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Pi Approximation Day

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Pi Approximation Day, I hope you’re ready for your big Pi Approximation Day party tomorrow. You might have observed Pi Day on March 14. It gets its name from 3.14, the first three digits of the ratio of a circle’s circumference to its diameter. Always on the lookout for excuses to eat pie, some geeky math types also celebrate the number on July 22. The fraction 22âپ„7 has a value of 3.142857, so it has the same first three digits as pi.

Both 3.14 and 22âپ„7 are approximations of pi, so the two days deserve the same title. In fact, 22âپ„7 is closer to pi than 3.14 is. So if you’re an aspiring pedant, you can choose to celebrate July 22 as Pi Day and March 14 as Not Quite as Close to Pi Day. (Either way, you’ll enjoy more pie.) But what does it mean to be an approximation of pi-and why does it matter?

Pi is irrational. That is, the decimal expansion never ends and never repeats, so any number of decimal places we write out is an approximation. (Of course, we can write the number exactly using just one symbol: د€.)

Each decimal digit we know makes any computation involving pi more precise. But how many of them do we actually need for sufficient accuracy? Of course it depends on the application. When we round pi to the integer 3, we are about 4.51 percent off from the correct value. If we use it to estimate the circumference of an object with a diameter of 100 feet, we will be off by 4 ½ feet. When we add the tenths place, and use the approximation 3.1 for pi, our error is only about 1.3 percent. The approximation 3.14 is about ½ percent off from the true value, and the fairly well known 3.14159 is within 0.000084 percent.

If you were building a fence around a giant circular swimming pool with a radius of 100 meters and used that approximation to estimate the amount of fencing you would need, you would be half a millimeter short. Half a millimeter is tiny compared with the total fence length, 628.3185 meters. Being within half a millimeter is surely sufficient, and the tools you are using to make the fence probably introduce more uncertainty into your structure than your approximation of pi.

What about something with higher precision standards over much larger distances? 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.

Pi appears most often in formulas involving circles or periodic motion, and it infiltrates some fundamental physical constants. These constants appear all over physics: masses of elementary particles, the number of molecules in a volume of a gas, the forces holding matter together, and so on. (Pi itself is not considered a fundamental physical constant.) The fine-structure constant, or “coupling constant,” which measures the strength of the electromagnetic force that governs how electrons and muons interact with photons, involves pi, and the permeability of free space, which describes how a magnetic field forms in a vacuum, is 4د€×10-7. It is important to know highly accurate values of the fundamental constants to make good predictions of phenomena involving physics, and the experimental determination of the constants can even help improve our understanding of the physical laws that govern the universe.

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