ANY application). As far as realistic applications go, anything beyond this is pretty meaningless. As A.T. points out, it's not a length, but a ratio. Pi continues to be calculated for the purposes of applied mathematics. I'm not a mathematician, but I do believe that fundamental applications and methods in math were developed through the process of coming up with all sorts of different algorithms to calculate Pi. We do it because we can and because it is fun, and we learn more about mathematics in the process. Not for any realistic physically applicable reason.

Just for reference, a calculation of a circle's area using Pi to 5 digits gives a result that is only off by 0.0000844664% from the same calculation using Pi to 12 digits (adding more digits to Pi is not going to move that '8' any closer to that decimal point). The only field where this would equate to a significant distance would be astronomy. When was the last time someone told you that the nearest star to Earth was 4.3266854126885933575 light years away (I made up every digit after 4.3!)? They don't! That's not for your sake. It's just the nature of astro physics. It's a science of estimations, so there is never a need for Pi to be accurate to more than just a few digits - even on the grandest scale.

Counterexample: GPS. If you don't want Google Maps to put you on the wrong road, your position needs to be accurate to within a few tens of meters. The satellites are ~20 million meters away. If you perform your orbital calculations using only 5 digits of pi, you will not get anywhere near the required accuracy. Military applications require even greater accuracy.

Indeed, the GPS specification even defines the exact value of $\pi$ which should be used:

Parameter Sensitivity. The sensitivity of the [satellite]'s antenna phase center position to small perturbations in most ephemeris parameters is extreme. The sensitivity of position to the parameters $(A)^{1/2}$, $C_{rc}$, and $C_{rs}$ is about one meter/meter. The sensitivity of position to the angular parameters is on the order of $10^8$ meters/semicircle, and to the angular rate parameters is on the order of $10^{12}$ meters/semicircle/second. Because of this extreme sensitivity to angular perturbations, the value of $\pi$ used in the curve fit is given here. $\pi = 3.1415926535898$.

 

[CWatters]

There is no realistic applied use of Pi that has more than just 5 significant figures or so (3.14159 - rarely is more than this necessary for ANY application).

That sounded a bit low to me so I did some googling. I found...

http://blogs.scientificamerican.com/observations/2012/07/21/how-much-pi-do-you-need/

I asked a NASA scientist how many digits of pi the agency uses for its calculations. Susan Gomez, manager of the International Space Station Guidance Navigation and Control (GNC) subsystem for NASA, said that calculations involving pi use 15 digits for GNC code and 16 for the Space Integrated Global Positioning System/Inertial Navigation System (SIGI). SIGI is the program that controls and stabilizes spacecraft during missions.

...but I note it said "use 15 digits" not "need 15 digits". Perhaps it does?

 

[willoughby]

Counterexample: GPS. If you don't want Google Maps to put you on the wrong road, your position needs to be accurate to within a few tens of meters. The satellites are ~20 million meters away. If you perform your orbital calculations using only 5 digits of pi, you will not get anywhere near the required accuracy. Military applications require even greater accuracy.

Indeed, the GPS specification even defines the exact value of $\pi$ which should be used:

I said 'rarely' needed. Not 'never' needed. That being said, your point doesn't invalidate the spirit of my argument. There is no realistic application for TRILLIONS of digits of pi. And RARELY is more than 5 (give or take) needed, and according to your quote, GPS satellites use 13 digits, so that certainly fits into the realm of 'give or take' from 5 when the alternative is 5 trillion digits. You know?

And just FYI, if you calculate the radius of a circle that has a circumference of ~165 million meters (about the circumference of the orbit of an average GPS satellite), the difference between the calculation using 5 digits of pi vs 13 digits is about 20 meters. So, it's not as far off as you would imply.

 

[willoughby]

That sounded a bit low to me so I did some googling. I found...

http://blogs.scientificamerican.com/observations/2012/07/21/how-much-pi-do-you-need/



...but I note it said "use 15 digits" not "need 15 digits". Perhaps it does?

I'm sure it does 'need' those digits. I regret making that comment. I said 'rarely' needed. Not 'never' needed, and as I've explained to others, In the spirit of my argument, 15 digits still makes my point. Pi has been calculated to 5 TRILLION digits. THAT is what I mean when I say there is no realistic application. Perhaps I undershot with 5, but like I said, even 15 makes my point. Thanks for looking that up though!