14
Mar
10

On Truth and Pie Pi

How close to reality does something have to be for one to consider it a “fact?” We often approach ideas as if an absolute “truth” exists, but more often than not, what we find is that different explanations can be correct for any value of “true.” Pi can be approximated as 3.14, for basic mathematical problems, but this will not yield an absolutely accurate number. Utilizing pi as “3.14159” will give a far more accurate number, but it is still not absolutely true. The number of digits of pi needed to approximate a number is the same as the number of orders of magnitude your measurements span. For example, if you are calculating the circumference of a perfect circle, and you measure, to the nearest millimeter, a radius of 5,000.000 meters, this means your measurements span 6 orders of magnitude. Let us calculate, then, the circumference with various approximations of pi:

pi=3 (not used)
C=2*pi*5,000.000
C=30,000.000 meters

pi=3.1 (not used)
C=31,000.000 meters

pi=3.1415 (should actually be rounded to 3.1416)-but it is often used
C=31,415.000 meters

pi=3.1415927
C=31,415.927 meters (see how much closer 3.1416 would have been?)

pi=3.14159265358
C=31,415.9265358 meters

Remember that we measured only to the nearest millimeter, so the closest reliable measurement of the circumference we could hope for would be to the nearest millimeter, or 31,415.927 meters. Let’s suppose, then, that you wanted to calculate the diameter of the solar system (and assuming it is a perfect circle, which it isn’t) to the nearest ångström-and we measured the diameter to this degree. The radius of the solar system (~5913520000000 m) is ~5.91352×1012 meters and an ångström is 1 x 10-9 meters, these measurements span 21 orders of magnitude, meaning we would need 21 digits of pi after the decimal to calculate to this degree of accuracy. Let me demonstrate with some impossibly accurate measurements of the solar system radius:

r=5913522139576.317692391 meters
2r=11,827,044,279,152.635384782 meters
Let’s calculate the circumference:
pi=3.14159
C=37155724036943.12780000

pi=3.14159265
C=37155755378610.467552960

pi=3.1415926535898
C=37155755421067.191106263

pi=3.141592653589793238463
C=37155755421067.111137262

pi=3.14159265358979323846264338328 (for absurdity)
C=37155755421067.111137261471705979

So, how many digits of pi do you need to accurately measure the circumference of the solar system to the nearest ångström (if you’ve measured the radius in ångströms)? 21

How close do you want to get to the truth?

I’d be happy to know something to the nearest 7000th decimal

HAPPY PI DAY!


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2 Responses to “On Truth and Pie Pi”


  1. March 15, 2010 at 12:35 am

    This is funny. Insightful and a bit challenging, but truly funny. It does underline that the idea of “proof” or adequate evidence might have guidelines but ultimately will reside with the individual and their choices.

    There are multitudes for whom pi has no real meaning. It is just not a direct part of their lives. Those people who can’t see the value of things like algebra or even history in their education and post-education lives won’t generally understand the importance of pi.

    Indirectly, of course, pi is important in all of our lives. I don’t know to what degree of accuracy. That is determine largely by application. I, myself, have no real inkling as to how often calculations made using pi truly impact my life. However, I can at least recognize my lack of real knowledge and even take some steps to educate myself regarding the many applications of pi.

    Pie, on the other hand, I know directly. I just ate a piece a few moments ago. Cherry, a favorite since childhood.

    Please continue to challenge us. This is interesting. I now have something to think about and learn.

    Mike

  2. 2 jaredcormier
    March 16, 2010 at 10:22 pm

    Thanks Michael! As you read or type, you are, actually, utilizing something which was developed because of pi! Coulomb’s law was essential in the formulation of the theory of electromagnetism. Any formula which involves trigonometric calculations utilizes it as well.


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