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slipperyfish10

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In summary, the conversation discusses the concept of precision in mathematical calculations, specifically in regards to Pi and the Planck length. It is explained that Pi is a transcendental number and not a physical length, and its value is determined by a power series rather than physical measurements. The Planck length is mentioned as a unit for very small distances, but it is not the smallest possible length. The conversation concludes by mentioning the importance of mathematicians in developing powerful mathematical tools and the potential need for more than 5 significant figures in certain calculations.

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slipperyfish10

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A.T.

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willoughby

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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.

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.

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slipperyfish10

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

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craigi

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slipperyfish10 said:

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.

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DrClaude

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Some fundamental constants are known (measured / determined) to much higher precision than that. For instance, the electronwilloughby said:There is no realistic applied use of Pi that has more than just 5 significant figures or so.

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

That is a common misconception. There is no reason why something couldn't be smaller than Planck's length.slipperyfish10 said:if nothing can be smaller than Planck length

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sophiecentaur

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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.

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willoughby

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

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sophiecentaur

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Meir Achuz

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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.

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Meir Achuz

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The Planck length is not the smallest possible length. It is just used as a unit for very small distances.

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dauto

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willoughby said: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.

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slipperyfish10

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DrClaude

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Don't be sorry. If you can't ask such a question here, where can you ask it? PF is a place for learning.slipperyfish10 said:

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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.willoughby said:

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.

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##.

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CWatters

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willoughby said: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?

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willoughby

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jbunniii said: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.

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willoughby

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CWatters said: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!

The Planck length is the smallest measurable length in the universe, and is approximately 1.6 x 10^-35 meters. It is derived from fundamental constants such as the speed of light and the gravitational constant, and is considered the scale at which the laws of physics as we know them break down.

The Planck length is related to pi through the Planck units, which are a set of natural units that use fundamental constants to define basic physical measurements. The Planck length is one of these units and is equal to the square root of pi times the Planck length squared.

Currently, the Planck length is beyond the capabilities of our technology to measure. It is so incredibly small that it is well below the resolution of any current scientific instruments.

The Planck length plays a crucial role in theories of quantum gravity, which attempt to reconcile the theories of general relativity and quantum mechanics. It is believed that at the Planck scale, spacetime becomes discrete and quantum effects become dominant.

According to current theories, the Planck length is considered to be the smallest possible length. However, some theories suggest the existence of a minimum length, known as the "holographic principle", which would mean that there is a limit to how small a length can be.

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