Pi314CA
Active member
How Much Pi Do You Need?
By Evelyn Lamb | July 21, 2012
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 22⁄7 has a value of 3.142857, so it has the same first three digits as pi.
Both 3.14 and 22⁄7 are approximations of pi, so the two days deserve the same title. In fact, 22⁄7 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: π.)
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 phttp://www.carriefans.com/forums/newthread.php?do=newthread&f=49i 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.
...
It seems that we know, and strive to discover, many, many more digits of pi than we need for any practical application on Earth, or even in the part of space we can hope to get to right now. I guess the endlessness of the decimal representation just fascinates people. Haraguchi, the pi reciter, told The Japan Times that his memorization of pi is part of his quest for eternal truth. For some, it is probably a challenge: How far can I go? We want to push our limits, and memorizing pages of numbers seems pointless until we give it the halo of pi.
...
Link: How Much Pi Do You Need? | Observations, Scientific American Blog Network
By Evelyn Lamb | July 21, 2012
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 22⁄7 has a value of 3.142857, so it has the same first three digits as pi.
Both 3.14 and 22⁄7 are approximations of pi, so the two days deserve the same title. In fact, 22⁄7 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: π.)
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 phttp://www.carriefans.com/forums/newthread.php?do=newthread&f=49i 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.
...
It seems that we know, and strive to discover, many, many more digits of pi than we need for any practical application on Earth, or even in the part of space we can hope to get to right now. I guess the endlessness of the decimal representation just fascinates people. Haraguchi, the pi reciter, told The Japan Times that his memorization of pi is part of his quest for eternal truth. For some, it is probably a challenge: How far can I go? We want to push our limits, and memorizing pages of numbers seems pointless until we give it the halo of pi.
...
Link: How Much Pi Do You Need? | Observations, Scientific American Blog Network
... twice! But it was not a quest for eternal truth.
